ICI, \^. Vi 11 i \ N O f'-, -|"^^- H np A CHi \ I .n \n\u\x^' KO. ' ^^ m, Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924032183117 arY878 *^°™*" ""'^'^''^ "-'"rary ^°ilii?nli'i?fi'ili,?,?.!, ..^I^wing. and sketching olin.anx 3 1924 032 183 117 V? Topographical Drawing AND Sketching, Including Applications of Photography, BY LIEUT. HENRY A. REED, U. S. ARMY, Assistant Professor of Drawing, U. S. Military Academy, West Point, N. Y. NEW YORK: JOHN WILEY & SONS, 15 AsTOR Place. 1886. A'ZZ^^S ORNCLL \ LIBRARY y Copyright, 1886, By JOHN WILEY & SONS. PREFACE. 'T'HE writing of this book was first suggested by the fact that there was no native work which fully treated of and illustrated rapid methods of hill-shading; and it is now writ- ten not only to explain these and other methods now used either separately or in conjunction with them, but also to present the subject of topographical sketching in a form suited to a beginner. The methods of both drawing and sketching as now practised in the principal topo- graphical schools, and there considered the best, are described in detail. The writer has searched every available source, both foreign and native, for new informa- tion, and desires especially to acknowledge that gained from the publications of the U. S. Coast Survey, the " Memorial de TOfificier du Gdnie," and from the recent works of Lehagre, Maes and Bertrand of the French schools, and of Colonels Richards and Roberts of the British Army; and also to express his indebtedness for valuable advice, assistance, and opportunities afforded by his friend. Professor Charles W. Larned, Department of Drawing, U. S. M. A. West Point, N. Y., June 23, 1886. CONTENTS. TOPOGRAPHICAL DRAWING. PART I.— INTRODUCTORY, INSTRUMENTS AND MATERIALS, " PRACTICAL RULES AND SUGGESTIONS, SCALES AND PLOTTING. PARAGRAPH I. Introductory, PAGE I Section I. — Instruments and Materials. 2. Pencils, Kind Used, and Pointing of, . . . . . 3. Pens, Common and Crowquill, and Holder, 4. Pens, Drawing- or Right-line, Form and Use, and How to Repair Points of, 5. Instruments for Drawing Straight Lines, ..... 6. Straight-edges, Kinds Required, and Use of, . 7. Triangles, Kinds and Use of, . 8. T-Square, Form and Use of, ...... . 9. Parallel Rulers, Forms and Use of, ..... 10. To Grade Intervals, ........ 11. Section-Liners, General Form of, ...... 12. Instruments Necessary for Drawing Curved Lines, .... 13. Dividers, Description and Use of, . 14. Dividers, Spring, Description and Use of, .... . 15. Irregular Cui-ves, Description and Use of, . 16. Beam-Compass, Description and Use of, .... . 17. Curve-Pen, Description and Use of, . . . . • . 18. Dotting, Border and Road Pens, Description and Use of, . 19. Instruments for Measuri>ig Lines and Angles, .... 20. Scale of Equal Parts, Description and Use of, .... 21. Spacing-dividers, Description and Use of, . ... 22. Protractors, Kinds ; and The Rectangular Protractor, Description of, 23. Protractors, Semicircular, Description of, . 24. To Protract an Angle, ........ 25. To Plot a Bearing, ........ 26. Vernier Protractor, Description and Use of, . 27. Protractors, Crozet's and Tabarant's, ..... 28. Protractors, Test for Semicircular, ...... 29. Sector, Description and Use of, . . . . . 30. Drawing-paper, Brands, Sizes and Kinds of, ... 31. Tracing-paper and Tracing-cloth, Description, Use and Manufacture of, 32. Transfer-paper, Description, Use and Manufacture of, ... 33. Cross-section and Profile Paper, Description and Use of, 34. India Ink, Qualities and Preparation of, .... . 35. Other Materials Required, . .... I 2 2 3 3 3 3 4 4 4 4 4 5 5 5 6 6 6 6 7 7 7 7 7 8 8 10 II II II II 12 VI CONTENTS. Section II.— Preparation of the Paper, Drawing-boards and Practical Rules and Suggestions. 36. Preparation of the Paper, and the Plain Drawing-board, ..•••• 37. Stretching the Paper, .....•••••' 38. Stretching frames, or " Stretchsrs," ....•■••• 39. To Mount Paper on Cloth, .....•••••' 40. To Join Sheets of Piiper, .....••••• 41. The Room, Light, etc., ......••••■ 42. Erasures, How to Make, .....■••••• 43. Rules for Line Drawing, ......••••• PAGE 12 13 13 13 14 14 15 15 Section IIL— Scales and Plotting. 44. Scale of the Map, Definition of, and Rule for Conversion of Distances, 45. Scale of Distances, Construction and Use of, . . 46. Diagonal Scale, Construction and Use of, 47. Vernier Scale, Construction and Use of, . 48. Vertical and Time Scales, Construction and Use of, . 49. To Divide a Line into Equal Parts, .... 50. Examples of Scale-Construction, .... 51. Inferior Limit of Scale-Measurement, .... 52. Choice of a Scale, Conditions Governing the, 53. Scales Used in Different Countries, .... 54. Plotting, General, ..... 55. Plotting of Points, Lines and Angles, 56. Defining the Limits of the Map and the Direction of a Meridian Line. 57. Plotting the Triangulation, ..... 58. Plotting the Triangulation, Rapid Methods of, ... 59. Plotting of Traverse Courses by Rectangular Coordinates, 60. Plotting of Traverse Courses by the Use of a Single Meridian Line, 61. Plotting of Traverse Courses by the Use of Several Meridian Lines, 62. The Trigonometer and its Use in Plotting, 63. Closing a Plot, and Error Sheets, ..... 64. Plotting a Vertical Section, 16 16 17 17 18 18 19 20 20 21 22 22 22 23 =5 25 26 24 26 27 27 PART II. -PLAIN TOPOGRAPHICAL DRAWING. 65. Division of tlie Subject, ...... ... Section I. — Conventional Signs. 66. General Description, ... ....... 67. Shading of the Signs, ....... 68. General Remarks, Number and Sizes of the Signs, etc., ..... 69. Arrangement of, in the Different Plates, . . ..... 70. Conventional Signs — Land, . , . , . . 71. Conventional Signs — Land and Water, ... Survey-Points and Lines, Boundaries, Enclosures, Communications and Buildings, Marsh, Water, Soundings, etc. ; Artificial Features pertaining to Streams. 72. Conventional Signs— Miscellaneous Military Signs, and Signs for Deciphering Maps, 73. Conventional Signs— U. S. Coast Survey, ....... 74. Conventional Signs, Rapid Method of Making, 29 29 29 29 30 30 34 35 35 CONTENTS. Section II. — Representing the Details. 75. Pencilling the Details, 76. Inking the Details, VII PAGE 36 36 Section III. — Representing the Configuration or Surface-forms of Ground. 77. General Methods, ........... 78. 79- 80. 81. 82. 83- 84. 85. 86. 87. 90. 91. 92. 93- 94- 95- First Method — By Contours. Contours, Equidistances and References, . ...... Degree of Declivity, How Determined from the Contours, .... Scale of Inclination, Construction and Use of, ..... . Horizontal Equivalents, Table Giving Ratio of Height to Base for Slopes from i° to 60°, Reduction of Surface-measurements to the Horizon, . ... Same, Goulier's Scale for, . ...... Same, for Stadia Measurements, ..... Same, when Difference of Level is Known, . ... Plotting the Contours, Remarks on. Same. Case I. — When the Surface-line is of Uniform Inclination, Same. Graphical Method, ..... ... Same. Case II. — When the Surface-line is not of Uniform Inclination, Same, Scale of Slopes for, .... .... Drawing of Contours, Marking the References, etc., ..... Intermediate and Auxiliary Contours, ....... Study of a Contoured Map, ........ Construction of a Vertical Section from a Contoured Map — The Sectograph, Intervisibility of Sta- tions, Elevations, ....... To Lay Out a Road on a Contoured Map, ... Second Method — By Contours Combined with Hill-Shading. 96. The Three Systems, 36 37 37 37 38 38 39 39 39 40 40 40 41 41 42 42 42 43 44 45 hill-shading. 97. Vertical Illumination, .... ..... 45 98. Oblique Illumination, ........... 46 99. Choice of Illumination, ........... 46 100. Definition of Terms used in Hill-shading — Normals, Zones, Hachures, Lines of Greatest Descent, Auxiliary Contours, Guiding-lines, Scales of Shade and Working-scales, ... 46 loi. Preliminary Work Required in the Application of the Different Systems, ... 48 102. The Horizontal System with Vertical Illumination. — Standard Scales, and the Operation of Shading, ... ....... 48 103. Desprez' and Mandrot's System, .......... 49 104. The Horizontal System with Oblique Illumination, ....... 49 105. The Vertical System with Vertical Illumination, ....... 49 106. Lehmann's Scale of Shade and Working-scale, ....... 49 107. The Austrian Scale of Shade and Working-scale, . . . . . . -Si 108. The U. S. Coast Survey Scale of Shade and Working-scale, ..... 52 109. The English Modification of Lehmann's Scale, . ...... 53 no. The Danish Scale of Shade and Working-scale, ....... 53 III. MiifHing's Modification of Lehmann's Scale, ........ 53 Viii CONTENTS. PAGE 112. The French System of Hill-shading, . ....... 53 113. Col. Bonne's Scale of Shade and Working-scale, . ,...•• 53 114. French Commission Scale of Shade and Working-scale, ....•• 54 115. The Operation of Shading — Vertical System with Vertical Illumination, . . ■ -55 T16. Necessity for Contours in Hachured Maps, . ....•• 5" 117. The Operation of Shading — Vertical System with Oblique Illumination, . . • ■ 5^ 118. Relative Merits of the H. and V. Systems, . . •■•56 119. Rock-surfaces, Representation of, . . . . . • • 57 120. Contours Combined with Brush Hill-shading. French Syste/n with V. Illumination: a. Scale of Shade; i. Working-scale; c. Application of the Working-scale; d. Tracing of Curves of Equal Shade; e. Case of Strongly Marked Irregularities of Surface;/. Features Requiring no Shade; _§-. Practical Operation, ... . . ■ • • 57 121. The same. Oblique Illumination, .... ... 60 122. Brush-shading, Ordinary Practice of — Materials Required, Preparation and Operation, . 61 123. Pencil Hill-shading, Description of, .......-• 62 Section IV. — Finishing the Map, including Lettering and Ornamentation. 124. General Requirements, .......... 63 125. The Lettering — Forms, Sizes and Disposition of, ....... 63 126. The Title, Arrangement and Disposition of, ........ 67 127. Construction of Letters, ... ....... 68 128. Scales, Compass, Border and Framing, ........ 68 PART III.— TOPOGRAPHICAL DRAWING IN COLORS. Section I. — Materials, Rules for Working in Colors, Preparation and Laying of Tints. 129. Materials and Instruments, ..... .... 70 130. Rules for Working in Colors, . . . . . . . . . . 71 131. Preparation of Tints, ........... 71 132. Laying of Flat Tints, ........... 72 133. Laying of Graded Tints, .......... 72 134. To Blend Tints, ........... 73 135. Stippling and Dragging, and the Durability of Colors, . . . . . -73 Section II.— The Conventional Tints and their Application. 136. Tints Employed for the Different Features, .... . . 74 137. The Operation of Tinting, ■•••-..... 76 138. The French System, Conventional Tints, and Rules Observed in Color- drawing, . 76 PART IV.— COPYING, REDUCTION AND ENLARGEMENT OF MAPS AND MODELLING. Section I.— Copying, Reduction and Enlargement of Maps. 139. General Purposes and Preliminary Work, ... .... 70 140. Copying to the Original Scale— I. Geometrically; II. By Pricking; III. By Tracing; IV. Copying on Glass ; V. By Transfer, . . •-.... 79 Reducing- and Enlarging. 141. Generalization of Detail, •••••-.... 81 142. Different Methods— Method by Squares, •■...... S'' 143. Proportional Dividers, Description and Use of, •••... 8^ CONTENTS. ix PAGE 144. The Angle of Reduction, Description and Use of, ...... 83 145. The Pantograph, Description and Use of, . . . . . • • -83 146. The Eidograph, Description and Use of, ....... . 85 147. Copying by Photography, Preparation of the Subject to be Copied, The Lens, Relative Positions of Ground Glass and Subject, Lighting the Subject, General Considerations, Blue Process, Black or Red Lines on White Ground, ........ 86 148. Modelling — Purposes of Models and Details of Construction, ..... 90 PART v.— PROJECTIONS FOR MAPS OF LARGE AREAS. 149 General Considerations, .......... 93 150. Cylindrical Projections — L Rectangular Projection ; IL Projection with Converging Meridians, 93 151. Conical Projections — L Simple Conic ; IL Bonne's; III. Polyconic, including Rectangular, Ordi- nary and Equidistant, .......... 94 152. Plotting the Triangulation, .......... 96 TOPOGRAPHICAL SKETCHING. PART VI.-SKETCHING-INSTRUMENTS, METHODS AND EXAMPLES. Section I. — Instruments, Locating Points, Measurement of Distances and Heights, the Skeleton. 153. General Considerations, .......... 97 154. Sketching-instruments— a. Hand-compasses;^. Hand-levels and Clinometers; c. Pencils, etc., 97 155. Scales, Plotting and Signs, .......... 100 1 56. Locating Points, ........... 100 157. Measurement of Distances — a. By Pacing, the Pedometer; b. By the Odometer; c. When Mounted ; d. By Eye Estimation ; e. On Slopes, ....... 100 158. Measurement of Heights — a. Graphical and Trigonometrical ; b. By the Aneroid Barometer, 103 159. Constructing the Skeleton — I. General Considerations; II. By Triangulation ; III. By Travers- ing; IV. By Traversing Combined with Radiation and Intersection; V. By Parallels and Perpendiculars ; VI. By Alignments and Prolongations, ..... 106 160. Filling in the Details — General Considerations ; Form (I.) of Note-book for Plotting the Sketch in the Field ; Form (II.) of Note-book Used when the Plotting is not Done in the Field, . 108 161. Features to be Sketched or Otherwise Described, . . . . . . .110 Section II. — Representing the Configuration of the Surface (or Hill-sketching). 162. General Considerations, . . . . . . . . . . .111 163. Case of a Detached Hill, ...... . m 164. Case of a Number of Elevations and Depressions, . . . . . . .112 165. Parts of Contours, Sketching of, from Distant Points — I. From Profile Views ; ll. By Vertical Angles, . . . . . . . . . . . .112 166. In Mountainous Districts, Horizontal and Profile Sketches, . . . . . • 113 Section III.— Sketching Without Instruments. 167. General Considerations and the Plane-table Method; the Cavalry Sketching-case; Example of Rapid Eye Sketching, . . . . . . . ' . , . 114 168. Landscape Sketching, . . . . . . . . . . .118 169. Itinerary, Form of, . . . . , . . . . . .118 CONTENTS. PART VII. PHOTOGRAPHY APPLIED TO TOPOGRAPHICAL SKETCHING. PACE 170. General Considerations, .......•••• "9 171. First Method, with the Ordinary Camera— a. Description of Camera; b. Principles of the Method, and Measurement of Horizontal and Vertical Angles ; c. Determining the Horizon and Focal Distance; d. Plotting of Points, . . . . • • • • i'9 172- Application of this Method— «. The Field-work;/. The Plotting;^. To Ascertain Diflferences of level ; h. To Determine the Horizon of a Print by Vertical Angles ; i. Plotting of Levelling Measurements, Form of Record, etc.; k. Choice of Scale, Degree of Accuracy Attainable and Selection of Stations, .......... 121 173. Second Method — By Horizontal Views ; Description of the Camera Used, the Field-work and Plot- ting, ............ 123 TABLES. Drawing-paper, Brands, Kinds and Sizes of, ........ 10 Topographical Scales Used in Different Countries, ....... 21 Trigonometrical Functions of any Arc or Angle in Terms of those of an Arc or Angle less than 90°, . 24 Horizontal Equivalents — Ratios of Height to Base for Slopes from 1° to 60°, ... 38 Letters — Forms and Heights (Belgian School), ........ 65 Differences of Level, Measurement of, in Mountainous Tracts, ...... 103 Differences of Level, Measurement of, with Aneroid (U. S. C. Survey), ..... 105 In Appendix. Table I. Measures of Different Countries in Metres and English Units. ... 125 II. For Mutual Conversion of Metres and English Units. . . . 125 III. Natural Sines, Cosines, Tangents and Cotangents. .... 126 IV. Meridional Arcs. ......... 128 V. Coordinates for Polyconic Projection. . . . . 12S VI. Length of Degrees of the Parallel. ... . . 129 " VII. Airy's Table for Measurement of Heights with the Aneroid Barometer. . . .129 ILLUSTRATIONS. TOPOGRAPHICAL DRAWING. Plate I. — Instruments and Scales. Figure I. a. Straight-edges; b, T-square; c, Triangles; d. Parallel Rulers — Rolling and Sliding. 2. e and /, Dividers; g. Pencil -point ; h. Pen-point; /, Extension-bar; k. Spring Dividers and Bow-pen ; m, Beam-compass ; n. Points for same. 3. Curve Pen. 4. Dotting-pen. 5. Border and Road Pen. 6. Dotting-pen. 7. Protractor, Semicircular. 8. Triangular Scale of Equal Parts. 9. Protractor, Rectangular. 10. Sector. 11. Protractor, Vernier. 12. Scales of Distance. 13. Diagonal Scale. 14. Division of a Line into Equal Parts. 15. 16, and 17. Verniers — Direct, Retrograde and Circular. Plate II.— Plotting. 18 and 19. Angles, Construction of, with Sines, etc. 20, 21, 22, and 23. Plotting by Rectangular Coordinates. 24. Plotting by Single Meridian Line and Protractor Sheets. 25. Plotting by Several Meridian Lines. 26. Closing a Plot. 27. Plotting of a Vertical Section. Plate III. — Conventional Signs — Land. Plate IV. — Conventional Signs— Land and Water. Plate V. — Conventional Signs —Miscellaneous, including Military Signs, and Signs for Deciphering Maps. Plate VI. — Conventional Signs employed in Maps of U. S. Coast Survey. Plate VII.— Contours, and Plotting of Same. 28. Contours, Equidistances and References. 29. 30, and 35. Scales of Inclination. xii ILLUSTRATIONS. Plate VII — Continued. 31. Reduction of Surface-measurements to the Horizon. 32. Goulier's Scale for same. 33 and 34. Plotting of Contours— Surface-line of Uniform Inclination. 36. Same, — Surface-line not of Uniform Inclination. 37 and 38. Same, Scale of Slopes for. Plate VIII.— Characteristic Surface Lines and Forms. 39. Study of Contours, etc. 40. Section from Contoured Map. 41. The Sectograph and its Application. 42 and 43. Cols, Geometrical and Natural. 44 and 45. Normals. 46, 47, 48 and 49. Lines of Greatest Descent. Plate IX.— Examples of Hachure Hill-shading.— Horizontal and Vertical Sys- tems, Vertical and Oblique Illumination. Plate X.— Scales of Shade and Working-scales. 50. Lehmann's Scale of Shade. 51. Same, Austrian Modification of. 52. Working-Scale for same. 53. U. S. Coast Survey Scale of Shade. 54. English Scale of Shade (for standard, see Fig. 59). 55. Danish Scale of Shade. 56. General Miiffling's Scale of Shade. 57. Colonel Bonne's Scale of Shade. 58. French War Department Scale of Shade. Plate XI. — Hachure Hill-shading. 59. English Standard Scale of Shade for -ysWs' and an equidistance of 25 feet. 60 and 61. Hachuring, Horizontal System. 62 and 63. Hachuring, Vertical System. Plate XII. — Examples of Hill-shading and Rock- representation. — U. S. Coast Survey and French Systems. Plate XIII. — Brush Hill-shading. 64. Scale of Shade, French System. 65, 66 and 67. Curves of Shade, Tracing of. Plate XIV.— Map Lettering.— U. S. Coast Survey. Plate XV.— Conventional Tints. Plate XVI.— Examples in the Use of Color— Map and Sketch. Plate XVII.— Example in the Use of Color— French System, Fart of Algeria. Plate XVIII.— Copying, Reduction and Enlargement of Maps. 68. Three legged Dividers. 69 and 69a. Copying by Squares. 70. Construction of Similar Rectangles. 71 and 72. Construction of Proportional Squares. ILLUSTRATIONS. xiii Plate XVIU.~ Continue J. 73. Proportional Dividers. 74. Angle of Reduction . 75. Pantograph. 76 and 77. Pantograph, Principle of Construction of. 78 and 79. Pantograph, another Form, and Principle of Construction of. 80. Micrograph. 81. Eidograph. Plate XIX. — Projections for Maps of Large Areas. 82. Rectangular Projection. 83. Projection with Converging Meridians 84. Simple Conic Projection. 85 and 86. Simple Conic Projection, Application of Rectangular Coordinates to. 87. Equidistant Polyconic Projection — Graphical Construction of. TOPOGRAPHICAL SKETCHING. Plate XX. Sketching-instruments. 88. Rectangular Box-compass. 89. Prismatic Compass. 90. Hand-level. 91. Abney's Reflecting Level and Clinometer. 92 and 92a!. Pedometer 93. Odometer. 94 and 943. Aneroid Barometer. Plate XXL — Topographical Sketching, Methods of. 95. Subject of Sketch, with Skeleton and Traverses. 96. Note-book — Form L, for Plotting in the Field. 97. Note-book — Form IL, for Subsequent Plotting. 98. Triangulation Notes, Record for. 99 and 100. Extending the Triangulation from the Base, loi. Traverse. 102. Alignments and Prolongations. Plate XXIL— Topographical Sketching, Methods and Examples of. 103, 104, 105, io6 and 107. Contour Sketching. 108. Profile Sketch. 109. Horizontal Sketch. Plate XXHL— Eye and Landscape Sketching. no. Rapid Eye Sketch. Ill and 112. Landscape Sketches. Plate XXIV. — Photographical Sketching. 113. Camera, Ordinary Type of. 114. Print. 115. Determining the Focal Distance. 116. 117 and 118. Plotting from Photographs. 119, Camera for Horizontal Views, ILL USTRA TIONS. DIAGRAMS IN TEXT. PAGE Pencil-points and Pen-holder, .......... i Drawing or Right-line Pen, .......... 2 Grading of Intervals, ......... . . 4 Irregular Curves, ............ 5 Interpolation of Heights with Sector, ......... 9 Drawing-boards, ...... ..... 12 Oblique Intersections, . . . . . . . . . . .15 Trigonometer, .......... . . 27 Elevations, ............. 44 Title, Arrangement of, .......... . 66 Names of Features, Position of, .......... 67 Brushes, Sizes used, ... ....... 70 Flat Tints, Laying of, ........... 72 Tints, Grading of, ... . ....... 73 Tints, Blending of, ............ 73 Dragging, Manipulation of Brush in, ........ . 74 Graphical Measurement of Heights, ......... 103 TOPOGRAPHICAL DRAWING. PART I. INSTRUMENTS AND MATERIALS-PRACTICAL RULES SCALES AND PLOTTING. AND SUGGESTIONS- 1. Introductory. — Topographical Drawing is the art of representing graphically the natural and artificial features of a limited portion of the earth's surface. The positions and dimensions of these features are determined by topographical surveying and sketching, and the drawing is made from the records and notes of these operations. The principal characteristic of this kind of drawing is the representation of the relief of forms. While in other maps, horizontal distances and dimensions only are given, in a topographical map heights also are represented, and conventional methods are employed to produce an effect similar to that which the features themselves would present to the eye if viewed from points vertically above them. Topographical Signs are conventional symbols, or figures, employed to represent the different features. In Plain Topographical Drawing (Part IL) these signs are in black and white, while in Colored Topographical Drawing (Part in.) various colors and tints are used to ex- press them. Section I.— Instruments and Materials. The increased cost of the best instruments is fully repaid by the time and labor saved in their use. 2. Pencil. — Lead-pencils corresponding to Fa- ber's Siberian Graphite, of H B, H, 4 H, and 6 H grades, are suitable for the purpose : the two latter grades for plotting, marking points of division of C"^ lines, and other fine work; and the former for sketching purposes, and for drawing some of the signs, such as trees and stone walls, which require a certain degree of emphasis, especially on maps drawn to a large scale. Lead of good quality will produce a smooth, even line without requiring pressure sufficient | to indent the paper. A sharp well-shaped point is required for good work. The accompany- TOPOGRAPHICAL DRAWING. O rM in- figure shows the "cone" and "chisel" points; the latter being used, on account of its du^'rabHity, for drawing straight lines, and the former for all other purposes. To form the conical point, the pyramidal point a is formed f^rst, and then trimmed to the shape b. The side and end view of the chisel-shaped point is given at c. Emery-paper or fine sand-paper, tacked to a flat surface, is very handy for keepmg a pencil sharp; but after its use the point should be made smooth by rubbing it flatwise on paper, before applying it to the drawing. 3. Common and Crow-quill Pens.-The common pen is used for all free-hand ink-drawing with exceptions, that for long smooth curves the curve-pen (par. 17), and for the finest lines the crow-quill, are substituted. It should be smooth-pointed and elastic. Gillott's " 303" and "404" are good standards for general use— the latter for the coarser work. The crow- quiU is a much smaller pen, with a very fine point. To promote steadiness, the lower part of the pen-holder should be reinforced, or enlarged, as shown at d in the preceding figure, and the pen inserted about half its length. It is held as in ordinary writing. Even with the best of ink, the pen must be wiped very often to prevent clogging. A fine-pointed pen is in good condition when with the pressure due to the weight of the pen and holder, the latter being held at the tip, or resting in the hollow of the thumb and forefinger, the point will describe a con- tinuous hair-line. 4. Drawing or Right-line Pen. — The different parts of this instrument are shown in the accompanying figure. Convenient sizes are 6 and 4J inches respectively — the smaller for fine lines. A drawing-pen consists of two steel points, called " nibs," attached to an ivory or ebony handle ; one nib is hinged at its base to facilitate cleaning the inner surfaces : a small block at the base tends to spread the nibs, which can be adjusted for lines of different breadth by means of the miiled- head screw shown in the figure. In the larger sizes, the handle, which may be unscrewed, is provided at its lower extremity with a needle-point, of use in marking points, erasing and transferring. German silver nibs are best for colored inks and water-colors. In use, it is filled by holding it vertically, nibs uppermost, and in- serting near the points a common pen or a piece of paper charged with ink ; the ink will usually be drawn upward to the points by capillary action, otherwise the pen or paper is passed between the points. The quantity of ink required is soon observed in practice, and there should be none on the outer surfaces of the nibs. A straight-edge or a curve is always used in connection with it. In drawing a line, the pen is held perpendicularly to the surface of the paper, with the ex- ception of a slight and uniform inclination in the direction of the movement, the points pressing equally upon the surface, and the nib opposite the milled head very lightly against the guiding edge. The line is drawn by a continuous movement, and as rapidly as the flow of the ink and the formation of an even line will permit. If the points become clogged, a piece of paper or a pen-point is passed between them ; and if this does not clear them, the nibs should be opened, cleaned and refilled. The pen should be cleaned when put away. @ INSTRUMENTS AND MATERIALS. 3 The points become dulled by use, and require resetting by an instrument-maker; but, lacking this resource, they may be put in order as follows : — First, make the nibs of equal length and of a rounded shape by drawing lines, so to speak, on a fine-grained whetstone ; then, holding the pen at an angle of about 20° with the surface of the stone, and applying the outer surface of each nib in succession, sharpen each point until, by reflected light, the dulled edge is no longer perceptible, and at the same time the nibs are of equal length and the points are not so sharp as to cut or scratch the drawing. 5. Instruments for Drawing Straight Lines. — The principal instruments used with pen or pencil for drawing straight lines, are the straight-edge or ruler, triangle and T-square. These are of steel, rubber, or hard fine-grained wood ; and should have straight, carefully finished edges, and lie flat upon the paper. Those of wood or rubber are liable to warp ; the edges should be frequently tested for accuracy, which is readily done by placing the edge to be tested in contact with a steel straight-edge, and holding the instruments between the eye and the light. Another method is to rule a fine line and prolong it with the same edge about half its length ; this edge, applied to different parts of the line, and on each side of it, should coin- cide with it. For fine work, these instruments should not exceed one tenth of an inch in thickness. Although metal rulers and triangles retain their shape and edges, they are more liable to soil the paper than those of wood. 6. Two straight-edges {a, Fig. i, Plate I.), of fifteen and thirty inches in length respec- tively, are enough for ordinary purposes. As a precaution against blotting, to keep the points of the pen from contact with the straight-edge, it is desirable to have one edge bevelled ; this edge, in inking, to be next to the drawing. A right line of greater length than the straight-edge may be drawn by placing two straight-edges together flat on the paper, with their edges overlapping; or by marking points exactly underneath a thread stretched the required distance, and then joining the points with the straight-edge. 7. Triangle (c, Fig. i). — The triangle is used in conjunction with the straight-edge for drawing parallels and perpendiculars. The open or " frame-pattern," shown in the figure, does not cover up so much of the work as the solid one, is handier, and if of wood or rubber, is less liable to warp. The smaller angles are usually either 45° and 45°, or 30° and 60°. To test the accuracy of the right angle, place either side, as al>, coincident with a fine right line, and draw a line along ac ; revolve the triangle about the side ac, make ad again coincide with the right line and draw another line along ac: the two lines thus drawn should be parallel. To draw Parallels : Place a side against a straight-edge, and draw a line along either of the other sides ; without moving the straight-edge, slide the triangle along the desired dis- tance, and draw another line along the same edge as before : the two lines thus drawn will be parallels. It is apparent that to draw lines perpendicular- to each other the longest side is placed against the straight-edge. 8. T- Square {b, Fig. i). — This instrument in its handiest shape consists of a thin " blade" and a thicker adjustable " head." The upper half of the head is firmly fastened to the blade, with its inner edge perpendicular to the edges of the latter, and its lower surface flush with TOPOGRAPHICAL DRAWING. that of the blade. The lower half is the duplicate of the upper, and by means of the clamp- screw, shown in the figure, its inner edge can be fixed at any desired angle with the edges of the blade. In using it, this inner edge, adjusted as required, is pressed against the edge of the drawing-board, while the blade lies flat upon the drawing. Under these conditions the square is moved over the drawing, and any number of sets of parallels can be drawn along its edges, or those of a triangle used in conjunction with it. If the edge of the drawing- board is not straight, a thin straight-edge can be fastened to it. The right angles between the edges of the upper head and blade are tested as described for the triangle. 9. Parallel Rulers {d, Fig. i).— Of these there are two kinds— the sliding and the rolling. The former consists of two straight-edges joined near their extremities by two metal strips of equal length, which are pivoted at the points of attachment, so that when the straight- edges are separated they remain parallel. The other ruler is a straight-edge of considerable weight provided, near its extremities, with rollers of equal diameter, movable about the same axis. The manner of using them is apparent. On account of the play at the joints, and of the axle, these instruments are not suitable for accurate work. 10. To grade Intervals.— The following is a simple method of drawing parallels with equal or regularly graded intervals, such as are required in section-lining and in filling in out- lines of buildings, etc. Using a triangle and straight-edge of the same thickness, the latter graduated along one of its edges into, say tenths of inches, place them as shown in the figure, with the index i coincident with one of the divisions of the straight-edge. If now the triangle be moved along the latter, and lines be drawn along ac when the index reaches each of the divisions in succession, it is apparent that the parallels will be equally spaced. If cab is go" and acb 30° : since sin 30° = |^, «^ = Ibc ; therefore with the above divisions the intervals are each ^'^ of an inch : or if ^3 is -J^, J .... of be, each interval will be aV' Ttr ■ • • • o^ ^" ''"'^^ '< ^""^ '^ ^^^ divisions are reduced to twentieths, these intervals become ^, -^^^ . . . . of an inch. It is readily seen that a regular gradation is ob- tained by increasing the distance passed over by the index, by one division for each parallel. 11. Section-Liners. — These are instruments devised for the purposes stated in the fore- going paragraph. A straight-edge is made to move over successive regular distances by means of springs and adjustable stops, the distances being gauged by attached scales and indices. The best forms would serve to produce an even gradation with little trouble ; but practice soon trains the eye to this kind of work, and for ordinary topographical purposes a triangle and a straight-edge will prove sufficient. 12. The necessary instruments for drawing curved lines are dividers, bow -pen and pencil, and an irregular curve. To these may be added bcam-compasscs for describing arcs of large radii, and the curve-pen. 13. Dividers. — This instrument (Fig. 2), shown at / with needle-point inserted, is fur- INSTRUMENTS AND MATERIALS. nished, in addition to the customary steel-points {e\ with pencil and pen points, and an extension-bar, shown at g, k, and i, respectively, which are inserted as needed and securely held in place by set-screws. The best instruments are of German silver, and have a steel friction-plate interposed between the branches or legs at the pivot, to prevent wear and secure smoothness of motion ; the necessary amount of friction being obtained by means of the horizontal adjusting-screws shown in the figure — the vertical adjusting-screws serving to bind the handle to the heads of the former. With the exception of the steel-points, used principally for spacing and for setting off distances, the different points are hinge-jointed, to admit of their being set per- pendicularly to the paper in describing arcs, which is particularly necessary in using the pen-point. In the absence of leads made especially for the pencil-point, serviceable ones can be cut from a hard-lead pencil. The " shouldered " needle, shown inserted, is more rigid and con- venient than a plain needle. A large radius is obtained with the extension-bar. In describing arcs, the handle is held between the thumb and forefinger, the point of the needle is kept in place with very light pressure, and the curve is described with a continuous movement. To prevent wearing large holes in the paper, when several arcs are described from the same centre, a " horn-centre" — a thin transparent piece of horn, with points on its under surface to keep it in place — is interposed between the point and the paper. 14. Spring-Dividers {k, Fig. 2). — This instrument, also called a bow-pen or pencil, is used for describing arcs of very small radii. In the kind shown at the top of the figure the legs are formed of one piece of steel, bent until the points are about an inch asunder, and are adjusted by means of the screw and nut. Of the other two, in the form shown on the left, the pen or pencil point revolves about a steel rod, which serves both as a handle and needle-point ; and as the rod is always kept perpendicular to the paper, and stationary, it is the handier form for describing very small arcs. 15. Irregular Curve. — This instrument has a variety of forms, two of which are shown in the accompanying figure. It is very useful as a guide to the pen or pencil point in drawing bends of roads and other accidental curves, giving a smoothness to the drawing not otherwise easily attain- able. The only difficulty attending its use is to avoid making angular junctions in the prolongation of a line. This is obviated by being careful to make the edge of the curve, at the beginning of the prolongation, tangent to the latter part of the line already drawn. 16. Beam-Compass {m, Fig. 2). — This instrument is used for describing arcs of greater radii than can be described with the dividers and extension-bar. As shown in the figure, it consists of two metal clamps which can be attached to a straight-edge or bar, and at any required distance apart. The clamps have sockets for pen, pencil, or needle points. Some kinds require special bars to fit the clamps, while in others, termed " portable beam-com- passes," the clamps can be attached to any straight-edge. The pattern (McCord's) shown in 6 TOPOGRAPHICAL DRAWING. the figure is substantial, and has exact means of adjustment afforded by the form of clamp, the part carrying the point being made to traverse the slide by means of the milled-head screw, while the points remain perpendicular to the paper. It is best used with a graduated bar made to fit it. A support furnished with casters is often used with a beam-compass to support the bar at its middle point. 17. The Curve-Pen (Fig. 3). — This differs from the right-line pen in having curved nibs attached to a rod which turns freely within the handle. It is particularly useful in drawing curves free-hand; the pen for this purpose being held with the handle perpendicular to the surface of the paper, and moved in the direction of the required curve — the point following the pencilled tracing. By means of a nut at the upper extremity of the handle, the latter may be clamped to the rod, and the instrument used as a right-line pen. 18. Dotting, Border, and Road Pens. — These may be termed labor-saving instruments, since by their use regular spacing and even width of line are attained by a single stroke, while the eye and hand are only engaged in following the right direction. The Dotting- Pen (Figs. 4 and 6) has a set of interchangeable wheels, with teeth or cogs corresponding to the arrangements of dots and dashes required. In the form (Fig. 6), the teeth act as cogs upon a lever which carries a pen-point at its outer extremity ; and motion is given to the cog-wheel by rolling the outer wheel, fastened to the same arbor with the latter, along the edge of a ruler. In the other form, the wheel is supplied with ink from a reservoir above, the teeth being applied directly to the paper. This instrument is also called a roulette. The Border and Road Pen (Fig. 5) is used for drawing either heavy or parallel lines. For the former, the space between the branches is filled with ink, and adjusted by means of the large milled-head screw ; for roads, the branches only are filled, and can be independently adjusted to different widths of Hne. It is useful in drawing any features represented by parallels. Some draughtsmen prefer the branches bent in the same direction, to facilitate following the edge of a ruler closely. 19. The principal instruments for measuring lines and angles, or for setting off horizontal distances, azimuths and bearings, are the scale of equal parts and the protractor. To a description of these is added that of the sector, mainly on account of its usefulness in findino- proportional parts. These instruments are made of boxwood, ivory, hard rubber, or metal. Horn and paper are also used for protractors and scales. 20. Scale of Equal Parts. — A convenient form of this instrument is the " triangular scale of equal parts" (Fig. 8), so called from the shape of its cross-section. Its different edges are graduated to show 10, 20, 30, 40, 50, and 60 parts of an inch, and for finer work 80 and 100 parts. Those graduated the entire length of the edges arc most convenient. Distances are laid off directly with this scale. The " scale-guard " is a sliding attachment used with the scale to save time in finding the particular edge in use, and in repeating a measurement. In the use of paper scales, which are accurately engraved or machine divided, and similarly graduated, the variations in length due to heat and moisture, are sensibly equal to corre- sponding changes in the drawing ; besides, they are not so liable to soil the latter as scales made from other material. Scales of equal parts, graduated to other convenient units, as chains, metres, etc., are likewise readily obtainable. When much time is required for the completion of a drawing. INSTRUMENTS AND MATERIALS. 7 greater accuracy is probably attained by the use of a scale constructed upon the drawing itself. 21. Spacing-Dividers. — The steel points of dividers are of frequent use in comparing dis- tances, and in transferring them from a scale ; but when great nicety is required and succes- sive small distances are to be laid ofT, the spacing-dividers, which differ from the spring- dividers (/&, Fig. 2) only in having two steel points instead of one, are very useful. 22. Protractors. — These instruments, in general use in plotting, are for the direct measure- ment of angles. The usual forms are the rectangular, semicircular, full circle, and vernier. The rectangular protractor is graduated on both faces, the front face (Fig. 9) having three edges graduated into degrees or half-degrees, the third edge being the diameter of the circle of graduation, with its centre marked as shown. The divisions are commonly numbered from 0° at the left, around to 180° on the outer edge, and repeated in the reverse direction on- an inner line. This face also contains scales of equal parts of an inch, and a scale of chords marked " CHO" or " C." The other face contains like scales, with the addition of a diagonal scale of equal parts. The Abbot Protractor, of rectangular shape, has the edges of each face graduated to half- degrees, numbered from left to right; on one face from 0° to 180°, and on the other from 180° to 360°. It contains a scale of tenths of inches, and one of hundredths of a foot. It is particularly handy in field-sketching. 23. The Semicircular Protractor is of two forms — either with the graduation extending a few degrees beyond the diameter, or with the diameter and outer edge coincident. In the latter case a straight-edge can be utilized in securing exact coincidence of the diameter with a given line, by first adjusting the straight-edge to the line, and then placing the diameter against it. The circular protractor is simply an extension of the graduation to 360°, covering an entire circle. 24. To protract an Angle with either of the above forms : The instrument is laid flat upon the paper, and held securely in position ; a line is drawn along the diameter, and points are marked upon the paper exactly beneath the centre, and the degree-division corresponding to the required angle ; then a right line joining these points will make the required angle with the first line drawn. To draw a line through any point of a given line, which shall make a required angle with the line : the diameter and centre are first placed coincident with the line and point respectively, and the operation is then as above described ; or the centre and degree-division corresponding to the angle may be placed coincident with the line, and a line drawn along the diameter. To protract an angle with the scale CHO : From a given point of a right line as a centre, with the distance 0-60 as a radius, describe an arc intersecting the line ; from this point of intersection as a centre, with a radius equal to the distance from o to that division of CHO corresponding to the number of degrees in the required angle, describe an arc inter- secting the first arc ; the right line joining the point of intersection of the arcs with the given point will make the required angle with the given line. 25. To Plot a Bearing. — In plotting bearings taken with the " surveyor's compass," the quadrant in which a course lies is apparent from the designation of the bearing : this is also the case with the improved prismatic compass {a, par. 154); but a little difficulty arises at 8 TOPOGRAPHICAL DRAWING. first with other methods of graduation used in the prismatic compass, which the following simple rule is intended to obviate. With the compass-card graduated from 0° at the N. around to the right 360° : I. Denote the difference between the reading and 180° by + i? or — i?, according as the reading is less or greater than 180°. II. For -|- i?,use the direct graduation of the protractor, placing the protractor so that its centre and the degree division R shall be on a meridian line passing through or near the plotted station, and the diameter shall pass through it ; a Hne drawn along the diameter from the station to the left will coincide with the required course. For — R, use the reverse graduation and draw the line to the right. It is apparent that, with a compass-card graduated from 0° at the N. around to the right 180°, repeated in the same direction on the Other semi-circumference, the only modification in this rule is that the W. readings themselves are the — R values. With the second protractor described in par. 22, and the above o°-36o° graduated card, the rule applies to readings less than 180° ; but owing to the direct graduation of this pro- tractor from 180° to 360°, readings greater than 180° are subtracted from 540°. 26. Vernier Protractor. — With protractors already described it is difficult to estimate as closely as one fourth of a degree; consequently for accurate work the instrument represented in Fig. II is used, The arm, movable about the centre, carries a vernier, reading in some instruments to minutes. The beveled edge of the arm is in the prolongation of the radius passing through the o of the vernier, and the exact centre is indicated by the intersection of two fine lines marked on a piece of horn let into the instrument. In protracting an angle, the arm is set by means of the vernier, the 180° diameter is placed coincident with the given line, with the centre at the given point ; and using a sharp-edged pencil, the required line is drawn along the beveled edge. In some instruments the graduation extends over a full circle. 27. Among other forms of vernier protractors are Crozet's and Tabarant's. In the former the scale is let into a flat rectangular metal frame. By turning the scale about its centre, the diameter may be set at any required angle with an edge of the frame, which edge is then placed against a straight-edge coincident with the meridian or other line of direction, and the protractor is moved along it until the diameter passes through the desired point. The special feature of Tabarant's protractor is a metal parallelogram, articulated at the corners, hinged at one corner to an extremity of the diameter and connected at an adjacent corner with the other extremity of the diameter by an arc, of which the length can be made equal to the magnetic declination. The parallelogram, of which the edge joining the other two corners- is placed against a straight-edge, serves to enlarge the area of plotting for any position of the latter. 28. In a semicircular protractor, the diameter and outer edge when not coincident, should be parallel. To test this : Draw a line along the diameter, also a radial line to any degree-division ; then slide the protractor along the radial, the centre and this division remaining upon it : the outer edge should coincide with the first line drawn. 29. Sector {:^\g. 10).— This is of about the form and size of the single -folding foot-rule. Radials, termed sectoral lines, arranged in pairs, one of a pair on each arm, and variously divided, are engraved on each face. INSTRUMENTS AND MATERIALS. 9 The sectoral lines on one face are a pair of scales of equal parts termed the " line of lines," a pair of scales of chords, of secants, and of polygons, marked respectively L, C, S and POL. A scale of tenths of inches is marked along the edge, and one of hundredths of a foot on the outer edge. On the other face are sectoral lines of sines, of tangents up to 45°, and another of tangents from 45° to 75° to a lesser radius, marked respectively 5 and T; together with "Gunter's lines" of logarithmic numbers, sines and tangents, marked respectively N, S, and T. The solution of problems with the sector depends upon the principle that homologous sides of similar triangles are proportional, and a solution is termed simple or compound according as it involves the use of one or two pairs of lines. In the latter case, from the above principle, the two pairs used should make equal angles at the centre ; but in some instruments the angles formed by the line T, from 45" to 75°, and of secants, are equal to each other, but unequal to those formed by the other sectoral lines ; therefore when one of these circular functions is used in a compound solution, it must be expressed in terms of some other function. A distance measured from the centre on a sectoral line is termed lateral (/) ; and from a point on one of a pair to the corresponding point on the other, transverse if). The divisions of a sectoral line are contained between three parallels, and the points of the dividers are always applied to the inner parallel, or to the one that passes through the centre. To use the line L : i. To find a fourth proportional, as x, in the proportion a : 6 :: c:x; set off /= «; open the sector until, at the extremity of l,t-=^b; prolong / until it is equal to c; then at the extremity of this prolongation,? = x. 2. To bisect a right line, as A : Make tio — 10^ A, then ^5 — 5 = — , which for the sake of verification is laid off from each extremity of A. 3. A line is divided into an even number of equal parts by first bisecting it as above, then bisecting each half, and so on ; and into an uneven number by a simple application of the same principle ; e.g., to subdivide A into five equal parts : make t^ — $ =^ A, lay off /3 — 3 from each extremity, and then bisect the extreme parts. 4. To obtain a scale of ^-^-j-, or i inch =10 feet, for measurements of feet and tenths of a foot : Make tio —10=1 inch ; then ^3.4 — 3.4 = 3.4 feet ; and so on. 5. For copying drawings to a scale differing from the original ; e.g., to reduce a drawing, lineally, one fifth : Make ^5 — 5 = 3- li^s of the original ; then ^4 ^ 4 is its reduced length, and points are found in a similar way by intersecting arcs described with reduced radii from the extremities of this line as centres, and so on throughout the drawing. The lines L may be used for interpolating heights. Thus, in the figure, the heights or references of the plotted points a and 6 are 20 and 25 feet respectively, and the line al> is of uniform declivity ; to interpolate the (23') point. On the lines L make t$ — ^ =^ al> ; then ^3 — 3 = ac, which laid off from a gives the required point. ^"^Q e ^C^^'J The lines C are used to pro- tract angles to any radius, /60 — 60. For any angle of 60° or less, say 20° : With i6o — 6o describe an arc; subtend any portion of it by a chord = ?20 — 20 ; the radials drawn to the extremities of the chord include the required angle. If the angle is very small, say 2", with t6o — 60 describe an arc as before ; then from any point of it set off 2 lo TOPOGRAPHICAL DRAWING. two chords, as t\0 - 40 and t\2. - 42, differing numerically by the number of degrees in the required arc : the radials drawn to the other extremities of these chords will include the required angle. For an angle greater than 60° : Divide the given number of degrees into parts of 60° and less ; with t6o - 60 describe an arc as before, and on it lay off consecutively as chords the values of t corresponding to the parts into which the angle is divided : the radials drawn to the points of beginning and ending of the series will include the required angle. The lines POL are used to inscribe a regular polygon, the radius of the circle being t6—6; the sides of the polygon are /4 — 4 for a square, /$ — 5 for a pentagon The lines C may be used in a siinilar manner by taking t6o — 60 as a radius and t = —- as a chord, « being the number of sides required, and an exact divisor of 360. The sine of an angle corresponding to a given radius is found by making igo — 90, on the hnes S, equal to this radius ; the sine required is then t, between the numbers correspond- ing to the number of degrees in the given angle. A tangent is similarly found by first making ^45 — 45 = to the given radius, and using the upper lines T for angles greater than 45°. For secants, /o — O is first made equal to the given radius. Any other circular function may be found by using its equivalent in terms of those above given. This instrument is not much used, but is very handy for the above purposes. The use of " Gunter's lines" can be found in treatises on mathematical instruments. 30. Drawing -Paper. — Whatman's paper is generally used for topographical purposes. There are three kinds: the "hot-pressed" (H), which has a smooth surface, mostly used for fine-lined drawing; the "not hot-pressed" (N), which has a finely grained surface, and is in general use for map-work ; and the " rough" (R), which has a coarsely grained surface adapted to strongly lined work and drawings made to a large scale. The last two kinds, known also as " cold-pressed," are suitable for colored maps, and generally for brush-work drawing. The following table gives the names of the different brands of Whatman's paper, sizes in inches of whole sheets, and the kinds of each brand usually obtainable. Brands. Emperor , Antiquarian Double Elephant Atlas Columbier Imperial Sizes. 48 31 26J 68 X 53 X 40 X 34 X 26 34iX 23i 30 X 21 ^inds N R H N H N R H N H N H N R Brands. Elephant Super Royal Royal Medium . . . Demy Cap Sizes. 28 X 23 27iX igi 24 X igi 22 X 17 20 X 15 17 X 13 Kinds. H N H N H N H N H N H N It is also graded according to quality into " selected best," which is without imperfec- tions, and "common;" but there is little difference in these either in cost or texture. The most imperfect sheets are found in the outside quires of a bundle. There is also a very fine, expensive grade of Double Elephant and Imperial known as "extra weight." The Double Elephant, Imperial and Demy are the brands mostly in use for ordinary map-work. The water-mark " Whatman," or " Whatman Turkey Mills," with date of manufacture, is readily seen by holding the paper between the eye and the light ; and the face nearest the eye when the name reads aright, is that on which the drawing is usually made; but there is little difference practically, a good rule being to choose the smoother face. INSTRUMENTS AND MATERIALS. ii It improves with age, since tlie seasoning process diminishes its Hability to vary in size. It may be obtained for long maps in continuous rolls varying from 36 to 62 inches in width, and either muslin-backed to make it less liable to tear, or plain. A point to note in the use of roll-paper for valuable maps is that it expands four or five times more in the direction of its breadth than that of its length. 31. Tracing -Paper and Cloth. — These are very thin and semi-transparent, and are used principally for copying purposes. Tracing-paper is obtained in sheets from 13X17 to 30X40 inches in size, or in continuous rolls from 37 to 54 inches in width. It can be made by applying a mixture of one part of boiled linseed-oil and five parts of turpentine, thinly, with a sponge, to one face of common tissue-paper, the paper resting upon a flat smooth surface : the paper is then hung up to dry, and when the more transparent oil-streaks disap- pear, is ready for use. Tracing-cloth is to be had in continuous rolls, from 18 to 42 inches in width. 32. Transfer -Paper. — This is thin paper having for drawing purposes one face covered with black-lead or other similar material, so that by laying this face on a sheet of ordinary paper and passing a sharp point over the reverse face, the mark thus made will be duplicated on the other paper. It should be very thin for transferring fine lines. The manner of using it is described under "Copying." It can be made by sifting pulverized black-lead over a stretched piece of strong tissue-paper, dusting off the larger particles, and then rubbing the surface with cotton until smooth, or until it will not easily soil the surface on which it is to be used. The different-colored ochre-powders may be utilized in the same way. 33. Cross-section and Profile Papers. — These are used for drawing directly to scale, and for sketching purposes. The former, in sheets from 14X 17 to 16x22 inches in size, or in rolls 20 inches wide, is ruled in squares of one sixteenth to one fourth of an inch, or according to the metric system, of i mm. or greater ; each fifth, tenth, or other convenient line being ruled heavier, to facilitate reading the distances. Profile paper is similarly ruled, except that the vertical are usually one tenth of the cor- responding horizontal intervals. 34. India Ink in sticks or in liquid form is always used in topographical drawing. The stick variety is the best and is the only kind suitable for brush-work. It varies much in quality ; the best kind having a smooth surface, and showing a brownish hue after rubbing on a wet surface, and a bright, iridescent surface of fracture. Although any small clean dish will serve for preparing the ink, it is more convenient to use an ink-saucer, provided with a cover to exclude the dust, and a deep cavity to hold the ink and retard evaporation. To prepare it, a very little water is poured into the saucer and the stick is rubbed in it, the end being pressed firmly on the bottom of the dish, until a jet-black liquid results. The degree of blackness is readily determined by observing the shade produced in drawing a line on white paper and blurring it by rubbing while wet ; a magnifying-glass is very handy in this connection. The ink is best freshly prepared for each day's work. A few drops of ox-gall will make it flow freely. The fluid variety has the merit of being always ready for use, only requiring the addition of a few drops of water, alcohol, or ammonia, when too thick, and to be shaken occasionally; 12 TOPOGRAPHICAL DRAWING. but unless of the best quality, it is liable to " run" when the lines made with it are washed, and is therefore not safe to use for brush-work. It also varies in intensity. The few colors used in plain topographical drawing are described in Part III. 35. The other necessary materials and instruments are: drawing-pins, commonly called " thumb-tacks," sharp pointed, and provided with flat, thin heads so as not to obstruct the movements of other instruments over' the surface of the drawing; horn-centres (par. 13); a penknife, a fine, flat file and emery or fine sand-paper for sharpening pencils ; a finely grained stone for sharpening pen, and needle-points ; a magnifying-glass for ascertaining the condition of pen-points, the exact intersections of lines, etc. ; two or three sponges, the largest about 6 inches in diameter; blotting-paper, a rubber and an ink-eraser, and a drawing-board which is described in the following paragraph. Materials specially required for colored drawing are described in Part III. Section II.— Preparation of the Paper, Drawing -Boards, and Practical Rules AND Suggestions. 36. Preparation of the Paper and the Plain Drawing-Board.— Y ox some purposes the paper might be simply fastened along its edges with thumb-tacks to a flat smooth board or table ; but for an elaborate drawing with pen or brush, and for the sake of accuracy and ease in working, it is stretched smoothly upon and firmly fastened to a drawing-board. The latter is of well-seasoned wood, from three quarters to one inch in thickness, and at least half an inch larger each way than the paper for the drawing, and to prevent warping, I @ % § @ @ 9 / A 1 J / / ^ G :^ strips of the same thickness are tongued and grooved to the edges, perpendicularly to the fibre ; or, if but one face for use is desired, a strip may be screwed to the back near each end. For use with the T-square, the board shoul(J be rectangular. Straight smooth strips of hard-wood, against which the head rests, are sometimes fastened along the edges, but an ordinary straight-edge may be used for this purpose. A substantial single-surface drawing-board is shown at A in the accompanying figure. — Longitudinal cuts about one eighth of an inch wide and two inches apart extending half-way through the board, are made along the back, and a hard-wood strip is then screwed to the PRACTICAL RULES AND SUGGESTIONS. 13 back near each end ; all the screws except the centre ones passing through slots in the strip which are furnished with metal washers. By this arrangement the warping takes place mainly in the different sections separated by the cuts ; the play of the screw-heads in the slots permits of a general expansion or contraction without bending the strip, which could be of metal, and the drawing surface is practically unimpaired. 37. Stretching the Paper. — This operation so often proves a source of annoyance on ac- count of the edges loosening in drying, that the following invariably successful plan is given in detail : a. Prepare some hot glue in the usual way ; light-colored or clarified glue is the best. b. Expand the paper by soaking it in water until it is limp, or will not spring back when bent. c. Let it drain till the water drops slowly from it ; then place it back up on a clean flat surface, and dry the edges a little more by applying blotting-paper along them. d. With a stiff brush — a " sash-tool " is the best— apply the hot glue rapidly along the edge for a space one half to one inch in breadth, according to the size of the sheet. e. Take the paper by its diagonally opposite corners, and place it, face up, evenly on the drawing-board, which should be dry and free from dust ; and with the finger-tips press each edge rapidly and firmly to the board, passing them from the middle point of the edges out- ward to the extremities. /. Place the board in a horizontal position, face up, to dry exposed to the ordinary tem- perature of the room. Some draughtsmen prefer to expand the paper by laying it on a flat, clean surface, and applying a wet sponge to the back ; and this plan would be more convenient with very large sheets. In using cold glue or paste, it is necessary in drying to secure the edges with thumb- tacks or by tacking a strip of wood along them, and safer to keep the central portion of the sheet damp until the edges adhere firmly. In expanding with a sponge the edges may be turned up to keep them dry. 38. Stretchmg-Frames. — A very expeditious way of stretching paper is to use the frame shown in section at B in the preceding figure. The expanded papery is laid upon the solid back b, the hard-wood frame / is then pressed down over the edges of b and secured in place by buttons attached to the lower surface oif. The rabbets should be so proportioned that the paper may be flush with the upper surface of/" when the frame is closed. For a brush-work drawing not exceeding about 2X3 feet in size, the most conven- ent form is a stiff, rectangular, open frame, on which the paper is stretched as described in par. 37. The open space permits the paper to be dampened on the back, so that in extend- ing the shades or tints the edges of the latter will not dry too rapidly. To prevent indent- ing the surface of the larger sheets, the hand should rest upon a support extending across the frame. With a support for the back of the paper while working, sheets much larger than 2X3 feet may be used with this frame. 39. Mounting Paper on Cloth. — The cloth used is bleached muslin, which is first attached to a stretcher, and the paper then pasted to it. A stretcher for small drawings, not exceeding about 20 X 30 inches, is a flat, rectangular, wooden frame {C, preceding fig.), the sides and ends of which are about four inches wide and three eighths of an inch in thickness. For large 14 TOPOGRAPHICAL DRAWING. drawings these dimensions are increased and cross-pieces are necessary to give sufficient strength and stiffness to resist the tension of the paper in drying. The inner upper edges^, ^, of the frame are rounded to prevent creasing the paper; and the cross-pieces c are let into the back at suitable intervals, so as to just reach the rounded edges, the rabbets r, r, not extending more than one third across the sides. To stretch the cloth, it is laid upon a smooth, flat surface, the stretcher is placed face down upon it, and the cloth is drawn tightly over the ends and tacked to the back of the frame. The middle points are tacked first ; the tacks are placed opposite each other and not more than three inches apart, and it is best to fold the edges of the cloth in tacking them ; the sides are then fastened in the same way. The paper is now placed face down on a cloth-covered table and expanded with sponge and water; the surplus water, or little pools, are then removed and cold flour-paste is thoroughly applied with a large brush, the stretched cloth is then placed upon the pasted surface and the back of the cloth is well rubbed to make it adhere closely. The stretcher is then reversed, the edges of the paper are rubbed down, and any air-bubbles are pricked v/ith a fine needle. In order to use the T-square, straight strips are fastened to the rough cloth edges. Ti'acings should first be mounted on paper. 40. To Join Sheets of Paper. — Each of the meeting edges is first beveled by placing it along the edge of a table and rubbing it down with fine sand or glass paper, making it very thin at the extremity. The edges are then glued together, rubbed down, and placed between straight-edges or flat surfaces, and under pressure, to dry. For additional security, a paper strip may be glued to the back of the joint. If joined sheets are to be stretched, it is best done before the joints are quite dry ; a sponge is used in dampening and the joints are carefully avoided. The joints in tracing- paper are made very narrow, and after being secured, a strip of paper is laid over them on each face, and they are left to dry, either rolled tightly, or upon a flat surface and under pressure. 41. The room used for drawing purposes should be well lighted. In an upper story the light is less liable to obstruction, and besides, the paper is less exposed to moisture. If it has a southern exposure, white window-shades are needed. The drawing-table, arranged for a sitting or a standing position, according to the size of the drawing and the convenience of the draughtsman, is placed near the window, and the board is so arranged upon it as to receive the best light available. A light from the left and front, striking the paper at an angle of about 45° with the horizontal, and an inclination of the board, when not very large, of about 5° downward toward the draughtsman, are favorable conditions. When gas-light, which is very objectionable, is used, the burners should be about two or three feet above the table, furnished with reflectors, and project horizontally so as not to cast a shadow upon the paper. For extensive work, running water, or a good supply of this very necessary article, with large receptacles for expanding paper and other uses, should be available. The paper must be kept as clean as possible, and for this purpose should be dusted each time before beginning work, and the straight-edges and triangles wiped clean; while at work, white paper should be interposed between the hand and the drawing, and the latter covered with white paper or cloth when the day's work is done. PRACTICAL RULES AND SUGGESTIONS. 15 It is advisable, in filling in the details, to expose only so much of the drawing as is needed for the time being. 42. Erasures must be carefully made. To erase pencil-lines : Remove as much lead as possible by rubbing bread-crumbs on the paper; then use an edge of the rubber eraser — a sharp wedge-shaped piece cut from the rubber is very handy for this purpose. To remove ink-lines : Apply blotting-paper and wait till dry ; then, if fine, use a needle-point, otherwise glass-paper or a very sharp knife or ink-eraser, keeping the blade perpendicular to the paper and moving it in different directions, the paper resting on glass or other smooth surface ; the rubber eraser is then applied, and the abraded surface afterwards rubbed smooth with the thumb-nail or with an ivory handle. An ink line or blot can be covered with Chinese white. If the drawing is to be shaded or tinted with the brush, neither the knife nor rubber is used for erasing ; pencil-lines are removed with bread or, like ink-lines, are washed out with sponge and water. As precautions against blotting, the ink-saucer is kept at the side of the drawing, the quantity of ink in the pen should not be more than is retained when the latter is lifted sud- denly over the saucer, and the outside of the nibs must be free from ink. 43. Rules for Line Drawing. — Lines are classed as dotted, broken, and full. A dotted line is a succession of equally sized dots with equal spaces ; a broken line, a series of dashes of equal length, equally spaced ; and a full line is simply a continuous line. For convenience of description, lines are graded as fine, medium, and heavy ; the first grade are as fine as can be drawn and be clearly visible ; the second are twice the breadth of the first ; and the third, twice the breadth of the second. In large maps drawn to a large scale these breadths are increased, while still retaining their relative values. 1. All construction-lines are fine, are drawn with as little pressure as possible ; and to show points of intersection distinctly, are prolonged a little beyond them. 2. All straight lines should be drawn with the straight-edge and right-line pen ; the lat- ter being moved along the upper or farther edge of the former, and in a direction from left to right. If drawn free-hand, the direction is the same or towards the draughtsman. In pro- longing a line, a slight interval is left between the original and the added portion, which, though hardly noticeable, can be afterwards filled with a common pen. 3. In laying off consecutive distances, to prevent cumulative errors, each point of divi- sion is located by measurement from the initial point ; or the total distance is laid off, and then subdivided. 4. Right angles are preferably determined geometrically with dividers and intersecting arcs ; but if with other means, the accuracy of the latter should be carefully tested. 5. Intersections of lines meeting at a very oblique angle can be determined, as shown in the figure, by assuming a point a on the bisecting line ab, and locating b so that bd, perpendicular to cd, shall be equal to rtc, also perpendicular to cd\ then s, the middle point of ab, is the required point. 6. When practicable, lines are drawn from, rather than to a given point ; tangents are drawn to arcs rather than the reverse, and the points of tangency are geometrically con- x6 TOPOGRAPHICAL DRAWING. structed. A point is marked when determined, by enclosing it in a very small circle, or, if a triangulation point, in a triangle ; a square figure is used to enclose a stadia survey point. 7. To prevent making ragged lines with the pen, it should be tried on a separate piece of paper or on the margin of the drawing each time it is refilled, or after any delay in the work, before applying it to the drawing. 8. Uniformity of line for like details in any map is necessary for clearness and to pro-, duce a pleasing effect ; and this is obtained by using a pen of such capacity that with Hght, even pressure, the ink will flow freely from it, and by pressing the nibs equally upon the paper. Section III.— Scales and Plotting. 44. Scale of the Map. — The representations on a map or plan bear a constant relation or ratio to the actual forms, dimensions and distances ; and this constant ratio is termed the scale of the map. This ratio could be assumed arbitrarily; but for convenience in calcula- tion, and to accord with the decimal division of scales of equal parts used in plotting, it is represented by a fraction of the form — , termed the representative fraction {r.f.), having unity for its numerator, and for its denominator some multiple of 10. If d and D represent re- spectively corresponding plotted and actual distances, expressed in units of the same denomi- d . . d \ . D nation, the ratio is -pr! dividing each term \>y d.-^f:: = — ; from which <^= — , and D = d y, D Dm m m: from which expressions, either the actual or plotted distance being given, the other is obtained ; therefore, to convert an actual into a plotted distance, multiply it by the representa- tive fraction ; and to convert a plotted into an actual distance, divide it by the representative fraction. For example : D, measured on the ground, is 375 feet, and r. f. = 5-^- ; then d = ^Vt f^^t = 3 inches: or, in the other case, d being given, then Z> == 3 X 1500 = 4500 inches = 375 feet. To ascertain the scale of a map, it is only necessary to place any plotted distance d with its corresponding actual distance D expressed in units of the same denomination, under the d . I . form of the fraction -=r, and reduce this fraction to the form — , or one having unity for its D m ^ numerator. 45. The Scale of Distances is used for mutually converting actual and plotted distances without calculation, and is constructed as follows : The data required are the scale of the map, or r.f; the units of measure, whether feet, yards, metres ; and the "reading" desired, whether thousands, hundreds, tens, or single units. Assuming r.f. = ts'tt' units of feet, and the reading, hundreds and tens of feet : Draw a right line — the fine upper line shown in Fig. 12 — and, since i inch on the map represents an actual distance of 1200 inches = 100 feet, divide this line into inches, subdivide the left-hand interval into tenths of inches, and mark and number the different points as shown. The part to the left of o is termed the " extension" of the scale, and its subdivisions secondary, or minor, in contradistinction to the primary or main divisions of the scale. The unit employed should always be shown, and is usually placed after the last or right-hand number, or as shown in the figure. The scale is completed by adding a heavy line below and the ;•./. above, as shown. SCALES AND PLOTTING. 17 This scale might be designated a " scale of i inch to 100 feet," written in place of or in addition to the r.f. If a map is to be reproduced by a photographic process, it is safer to omit the r.f. A corresponding scale of metres is also shown in the figure. Two or more scales having the same r./"., but different units, placed upon the same map, are termed "comparative scales ;" and as the early general adoption of the metric system is possible, it is advisable to have scales of English units and metres on all important maps. Owing to changes in the size of a drawing, due to heat and moisture, and to shrinkage when cut from a stretcher, a scale of distances should be constructed upon the map when the drawing is begun. This is more particularly necessary in important maps requiring consid- erable time for their completion. (See also par. 128.) 46. Diagonal Scales. — In using the scale just described, distances less than those given by the secondary divisions have to be estimated, and some means for their exact measure- ment are necessary. This is afforded by the diagonal scale, which is constructed as follows : Using the same data as in Fig. 12, construct the scale of distances represented by the lower line in Fig. 13. At the o point erect a perpendicular of convenient length, and divide it into ten equal parts ; through the points of division draw parallels to the line of the scale of distances, and erect perpendiculars at the 100, 2CX), 500 points. In the rectangle thus formed on the " extension," subdivide the side opposite the extension into ten equal parts, and join its points of division with the alternate points of division of the extension, by the diagonals, as shown. It is evident from inspection that each diagonal in its ascent gains a total distance of -^^ of an inch, or 10 feet, to the left of a vertical, and therefore a distance of I foot to the left, between each two consecutive horizontals. For convenience of application, the horizontals are numbered from below upward as shown. To set off 308 feet : The diagonal through o of the scale of distances having gained a distance of eight feet to the left, at the horizontal numbered eight, extend the points of the dividers on this horizontal, from the 300 vertical to the first or nearest diagonal ; for 347 feet, extend them on the seventh horizontal, from the 300 vertical to the fourth diagonal; and so on. To show furlongs with an extension reading to single miles, only eight horizontals would be necessary ; and similarly, the diagonal construction can be applied to any other unit. This scale and the scale of distances used in the construction of the map are sometimes called " Scales of Construction." 47. A vernier is a small scale of equal parts used for reading fractional parts of the divi- sions of the scale to which it is attached. Referring to Fig. 15, the vernier Vis equal in length to 9 divisions of the scale S, and is divided into 10 equal parts. Suppose 5 to be a scale of tenths of inches ; then the difference in length between a division of S, or -^^ of an inch, and a division of V, or ^ of ^ of an inch, is ^-^ — yf^ = 3-^ inch, which is called the " least count of the vernier," and is the smallest fractional part that can be read by it. The least count is numerically equal to the value of one division of the scale divided by the number of divisions of the vernier. Algebraically considered, let s = length of one division of S, v =: length of one division of V, and let m of the former divisions equal in length m-\- \ of the latter : then ms = in , , , m s (m -[- 1) v; v = 1 — s; and s — v, or the least count, — s ; — j = j — , as above. 1 8 TOPOGRAPHICAL DRAWING. The o of the vernier is the index of the scale : referring to the figure, if V, o and S, 7 coincided, the reading would be 7 inches; but in its present position the fractional part between o and 7 must be added to seven inches. If V, i coincided with S, 7.1, the addition would be just equal to the least count, or ^ inch ; if F, 2 coincided with S, 7.2, it would be yf^; but, as shown, V, 7 coincides with 5, y.J, and the reading is 7. +TT7r o"" 7-<^7 inches; therefore to read the vernier: If its O division coincides with a division of S, the latter gives the required reading; if not, read the next preceding division of S; find the F division that coincides with any 5 division, multiply its number by the least count, and add the product to the 5 reading. If several divisions of S and F appear coincident, use the middle one ; if none are coincident, and a division of 5' is midway between two of V, the true reading corresponds to the point midway between them. Verniers are classed as direct, to which class the one just described belongs, and retro- grade. The number of divisions of a direct vernier is greater by one, and of a retrograde vernier less by one, than the number of divisions of the line or arc which they respectively cover. The divisions of the former are numbered in the same order, that is to say, the num- bers increase in the same direction, as those of the scale to which it is attached ; and the reverse usually obtains in the latter class. The retrograde form is used for convenience when the vernier cannot be moved a dis- tance, at least equal to its length, beyond the extremity of the scale. Thus in Fig. 16, read- ings are obtained up to 12 inches, without the index passing that division; the reading as shown is 11.97 inches; the index having passed the 11.9 division of S, and V, 7 coinciding with an 5 division. Fig. 17 represents a vernier for reading fractions of a degree. The limb is graduated to J", as in the common vernier protractor, and since 15 V divisions cover 14 S divisions, the i° least count is — - = i minute. For convenience in reading in either direction, the vernier is made of double length, the graduation is extended in each direction from the index, and it is called a double vernier, in contradistinction to those already described. The reading as shown is 22° 37', the index moving to the left having passed the 22f° division of 5 ; and V, 8, to the left of the index, coinciding with an 5 division. This reading is obtained directly from the right-hand vernier. Considering 20 and 25 of S interchanged, and the measurement as made to the right, the left-hand vernier is used, and the reading is 22° 23'. 48. Vertical and Time Scales. — The vertical scale, or scale of heights, is generally greater than the scale of distances, and the proportional increase is termed " the exaggeration of the scale." This is made necessary because differences of elevation are less apparent than hori- zontal distances; and besides, in levelling, very minute differences of elevation are often required. The usual exaggeration is 10 ; e.g., with a scale of distances of -^^, the scale of heights would be y-J-j-. Time-scales are used to measure distances by the time required to traverse them ; the r.f., rate of motion, and the desired reading arc the data required. They arc used in rapid sketch- ing, and for plotting maps from itineraries, the intervals of time required in passing from point to point being noted (see g, par. 50). 49. To Divide a Line into Equal Parts.— ¥\g. 14 illustrates a convenient method of divid- ing a line into equal parts. To divide .4^ into five equal parts: Draw yiC, and beginning SCALES AND PLOTTING. 19 at A, set off with dividers or measuring-scale the five consecutive equal distances Aa, ab de; join the last point of division e with B, and draw dd' , cc', parallel to CB: the points of intersection of these lines with AB are the required points of division. CAB should be of such a size as to prevent very oblique intersections with AB. This construction depends upon the principle that like parts of similar triangles are proportional to each other. In a like manner, to subdivide an interval between two parallel lines into equal parts, other than those marked on the scale : Place the scale obliquely, so that the required number will be included between the lines ; mark the points, and draw lines through them parallel to the given lines. 50. Examples of Scale-Construction. — a. Required a scale of 6 inches to one mile to read thousands, with subdivisions of hundreds of feet : 6 _6 I _ No. of inches in one mile ~ 63360 ~ 10560 "" '^' ' and for a total representation of 5000 feet, 6000 X rcrinr = 5-68 + inches; a sufficiently close approximation is obtained by using 5.68 (see par. 51). A right line 5.68 inches in length is then subdivided into five equal parts, each part representing looo feet, and the extension also subdivided into ten equal parts will show hundreds of feet. The total length could also be obtained from the proportion, 5280 (No. feet in one mile) : 5000:16 : total length of the scale. b. Required a scale of four inches to one mile to read thousands and hundreds of yards, ^■f- = 15840 - F^'' ^ t°tal representation of 3000 yards, 1760 (No. yards in one mile) : 3000 ::4: total length = 6.82 inches, very nearly. Divided into three equal parts, each part will represent lOOO yards, and each of the ten subdivisions of the extension will represent 100 yards. c. Given r.f. = -s-oTnr ! t° construct a scale with primary divisions of 100 feet each : 100 feet = 1200 inches, and 1200 X -jinnr = -4^^ inches = length of a primary division. 1000 feet is represented by 4.166 inches, which is laid off and divided as already described to show hundreds and tens of feet. d. Given a map with no scale or r.f. upon it ; to find the r.f. : Measure an actual dis- tance, and its representation on the map; and place these measurements, reduced to the same denomination, in the form of a fraction with the plotted distance as the numerator, and reduce to the form — : a scale can then be constructed as already described. If the superficial area only of the surface represented is known : Let the map be 12 inches square and represent 10 acres; to find r.f.: \oA = 48400 square yards; 12 inches therefore represents V48400 = 220 yards = 660 feet; .-. r.f. = ^, from which the desired scale may be constructed. If this map were 8 X 12 inches in dimensions, then one square inch would represent 504 square yards, and r.f. = ■^, approximately. e. As already indicated, comparative scales have the same r.f, and different units of measurement. A French map has a scale of French leagues attached, and the r.f. is omitted. By measurement, the distance o — 30, or 30 leagues, on the scale = 2.5 inches. A French league = 4262.84 yards. To construct a comparative scale of English or statute miles : 30 French leagues = 127885.2 yards, 100 miles = 176000 yards; therefore 127885.2 : 176000 :: 2.5 : the number of inches representing 100 miles = 3.44 inches very nearly ; and a right line of this length is divided and subdivided as already explained. 20 TOPOGRAPHICAL DRAWING. f. Having subdivisions of sixths of degrees, to construct a direct vernier to read minutes: s ' J — z^ = T^L- = ; — = — ^7 — ; therefore ;«+ i = 4 X 60 = 10, or 10 F divisions cover 9 5 ^^ m -{- \ m -{- \ 16 divisions. For a retrograde vernier, v —s = ; ;;^ — i = 10; and 10 F" divisions cover II S divisions. For a direct vernier reading to 20 seconds, s — v = 20" = j — ; therefore m -\- 1 = -^"J- = 30; and 30 /^divisions cover 29 5 divisions. g. The rate of motion is three miles per hour; to construct a scale of five miles to one inch {r.f. = -gxoVmr)' *^° show distances traversed in five minutes: Let x inches represent the distance travelled in one hour ; then 316800 : i :: 3 X 5280 X 12: x ^ -^-^ inch; therefore the distance traversed in five minutes is ^V o^ t^ = xir hich, and the scale is constructed accordingly. k. Given a scale of 500 paces to the inch, the average pace being 29.63 inches, to con- struct a scale of yards ;r._/. is 14 ^15, and to read 500 yards the length of the scale is 500 X 36 X i^li j = 1.21 inches, -J- of which represents 100 yards. i. To construct a scale of paces reading to looo paces, and showing 10 paces; the aver- age pace being 31 inches, and r.f. ^whs- Its total length is fflfg- = 2.935 inches, which is subdivided into ten equal parts representing 100 paces each, etc. 51. Inferior Limit of Scale Measurement. — The smallest distance that can be estimated by the eye, or that can be measured with dividers, may be assumed as o.oi of an inch ; and this distance is the probable error which would be made in plotting, or in measuring dis- tances on the map. Denoting this by e, the scale of the map being — , the approximation to m the true distance is e X m ; and in plotting with the instruments ordinarily used, distances less than this cannot be exactly represented, e.g. If r.f. = -g-^-jnr' e X ?« = 5C inches; if ^■/- = -BTTiTF' e X m = c^o feet; which distances are therefore the inferior limits of their respective scales. 52. TAe principal conditions governing the choice of a scale are — The smallest measured detail required upon the map should be clearly represented. To economize time, the smallest scale consistent with the above condition should be chosen. The actual and plotted distances should be readily convertible. As to the first condition, since an application of the formula e X m (par. 51) determines the least dimension that can be clearly represented according to any given scale, it is a simple matter to conform to it. The second condition is particularly applicable to maps of laro-e extent, or those covering large tracts containing much detail. The third condition would be perfectly fulfilled by the adoption of the metric system of measurement ; but since English units are in general use in the United States for surveying purposes, the best alternative is to apply the decimal system to them. The scale might then be represented by either one of the following fractions : ^V, rk- toVt; xio- ttsW Tuiw : irU- loW- -»W. affording an ample range for all purposes. Actual measurements could be made with the 50 or 100 feet chain, with tape or rod graduated into feet and decimals of feet, making reductions to feet for plot- ting unnecessary ; and the plotting would be done with a scale of equal parts of one foot, decimally divided. Some of the advantages of a simple system like this are saving of time, and less Hability to error in calculation, a tendency to uniformity of scale in maps made for like purposes, and the facility in reading maps which this uniformity would give. SCALES AND PLOTTING. S3. The following scales are used in different countries for topographical purposes ; IN GREAT BRITAIN. Representativb Fraction. For what used. jre^TO (i in. to I mile) TsirijUin. to i mile) The smaller Ordnance maps, surveys of provinces for both civil and military purposes; suited to maps of explorations. Smallest scale permitted for deposited plans of proposed works for military road reconnaissances. ttteotC^ in- to I mile) Larger Ordnance maps. Just large enough to show roads, buildings, and other impor- tant objects distinctly in their true proportions, and at the same time embrace a plan of a considerable extent of country. Best adapted for selection of lines for engineering works, for parliamentary plans and preliminary estimates. Suit- able for military positions. Decimal scale suited to same constructions. Smallest scale permitted for "enlarged plans" of buildings, and of land within the curtilage. Suited to working surveys and land plans of great engineering works. For plans of part of the Ordnance Survey from which above Ordnance maps are reduced. Suited to land plans of engineering works and of estates. wW(40oft. to I in.) nrerr ^VsC^ooft. to I in.) T^(iooft. to I in.) Smallest scale prescribed for land or contract plans in Ireland. Suited to plans of towns when not very intricate. TsW (6° '"■ '° I mile). . . . „i Ordnance plans of same. Decimal scale suited to same. Ordnance plans of the more intricately built towns. Decimal scale suited to same. For Plans of Railways. ^ (120 in. to I mile) 1 Minimum scale for plans and sections. Minimum vertical scale for sections. Minimum scale for cross-sections and sections of road alterations. „i, IN FRANCE. Representative Fraction. For what used. 1 For the new engraved map of France. For the surveys from which the above map was made. For surveys of large extent, military reconnaissances of frontiers, and the encamp- ment of an array. Plan of a place with its environs, of a canton, special reconnaissances. Surveys of a town with a large extent of the surrounding country, plans of battlefields. Surveys of cities, routes, and defensible positions. For the front of a fortification, with its outworks. loioo 1_. winr i_ IN THE UNITED STATES. Representative Fraction. For wjiat used. For small areas, such as farm surveys. For topographical surveys in general. For topographical surveys of large areas. For general maps covering a large territory. TTiirn!-' TBTTinr' isrireTr 26005-. 4o5o8i FliTiTrir aoootnr '° 4 o^o 22 TOPOGRAPHICAL DRAWING. IN BELGIUM. Representative Fraction. For what used. 1 For mineral concessions. ^, , , , , ■ These are also used for farm surveys. For projected railways. For topographical plans. For topographical maps. 54. Plotting consists in locating on paper and in their true relative positions, the points, lines and angles determined in the field. Topographical maps are usually confined to the representations of areas of the earth's surface so small that in their construction the earth's sphericity may be neglected ; therefore the methods of plotting used in plane-surveying are appropriate. The methods employed for representations of larger areas are described in Part V. The usual sequence observed is to first define the limits of the map and the proper direc- tion of a meridian line, then to plot the triangulation, traversing and levelling, in the order given. 55. Plottitig of Points, Lines and Angles. — A point is plotted by assuming its position, by determining its direction and distance from a given point, or by the intersection of given lines. Its position is indicated as described in 6, par. 43. A right line of known length and unknown position is plotted by assuming a point for one extremity, and, with a straight-edge as a guide for the pencil-point, drawing a fine line accurately through the point ; the other extremity is then determined by the application of the scale of distances ; or the exact length may be laid off from the assumed point with the scale, and the line drawn afterwards. In either case the extremities must be indicated exactly opposite the proper scale-divisions. The manner of plotting or protracting angles with the protractor and with the scale of chords has been described in pars. 24 and 25. A table of natural sines, cosines and tangents (see Table III., Appendix) is frequently used for this purpose, e.g., To plot an angle A : Make AB (Fig. 18, Plate II.) = cos A to any convenient radius ; at B erect the perpendicular BC = sin A to the same scale; draw AC: then BAC is the required angle. Or make AB = cos A as before, then from B and A as centres, and with radii corresponding respectively to the sin. and radius reduced to the same scale, describe arcs intersecting at C. The natural sines can be used as a scale of chords, since the chord of an arc which meas- ures an angle as @ = 2 sin \@,. Thus, in Fig. 19, with D on DE as a centre, and a con- venient radius R, describe the arc Ea ; find from the table the sine of \®, multiply it by 2R, and with the product as a radius, and j5 as a centre, describe an arc intersecting the first at F. FDE is the angle required. To use the natural tangents: Set off .<4 5 (Fig. 18) any convenient number n of inches in length; erect the perpendicular j5C = k X tang yi, and join A and C. CAB is the angle required. 56. Defining the Limits of the Map and the Direction of a Meridian Z?'«r.— The total length and breadth of the map can be readily determined from the record by calculation : e.g., the sum of the Eastings and Westings is the horizontal, and of the Northings and Southings SCALES AND PLOTTING. 23 the vertical dimensions ; then assuming, when practicable, the top of the paper as N., the rect- angle to enclose the plot, and of which the sides are the inner lines of the border, is constructed as follows : Through the middle point of the paper, determined by its intersecting diagonals, draw two right lines perpendicular to each other, one of them at least parallel to an edge of the draw- ing-Loard. From the middle point set off half the length of the map in each direction on one of these lines, and half the breadth in each direction on the other, using the longer line for the greater dimension ; then right lines through the extremities of these distances and parallel to those already drawn will be the limiting lines required. A marginal space, as broad as may be desired, between them and the cutting-out line, may then be defined, and the part outside of this used for trying the pen, etc. Until after considerable practice in plotting, it is best to plot the triangulation, and at least the courses of the traversing, on paper other than that intended for the map. The limits of the map are at once shown, the inner lines of the border are drawn, or their posi- tions with reference to a meridian line marked, and the work is properly adjusted upon the map-paper and transferred, usually by pricking through the angular points (II., par. 140). By this plan the surface of the map-paper is to a great extent kept unimpaired, and there is no liability to an unfavorable disposition of the plot. As already indicated, the vertical edges of the border are usually parallel to a true merid- ian line, with their upper extremities toward the N. This disposition may be found incon- venient, especially on very large maps which for convenience in reading always require their greater dimensions to be placed horizontally ; but if practicable N. should be either toward the upper or the right-hand edges of the map. In any case the N. point should be in the upper quadrants. 57. Plotting the Triangulation. — Since the triangulation is the most important part of the field-work, the more exact methods are used in plotting it ; and of these a system of rectan- gular coordinates is the best, because each survey-point is independently fixed, and cumula- tive errors are thus avoided. In Fig. 20, the right lines XX' and YY' perpendicular to each other are termed the coordinate axes, and distances from either axis measured on the other, or on lines parallel to it, are the coordinates. Coordinates parallel to XX' are designated by the letters x, x', or x^,x° ; and those parallel to YY' hy y,y' or _;/% ^ Co- ordinates measured in opposite directions from either axis are distinguished by contrary signs : e.g., if the coordinates oi P are ■\- x 2s\A -\- y, those of P' are -\- x and — y\ of P" , — x and ■\- y The field record gives at least the length of the base, its azimuth, or direction with refer- ence to the meridian, — usually measured from S. around through W., — and at least two angles of each of the different triangles. The base being reduced to the horizon, and the sides of the triangles computed, the triangulation may then be plotted as follows : The axes NS and WE (Fig. 22), meridian and east and west lines respectively, are drawn through the plotted extremity A of the base AB. x, x' coordinates are measured parallel to WE\ -f in the direction AW, and — in the direction AE. y, y' coordinates are measured parallel to NS; -j- in the direction AS, and — in the direction AN. The origin, or initial point of arcs, is taken in AS, so that the measurement of azimuths will be in the direction S.W.N. E, to agree with field-azlmuthal observations. Since the length of the base and its 24 TOPOGRAPHICAL DRAWING. azimuth z (90° + @; see accompanying table) are given, the coordinates of ^ are ;tr = ^^ = AB cos @, J/ = Ab' = — AB sin @ ; and by laying off these distances from A on the corre- sponding axes, and drawing lines parallel to the axes through the extremities b and b' as shown, the intersection of these lines fixes B. Instead of drawing parallels, any point as B is usually fixed by the intersection of two arcs ; one described from ^ as a centre and a radius = j^, and the other from b' and radius ^x. TABLE GIVING THE TRIGONOMETRICAL FUNCTIONS OF ANY ARC OR ANGLE IN TERMS OF THOSE OF AN ARC OR ANGLE LESS THAN 90°. Angle. sin. COS. tan. cot. sec. cosec. 90° + ® cos @ sin @ — sin @ — cos @ — cos @ — sin @ — sin @ — cos @ — cos @ — sin @ sin @ cos @ — cot @ — tan @ tan @ cot @ — cot @ — tan @ — tan @ — cot @ cot @ tan @ — tan @ — cot ® — cosec @ — sec @ — sec @ — cosec @ cosec @ sec @ sec @ cosec @ — cosec @ — sec @ — sec @ — cosec© 180° — @ 180° + ® 270° — @ 270° -|- @ 360° — @ To find C: AC is given ; z', the azimuth of y^C is ^'-l- C^5 ; and the coordinates of C are x' = Ac = AC cos @ ; y = Ac' = — ^C sin @ ; and C is fixed as before by intersecting lines through c and c'. If through any point as C, right lines //' Cc, are drawn parallel to the axes, the sides of the triangles meeting at this point form the hypothenuses of right- angled triangles, of which the angles can be deduced from the azimuths, and the coordinates determined : e.g.; z", the azimuth of CB, is z'— {ABC + CAB) ; 2'", the azimuth of CD, is z"-{- BCD The coordinates of D are x" = Ad=x' -\- CI, y" = dD = dl-\- ID= y' -\-lD; and since, in the right-angled triangle DCl, CD is known, and DCl= z'" — go° = s (see figure), Cl = CD cos J, and ID = CD sin j; therefore x" = x' -^ CD cos s, and/' = — {y' -\- CD sin s) To facihtate the plotting, the coordinates may be previously calculated and arranged in tabular form as shown below, the prehminary plotting affording an opportunity for this pur- pose ; and if either axis is intermediate in a chain of triangles, the accuracy of this computa- Stations. Coordinates (in feet). + ^- — jr. +y- — y- A 450 •500 1385 i68o 1760 "75 370 1740 900 1300 B C D G H tion can be verified for the coordinates measured in the direction of this axis, by first com- puting the coordinates of the points on one side of the axis, and then of those on the other ; the sums of these separate values should be. equal. Thus, in Fig. 23, measuring from A in the direction AF, the sum of the coordinates of SCALES AND PLOTTING. 25 C, E and F is evidently equal to the sum of the coordinates of B, D and F. This would find its application in plotting secondary triangles, the line AF representing a side of one of the primary triangles. In the foregoing description the coordinates are deduced from the azimuths and the lengths of the sides. To reverse the operation, which is sometimes found necessary, the following formulae from analytical geometry are used : d= V{x' — xf + (j' — jif, and tang @ = X — X in which afis the distance between two points of which the coordinates are x, y, and x' , y' , respectively, and @ is the angle included between the right line joining these points and the meridian. 58. Simple and rapid methods of plotting a triangulation when great accuracy is not required, are either by intersecting arcs described with the calculated sides as radii, and thus consecutively locating the different vertices ; or having given the different angles, by pro- tracting them in their relative positions at the extremities of the sides thus consecutively determined — in either case beginning with the plotted base, or a plotted side of one of the triangles. The latter method would generally be used in plotting a sketch in which the principal points were located by a system of triangles. The former method could best be applied in the case of well-conditioned triangles, of which none of the angles are less than 30°, since the points of intersection could then be clearly determined. 59. The plotting of the courses of compass-traverse surveys by rect. coordinates is a simple operation. Thus in Fig. 21 the meridian line NS drawn through the plotted starting-point A, and the line WE perpendicular to it, are the axes; and to conform to the usual order, -\- values of x are measured from NS to the right, and — values to the left ; -)- values of y from WE upward, and — values downward. The letters of a bearing determine the algebraic signs of the coordinates of either extremity of a course referred to axes parallel to the principal axes NS and WE, drawn through the other extremity. Thus the bearing of AB is 7V/J° W, and for B, X is — , and y -\- ; for C, x' is — , and y' -\- ; for D, x" is -|-, and y" -\- ; and generally, for NE bearings, x and y are -(- ; for 5^ bearings, x andj are — ; for NW bearings, ;ir is — , and y -j- ; for SE bearings, x is -|-, and y — . The coordinates required are x, x^, x° y,y'',y^ For B, X = — AB sin /?, and y = AB cos /S ; denoti'ng by /?' and yS" the respective bear- ings of BC and CD, then for C, xf" = -- {x-{-x') = — {x-\-BC sin ^[), y^ =y+y' = f + BC cos /3' ; iorB.x" =: - X - x'-\-x" = - {x^ - CD sm ft"), y" = y -\- y' + y" = y^ -\- CD cos ft" ; 26 TOPOGRAPHICAL DRAWING. and in general, the principal coordinate of any point is that of the next preceding point ± the coordinate of the required point referred to the latter as an origin. For convenience in plotting, the coordinates of the different points are first computed and arranged in tabular form (par. 57) ; and in a closed survey the accuracy of the computation is verified when the final values of x and y, the coordinates of the starting-point, are each o. 60. The courses of compass-traverses, or of other traverses in which the angles are measured from a fixed line of direction, may be plotted as in Fig. 24. The protractor is placed in a central position on the sheet, with its 0° and 180° divisions coincident with a meridian line. The angles made by the different courses with the meridian are then marked on the sheet opposite the corresponding divisions of the protractor. If radii Ob, Oc are drawn to these points they will make angles with NS equal to the corresponding bearings ; thus, NOb — bearing of AB, SOc = bearing oi BC Therefore A is first fixed; then, with ruler and triangle, a line is drawn through A parallel to Ob, and B is fixed by limiting the course AB to its proper length. In the same manner a line is drawn through B parallel to Oc, and limited to its proper length, fixing C; and similarly for the rest of the courses. The last course plotted should end at ^. A central position for the protractor is preferable, because then no parallels are trans- ferred to a very great distance from the radii. Paper protractors of large size are much used for this purpose, being temporarily fastened in the required position ; also " protractor-sheets," — full sheets of paper with protractors printed on them. A handy means of protracting angles and setting off distances at the same time, as in the plotting of stadia-measurements, when a paper scale is not used, is to glue a piece of paper to the lower surface of the scale of equal parts, and fasten the o point of the scale-edge used to the vertex by a fine needle. This edge can then be turned about the needle at pleasure, and much time saved that would otherwise be spent in adjusting it. 61. The courses may be plotted as shown in Fig. 25. At G, the starting-point, the bear- ing NGH is plotted; the point H \s then fixed by measurement, and a second meridian line is drawn through it as shown, and used to plot HI, and so on to the end of the traverse. This method is preferable to the last in a map of large extent, where it is difficult to transfer the bearings from a single meridian line. The T-square is handy for the purpose, since in its different parallel positions its edges show the direction of the meridian lines ; and the semi- circular or rectangular protractor, with its diameter resting against the edges, is readily used to plot the bearings at the different points. In this, and in the method described in the preceding paragraph, an error committed in plotting a bearing affects all the following courses. This method involves an additional chance of error in drawing the several meridian lines ; and neither of the methods should be used, except for the location of minor details, or when frequent checks can be had upon the work. Either, however, is well adapted to the plotting of sketches where an approximation only to the truth is desired. 62. The Trigonometer is an instrument, illustrated in the accompanying figure, used for the mechanical solution of trigonometrical problems. SCALES AND PLOTTING. 27 9 — 1 — iia — I .^A L -ya / > S W AB is a thin metal plate 15 inches square, containing 100 small equal squares arranged and numbered as shown, each being subdivided into 100 others, omitted in the figure. The unit of graduation of the arm CD, pivoted at C, is a side of one of the small squares, and is also sub- divided into 10 equal parts. The outer scale in the original shows quarter degrees. To find the latitude and departure for any course as CE, indicated by the arm-graduation, turn the arm until its edge cuts the degree-division of the bearing; then / and d are the latitude and departure respectively. For the radius CF, d and e are respectively the tangent and cot. of the arc given by the outer scale, and similarly for the other circular functions. 63. Closing a Plot.- — A traverse should close either upon its starting point or upon some other determined point of the survey, according to the plan adopted in the field-work. If it does not close, the plotting should be gone over; and if still in error, the fault evidently rests with the field-work, and a new survey is required. If the error is inconsid- erable as referred to the purpose of the map, it can be distributed among the courses in pro- portion to their lengths, or by the method illustrated in Fig. 26. In plotting the traverse ABC the last course ends at G instead of at A, showing the error GA. To close the plot, join A and G, and through the other stations draw parallels to AG, as shown. The cor- rection for each course is then obtained as follows : For AB, the sum of the courses : AB :: GA : X = Bb, which is laid off from B in the direction GA. Ab is then the corrected position of the first course. For BC, the sum of the courses : AB -\- BC :: GA : x =^ Cc, and be is the cor- rected position of the second course ; and similarly for the rest of the courses. The first term of the proportion is the sum of all the courses ; the second term is the sum of the course under consideration and the preceding courses ; and the third is the total error. A graphical construc- tion is to set off the courses consecutively on a right line ; erect perpendiculars at the extremity of each course ; set off the total error on the perpendicular at the farther extremity of the last course, and join its outer extremity with the initial point by a right line. From the similar triangles thus formed the corrections for the different courses will evidently be the corre- sponding perpendiculars included between the two right lines. When it is found necessary to resurvey a line on account of an error evidently in the field-work, an error-sheet is used. It is of various forms, but should show clearly the location of the line in error, its recorded and scale lengths, and where the record of its survey can be found. Some indication of the source of error should if possible be furnished the surveyor. 64. Plotting a Vertical Section. — The levelling record is given below. The references, or heights of the stations above the datum-plane, are obtained by adding a rise or subtracting a fall from the next preceding reference. On the right line MN (Fig. 27) set off from M the " distances" as given in the record. Erect the perpendiculars as shown, and set off from MN, on these lines, distances corresponding to the "references." 28 TOPOGRAPHICAL DRAWING. Stations. Bearings. Distances. Feet. Readings. Differences. Refer- ences. Feet. Remarks. B. S. F. S. Rise +. Fall -. A B 50B 6 9.28 7.92 I 36 100 . 00 101.36 Datum-plane. 100 fee: below A. B C 694 3 6.54 11-43 4 89 96.47 C D 427 5 12.65 5-25 7 40 •■ 103.87 D E 486 4 10.37 3-58 6 79 ■• 110.66 2116 8 38.84 28.18 28.18 15 4 10 55 89 66 4 89 100.00 10.66 10.66 The line AB E, through the outer extremities of these distances, is the required section or profile of the surface. It is more exact to set off all distances from M, adding each to the sum of those preced- ing. The vertical scale is ten times as large as that for the horizontal distances, for reasons already given ; and it is apparent in this case that the differences of elevation would be barely visible if it were not so increased. The j-square is convenient for erecting the perpen- diculars ; but the use of section or profile paper (par. 33) obviates the necessity of drawing them, and it is generally used for this work. The plotting of contours, and the construction of sections and elevations from contoured maps, are described in Part II. CONVENTIONAL SIGNS. 29 PART II. PLAIN TOPOGRAPHICAL DRAWING. 65. As may be inferred from the heading, Part IL is a description of the construction of a plain or uncolored topographical map. The triangulation and traverses, together with such other reference lines and points as may be necessary to locate the different features having been plotted, topographical drawing properly begins ; and the subject is naturally divided into — I. A Description of the Conventional Signs. IL Representing the Details, in. Representing the Configuration or Surface Forms of Ground. IV. Finishing the Map, including Lettering and Ornamentation. Section I. — Conventional Signs. 66. The signs or symbols used to represent the different features are intended to be of such forms as to be easily and rapidly made and readily understood. Those used for the commonest features, such as cleared land, forests and streams, are practically identical in all schools of map-drawing; while other signs for special features vary in different countries and with the taste or skill of different draughtsmen. It would be very advantageous to have the same sign for the same feature on all occa- sions, so as to prevent the possibility of mistake — a condition which is well appreciated in the military service, where a commander's knowledge of the country in which a campaign is to be conducted, or of the ground where a decisive battle may be fought, may depend upon sketches hastily made, and in which the signs are necessarily roughly drawn. The signs are made to resemble approximately the features or objects themselves, as the latter would appear from a point of observation located above them, or in some cases as if viewed in elevation. 6"]. Shading of the Signs. — In most topographical drawings, the effect of light and shade is added to produce relief. The source of light is regarded as above and to the left of objects ; or, more definitely, the rays of light are parallel to a ray making an angle of 45° with the hori- zontal plane, and of which the projection upon the drawing bisects the lower right or upper left angle of the border. Objects extending above or below the general surface are shaded correspondingly ; the former by a heavier outline on the side remote from the source of light, and the latter on the side nearest it. The positions of shadows when represented are simi- larly determined. The present tendency toward simplifying the signs is to omit both shades and shadows unless, as in the case of a colored drawing, the latter may be needed to distinguish certain features. 68. A multiplicity of signs tends to confusion, and taxes the memory unnecessarily; names or verbal descriptions being much better suited to the purpose. 30 TOPOGRAPHICAL DRAWING. It may be remarked that a strict adherence to the forms given is not necessary for the production of an intelligible map. Still the aim should be to make a map that shall also please the eye : clearness should be combined with tasteful execution, evenness, regularity, and uniformity in repetition ; and to assist in attaining this end, a detailed description of the manner of making the signs accompanies the plates, and the faults most frequently com- mitted in drawing them are pointed out. No exact rule can be laid down as to the sizes of the signs. In maps drawn to a small scale, some which individually represent objects measured in the field are, for the sake of clearness, necessarily much exaggerated above their scale dimensions. They are usually drawn too large. As a general rule, they should be just large enough to be easily read; but if the scale is very large, they may be increased in size proportionately, to lessen the number of repetitions and thus diminish the amount of work. 69. For convenience of reference, the important signs are represented in the different Plates in the following order : Plate III. Conventional Signs — Land. Plate IV. Conventional Signs — Land and Water. Plate V. Conventional Signs— Miscellaneous, including Military Signs and Signs for Deciphering Maps. Plate VI. Conventional Signs employed in Maps of the U. S. Coast Survey. PLATE III. 70. Conventional Signs — Land. Cleared Land. — Groups of fine dots and dashes, S to 7 in each group or tuft, are dis- tributed over the tract evenly, but not in rows ; the base of each tuft parallel to the lower edge of the border of the map. As shown at a, each tuft begins and ends with a dot; while the intermediate dashes, increasing in length to the middle one, which is vertical, radiate from a point below the middle of the base at a distance from the latter of about half its length. The effect desired, viewed at a short distance, is that of a flat tint ; and when the lack of this is due to uneven distribution, the fault may be remedied by filling the larger spaces with groups of dots. The effect is more pleasing when the vertical dimensions of the tufts are slight, and the dashes are curved outward from the middle one as shown at b. The usual faults consist in making the signs too large, in giving the bases an upward in- clination, and in placing the tufts in rows. If the straight edge of the paper used under the hand for protecting the drawing is kept parallel to the lower border of the map, it will serve as a guide for the direction of the bases ; and by first placing the tufts here and there irregu- larly, at wide intervals apart, and then filling the intervals in a similar way, the making of rows can be avoided. Cultivated Land. — Alternate parallel rows of dots and dashes corresponding to furrows are used. The dots and dashes are evenly spaced, of uniform size, and close together. The spaces of any row of dashes are opposite the dashes of adjacent rows, and the rows of adjoin- ing fields are oblique to each other. The rows arc drawn with a ruler and triangle, the paper being so disposed that the lines may be drawn from left to right along the edge of the triangle. If more convenient, the rows of dashes may be drawn first, and the rows of dots interlined. CONVENTIONAL SIGNS. 31 The principal fault is irregularity in lengths of dashes and in spacing, which, for a be- ginner, makes the use of a scale of equal parts necessary. Sandy Gravel and Mud. — For isolated sand-tracts, the dots are close together and evenly distributed, as at c. Along shores a good effect is produced by diminishing the sizes of the dots and increasing the intervals as the distance from the shore-line increases, as at d. In tidal waters, for the portion of the shore uncovered by the ebb, rows of dots are drawn par- allel to the shore-line, as at e. Sand-dunes are represented as at/. Gravelly tracts are repre- sented by the sign for sand, with coarse dots, and very small, curved and angular outlines, representing stones, distributed throughout, as at^; and mud, by short strokes parallel to the lower edge of the border and in sets, as at h. Orchard. — Small, curved outlines, representing the trees in plan, are regularly distributed within the enclosure. These are first located by pencil-dots placed at equal intervals. Each tree is shaded by slightly emphasizing the outline and adding a few interior curves on the side remote from the source of light. The shadow beginning at the outline is of an oval shape, with the longer axis parallel to the projection of a ray of light, and is made up of parallel straight lines close together and perpendicular to this projection, as at i. The faults consist in making the signs too large, angularity of outline, uneven shadows and in extending the latter beyond the projections of rays tangent to the outline. At i in the figure, is shown the order, indicated by the numbers, in which the curves of the outline are conveniently drawn — i and 2 from right to left, and 3 and 4 from left to right ; and at k, a method of spacing the trees and assuring the direction of the shadow : the hori- zontal lines are drawn first, at equal intervals apart, one set at least being parallel to a side of the enclosure ; the oblique lines are then drawn through the points of intersection, and parallel to the projection of a ray. In making the shadows, if a straight edge of the paper used to protect the hand is kept parallel to the projection of a ray, and so that the shadow-lines may be drawn toward the draughtsman, the proper direction of these lines is readily observed ; and by preserving equal intervals and using the same number of lines for each of the shadows, the latter will be of uniform length. On maps drawn to a small scale, the shadows are fre- quently omitted. Woods. — But two signs are generally used, representing respectively deciduous and ever- green trees. The former sign is similar to that for orchards, except that the sizes and dis- tribution of the signs are irregular. Projecting boughs of isolated trees, and portions of clumps or large masses, should have rounded forms. Angularity of outline, and regularity in size and distribution, are to be avoided. The strokes should not cross each other, nor loops be formed. Shadows are not necessary, especially in a close growth such as a forest ; but it is best to shade the outlines as already described. At /, are various forms of clumps ; at m, evergreens ; at n, is a special sign for oak ; and at the lower right-hand corner, these signs with that for underbrush interspersed, are shown. Alkaline Flat and Saline Bed. — The former is represented at /, the heaviest dots forming the outline, and defining a space which is left white. The latter, q, is represented by its outline filled with rows of short dashes perpendicular to the lower edge of the map-border. Vegetable-Garden. — The enclosure is divided into rectangular or triangular figures, separated by narrow spaces representing paths ; and the figures are then filled with the sign 32 TOPOGRAPHICAL DRAWING. for cultivated land drawn to a small scale. For flower-gardens, the sign for cleared land and woods, drawn to a very small scale, are used instead of that for cultivated land. Park and Cemetery. — The enclosure is divided into regular and irregular figures, with narrow spaces bounded by parallel lines for the roads and paths; the figures are then filled in with the signs for cleared land and woods. The distinguishing features for a cemetery are the headstones, as shown. Signs for different field-products., suited to maps drawn to a large scale, are also given. PLATE IV. 71. Conventional Signs — Land and Water. — Survey points and lines are represented by the signs numbered from i to 11 inclusive. 1. Triangulation point. 5. Stone landmark. g. Monument. 2. Plane-table " 6. Wooden " 10. County boundary. 3. Common survey " 7. Mound " n. State " 4. Signal-tower. 8. Tree " In addition to these, a rectangle containing a dot is used for a stadia survey point. The signs for boundaries should be of such weight and form as to be readily distinguished from the others. Enclosures : 12. Rail or " worm" fence. 14. Board fence. 16. Stone wall, rough. 13. Picket fence. 15. Stone wall, with coping. 17. Hedge. For 12, the angles should be obtuse and the panels of equal length. The large dots of 13 and 14 represent the posts, and are equally spaced; 15 is shaded according to the general rule (par. dy); and 15, 16 and 17 are more clearly distinguished when the shadows are drawn : the shadow-lines should be perpendicular to the projection of a ray. Communications : 18. Path, foot. 22. Road, main. 26. Telegraph. 19. " mounted. 23. " paved. 27. Embankment. 20. Road, undefined. 24, Railroad, single track. 2S. Cutting. 21. " minor. 25. " double " 29. Tunnel. The distinction made between 21 and 22 is that the former consists of fine lines, and the latter of medium lines with a greater interval. In maps to large scales, these lines are replaced by signs for the features which enclose the road. The English convention is to represent fenced roads by full hnes, and unfenced roads by broken lines. In the sign for railroads, the number of lines is increased correspondingly for a greater number of tracks. Horse-railroads, tramways, etc., are represented by medium lines, single or double, through the middle of the street. In 26, the posts, or longer lines, are perpendicular to the edge of the road ; and if no road exists, the bases should be connected by a fine line. An embankment is represented by hachures alone ; while a cutting has an outline drawn along its outer edges. The direction of the road in 29 is indicated by a heavy dotted line, and other subter- ranean passages are indicated in like manner. CONVENTIONAL SIGNS. 33 Parts of communications, the positions of which are not fully determined, are repre- sented by dotted or broken lines. Buildi?tgs : 30. Building, wooden. 31. . " masonry. 32. Church. 33. Light-house. 34. Windmill. 35. Settlement. 36. Village. 37. Town. 38. City. The main distinction observed in representing buildings is as to the material used in their construction. In Plain Topographical Drawjng, wooden buildings are represented by fine, and buildings of masonry by heavy, outlines ; and in either case the interior space is usually filled with fine, parallel lines, close together. These filling-in lines should be parallel throughout the map, and at the same time diagonals of the figures. Therefore, when the buildings face different points of the compass, a direction for these lines is chosen that will make them diagonals of the greatest number of buildings. If it is desired to make a distinc- tion between brick and stone buildings, use very heavy outlines for the latter. The outlines may be shaded, but in black and white maps shadows are not usually drawn. In maps drawn to a very small scale, no distinction can well be made as to material ; and to make them clearly visible, all buildings are represented by soHd black figures. Signs pertaining to Water: 39. Well. 54- Aqueduct. 69. Direction of current. 40. Spring. 55- Head of navigation. 70. Ford, foot. 41. Pond. 56. Buoy. 71- " wagon. 42. Marsh, fresh. 57- Rocks, bare. 72. Bridge, foot. 43. " salt. 58. " sometimes bare. 73. " wooden. 44. Brook. 59- sunken. 74. ' ' draw. 45. Creek. 60. " dangerous. 75- " truss. 46. Practicable for all arms. 61. " position uncertain. 76. stone. 47. " " inf'y and cavalry. 62. Bar. 77- " pontoon. 48. Impracticable. 63. Shoal. 78. " suspension. 49. River. 64. Fog-bell, 79. Ferry, rowboat. 50. High and low water. 65. Light-ship. 80. " steam. 51. Dam. 66. Anchorage. 81. " rope. 52. Falls. 67. Wreck. 82. flying. 53. Canal, with locks. 68. Surf. Pond. — The outline properly shaded is filled with fine lines, with small equal intervals and parallel to the lower edge of the border of the map. Any unevenness in the lines or irregularity in the spaces is particularly noticeable, and must be carefully avoided. Marsh. — This is a combination of the signs for pond and cleared land. The former is drawn first. For fresh-water marshes, irregular spaces are left in the ruling, and these spaces are afterwards filled in with the sign for cleared land : the latter sign is also drawn here and there at irregular intervals, between the ruled lines. The proper outline of each space is produced by not having consecutive lines terminate in the same vertical. For salt-water marsh, no spaces are left in the ruling. Reclaimed marshes are indicated by the addition of confined streams or ditches. If it is desired to distinguish between a marsh, which may be sometimes dry, and a swamp, for the former use rows of dots here and there in place of the full lines. Brook or Rivulet. — An irregular line following its course, fine at the source, and increas- 5 34 TOPOGRAPHICAL DRAWING. ing slightly in breadth to the mouth. Frequent curves near the source are characteristic of small streams, especially in rugged tracts ; and whip-lash, snaky-looking lines should always be avoided. An important consideration in regard to streams, particularly in a military point of view, is the practicability of crossing them ; and a conventional sign to express this condition, used for sketching-purposes in the English military schools, is shown. River, or Large Body of Water. — The space between the shore-lines is filled with parallels termed " water-lines." The shore-lines are drawn first, then a water-line next to each shore- line, carefully conforming in its direction to the various projections and indentations of the latter. Another water-line is then drawn parallel to the first, and so on. The final line or lines will thus be in the middle or axis of the stream, and proper junctions corresponding to the varying widths will be formed. To produce the best effect, the water-lines and intervals are both graded so that the finest line and broadest interval shall be in the axis of the stream ; but, as it is very difficult, especially for a beginner, to preserve both gradations, the lines are usually all made fine. The irregularities are lessened in consecutive lines, so that in a broad stream the curves of the central lines are quite smooth. In drawing a water-line, the preceding one should be at the left hand, the line drawn towards the body, and the paper so disposed as to facilitate this operation. To preserve the proper interval, it is better to keep the eye on the interval rather than on the line. The general rule for shading is sometimes applied to shore-lines. The direction of the current is indicated by an arrow, as shown. In tidal waters, two sets of shore- and water-lines may be used as shown, one set corre- sponding to high- and the other to low-water mark. The high-water shore-line is made the heaviest. Y ox: Rapids and Torrents (XII. Plate VI.), the water-lines are broken into short, curved lengths ; and for Whirlpools they are given spiral shapes, still preserving the local gradation of the intervals. For Canal-locks, the V-shaped figures have their vertices up-stream. The anchor for anchorage is pointed up-stream. Sand-shoals and Bars. — A dotted contour-line defining the boundary. If permanent, the depth of this line is noted upon it. Mud banks. — Two sets of dotted lines crossing each other obliquely. For sunken rocks, the depth in feet is noted in the attached circle. Rock-formations of considerable extent are drawn as described in Section III. Soundings are represented by numbers giving the depth in feet, as shown near the bottom of the plate. The i-, 2-, and 3-fathom curves may be indicated by dotted lines, the dots being respectively continuous, in pairs and in groups of three. The direction of the current should alwa3's be indicated at fords. The flared ends of the sign for a bridge indicate the approaches or abutments. The bases of the piers are usually pointed at the up-stream extremities. PLATE V. 72. Conventional Signs— Miscellaneous.— As to the military signs, the side on which the blackened portions of those for infantry and cavalry, in line, is placed, indicates the front or CONVENTIONAL SIGNS. 35 direction of the movement ; and in the sign for artillery, the short arm of the figure projects towards the target. The direction, in march, is usually indicated by an arrow-head placed at the head_of the column. The heavy line in the signs for fortifications represents the interior crest ; and the width of the interval between it and the adjacent fine line on the exterior, the thickness of the parapet. On maps to a very small scale, fortifications are represented by a single heavy line, giving the position of the interior crest. The hachured slopes of trenches, etc., are on the right, left and front. The 40 signs " For Deciphering Maps" in the plate, are introduced for the service they may afford in reading old maps (see par. 68). At least 800 others have been in use within a comparatively recent date, but those given are more commonly seen. Their names are : I. Mill, water-power. 15- Rope-walk. 29. Sentinels. 2. ' ' steam-power. 1 6. Quarry. 30. Vedettes. 3. Saw-mill. i7- Chapel and church. 31. Indians, with firearms, 4. Ship-mill. 18. Custom house. 32. " without " 5. Powder-mill. 19. Post-office. 33- Gold. 6. Cotton factory. 20. Hotel. 34. Silver. 7. Woollen " 21. Inn. 35. Copper. 8. Sugar " 22. Railroad station. 36. Mercury. 9. Silk 23. Telegraph " 37. Lead. 10. Indigo " 24. Mineral spring. 38. Tin. ir. Iron-works. 25- Gate, turnstile. 39. Iron. 12. Glass-works. 26. Weir. 40. Coal. 13. Brickkiln. 27- Buoys. 14. Limekiln. 28. " mooring. PLATE VI. 73. Conventional Signs — U. S. Coast Survey. — The signs in this plate are typical examples photo-engraved from the present standard topographical drawings of the U. S. Coast Survey. The different features represented in the rectangles are as follows : I. Oak woods and orchards. II. Rice-dikes. III. Abraded rock-faces, granite. IV. Granite cliff — crest, face and talus. V. Town and pine woods. VI. City, villas, salt and fresh marsh, and docks. VII. Arroyos. VIII. Soft stratified rocks and gulches. IX. Railroad, mid-river sands and common rock escarpments. X. Railroad, tunnel, mid-river drift, canal and dam. XL Eroded drift-banks, with boulders set free. XII. River-torrent, eruptive rocks and basaltic escarpments. 74. An ingenious and rapid method of printing the signs by hand, due to J. A. Ocker- son, Am. Soc. C.E., is described in the Report of the Mississippi River Commission. It consists in electrotyping a tasteful arrangement of signs for any feature on a very thin copper plate, about 3X14 inches in size, backing the plate with rubber, and wrapping it around a wooden cylinder of corresponding dimensions. The cylinder revolves about a handle passing through its centre and projecting at the ends. The plate is inked at each revolution by means of an ink-roller — the best quality of German printing-ink being used. Due pressure is applied to the handles, and the cylinder is rolled slowly over the map. Impressions are readily joined by placing a strip of paper along the edge of the part already printed, thus permitting the cylinder to overlap as much as may be desired. To prevent any sign or name upon the map from receiving an impression, a pencil made 36 TOPOGRAPHICAL DRAWING. of common starch is drawn over it once ; and after the printing is finished, the starch is easily- brushed off. Electrotypes are made for the different signs to be used in a map of any particular tract. The time required for printing is at most one tenth of that spent in drawing by hand, and the process is exceedingly well adapted to maps of large extent, and to all which do not require a fine grade of lines. This process is also extended to include the lettering, as described in par. 127. Section II.— Representing the Details. 75. Pencilling the Details. — The positions of the details are first carefully determined by hues or dots in pencil, lightly drawn ; the shading, shadows and filling-in of outlines being omitted. Thus the edges of roads, the shore-lines of streams, are accurately represented by fine lines drawn through the plotted points, and following their various bends and curves, as given in the field-notes ; buildings, fields, forests and all features of which the boundaries are given by outlines ; orchards, by outlines with properly spaced dots indicating the positions of the different trees ; and so on, until all the features are so completely located that no doubt or delay can arise, nor liability to erasure be incurred, in the subsequent inking. Tlie following order to be observed in the pencilling is found from experience to facilitate the work : I. The communications; 2, the streams, beginning with the principal one; 3, the cities, towns, villages and detached buildings; 4, the enclosures; 5, the marshes, woods and remain- ing features ; 6, the contours (par. 86). To avoid mistakes, it is well to indicate verbally, or by a few hastily drawn signs, the kinds of wood, field productions, material, etc., within the various outlines as soon as drawn. The location of rocks should be shown, so that the contours in ink may not be drawn through them. A space may also be outlined for the scale, compass and legend. 76. Inking the Details. — The construction-lines, and all lines not needed to fix the posi- tions of the features or in finishing the map, are then erased by means already described ; and those left should be reduced as much as possible consistent with visibility. The drawing is then ready for inking; and in this operation it will be found convenient to preserve the same order as prescribed for the pencilling. It is advisable to begin with such a part of the drawing that, in the progress of the work, lines already inked may not be defaced by rubbing, or by leaning upon the drawing. Section III. — Representing the Configuration or Surface-forms of Ground. •jj. There are two general methods of representing hills or mountains and surface- undulations, viz. : I. By contours. II. By contours and hill-shading. Surface-forms arc also represented by hill-shading alone ;. but the contours are first sur- veyed and plotted in pencil, so that, although the contours may not appear in the finished map, this operation virtually resolves itself into Method II. It may be remarked, however. THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 37 that in the representations of sh'ght changes of form and in field-sketching, hill-shading alone is frequently used. Method I. has the advantage of simplicity, and the topographical signs are never obscured. In Method II., the elevations and depressions are more forcibly presented to the eye ; and if the shading is so light as not to obscure the signs, the addition of the relief thus obtained is evidently an advantage. Method I.— By Contours. 78. Contours, Equidistances and References (Plate VII.). — The intersection of the surface of the ground by a horizontal plane is a contour (the term is also applied to its projection upon a map). Shore-lines of still water are familiar illustrations of contours; and if they should be determined for an elevation, by survey at each considerable rise of water, and all projected vertically upon a horizontal plane, it is obvious that by giving to each of the pro- jections a number representing its vertical distance from this plane, the general shape or con- figuration of the surface would be at once shown. It is also apparent that the less the rise of water between consecutive measurements, the greater would be the number of shore-lines, and consequently of projections, and the more complete would be the representation. In Fig. 28, the hill, of which DSP is a profile, is cut by the horizontal planes, of which aa' , bb' , and ccf are the traces, and a!' , b" , and c" are the projections of the contours thus de- termined, upon the horizontal plane Z''C7'' ; the numbers 40, 80 and 120 giving the heights of a", b" and c" above D'CF. Contours are determined at equal vertical distances apart. Their heights are all referred to a single horizontal plane called the datum plane. These heights are indicated by numbers, called references, placed upon the contours, and the equal vertical distances are called equi- distances. Thus, in the figure, D'CP' is the datum plane, its reference is o, and the equi- distance is 40. If the datum plane is intermediate in a system of contours, the — sign is pre- fixed to the references of contours below it. It is apparent that the greater the degree of declivity, the closer are the projections, — compare D'a" and b"c", the projections of Da and be respectively, — which condition enables the observer to see at once upon the map the relative steepness of different slopes. 79. The slope between consecutive contours, along a normal or line perpendicular to both, is considered uniform ; therefore the degree of declivity can readily be determined from the projections upon the map — e.g., in the figure (drawn to a scale of -j-sVir), the degree of declivity of a"D' is the angle aDd of the right-angled triangle, of which Dd^?,o feet, as determined by the scale, is the base ; and the altitude ad = 40 feet, is the equidis- tance. This angle may be constructed and measured with a protractor; or determined trig- ad onometrically from the relation tang. aDd = jj-,= ^^^ = ^,.-. aDd = 2y°, very r\c^\\y;hnt the simplest means is a scale of inclination described in the following paragraph. 80. The Scale of Inclination (Fig. 29), also called a scale of horizontal equivalents, is con- structed as follows: — Draw AB and AC perpendicular to each other; through^ draw the radials making with AB angles of 5", 10°, 15°, 45°, as shown; set off ^^ equal to the equidistance according to the scale of the map, and draw^/" parallel to AB. The parts of 38 TOPOGRAPHICAL DRA WING. ^/"intercepted between^ and the points of intersection of the different radials are the cotan- gents of the angles to the radius Ag—z.g., in Fig. 35, the tang, and cot. of 25° are .5^ and Ce respectively ; and of 30°, Bd' and Ce' ; and these are the projections of the inclined lines Ae and Ae' respectively. Similarly, in Fig 28, D'a" = Dd is the cot. of VaD (the angle of de- pression of aD), to a radius corresponding to the equidistance ad, and it is also the projec- tion of Da. The application of the scale is thus apparent ; e.g., to ascertain the angle of in- clination, or the degree of declivity along a normal as D'a", between two consecutive con- tours ; — with the dividers, set off from ^, (Fig. 2g), gd=D' a" ; the position of <^ between the 25" and 30° radials indicates a slope of 27°, as in par. 79. To ascertain intermediate slopes and those exceeding 45°, it is only necessary to add the corresponding radials. By cutting it out along^, the scale can be applied directly to the map. For another form of this scale see close of par. 81. 81. Since Ce, Ce' (Fig. 35), the horizontal projections of Ae, Ae', , are the cotan- gents of the vertical angles 25°, 30°, ; a table of cotangents gives at once the ratio of the height to the base, corresponding to a unit of distance and to any angle of inclination ; and the following table thus obtained is of general use : RATIO OF HEIGHT TO BASE, AND LENGTH OF BASE CORRESPONDING TO THE EQUI- DISTANCE, FOR SLOPES FROM 1° TO 60°. Ratio of Base for Ratio of Base for Slopes. height to base (approxi- maie). I unit of lieight(col of slope). 5 units of height. 10 units of height. Slopes. height to base (ap- proximate). I unit of height (cot of slope). 5 units of height. 10 units of height. 1° T to 57 57-29 286.45 572.9 15° I to 3.7 3 73 18.6 37-3 2 I to 29 28. 64 143.2 286.4 16 I to 3.5 3-49 17-5 34-9 3 I to 19 19.08 90.4 190.8 17 I to 3.2 3 27 16.3 32-7 4 T to 14 14.30 71-5 143 18 I to 3 3.08 15-4 30.8 5 I to II 11.43 57-2 "4-3 19 I to 3 2.9 14.5 29 6 I to 10 9.51 47-5 95-1 20 I to 2.7 2.75 13.7 27-5 7 I to 8 8.14 40.7 81.4 25 I to 2 2.14 10.7 21.4 8 I to 7 7.12 35.6 71.2 30 I to 1.7 1.73 8.6 17.3 9 I to 6 6.31 31-5 63.1 35 I to I .4 i.43 7-2 14.3 10 I to 6 5.67 28.3 56.7 40 I to 1 . 2 1. 19 6 II. 9 II I to 5 5 14 25.7 51-4 45 I to I . I 5 10 12 I to 4-7 4-7 23.5 47 50 I to 0.8 0.84 4.2 8.4 13 T to 4.3 4-33 21.6 43-3 55 I to 0.7 0.7 3-5 7 14 T to 4 4.01 20 40.1 60 I to 0.6 0.58 2.9 5-8 Lengths of base corresponding to heights or equidistances not given in the table, are readily found by multiplication. Thus, for an equidistance of 15 feet and a slope of 6°, the base is 3 X 47-5 = 142.5. or three times the base for 5 feet ; for an equidistance of 50 feet and a slope of 19°, the base is 145.0, or 5 times the base for 10 feet, etc. The length of base corresponding to a unit of height is roughly obtained by dividing 60 by the number of degrees representing the slope ; e.g., the base for 10° = |^- = 5 ; for 15° it is 4, for 20", 3 A scale of inclination for any equidistance can also be constructed by use of this table, as follows: — From G on the right line GF {¥\g. 30), set off, according to the scale of the map, Ga, Gb Gk, corresponding respectively to the bases for the different slopes and the given equidistance, and mark the points of division as shown. 82. The reduction of surface-measurements to the horizon is of frequent application in the plotting of contours. A case often presented is : given the angle of elevation CAB (Fig. 31X or of depression VGA, {ox VaD, Fig. 28), and the distance AC\\.o find ^.g. Let @ rcpre- THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 19 sent the vertical angle, and h, a and b the distances AC, BC and AB respectively; then 3 = /; cos @ — e.g., @ = 15°, of which the cosine 130.9659, and h is 320 feet; then the horizontal equivalent b = 320 X 0.9659 = 309 feet. If the vertical angles are very small, say less than 10°, it is better, for great exactness, to use the sine of the angle in the computation, since the cosines of small angles differ but little from each other in value. Thus, referring to the figure, Ii — b ^h — h cos @ = k (i — cos @) ; and since i — cos @ = 2 sin^ 2>^ = ^ (^""^ ^^^^ — )• This reduction may be made graphically by constructing a right-angled triangle in accordance with the given conditions. Taking the foregoing example ; at A, on the right line AB, construct the angle @ = 15", and make AC ^ 320 feet, according to the scale; from C, let fall the perpendicular CB, and find the length of AB from the same scale. 83. The handiest means, however, for making this reduction is Gouliers scale, which is constructed as follows : Let AB (Fig. 32) be the scale of the map ; from C, distant from AB at xAB least , as a centre, describe the arc of a circle tangent to the scale, as shown ; and begin- ning at the tangent point, subdivide the arc into degrees. Join C and the points of division of the scale by right lines, and through the degree divisions of the arc draw parallels to the line of the scale, terminating and marking them by numbers, as shown. Each of these parallels is a scale of reduction for the slope indicated by the number of degrees at its extremity — e.g., the scale being ig^uo , EF is the reduction of 120 feet measured on a slope of 15°; for from the similar triangles CEF a.nd CDG, EF : DG :: CE: CD; denoting the radius CD by R, CE = R cos 15°, .\EF= DG cos 15 = 120 ft. X cos 15°. Therefore, in the use of this scale, the reduction is obtained at once from the parallel corresponding to the degree of declivity. This scale, like others of frequent use, can be engraved with great exactness on metal. 84. A similar scale is adapted to the reduction of distances obtained by stadia measure- ments to their horizontal values. The rod being vertical, the angle of inclination of the optical axis to the horizon represented by cj), and the reading by L ; then the horizontal dis- tance H ^^ L cos" 0. To construct the scale, draw CD as in Fig. 32, perpendicular to and equal in length to at least f AB. With the middle point of CD as a centre, describe an arc tangent to AB; beginning at D, mark every second degree up to 72° ; draw the parallels and radius, as in Fig. 32, through the second degree divisions, and from C respectively ; and mark each of the parallels at its extremity, by a number representing one half the number of degrees corresponding to the division of the arc through which it passes. Denoting the radius by R, any distance, as EF: DG :: R -\- R cos" angle C, .'. tang B > tang C, and A'B must be less than A'C; therefore A'B is perpendicular to BC, and consequently to c'c' ; and from a well-known proposition of geometry, AB is also perpendicular to c'c'. Therefore in projection, as well as upon the natural surface, a line of greatest descent meets or cuts the contours perpendicularly, and is consequently normal to them. Since water-part- ings and watercourses are also normal to the contours, they are included in this class, being lines of greatest descent having a less inclination than others in their vicinity. In the horizontal and vertical systems, hnes of greatest descent determine the directions of the hachures; it is therefore essential that they should be correctly drawn, and the water- partings and watercourses described in par. 93, together with auxiliary contours, are used for this purpose. No difficulty usually attends the drawing of a water-parting or a watercourse, since it is simply necessary to begin it at the summit of a salient or re-entrant, as the case may be, and produce the line so it shall cut the contours at their points of greatest curvature, and at the same time be normal to them. Thus, in Fig. 47, the water-parting SF is so drawn ; but when the contours are widely separated and oblique to each other, it is difficult at first sight to determine the directions of lines of greatest descent, as from a, a', ; therefore lines called auxiliary contours are interpolated, as shown in Fig. 48. Since the sections of the water- parting SP, included in the different zones, increase in length progressively from the summit to the base of the slope, they should be correspondingly subdivided to obtain the required equidistant points for the auxiliary contours. The lines S'P' and S"P", separating the more rounded portion of the salient from the slopes represented by contours sensibly parallel, are first drawn and their sections are subdivided, as in the case of SP: the auxiliary contours are then drawn through the corresponding points of subdivision, as shown in the figure. The smaller zones thus formed may be further subdivided if desired, until no doubt can exist as to the direction of the lines of greatest descent, which are then drawn as required. After some practice, the operation may be simplified by subdividing the water-parting or watercourse only as above described, and then drawing the auxiliary contours by eye, so they shall pass through these points. In the case of a re-entrant or ravine these lines are drawn in a similar manner. A col. Fig. 49, presents the only case where the drawing of these lines appears difificult. Suppose the lowest point C of the ridge SS', and also of the col itself, to be an equi- distant point ; then the contours with the same reference as this equidistant point, of all the slopes, will meet at this point, and these contours will conform sensibly to the lines cc and c'c'. If C is not an equidistant point, the directions of the adjacent contours will conform closely to the directions of cc and c'c', and their distance from these lines can be readily estimated. Therefore, these lines are drawn first in their approximately true position ; the water-partings SC, S'C, and the watercourses CB, CA, are then drawn ; and the sec- tions of the latter contained in the different zones, and between C and the nearest con- tours, are subdivided as described in the cases of spurs and ravines. Auxiliary contours are 48 TOPOGRAPHICAL DRAWING. drawn through the points of subdivision, as shown in the figure, and the directions of the lines of greatest descent are at once determined. It may be observed that when C is an equi- distant point, the solitary case of contours meeting upon a natural surface is presented ; and this case and that of a vertical surface are the only instances of contours meeting or appar- ently intersecting in projection. e. Lines of greatest descent and auxiliary contours are called guiding-lines, because, as heretofore stated, they determine the directions of the hachures. /. A scale of sliade is a means for determining the intensity of shade appropriate to a given slope ; and a working scale is a guide to the proper distribution of the shade, in the form of hachures or with the brush, in accordance with the scale of shade. Since these scales differ in the different systems, they are separately described in connec- tion with these systems. loi. Preliminary Work required in the Application of the Different Systems. — The contours having been plotted in pencil — I. Construct a scale of inclination (par. 8o) for the given equidistance. II. Construct a working-scale (described with each system) suited to the scale of the map. III. Draw the guiding-lines [e, par. lOO), if the map is to be hachured, so that no doubt can exist as to the proper directions of the hachures. These may be drawn as the work of shading progresses, and are in pencil. 102. An example of the Horizontal System with Vertical Illumination is given in Plate IX. This is commonly known as the English system, and is now in use in the English schools of topography (see also par. 109). The standard scale of shade and a working-scale, used in the English government sur- veys, are represented in Plate XL, Fig. 59. The former is in the column headed " Scale of Shade for Angle given ;" each set of hachures, with its intervals, showing the relative amount of black and white for the slope designated below it, as well as the exact breadth of hachure and interval to be used in representing this slope upon a map drawn to any scale and with any equidistance. The working-scale, in the two outer columns, shows the number of these hachures, with their intervals, to be used in the different zones for a map drawn to a scale o i^^\^^ , or 6 inches to I mile, and an equidistance of 25 feet. For an equi-distance of 50 feet and the same representative fraction (xrrrir) the number of hachures for a 35° zone would be 4, instead of 2 as in the figure ; for a 20° zone, 6 To find the number of hachures for the different slopes for any representative fraction and equidistance, apply the corresponding scale of inclination (par. 80) to the edge of the scale of shade ; and the number of hachures in the 35", 30° sets, covered by the 35", 30° divi- sions of the scale of inclination, is the number required. For slopes between 35° and 45", the hachures given for 35° are used, the breadths being increased, while the number remains the same ; and beyond 45°, vertical hachures or dashes irregularly placed are used. The inner column, headed "Approximate Gradients," gives the tangents approximately, instead of the degrees of inclination. The lengths of hachures are not fixed, varying between ^L and \ of an inch, the finer hachures being made the longer. The operation of shading, following the preliminary work (par. lOl), is governed by the following rules : THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 49 The working-scale is applied to the different zones where the slope changes, and a few strokes of the proper breadth and in the right direction are made — preferably with the pencil at first — to serve as guides in the hachuring. The hachures conform in direction to the auxiliary contours, and are invariably drawn perpendicularly to the lines of greatest descent ; as at mm, Fig. 60, and not as at nn. Beginning at the summits and working downwards, the hachures are drawn in sets which are blended into each other, and which break intervals so as to prevent the existence of white streaks, as at 00. The sets should not be drawn in bands around the elevation. The hachures are drawn towards the draughtsman, the work extending toward the right, so the pen will not hide the last hachure drawn, and the space can be readily estimated — the paper being so disposed as to facilitate this operation. Each hachure is of the same breadth throughout, and should be drawn freely and firmly. Much care and deliberation are required at first, but freedom of execution will come from practice. If the contours are to be in the finished map, they are drawn in red, as described in par. 91 ; or if in black, in broken and dotted hnes, to distinguish them from the hachures. 103. Desprez System is the same as the English system, except that the contours are always required in the finished map. In Mandrofs System, employed in Switzerland, no hachures are used, the shading being effected by increasing the breadths of the contours in direct proportion to the angles of incli- nation ; and where a brown color, instead of India ink, is used for the contours, the details are not obscured, and a pleasing effect is produced. Some of the fine Russian maps of Sebasto- pol were constructed, under the direction of General Todleben, according to this system. 104. The Horizontal System with Oblique Illumination is also illustrated in Plate IX. The method of procedure is the same as with vertical illumination, except that the gradation of the hachures is in accordance with the principles given in par. 98. As before stated, this kind of representation produces a very striking effect when the surface is very rugged. 105. The Vertical System with Vertical Illumination is illustrated in Plates IX. and XII. There are several varieties of this system of shading, the most important being Lehmann's, commonly known as the " German System," and its modifications, and the French System ; in all of which the hachures are drawn coincident with lines of greatest descent (par. 100). Except in the French System, and for slopes from i" to S'' in the use of Lehmann's scale, the hachures are drawn a certain number to an inch, measured perpendicularly to the hnes of greatest descent throughout the same map; while their lengths are not fixed, but are gen- erally, as in the horizontal system, increased for the gentler slopes. The mechanical operation of drawing hachures, including curving, blending, etc., is the same for all, and is described in par. 115; but their distribution is in accordance with the different scales of shade and working-scales. Scales of Shade and Working-scales.— Plate X. 106. Lehmann's Scale of Shade and Working-scale.— Reiemng to the general rule for shad- ing with vertical illumination (par. 97), n° in this scale is 45°, and the intensities of shade 7 5° TOPOGRAPHICAL DRAWING. for the different slopes from o° to 45°, may be denoted by the numbers o, I, 2, 3, 45.— o representing white, and 45 black. Considering only the slopes of 5°, 10°, 15", 40°, differing by 5°, the intensities of the corresponding shades are then l, 2, 3, 8; therefore, to construct the scale of shade, the space between the parallels AB and CD, Fig. 50, is divided into ten equal rectangles marked o", S°, 45°, each of which is subdivided into nine others by vertical lines as shown. One subdivision of the 5° rectangle is then blackened, 2 of the 10°, 3 of the 15°, .... 9 of the 45". The ratio of black to white is thus shown to be : For a 5" slope, i : 8, or f ; for a 10° slope, 2 : 7, or |^ ; for a 15° slope, 3 : 6, or f ; for a 20° slope, 4:5, or | ; for a 25° slope, 5 : 4, or | ; for a 30° slope, 6 : 3, or f ; for a 35° slope, 7:2, or f; for a 40° slope, 8 : l, or f; and this ratio for any slope is obtained from the expression 45^ representing the 15 1 6" angle of inclination : e.g., the ratio for 15" is ■ s > s. ^ 45 - IS An exception is made for slopes less than 5°, the number of hachures being diminished as follows: For the same map-space which for a slope of 5° is covered by five hachures, four hachures only are used for a 4° slope, 3 for 3°, 2 for 2", and I for i". To construct a working-scale, it is usually necessary to first decide upon the number of hachures required per inch. This number varies generally with the scale of the map, as many as 75 being used for maps drawn to very small scales, while 40 to 60 is the usual number. In Lehmann's standard working-scale, however, this number is fixed by making the breadth of the hachure for a 5° slope -|- of a millimetre (Maes) ; and the breadth of hachure for any slope exceeding 5° is readily deduced from the proportion H : S :: @, : 45°; in which H is the breadth required, 5 is the space between the axes of the hachures — in other words, a unit of the number per inch — and @ is the angle of inclination. A method of constructing the standard working-scale is shown in the lower part of the figure (50). The main divisions of the scale of shade are made 16 mm. (about -f-J- inch) in length, four parallels are drawn below CD, as shown, and the verticals defining the main divisions of the scale of shade are produced to EF. In the first row of rectangles thus formed below CD, the black correspond- ing to the different slopes is subdivided into two equal parts and distributed as shown ; these parts are again subdivided and distributed in the following row ; and so on to EF, in which the black and white is in the form of hachures with spaces of i mm. In following this method of construction, to obtain any required number per inch, make the main divisions of the scale of shade of corresponding length, and, if necessary, change the number of rows used to distribute the black; e.g., for 20 per inch, the main divisions should be 0.8 inch in length ; for 30, |-f In practice, the breadths of hachures and intervals intermediate to those given in the working-scale are estimated by eye. To make use of this scale, it is cut off along the line EF, the scale of inclination (par. 80) is applied to the contours, and the hachures corresponding to the slope thus determined are copied from it — the edge EF he'mg placed just above the contour for this purpose. The 40° hachures, increa.sed in breadth, are used instead of solid black for the 45" slopes. THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 51 It may be observed that contours in Lehmann's original system simply gave the general outline of the configuration, without indicating absolute heights, and thus merely served as guiding-lines for the hachures. 107. The Austrian Scale of Shade and Working-scale . — On account of the total obscura- tion of details frequently produced in the representation of mountainous tracts by the use of Lehmann's scale, the Austrian school increased its range by making « = 50 (par. 97), for slopes from 5" to 45° inclusive. The intensities of shade for the different slopes, 5°, 10°, 15", 45°, are thus represented by the numbers i, 2, 3, 9, and the scale of shade could be con- structed in a manner similar to that adopted in the construction of Lehmann's ; but this school chose to represent in the different rectangles the relative amount of black and white, for the different slopes, in spaces corresponding to intervals between consecutive contours, and the scale (Fig. 51) is thus constructed. From G on ^H, lay off consecutively, and to any convenient radius, the cotangents of 5°, 10°, 15° 45", which is readily done by use of the scale of inclination (par. 80), and erect perpendiculars at the extremities of these distances ; divide one of these perpendiculars, as Gg, into ten equal parts, and through the points of division draw lines parallel to GH, as shown. The main divisions of the scale, marked 5°, 10°, 45°, will thus be divided into ten equal parts. One part of the 5° division is then blackened, two parts of the 10°, 9 of the 45°. The ratio of black to white is thus shown to be : For a 5° slope, i : 9; for a 25" slope, 5:5; for a 10° slope, 2:8; for a 30° slope, 6:4; for a 15" slope, 3:7; for a 35" slope, 7:3; for a 20° slope, 4:6; for a 40° slope, 8:2; @ and for slopes of 45° and upwards, 9 : i. This ratio for any slope between 5° and 45" is obtained from the expression _ @ representing the angle of inclination. In the plates issued by the Polytechnic School at Carlsruhe, @ = 50. The same exception as in Lehmann's scale is made for slopes between 1° and 5°. These ratios and the consideration that the lengths of hachures are purely arbitrary, render the construction of the working-scale a very simple matter. For clearness of illustra- tion 20 hachures per inch are assumed, and the working-scale is constructed as shown in Fig. 52. Five hachures for each slope are a sufficient number to serve as guides in the dis- tribution of the shade ; therefore on IK, 2\ inches in length, construct 9 equal rectangles and mark them 5°, 10°, 45°, as shown; subdivide each rectangle by fine vertical lines into five equal parts, and blacken each part in accordance with the scale of shade. The last opera- tion is conveniently performed as follows : Since the breadth of the black for 5° in the scale of shade is^ of the breadth of the 5° rectangle, the hachure for this slope vs,^ oi ■^ — ^-5- of an inch; therefore the fine vertical lines already drawn in the 5° rectangle of the working- scale represent the hachures for this slope. Similarly, the intervals between the hachures of the 45° rectangle are made as narrow as possible. The hachures of the 25° rectangle are then 52 TOPOGRAPHICAL DRAWING. constructed by blackening \ of each vertical subdivision; those of the 35", by blackening |; of the 15", i; and the breadths of the hachures for the other rectangles are then proportioned between those already drawn. The blackening in each case should begin at the vertical lines of subdivision and extend in the same direction throughout the scale. The breadths of hachures for slopes intermediate to those given in the scale are esti- mated by eye. The scale is completed by producing the main lines of subdivision downwards, a distance from /AT equal to the lengths of the horizontal equivalents for the corresponding slope (par. 80), and cutting it out along the hnes /^and LM \ or a separate scale of inclination may be used. It is applied as described at the close of par. 106. 108. The U. S. Coast Survey Scale of Shade and Working scale. — A modification of Lehmann's scale, by which its range was increased to include a slope of 75°, was adopted by the U. S. Coast Survey in i860. For Coast Survey maps, the ratios of black to white for the different slopes are as given in the first two columns of the adjoining table. Lehmann's ratios from 5° to 25° inclusive are thus retained; from 25° to 40°, the scale follows the curve of natural sines ; but beyond this, the required intensity of shade is produced in hachuring by retaining the same interval as for 40°, while increasing the 40° space 25 per cent for 45°, 50 per cent for 55°, 75 per cent for 65° and 100 per cent for 75". For slopes less than 5°, the breadth of hachures is the same as for 5° ; but the 5" space is increased 25 per cent for 4°, 50 per cent for 3", 75 per cent for 2°, and 100 per cent for i". The last column of the table gives the number of hachures per inch employed by the engraver for the published maps of -g-jj-J-or- The number per inch for manuscript maps is: No. of Slope. Ratio of black to white hachures per in. 1° I 21 50 2 I 18 57-1 3 I 15* 66f 4 I I2t 80 5 I 8 " 10 2 7 15 3 6 20 4 5 \ 100 25 5 4 30 3 2 35 7 3 40 4 45 5i 80 55 61 66f 65 7f 57-1 75 9 50 For the scale of xo-g-o-j-, 40 ; for the scale of ^u^ttt' 5° ; for the scale of ^f^rwrr- 65 ; and for the scale of ^-0-r(nr' ^O- The model of ^-j-J^^o"' ^'^'^ ^^ other maps to very small scales, are reduced for the engraver from the manuscript maps, by the processes of photography (par. 147). From the above data a working-scale is thus constructed. For illustration, for the slopes from 5° to 40° inclusive, 5 hachures only are represented and 20 per inch instead of 100. On NO (Fig. 58), 2 inches in length, construct 8 equal rectangles, marking them 5°, 10°, 40°, and subdividing each by fine vertical lines into 5 smaller equal rectangles, as shown. Blacken the latter in accordance with the corresponding ratios of the scale of shade, the black extending from the vertical lines in the same direction throughout the scale. From C toward the left, and from i\^ toward the right, the breadths of the smaller rectangles corresponding to the other slopes — in other words, the spaces — are increased in regular order 25, 50, 75 and 100 per cent, in accordance with the above table. The lines of subdivision from N to Q, and the intervals from O to P, are therefore made very narrow ; the former, for a working scale con- structed in accordance with the table, being yy',,^ (the breadth of hachure for 5°), and the THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 53 latter ^^ of an inch (the interval for 40°). The proportionate widths for the scale (Fig. 53), are -j^-j and yJ-j of an inch respectively. 109. A modification of Lehinann's scale of shade, in use in England for ordinary purposes, has the advantage of simplicity and is very easily applied. The ratios of black to white are : For a slope of 15°, 1:2; for a slope of 30°, 2:1; for a slope of 22^°, i : i ; for 0° white ; and 45", black. According to this convention, the working-scale is constructed by assuming for the medium slope of 22J a breadth of hachure best suited to the work in hand ; the breadths for 15° and 30° are then fixed by the above ratios, and those for intermediate slopes are estimated by eye. Fig. 54 is a working-scale of 20 hachures to the inch, constructed in accordance with these ratios. no. The Danish method of hill-shading, as illustrated in a portion of the " Map of Den- mark and Schleswig-Holstein," resembles Lehmann's to some extent. This map is con- structed to a scale of -j-u-Jinr ^"^^ reduced to -g-n-oTTF- The principal feature of its construction consists in varying the equidistances according to the steepness of the slopes ; so that the equidistance employed for slopes of 14°, or \, and less, is doubled for slopes from 14° to 27°, or \, and quadrupled for slopes from 27° to 45°, or i. No shading is used on slopes less than 14°, while the intensity of shade increases regularly from 14° upward. Fig- 55 's ^ working-scale for a map of xwrnr- with equidistances as shown. In each of the rectangles marked 1°, g°, 10" and 14°, four lo-ft. equidistances are given ; from 14° again to 27°, two 20-ft. equ-idistances ; and from 27° again to 45°, one 40-ft. equidistance. The finest hachures are used for the 14° slope. The breadths of the hachures increase reg- ularly up to 45°, and the same number of hachures per inch is employed for all the rectangles of each group, as shown in the figure. 111. General Muffling, of the Prussian army, modified Lehmann's scale conformably to the Austrian system (par. 107), and in addition introduced the style of hachuring shown in Fig. 56. Slopes of 45" and upward are represented by the hachures for 45° ; slopes of 40° to 45", by those for 40°, and so on down the scale ; the slopes intermediate to those given in the scale being represented by corresponding gradations. The advantages claimed are that no previous practice in hachuring is necessary to pro- duce an intelligible map, and that a notable error in reading the map is impossible. 1 12. The French System of Hill-shading. — The main difference between this and the other forms of the vertical system just described, is in the method of separating the hachures. Two varieties of this system are in use in the French School, viz., Colonel Bonne's and that adopted by the Commission of 1828. 113. Colonel Bonne's system, now employed in the French War Department, comprises scales of shade for maps of ywfd"' "jwinr' ttouo'' ''"'^ "sttfito- The ratio of black to white for any slope is f X tang, of the inclination ; therefore the ratios for the different slopes are as follows : 54 TOPOGRAPHICAL DRAWING. For a slope of i", i for a slope of 5", i for a slope of 10°, i for a slope of 15", i for a slope of 20°, i 38.0; for a slope of 25°, I : 1.3 ; 7.6 3-8 2.5 1.8 for a slope of 30°, i : 1.15 ; for a slope of 35", i : 0.95 ; for a slope of 40°, i : 0.8 ; for a slope of 45°, 3 : 2. The intensities of shade for the different slopes are uniform in the same map, but the breadths of the hachures diminish directly with the scale of the map. As to the intervals between hachures, Colonel Bonne appears to have imposed the condition that, for the same map, the hachures for all the slopes should become indistinct at the same distance from the eye (Lehagre). A working-scale constructed in accordance with the scale of shade is shown in Fig. 53. The slopes are designated by their tangents, and the horizontal equivalents (par. 80) are given upon the slips projecting below the groups of hachures. Thus ab is the hor. equiv. for an equidistance of 2.5 m., ac for 5 m., and correspondingly for the other slopes. To apply it, the scale, cut out along its outline, is so placed that the right-hand edges of the slips shall be normal to two consecutive contours; and the hor. equiv. being thus de- termined, the hachures immediately above the corresponding slip are copied upon the map. A proportional scale may readily be constructed for any equidistance or degree-graduation. 114. The Commission of 1828 adopted the following method of hill-shading (Lehagre): The contours are plotted in pencil, and must have a constant equidistance for the same map. The hachures follow the lines of greatest descent and, as a rule, are continuous in each zone — that is to say, they extend entirely across a zone from one contour to the other ; they also break intervals, so as to mark the positions of the contours when the latter are erased. They do not extend across roads, but are so drawn on either side of them as to give the roads the appearance of being cut out or erased from the general shaded surface. They are drawn across paths of gardens and parks. Slopes less than -^ (corresponding to the French "grade") are left white, except in cases where it is desirable to preserve the continuity of shade. Slopes exceeding 45° are expressed by special signs : if rock-surfaces, by figures resembling outlines of rocks ; if simply rugged with ledges here and there, by hachures with white spaces irregularly distributed between the zones, which are subsequently filled with horizontal and oblique strokes termed rock- markings. (See par. 1 19.) Between the limits of -^ and \ the relative breadths and intervals of the hachures are fixed by the following rules : 1. For hachures which upon a uniform slope are more than 2 mm. (0.078 in.) in length, the breadth is constant, and the interval is equal to \ the length. 2. For hachures less than 2 mm. in length, the space = |-mm., and the breadth of hachure increases as the length diminishes. To preserve a certain degree of uniformity in hachuring by different drauo-htsmen, the following conventions have been recently adopted (Lehagre, 1876). For the gentler slopes, the hachures for maps of -^\-^^ and less are made very fine, and a little heavier for xTj-lirT^. For the slope of \, the hachures and intervals are of equal breadth ; THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 55 for \, the breadth of hachure is twice that of the interval ; and between these limits and that corresponding to hachures 2 mm. in length, the breadths are graded by eye, — the space remaining constant. A working-scale in accordance with these rules and conventions may be constructed as follows : Assuming the scale of the map as TiroTnr> ^""^ *^^ equidistance 40 feet, first ascertain the limiting slope of the 2 mm. hachures. The equidistance reduced to the scale of the map is 0.05 in. 2 mm. " " " " " " 0.08 in. The tangent of the required slope is therefore f = 0.625, and the slope is 32°. From 32° to 45° the space is ^ mm. = 0.02 in.; and the space for slopes less than 32° is found , cot. of the slope X equidistance from the formula . 4 T^u r o ^u • 1-73 X 0.05 in. Thus, for 30 , the space is = 0.021 m.; for 25°, 0.022 in.; for 15°, 0.046 in.; for S°, 0.14 in.; for 20°, 0.026 in.; for 10°, 0.07 in.; for i", 0.70 in. These spaces are then laid off in groups, a convenient number in each group, as shown in the rectangles of Fig. 58 ; e.g., for 25°, lay off ab= 10 y. 0.022 in. = 0.22 in., and subdivide it into 10 equal parts. The hachures are then drawn in the spaces according to the modifications of the rules above given — the black extending from the vertical lines of subdivision in the same direction throughout the scale. Thus, since the ratio of black to white for the slope of f (26° 30', very nearly) is l, the breadth of hachure for 25° is a trifle less, and for 30° a trifle greater, than js the space; for 45°, the breadth is f the space, and the breadths diminish regularly from 45° to 30°, and from 25° to i" where the hachures are fine lines as already described. A scale of inclination (par. 80) may then be attached, as shown in Fig. sz, or constructed separately ; and with this and the scale of shade, the hachures for the different slopes are at once determined. 115. The Operation of Shading in the Vertical System with Vertical Illumination is illus- trated in the right half of Plate XI. The preliminary work (par. loi) having been performed, the hachuring is begun at the summit of the principal elevation, and the work is extended from left to right, as indicated at a, Fig. 62, so that the hachure last drawn shall always be at the left of the pen-point and the*interval readily observed. Having completed the upper row or band of hachures about the summit, the shading is extended downwards by a second row, and so on to the base of the slope. The other eleva- tions are similarly shaded, and finally the slopes of the general surface. In order to produce good work, certain rules must be observed in drawing hachures. 1. The hachures must follow lines of greatest descent. 2. They are always drawn towards the draughtsman, and the paper should be so dis- posed as to facilitate this operation. S6 TOPOGRAPHICAL DRAWING. 3. Shades of different intensities are blended into each other, and the limiting hachures of a slope are blended into the level surface. Thus, as in Fig. 62, the hachures for the slopes at b and c having been determined by the scale of shade, they are gradually increased in breadth in working between these points. Similarly, in working down the slopes, the rows corresponding to different degrees of declivity are blended by tapering the extremities of the hachures as along dd or ee. The upper and lower rows are also blended with the level sur- faces, as at/ and ^. The shading is sometimes blended at the bases of slopes by shortening alternate hachures, as shown at g' ; but an irregular outline produces the best effect. 4. The hachures of any row should begin at an imaginary line joining the lower extremi- ties of the preceding row, as at h — neither below it, as at i, nor above it, as at >^ ; and as a gen- eral rule they should break intervals, although no special effort should be made to this effect. 5. To avoid too much splaying of hachures, a free use of guiding-lines (par. 100) is neces- sary in hachuring the parts of slopes defined by sharply curved contours. As shown in Fig. 63, the hachures in the vicinity of mm' and lo' , are made quite short as compared with the breadth of zone, and the numerous guiding-lines assure their direction. At / and q is shown the manner of rounding sharp curves by occasionally shortening the hachures ; and at r, an application of the foregoing rules to the hachuring of a col, - — see also d, par. 100. In rounding sharp curves, hachures for steep slopes are also tapered, as at s. In the French system, in order to obtain the proper shade when the hachures diverge considerably, the spaces are measured on an auxiliary contour midway between the principal contours, as at t. For regular spacing of hachures, a roulette or small toothed wheel, the intervals between the teeth corresponding to the number of hachures per inch, is very convenient for marking off the spaces. The requisites for good work are a steady hand, good instruments and material, facility in drawing guiding-lines, in blending the different grades of shade along as well as across the zones and in preserving uniformity of shade for like slopes throughout the map ; all of which necessitates a careful observance of the rules and considerable practice. 116. A rigid adherence to the scales of shade in hachuring would evidently result in representing slopes strictly in accordance with their steepness; and with the working-scale attached to the map, it would be simply necessary to refer to it, to ascertain the exact incli- nation at any point ; but, aside from the great labor and skill required for such precise work, the reference itself requires a practised eye ; and for practical purposes and general use, maps so constructed, with hachures only to represent the slopes, may be said to be inappropriate. If, however, the contours are in the finished map, and hachuring is employed simply as an accessory to give relief to the forms of ground, exactness of representation is afforded by the contours, much latitude is permitted in hachuring, whereby the labor and skill required are very much lessened, and a map is produced which is not only correct in the delineation of forms, but also attractive in its nicety of detail. 117. Plate IX. affords an example of Hill shading by the Vertical System, zvith Oblique Illumination. The operation of shading is similar to that described in par. 115, with a further gradation of the hachures in conformance with the general rule given in par. 98. 118. The relative merits of the horizontal and vertical systems are difficult to determine. THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 57 It may be said that the former is harder to learn, for fine work it is slower of execution, and it has the disadvantage that communications, which usually follow the general level, are liable to be confounded with the hachures. In the latter, the hachures, as a rule, break intervals throughout the map ; therefore no continuous lines exist to cause confusion of detail, besides, the vertical system appears best suited to engraving, since the finest maps of recent date are engraved in accordance with it. 1 19. Representation, of Rock-surfaces. — The outlines of rock-surfaces and rugged local- ities are indicated in pencil during the plotting of the contours ; and in order that the par- ticular formations may be approximately represented, sketches or verbal descriptions of them should be found in the field-notes. Characteristic lines, extending in the general directions of the main combs and crevices, give an idea of the formation and furnish a basis or frame-work for the shading. Vertical or oblique illumination is used with either system of hill-shading. The latter is simpler, more effective, and when well executed is much more pleasing to the eye, the former requiring so much black on the steep surfaces as to rob other details of their values. The regular hachuring ends at the pencilled outlines above described, abruptly, if the hachured slope is steep, or with fine hachures touching the outlines here and there if this slope is gentle. The rock-shading is effected by groups of hachures, adjacent groups having different directions; and while these are being applied, a variety of harmonious forms, as well as the proper intensity of shade to bring them out, will suggest themselves. Fig. 39, Plate VIII., contains outline rock-formations. Plate VI., examples of rock- shading with vertical illumination ; and Plate XII., examples with both vertical and oblique illumination. It mars a map very much to employ criss-cross lines, or regular blocks of lines oblique- to each other, to represent this feature. The study and copying of good examples, or a little practice in sketching rock-forms in outline, and in constructing a rough plan of such outlines, would soon give facility in drawing pleasing signs for them. Variety of shape and a careful consideration of the direction of the light are essentials to good work. 120. Contours Combined with Brush Hill-shading (Plate XIII.). — This system, as employed at the French School of Application, is as follows: The same principle applies as in the horizontal and vertical systems, — viz., that the in- tensity of shade is proportioned to the degree of declivity ; it is therefore necessary to estab- lish a corresponding law of gradation. Vertical Illumination is first considered. a. Six elementary shades are used, having the intensities or values represented by the fractions |f , \\, i^, /j, ^ and ^\, corresponding to the slopes f, \, \, ^, gV and ^, which they respectively represent. These values taken in the above order are also designated by the numbers 6, 5, 4, 3, 2, and i. No. 6 is termed the normal tone or shade, and its value cor- responds to that of a series of parallel lines and spaces of equal breadth, as shown in Fig. 64. This figure also shows the order of arrangement of the different shades. The normal shade is obtained by trial with India ink, or other shading material used, and brush, — the correct shade being produced, when, at a short distance from the eye, it appears of the same intensity as the above series of parallel lines. No. 5 is obtained by adding to 6 a quantity of water equal to that it already contains, 4 is obtained from 5 in a similar manner, and so on to num- ber I ; in each case doubling the quantity of water used in the next preceding shade. 8 S8 TOPOGRAPHICAL DRAWING. b. A working-scale could be formed by filling these rectangles with the elementary shades prepared as already described ; but to conform to the method in practice, it is con- structed by superposed shades as indicated in " Superposition of Shades,"— see figure. The rectangles from i to 6 inclusive are shaded with No. i ; those from 2 to 6 inclusive, again with No. I ; from 3 to 6, with No. 2 : from 4 to 6, with No. 3 ; and so on, sensibly doubling the intensity for each increase in the degree of declivity. Although the shades for but few slopes are given in the scale, it is readily seen that by the addition of more or less water to the elementary shades, any intermediate shade required can be produced. c. The application of the working-scale is thus illustrated : Let a plane be moved tan- gent to the surface of the ground while preserving a constant inclination to the horizon, of say ^. The line of contact, termed a curve of equal shade, thus formed, would be repre- sented by shade No. 3, corresponding to the inclination of -^ ; while the inchnation on one side of this curve would be greater and on the other less than Jg-. The curve of equal shade -gJ^-, being similarly determined, and represented by No. 2, a zone is defined bounded by these two curves ; and it is apparent that in shading this zone, the shade should be graded from one curve to the other. Supposing that all the curves of equal shade, -gV) A' T^*"^ traced upon the map; then, conforming to the working-scale, all slopes steeper than -g^ would be covered with shade No. I, graded to zero at the edges of level ground; shade No. i would then again be applied to slopes steeper than -^, and graded to zero at curve -^\ No. 3, to slopes steeper than ^, and graded to ^, and so on for the rest of the slopes. The intensities of shade so produced conform to the scale, and a true representation of the configuration can thus be obtained. If irregular slopes intervene between these curves of equal shade, the shades correspond- ing to them are capable of exact determination as follows : Let ^ be such a slope ; the shade value required is -j^ + ^ + ^, and shades 3, 2 and I are correspondingly superposed. As a ready means of obtaining these different values the following table is prepared : Slopes. Superposed Shades. Shade Values. I I 2 3 ♦ s 6 ^ liV , , A ^ T^ ^ A iV A A A A i -h ih -h A A i ^ A i^ A A if \ A ih A A A H M \ irV ih A A A if M 14 To find the shade value for a slope of ||-, look under "shade values" for next higher scale value = ff, and add together the least number of " superposed shades," on the same THE CONFIGURATION, OR SURFACE-FORMS OF GROUND 59 horizontal line, that will produce y- = — ; or the required shade is produced by the superposition of 5, 3, 2, and i. It is observed that a slope of 45° is represented by the superposition of all the shades. The result, although darker than No. 6, is by no means black ; in fact, all the shades produced form simply a parallel series having certain relative values, as in many other forms of scales of shade. d. The next step is to trace the curves of equal shade upon the contoured map, which is readily effected from the consideration that the slope along a normal between adjacent contours is uniform, and that normals of equal length between adjacent contours indicate equal degrees of declivity. Thus, in Fig. 65 the normals n, «, being of equal length mark slopes of equal declivity about the watercourse ab, and the dotted line drawn through their middle points is a curve of equal shade. The points n' and n" are located approximately as follows: n' is evidently between c and d; and since the inclination is less rapid from the middle point of fd^ towards c than toward d, n' will be nearer the lower contour; and, simi- larly, n" is nearer e than/". The points n,n, can be located by measurement upon the map ; e.g., to find the curve of equal shade ^, — this ratio representing practically the tangent of 7", — take in the dividers the horizontal equivalent of 7" from the scale of inclination (par. 80), and mark by dots, as in the figure, the different points where adjacent contours are sep- arated by this distance. No sensible error arises from considering the normals as straight where they may be slightly curved. e. In par. ^ above, an undulating surface is assumed; but where the forms of ground are more pronounced, they are considered geometrically. Thus the spur (Fig. 66) may be regarded as a conoidal surface. Draw the parallels gh and //^tangent respectively to the 40 and 20 contours; the line hn joining the points of tangency is sensibly the line of contact of the plane of these tangents, with the surface, and the line op drawn through the middle point of lin, and perpendicular to the tangents, will have the same inclination as the plane. In practice an opening, qporst, is made in paper or cardboard, the parallel edges qp and or being separated by the perpendicular edge op = horizontal equivalent, according to the scale of the map, of the slope for which the curve of shade is desired ; and the middle point u of op is marked. This instrument is so applied to adjacent contours that qp and or are tangent to them, and ol appears equal \.o pm ; a point is marked immediately below u for one point of the required curve ; and so on for the other points. Curves are similarly determined for ravines or re-entrants, the instrument in any case being so placed that the convexity of the contours within the opening shall be towards the edges qp and or. f. The hill-features requiring no shade are a summit, the base of a cup-.shaped depression or hollow, and a col. The white spaces marking them are slightly exaggerated to make tliem more conspicuous ; and the shades beginning here are gradually increased to their proper values at the nearest curves. g. In practice the curves of equal shade, which are represented by light pencil-dots, are not traced with mathematical accuracy ; and after considerable experience in reading con- 6o TOPOGRAPHICAL DRAWING. toured maps, their location may be estimated by eye, or at least with the occasional use of a scale of inclination. The forms of ground should be previously carefully studied and the levels particularly noted. The shading corresponding to a given slope is not stopped abruptly at the extreme curves; but since slopes, except in rugged tracts, change their inclinations gradually, it is made to conform to this condition, and is carried beyond and blended into the local shade. Any of the elementary shades required for the relief may be used first ; but it is better to begin with one of the lighter, bearing in mind that a darker shade is never used until the one next preceding has been superposed once upon itself. The roads being left white separate the map into parts for shading, which is a great convenience on account of the rapid- ity with which the shading dries, and the consequent difficulty in large areas of properly blending the different shades. Intermediate shades are used as required. During the progress of the work the effect of the relief already obtained should be carefully studied, and the subsequent shades be increased or diminished in value accordingly. To avoid spotting, the paper should be neither worn nor scratched ; but if so injured, a light wash of size is applied before the shading is begun. If the paper has been submitted to strong pressure, light friction with a damp sponge will usually restore the grain, and it should be kept a trifle damp to prevent the washes from drying too rapidly. If, nevertheless, spots appear, they may, if dark, be removed with a sponge, or by careful rubbing with soft linen ; — a paper with a hole just exposing the spot being used to protect the map; — if light, by stippling (par. 135) with a brush nearly dry, or with a pencil. For shading, sepia may be used instead of India ink ; but the latter with a little indigo or cobalt added gives a better effect, particularly if colors are to be employed in representing the other features. 121. For Oblique Illumination, M. Goulier's law of gradation (par. 98) is applied. The relief is first given as described for vertical illumination (par. 120), but much lighter shades are employed ; e.g., the elementary shades i, 2, are used for slopes of -^-^, ^ respectively, instead of for ^, yg-, The slopes extending toward N. E., S. E. and S. W. are again shaded as in the first operation, and the shade is blended with the N. W. portions. The " normal tone" is then applied, as with vertical illumination, to the S. E. slopes, and blended with the N. E. and S. W. portions. The time required is about twice as long, and the maps have about the same intensity as with vertical illumination. Either India ink or sepia is used, but a brighter effect is produced by using the latter for the first shading, and the former for the other two. Certain colors may be used in the first shading, — sepia for tillable land, neutral tint (a mix- ture of red, blue and yellow) for rocky surfaces, neutral tint and burnt sienna for rocks, and Prussian blue for glaciers — the rest of this shading to be in India ink. Glacier shades are reduced to half values. Whether the illumination be vertical or oblique, the effect of aerial perspective is in- creased by treating the higher elevations with a light wash of burnt sienna, and the depres- sions with a light tint of cobalt,— the former at its lower edge, and the latter at its upper THE CONFIGURATION, OR SURFACE-FORMS OF GROUND. 6i edge, being blended into the slopes. This effect is intensified in the valleys by a very light wash of Chinese white tinted with cobalt. Finally, the contours are represented either in black or red. 122. Brush shading as ordinarily practised. — In the details just given is observed a care- ful adherence to the principles forming the basis of all art-representation, viz., working from the whole to a part and following a well-prearranged system ; and whatever method may be adopted, these principles should underlie it. The details are very valuable in affording cor- rect ideas of relief and of the exact means for producing it, and should be carefully studied. In view of the fact, however, that shading is now employed as an accessory to contours (see also par. 1 16), the operation becomes very simple. The materials required are stick India ink of good quality, dishes for mixing the shades, the brushes described in par. 129, water and blotting-paper. The open frame described in par. 38 is best adapted to the work. The outlines of all the details, as well as the contours, should be plotted in order to define the limits of the shade. For the best results the plotting should be in pencil only ; in lines fine as a rule, but sufficiently strong on the steeper slopes to be visible after the shade is applied. All superfluous lead is removed with bread, as described in par. 42, or by the application of water with a large soft brush or sponge. ' If it is desired to ink the contours and outlines of the details first, the drawing must be washed, before shading, with water and a large soft brush, until the ink ceases to run, the frame being held in an inclined position, face up, for this purpose, and the drawing finally flowed with clean water. It is then placed in a horizontal position and permitted to dry at the ordinary temperature of the room. In case the ink runs, the drawing should be washed again. Heavy lining must be de- ferred till the shading is finished. The ink, prepared as described in par. 34, is then diluted to produce the first shade, which should be dark enough when applied to be distinctly visible at a short distance from the map. When dry, the drawing is moistened on the back and disposed as described for laying a flat tint (par. 132), and is ready for the shading. The steepest and gentlest slopes are observed and their relative shade values decided upon, the number of intermediate values depending upon the differences of slope and the degree or strength of relief desired. Ordinarily from three to five shades are sufficient, mountainous tracts requiring the greater number. The first shade, prepared as above, is then thoroughly stirred with the brush and applied as described in par. 132, to all the slopes, beginning at the upper part of the map and work- ing across and downwards; and, as the work proceeds, is blended (par. 134) into the levels, which are left white. Roads and buildings, except in small scaled maps, are also left white. The drawing is then dried, artificial heat may be used to expedite the work, and any lack of flatness may be remedied by washing with soft brush and water ; but washing should never be attempted until the drawing is thoroughly dry. The back of the drawing is again moistened, and the second shade, of the same value as the first or of a slightly increased value, is applied to all but the gentlest slopes, working in •62 TOPOGRAPHICAL DRAWING. each case from the summit around and downward toward the base — the brush following the general direction of the contours, — and blending it at the edges into the first shade. The rest of the shades are then applied, in each case omitting the slopes corresponding to the preceding values, the intensity of each being judged of by eye, and so related to the others that the shade value decided upon at first for the steepest slope shall not be exceeded. In general a light relief, requiring the use of light shades throughout, is sufficient, and also prevents any obscuration of the other features. The quantity of liquid used in the brush is lessened for the smaller areas. The amount of moisture required in the paper to make the shade flow freely, and to facilitate blending by preventing too rapid drying at the edges, is soon learned from practice. It will also be found unnecessary in laying successive shades to wait each time until the paper itself is dry; but a good rule to follow at first is not to apply a shade, or, in colors, a tint, until the preceding one is thoroughly dry ; and if the area to be covered is of considerable extent, to always moisten the paper as above described before applying it. If the open frame is not used, the front surface of the drawing is moistened with a soft brush, and the surplus moisture may be absorbed with blotting-paper. The tendency at first is to use too strong shades. If greater precision is desired, the curves of equal shade {d, par. 120) may be lightly dotted in pencil upon the map, as shown in Fig. (}■], Plate XIII. Three shade values are here em- ployed to produce the relief. The first shade extends from summit to base; the second, from near the summit to the curve bb; and the third, from near the upper limit of the second to aa. The first shade is blended into the summit and the general level at the base, and the others into the zones outside of their limiting curves. With a little practice, however, the different slopes are readily separated according to steepness, into three or more groups, and the corresponding shades are applied as already described. A little Payne's gray, sufficient to produce a slightly bluish tint, causes the India ink to work well and produces a very pleasing effect. Indigo or cobalt may be used instead of Payne's gray. When the hill-shading is finished, the outlines already drawn are filled with the proper signs, as heretofore described (par. "]&). Rocky surfaces are represented with the tint used for the shading, strengthened when necessary, and applied as described in par. 135, and illustrated in colors in Plate XV., the char- acteristic pen-markings (par. 119) being added if desired. It is apparent that this method of hill-shading has two very important recommendations, viz., great saving of time and clearness of representation. It is therefore particularly well adapted to the production of manuscript maps. For reproduction by photo-engraving, it is only necessary to make a fair copy with wax crayon on grained paper for the engraver ; or, if by lithography, substitute lithographic chalk for the wax crayon. This preparation, however, is usually made by the engraver from the models furnished him. 123. Pencil Hill-shading. — A very rapid method of hill-shading, well suited to sketching- purposes, is now used for all English military maps and sketches. It resembles charcoal draw- ing, and is effected as follows : FINISHING THE MAP, LETTERING AND ORNAMENTATION 63 I The contours are plotted in crimson lake and the streams in Prussian blue, — in sketching, these features are drawn with a red and blue pencil, — the shading is then applied in accord- ance with the standard scale of shade (par. 102). A smooth, hard surface should be underneath the drawing. A soft pencil is applied to the steepest slopes ; the shade thus produced is then rubbed with a piece of chamois leather, folded into a pad, and is extended to the gentler slopes. The depth of shade is increased as required, by the use of the pencil, and diminished by the application of a rubber eraser, — sharp lights produced by the latter can be blended at the edges with the pad. The general relief is produced broadly at first, and the details are brought out after- wards as above described. To bring out any particular form sharply, a hole of the corresponding shape is cut in a piece of paper used to protect the rest of the drawing, and the rubber eraser is applied as above. To shade a minute feature, a pointed shape is given to the folded chamois leather. Rough patches of shade and pencil-marks can usually be smoothed by hard rubbing with the leather; but in case the pencil is too hard, scrapings from it are used rather than the pencil itself. Low levels receive a light shade, summits are left white, and spurs are shaded lighter than ravines. The shades should be so firmly rubbed in as to bear handling; but can be fixed by spray- ing with shellac and alcohol, or by floating upon a thin solution of isinglass or gum-arabic. Any coloring desired is deferred until the shading is finished. A variation of this method of shading consists in first plotting the contours very lightly in pencil, then indenting them with a sharp style, and applying the shade as just described. In this case the contours are left nearly white. Section IV.— Finishing the Map, including Lettering and Ornamentation. 124. To complete the map, after the shading and signs are finished, there are required : I. The lettering, including the title ; 2. The scale of distances; 3. The "compass," or indica- tions of the directions of the true and magnetic meridians; and 4. The border: all of which are usually added in the order given. The equidistance (par, 91) and the height of the datum-plane above some permanent bench-mark should also be noted ; and if the map is in relief, a scale of shade is usually attached, the latter being indispensable if the contours are not drawn. 125. The Lettering. — Roman letters and italics are now in general use for all topographi- cal drawings. Forms of Letters. — The general rule is to employ upright Roman capitals for names of the most important features ; inclined Roman or italic capitals for those next in impor- tance; Roman upright with initial capitals for the third in rank; Roman inclined, or italics with initial capitals, for the fourth ; and small italics for the fifth, or for the least important purposes and designations. The French convention is to assign these forms in the above order respectively to cities, towns, villages, hamlets and farms, or isolated buildings. 64 TOPOGRAPHICAL DRAWING. The general rule is modified in the construction of the U. S. Coast Survey maps, by always using inclined letters for water features ; and since inclined letters are more easily read and produce a more pleasing effect when names are placed obliquely with reference to the map- borders, as is generally the case for water features, it seems best to adhere to this modifica- tion. Sizes of Letters. — The largest letters are used in the title for the name of the tract or object represented, the smallest for explanatory notes ; and between these dimensions, the sizes are proportioned to the importance of the different features designated. According to the last condition, the largest of these intermediate sizes are used for the principal mountain-ranges and main divisions of the tract ; for the larger bodies of water, — oceans, bays and great lakes ; the next in size for the minor mountain-ranges and ranges of hills, the minor divisions of the tract and for the smaller bodies of water, — coves, small lakes ; the third in size for detached mountains, peaks, passes, ravines and other important hill features, cities, the larger towns, the principal communications and streams ; and the fourth for hills and hill features of minor importance, villages, settlements, farms, important buildings and constructions, including the minor roads, and for rivulets, brooks and ponds. Since a pond in one map may rank in importance with an ocean in another, it is evidently impossible to prescribe exact sizes of letters for the different features ; but the foregoing list will give a pretty clear idea of proper gradation. For the scale of ^o^oj, a good practical rule is to make the largest letters of the title — For a map of about 2X3 feet in dimensions — 0.3 in. in height ; " " " " " IX 1.5 " " " — 0.2 in. " " ; and proportionately for maps of other dimensions ; and since explanatory notes are always very small letters, the intermediate grades are readily assigned. Small letters, called " lower-case," are practically f the height of the corresponding capitals. The following table (from Maes) gives the forms and height of letters used in the Belgian School for maps of ordinary size drawn to scales as given. The heights are in decimillimetres ; In the column headed "Form," RU, RI, ru, r? and i represent, respectively, Roman capitals upright, the same incHned, Roman small upright, the same inclined, and italics. It -is pre- sented for the assistance it may afford in determining suitable relative heights of letters for the different features, when much precision is required. FINISH] NG THE MAP, LETTERING AND ORNAMENTATION. MAP LETTERS— BELGIAN SYSTEM. 6S Features. Abbeys Aqueducts Avenues Barracks Batteries Battle-fields Bays Brickyards Brooks Camps ^-^•M^;-.-.v::::: c^p". te,-.v.;:::: Cemeteries Ctiurches ( ist order Cities, \ 2d order ( 3d order Creeks Cross-roads Defiles Embankments Farms Fences Ferries and Fords Forests, 1 '"Sf, ' { small Forts Fountains ( large Gulfs, ■< medium ( ordinary , Hamlets Harbors , Heaths , Hills I large . . , Islands (sea), -j medium ( small. . , Islands (river) ( large Lakes, \ medium ( small Heights for Scales of Form. 1 1 ^irtmr eooo 10000 20 15 12 ri 20 18 12 ru 20 15 12 ru 20 15 12 n 15 10 10 I 25 20 15 ru 45 40 35 RU 15 10 10 I 18 15 )2 I 30 20 15 ru 35 30 25 RI 25 20 18 ru 40 35 30 RI 30 20 15 ru v>H 15 12 I 15 10 10 I 80 70 55 RU 65 60 50 RU 50 45 35 RU 30 25 20 ru 15 10 10 I 15 10 10 I 15 10 10 I J5 10 10 t 15 10 10 I 15 10 10 I 65 60 50 RU 55 50 40 RU 30 25 20 ru lO 10 10 I 65 60 50 RU 50 40 30 RI 30 20 15 ru 25 20 15 ri. 20 15 12 ru 30 25 20 ri 25 20 18 ru 70 60 50 RU 50 45 40 RI 30 20 15 ru 25 20 18 n 40 35 30 RU 35 30 25 RI 25 15 12 ru Features. Levels (references) Light-houses Manufactories Marshes Meadows Milestones Mines ( chain Mountains, •< secondary.. ( isolated. ... Parks Paths Plantations Posts, military Race-course Ravines Redoubts '^--.j^lu;;;;::::: ilst class 2d class. , 3d class. Roads (minor), j ^-|^; Roadsteads Sand-banks, |',^^f^,;;; Suburbs Towns Trenches v^"^y=- Umfn.:;;.;:: Villages, V'^'p ° ' ( ordmary.. . . Villas, V^^^f, ' ( small Woods, '^^S^, ' ( small Small isolated objects in general, bridges, houses, etc Heights for Scales of ToVir 1 SOOOU 10000 15 10 10 15 lO to 15 10 10 30 25 20 30 20 15 10 10 10 15 10 10 65 60 50 55 50 40 35 30 25 30 25 20 15 15 12 30 25 20 15 ID 10 15 10 10 15 10 ID 25 20 15 40 30 25 30 25 20 30 20 18 25 15 12 20 J5 12 28 18 15 18 15 12 40 35 30 35 25 20 25 20 15 40 30 25 45 35 25 15 10 10 45 40 30 30 25 20 35 30 25 30 25 20 30 20 15 20 15 12 40 35 25 30 25 20 15 10 10 Form. RU RI RI ru RI ru ru ru i RI ru ri RI RI i RI ru ru ru ru ru RI ru Disposition of the Letters. — Names should be so placed as to be easily read, show clearly the object designated and not obscure the signs. For Isolated Features, the names are placed, when possible, at the right or left of, and close to, the different objects designated, and parallel to the lower edge of the border, the spaces between the letters and words conforming to those of ordinary print. For Communications, the names are placed parallel and close to the edges or boundaries ; the spaces between letters remaining the same as in ordinary print, while the words are separated by intervals, which, for uniformity in the same map, may be taken equal to the length of the longest word there used for these designations. The bases of the letters are best turned toward the communication, and such part of the latter should be selected for the name that when the drawing is held in its proper position the letters will not be reversed; e.g., if the road extends upward to the right or downward to the left, or in a vertical direction, the name is placed on the left of it ; if horizontally, above ; otherwise, to the right. If a terminus of a road is within the drawing, its name maybe omitted in the desig- nation, — the word "from" or "to," preceding the name of the other terminus, being sufficient. TITLES AND BORDERS. [ SOUTHAMPTON, 2 3 1[ 2 1 [ 3 1[ NEW YORK Surveyed by James Atkinson, O.E. Scale rmiTB- 1886. 5[ PART OF « [U.S. MILITARY EESEEVATION, 4[ WEST POINT, N. Y. 10 3[ 4 3 [ Scale ^^. Surveyed by Samuel Green, C.E. 1877-8. Equidistance 20 ft. :=f=^ 66 FINISHING THE MAP, LETTERING AND ORNAMENTATION. 67 \i y Na-me \ / Name iiX / v^ f (0 4 \ For Streams, the same rules apply as for communications, except when their widths are at least twice the height of the letters, in which case the names are placed along the axis of the streams, independently of the direction of the current which is indicated by an arrow. The accompanying figure shows the proper position of names relative to lines of features having various directions. For Elongated Outlines, as in the cases of forests, marshes and bodies of water, the names are extended in the direction of the greater dimension and along straight or smoothly curved lines midway between the bounda- ries. The letters are not " extended ;" but, with the words, are so spaced that the name in each case will occupy nearly the entire length of the feature, — the spaces being proportional to those used in ordinary print. In certain cases, when names placed on the features would obscure them, or the scale of the map is very small, a legend, or list of the names, is placed in an unimportant part of the map, or out- side of the border; and corresponding letters or numerals attached to the different names and features serve as mutual references. These names should be horizontal, and arranged in one or more vertical rows. 126. The Title, as shown on the opposite page, consists of the name of the tract or locality represented, called the principal name, the location of the tract, the surveyor's name, date of survey and the scale of the map expressed by its representative fraction. The equidis- tance between contours may also be stated, and any explanatory notes it may be desirable to give may be arranged with the title or placed elsewhere upon the map. The Position of the Title is next to the border, usually in a corner of the map where it will interfere least with the details. A central position next to the upper or lower border may be selected, in which case the title is not enclosed in a rectangle. In maps where the entire space within the border is desired for the details, the title is placed outside of the border, the principal name at the top, midway, and the rest at the bottom of the map. The largest letters used in the map are reserved for the principal name, the next in size for any modification of this name, as " PART OF," " and vicinity," etc.; the name of the county or State in which the tract is located may be about f the height of the principal name : all of which are capitals. The words " Surveyed by ," " Scale " and the date are about \ the height of the principal name, and in Roman capitals and small letters ; while the explanatory notes are in italics, and of less size. The title should be symmetrically arranged with reference to a vertical line through the middle point of the space assigned to it ; which is readily done by sketching each line first on a separate paper, and then assigning its middle point to the vertical line, and working out- wards from it in the construction of the letters. A simple arrangement of plain letters is more effective, pleases the eye for a greater length of time, and is easier of execution than elaborate lettering. The figures at the left in the diagram give relative heights of letters and spaces suitable 68 TOPOGRAPHICAL DRAWING. for titles, and the arrangement of broken lines at the bottom of the plate shows the manner of gauging them. 127. The Construction of Letters. — Nothing mars a map more than poor lettering, and if the skill necessary to draw Roman or italic letters free-hand well is desired, it must be acquired through long and patient practice in copying from the best models. Two ways are therefore suggested to avoid the difificulty ; one is to trace and transfer the outlines of the letters by the means described in pars. 31-2, then ink and fill them in ; and the other, by far the easier, is to obtain the alphabets required in type metal (which can be done at a very small cost), " set them up" as desired, and, clamping them firmly in some simple device, or between flat surfaces with the hand, press the letters on a cloth covered with pow- dered black-lead, stamp the name in its proper place indicated by a pencil-line, and ink the impression with a common pen. Where many maps are made, it is of course preferable to obtain the clamps, rollers and ink necessary for regularly printing the names. Ockerson's devices (par. 74) are extended as follows to the printing of letters and figures. For letters, a stamp, working like an ordinary office-stamp, is used. The type com- posing any desired name is "set up" and clamped in a typeholder at the base of the pis- ton, and, after inking, is impressed upon the paper through an opening in the base of the frame. The outer edges of this base being rectangular, with two of them parallel to the line of type, serve as guides in locating the names. For numbers, three concentric revolving disks are used, each disk containing the t;n numerals in type, so that any combination of three figures is readily formed. This is particu- larly valuable for representing soundings. Plate XIV. gives specimens of lettering used in maps of the U. S. Coast Survey. For colored maps, a more pleasing effect is produced by the use of either " light-faced " or " open" capitals (see Plate XVI.) for the principal name. Although the spaces between letters and words may be ascertained by measurement of common print, the general rule for letters may be found useful ; which is to make the spaces between capitals and small letters the breadths of the intervals between the uprights of the letters H and n respectively. 128. Scales, Compass, Border and Framing. — In addition to the representative fraction given in the title, the scale of distances (par. 45) is placed in the vicinity of, or immediately below the title. For important maps a scale of metres, in addition to one of English units, should be constructed, the relative positions being shown in Plate I. The scale of shade for maps in relief may be placed in the vicinity of the scale of distances, or immediately below a lower corner of the border. The direction of true N., and the magnetic declination, or variation of the compass, should always be indicated. This is usually done by means of a figure called a " compass" conspicu- ously placed in some convenient part of the map, usually over the sign for water, where a large body of water is represented. A simple form is the most pleasing and useful ; and may con- sist of two intersecting lines of medium breadth, showing the directions of the cardinal points, the N. point being marked with the letter N, and of two or more finer lines indicating the intermediate points. A still simpler form (see Plates VIII. and XVI.) is an arrow several inches in length, say FINISHING THE MAP, LETTERING AND ORNAMENTATION. 69 10 inches for a map 2x3 feet in dimensions, pointing true N. ; with the variation of the com- pass indicated by a second shaft diverging, in the direction of the variation, from a point near the middle of the first, and terminated by a half arrow-head placed without the angle thus formed. The arrow-heads should be equidistant from this point of divergence, and be of light, curved lines rather than solid. The amount of variation may be noted along the shaft, or within the angle in the direction of an arc described from the point of divergence of the shafts as a centre — the usual wording being " Variation — ° E," or " W," as the case may be. A compass is of course unnecessary with parallels of latitude and meridians. The border should be rectangular in shape, and also of some simple combination of right lines. Some varieties are shown in the diagram of titles facing page 6"]. A general rule is to make its total breadth = -^^ of the shorter inner edge already drawn (par. 56) ; and when composed of a fine, inner line and a heavy, outer one, to make the breadth of the latter \ the total breadth. If the meridians and parallels of latitude are represented, an inner space is required for the degree-numbers ; and may be defined by a fine line, distant from the above about three times the total breadth above given. In the absence of meridians and parallels, it is advisable, in order to detect subsequent distortions of any part of the map, to construct a set of rectangles of fine lines parallel to the border lines, 100, 1000 feet, or i mile on a side, according to the scale of the map. The margin should be sufficiently broad to give a good relief to the border. Narrow flat mouldings of plain wood, varnished to show the grain, and with bevelled black-enamelled edges, make very pleasing frames for plain topographical drawings. 70 TOPOGRAPHICAL DRAWING. PART III. TOPOGRAPHICAL DRAWING IN COLORS. Section L— Materials, Rules for Working in Colors and Preparation and Laying of Tints. 129. Materials and Instruments. — For topographical drawing in colors, the materials and instruments required, in addition to those prescribed in Part L, Section IL, are water-colors, brushes, plates or saucers and tumblers for mixing tints and cleansing brushes. A right-line pen with plated or German silver nibs, on account of its freedom from corrosion, is superior to the steel pen for line-work in colors. The best stick India ink should be used. Paper suited to this work is described in par. 30. Moist water-colors in " pans" or " half-pans" are handiest, and Winsor & Newton's are always of good quality. The colors necessary are : Yellow Ochre (Y. 0.), Indigo (I.), Crimson Lake (C. L.), Gam- boge (G.), Prussian Blue (P. B.), Payne's Gray (P. G.), Sepia (S.) and Burnt Sienna (B. S.) ; to which may be added Brown Madder and Vandyke Brown. A tube of Chinese White will prove useful for touching up ragged edges of lines and for covering blemishes that cannot be erased. The colors should be kept in a covered tin box to retain their moisture, and so sep- arated from each other that, in supplying the brush, they will not be mixed. When too hard, a drop of glycerine will soften them. Four sable brushes are necessary — three small ones for applying the tints, and a larger one for washing and toning purposes. The accompanying figure shows the sizes for the small brushes, two of the larger size being used for general work, and the smaller for covering small areas, buildings, rock-markings, y (II [| l^^^^ a* etc. A good brush after a thorough wetting should preserve its elasticity, and show a fine point when the water is thrown out of it by a quick motion of the hand. A compartment in the tin box should be provided for them. L They must be washed in clean water before and after using them, never left standing in water, and they should be put away " pointed " by the quick motion above described. A flat sable brush, 2 to 3 inches in width, is best for washing and toning, and should be treated with the same care as the others. RULES FOR WORKING IN COLORS AND LAYING OF TINTS. 71 130. The following rules should be carefully observed in working with colors: I. The materials must be kept clean and free from dust. II. In preparing tints, the colors must be thoroughly mixed with the water, and the mix- ture well stirred with the brush at first, and each time the latter is refilled. III. Surfaces greater than 2 or 3 square inches in extent are best dampened before tint- ing, as described in " Laying of Flat Tints" (par. 132). IV. The surface to be tinted should be inclined at a small angle, 5° to 10°, so that the tint will flow freely along it and in the direction of its greatest dimension. The drawing should be left or placed to dry in the same position. V. The base of the brush or part next to the handle must not touch the surface of the paper in the operation of laying a tint. VI. A tint must not be retouched in any way until dry, either while laying it, or to cor- rect errors afterwards. When dry, small spots lighter than the general tint may be stippled (par. 135) with the point of the brush and a small quantity of color; and small spots darker than the general tint, by repeated alternating applications of water with the point of the brush, and of blotting-paper. Large inequalities must be removed by washing the entire tinted surface with the large brush, and deep markings require the use of the sponge. VII. Light tints must be used at first, at least until practice is had in estimating values. VIII. Repetitions of the same sign must be of a uniform tint throughout the map, and tints of different color should have like values. IX. Except as described in par. 122, all ink markings or lining must be deferred till the coloring is finished. 131. Preparation of Tints. — A single tint is prepared by rubbing the wet brush on the surface of the color and stirring it in a quantity of water somewhat greater than would be required to cover the surface to be tinted. The colors prescribed are transparent ; and since the white of the paper corresponds in effect to the white pigment used in oil-colors, a tint is made strong or light by the use of more or less of the color, or by varying the quantity of water. In preparing a dottble tint, or one made of two colors (connected by the sign -|- in the text), the lighter color is mixed first with the water, and the other then added until the re- quired tint is produced. Tints are sometimes superposed singly to produce a double tint, and when properly done the effect is more brilliant than if previously mixed. The following table may prove useful in showing how diflerent colors are produced : Thus red -)- yellow = orange ; purple -\- green = olive, etc. A neutral tint is a mixture of red, yellow and blue. Blue is a cold color, red and yel- low are warm colors, and are correspondingly added or used in greater quantity in a mix- ture, to produce cold and warm tints. Each of the primaries harmonizes with the combination of the other two ; yellow with purple, etc.; and the same is true of the secondaries — purple with citrine, etc. 72 TOPOGRAPHICAL DRAWING. A transparent color is made opaque by the addition of Chinese white ; so unless this effect is desired, the brush must be carefully washed after using this pigment. If hard or cake colors only are available, the tints are prepared by rubbing, as described for India ink (par. 34). 132. Laying of Flat Tints.— S'mce laying a flat or even tint is quite difificult for the be- ginner and is the most frequent operation in colored drawing, it is minutely described. Pure , color, clean water, a soft and elastic sable brush and a good paper surface are the requisites. Presuming the open frame (par. 38) in use, the paper is moistened on the back, and when the front surface feels cold to the touch or the paper begins to warp, it is disposed at an angle of from 5° to 10°, sloping towards the draughtsman. The tint prepared as above described is then thoroughly stirred with the brush ; the latter is then well filled, and, with the handle pointing toward the right shoulder, the operation is begun at the upper left-hand corner of the surface to be tinted. With light pressure and a motion just slow enough to permit covering, the tint is laid on in bands, as shown in the accompanying figure, from left to right along the upper edge, as indicated by the arrow, the point of the brush extending to the edge ; then down the right-hand edge a distance of about half the length of the brush ; then back to the left-hand edge, over- lapping the first band enough to cause the tint to flow freely from the latter into the second band, and so on to the lower edge of the surface. The pressure should be even throughout, and the motion continuous. The brush should be kept well filled until reaching a point as at a near the lower edge, when the quantity of liquid in it should be a little more than enough to finish. When the lower band is completed, the quantity re- [ maining is thrown out of the brush, as heretofore described, and the superfluous tint is removed from the drawing through capillary action, by moving the point of the brush back and forth along the lower edge ; or a straight edge of blotting-paper may be used for the purpose. It may be found easier at first to lay the upper band as described ; and the others by successive pats with the brush, following each other back and forth across the surface in the order shown in the figure, the tip of the brush remaining in contact with the surface. The result, when the paper is dry, should be a perfectly uniform tint without streaks or spots. If the open frame is not used, the front surface may be dampened with a sponge or brush and the tint applied when the surface ceases to glisten in observing it obliquely. In this case blotting-paper can be used to expedite the drying before the application of the first tint, but not afterwards. Artificial heat of a moderate temperature, or exposure to the sun's rays, may be resorted to. 133. To Lay a Graded Tint. — Having decided upon the relative values required at the two extremities of the surface, as at L and D in the accompanying figure, prepare the correspond- ing tints. For a single gradation, as from light at L to dark at D, lay a band of the lighter tint along the upper edge; then, adding some of the darker to the lighter tint, lay the next band with the mixture slightly overlapping the first, and so on, adding the darker tint RULES FOR WORKING IN COLORS AND LAYING OF TINTS. 73 A for each successive band until D is reached. The relative quantities required of each tint are easily found from practice. The result should be a tint increasing uniformly in intensity from L to D. If the gra- dation is not sufficiently rapid, when dry begin with a band of water at the upper end, add the tint as required, and re- grade. For a double gradation, as from dark at D to light at both L and L' , proceed as above from L to D and continue to L', reversing the process by adding water for the successive bands. 134. To Blend Tints. — If a single tint is to be blended into a white or colored surface, as at the extremities of a slope ; first, take up most of the superfluous tint at the edge to be blended, by passing the point of the brush back and forth once or twice ; then, working rapidly, lay a narrow band of water slightly overlapping this edge, and treat it the same as the final band of color ; apply water as before, and so on until the color fades into the white or colored surface. To blend tints of different colors, as in the sign for "brush- wood " (Plate XV.), suppose that the upper and lower halves of the rectangle in the accompanying figure are to contain differ- ent tints so applied that their line of junction shall be invisible. Beginning at A, lay one tint as far as the dotted line a, and, working rapidly, take up most of the superfluous color along this Hne ; then cleanse the brush quickly, fill it with the other tint, and beginning at the dotted line b, lay this tint to the bottom of the rectangle. It is more convenient to have a sepa- rate brush for each tint. 135. Stippling and " Dragging." — Stippling is the operation of laying a tint by dots placed close together, and is of use in covering spots or blemishes. The point of the brush and very little of the required tint are used. It is best to cover the space gradually with several sets of dots, waiting each time for the preceding set to dry before filling the intervals with another. " Dragging" consists in laying a ragged tint, by which irregular forms and " accidental lights" are produced. It is of much use in expressing rugged surfaces, rock-markings and shadows. The side of the brush and very little color are used, and the handle of the brush is so held as to make a very small angle with the surface of the drawing. Thus, as in the following figure, for " dragging" rugged or rock surfaces, the handle is held between the thumb and forefinger, while the hand is supported by the tip of the second finger. With gentle pressure, the brush is moved back and forth, or in one direction only, as indicated by the arrow, until the desired degree of ruggedness is expressed. The effect is heightened by a second or third appHcation. (See Rule VI., par. 130.) 74 TOPOGRAPHICAL DRAWING. As to their Durability, all water-colors undergo more or less change from exposure to light and atmospheric impurities. The most sensitive and therefore unreliable of the list given in par. 129 are Prussian blue and crimson lake, for which cobalt and light red may be advantageously substi- tuted. Indigo, Payne's gray and sepia are affected to a much less degree ; but, to insure their retention, should be applied with greater force than would otherwise be required. The other colors, although becoming somewhat darker from exposure, may be regarded as permanent. As a rule, the colors known as " earths" are the most stable. Section II. — The Conventional Tints (Plate XV.). 136. The different features are represented by colors, alone or in combination with the signs employed in plain topographical drawing. A color-resemblance in some cases assists the eye in recognizing them. Thus, water, trees and sand are represented by blue, green and yellow respectively. Except for slopes, the illumination is oblique (par. 98). The conven- tional tints are as follows: For Sand — Yellow ochre, a flat tint. Gravel may be shown by dots of burnt sienna. For iPF«/^r— Prussian blue, a flat tint. If a lower tone is desired for the map, use indigo instead of Prussian blue. Another form is given in the plate, and also in Plate XVII. For Cultivated Land. — Burnt Sienna, a flat tint, which, when dry, is ruled with parallel lines, as shown in the figure ; a right-line pen, ruler and triangle, and the same tint, or one a trifle stronger, being used for this purpose. For greater variety, alternate fields may be tinted, as shown, with P. G. -\- Cr. L., or B. S. -|- Cr. L., each ruled with tints of the same ; and all three may be used, care being taken to so arrange the different tints (see par. 131) as to produce a pleasing effect. If the ruled hnes are omitted, B. S. alone must be used. For Cleared Land — Indigo -|- gamboge, a warm flat tint (par. 131). For Underbrush — The tints for cleared and cultivated land, alternating in irregularly shaped patches, and blended as described in par. 134. To produce the best effect, observe the law of harmonious contrast (par. 131) as to which of these tints to place next to the feature already represented. For Marsh — Prussian blue, "dragged," as described in par. 135. The strokes of the brush should be made parallel to the lower edge of the map-border. The lower edges of the strips of land are then shaded with a strong tint of P. G. -j- Cr. L., applied with the tip of the brush, and invariably in the direction above stated. In the figure, the marsh is in a tract of underbrush, and, as in all cases, is superposed upon the sign for the latter. For Mud — The same horizontal strokes as in plain drawing, except that sepia is used instead of India ink. For Trees — Trees and clumps of trees are first outlined distinctly in pencil; and, for maps drawn to a large scale, a flat tint of gamboge is then laid over the entire surface within these THE CONVENTIONAL TINTS. 75 outlines (see " Trees, Analysis"). The shading is then effected with a green tint, stronger and cooler than that used for cleared land, and composed of Ind. -|-G., or Pr. B. -\- G., accord- ing as Ind. or Pr. B. is used for water. The shading-strokes are applied on the side remote from the source of light, and in a curvilinear direction, so as to bring out the rounded forms by properly defining the lines of shade. For the shadows, a tint of P. G. + Cr. L. is used, and applied as described for the shade, a rounded outline being given to the forms. The accidental lights produced by the operation of dragging (par. 135), applied to both shades and shadows, prevents a harsh effect. The pencilled outlines may be strengthened, if necessary, particularly on the shaded side. For maps drawn to a smaller scale, a flat tint of Pr. B. -f- G., warmed with a little sepia, is recommended for the entire surface within the carefully pencilled outlines (see " Trees, Masses"), the shadows and shading being omitted. This is well adapted to forests, when the trees are represented in large masses. This sign is always superposed upon that for the general surface. For Orchards — The outlines, regularly distributed, as shown in Plate 111.^ are tinted as above described. For Evergreens — The tints of green are cooler and darker ; the star-shaped figures (Plate III.) may be used instead of those here given. Buildings — If of wood, sepia; for masonry, Cr. L. The outlines of wooden buildings are drawn and shaded with India ink ; and of masonry with Cr. L., a very strong tint being used. To distinguish brick from stone, make the outlines of the latter the heavier. If de- sired, P. G. may be used for stone and light red for brick. For Roads — The pencilled outlines are filled with a flat tint of Y. O.; the edges being subsequently traced in India ink, as described for plain drawings. For a Path, Trail, or Ford — India ink, the same as in plain drawing. When paths are in outline, the latter is filled with Y. O. For Bridges — The outline is filled with Y. O., and the same distinction as to wood and masonry is observed as in " Buildings." Fences are represented with India ink, as in plain drawing ; Stone Walls by their outlines filled with Cr. L., and Hedges by the sign for " Trees." The shadows for the last two features are in P. G. -\- Cr. L. For Slopes (Plates XV. and XVI.) — Graded tints of P. G. + India ink, or cobalt -|- ivory black, enough of the first ingredient to produce a bluish tint, are applied as described for brush-shading (par. 122). The contours, in distinct medium lines, are drawn with strong Cr. L.; and the references with red or India ink, small fine-lined figures being employed. The shading should be a little darker than for plain drawings, on account of its obscura- tion to some extent by the superposed tints which represent the other features. The necessity of having the details to be left white clearly outlined in pencil, previous to the shading, is apparent. The clearest drawing is obtained by making use of the pencilled contours for defining the forms of ground, and deferring the drawing of the contours in color until the shading is finished and the other features are tinted. Brown ochre, or sepia -j- B. S., is suited to any hachuring that may be required. For maps of very large areas (see Plate XVII.), the shading is sometimes omitted. .76 TOPOGRAPHICAL DRAWING. For Rocks— 'it^id. warmed with a little B. S. ' This is first " dragged " (par. 135) over the shaded slope, thus producing the general forms or outlines shown in the middle rectangle at the right of the plate ; a second application, and a third if necessary, with a few touches of strong color here and there, will produce the effect shown in the lower rectangle. The three rectangles fully illustrate the different stages of the operation. Hachures in sepia are sometimes added, as for " slopes." Other features not above enumerated are represented by their outlines tinted, or filled with tints corresponding to the material of which they are composed. Iron, the only other element which may be required, as in the case of bridges, is represented b)' P. G. -|- Ind. -|- a little India ink. Features not otherwise clearly expressed are designated by name. Plate XVI. shows a combination of these signs in the shape of a map, and also the use of color in sketching, described in Part VI. 137. The following sequence is observed in the different operations : I. Plotting the triangulation and traverses (pars. 57 to 61), the contours (par. 86 et seg), outlining the details (par. 75) and the place for the title and legend (par. 126). If the plot- ting involves the use of many lines, it is advisable to use a separate sheet for this work and transfer it (see " Copying," par. 140) to the stretched paper, thus saving the surface of the latter just so much wear. As a rule, this work should be done in pencil, and upon its accuracy the reliability of the map will depend. II. Cleaning the surface (par. 42) preparatory to laying the tints. In this great care is required to prevent injuring the surface, and thus avoid the blemishes that would otherwise result in the tinting. III. The tinting, in the following order: The streams, roads, hill-shading, including rock- surfaces, finishing the contours in Cr. L. ; the different kinds of land, buildings, and the enclosures. IV. The inking of these details. V. Finishing the map as described in section IV (pars. 12410 128). Since India ink of an intense black is used, and the effect of black by juxtaposition is to subdue colors, the dif- ferent ink lines are made hghter than in plain drawing, and the letters are preferably of the light-faced, open, or skeleton type. 138. The French system. — The Conventional Tints are as follows: For Water — A light tint of Prussian blue ; the shore-lines are bands of a darker blue, graded outward to the general tint ; or the space is filled with regular water-hnes, as shown in Plate XVII. For Ditches — When the scale is small, a single line of Prussian blue or sepia, according as the ditch is wet or dry ; for large scales, the same as for "water." Marshes are represented by outlines filled with light blue ; the edges of the land are shaded with a darker tint of the kind used to represent the land, or with blue. For Cultivated Land — When the relief is not expressed, by white ; otherwise, by a flat tint of red ochre (" ocre de ru") — burnt sienna corresponds to this tint. For Cleared Land (meadows, etc.) — A bluish or cool green. For Trees — If isolated, by small rounded forms of warm green, for scales of -^-^ or greater ; and for smaller scales, dots of this tint sufficiently large and varied in shape to not be mistaken for dotted lines. THE CONVENTIONAL TINTS. 77 For Orchards — These signs are regularly distributed in rows, parallel to the longer sides of the enclosures. Shade-trees along roads, for scales of -^-^ and less, are placed just outside of the edges, at regular intervals and opposite each other ; but along streams, the trees on one bank are opposite the intervals on the other. The tints for brushwood and evergreens are as in Plate XV. For Slopes — If the drawing is orometric (the configuration being expressed by contours only), the contours and references are in burnt sienna (see Plate XVII). If in relief, hachured, the hachures are in brown ochre, the contours and references being as above. Rocky ledges are expressed by horizontal strokes of B. S. or neutral tint, or both. Artificial slopes, such as in terraces, fortifications, etc., receive a tint of India ink, graded from dark to light, from summit to base, and superposed with a tint of green or B. S., for embankments or cuttings respectively. The upper outlines are made heavier than the lower. With scales of -^^-^ or less canal and banquette slopes are not expressed ; and with scales of xTrFTnr ^'^ l^ss, levee and exterior slopes, of which the height is less than 2 metres, are repre- sented by single lines. For Buildings — If of masonry, outlines of Cr. L. or carmine, filled with a flat tint of the same. For wooden buildings, the outlines and interior flat tint are in India ink, the outUnes being shaded to make the material of construction more apparent and to give relief. For arched walls, or walls supported by columns, dotted outlines are used ; and for ruins, the dotted outlines, omitting the interior tint. For masses of buildings (city blocks, etc.), the general masses only, without interior de- tail. An exception is made in the large-scale drawings for the gardens, etc., belonging to im- portant houses, and for those which abut against the roadway. For public buildings, the outlines are made very heavy ; and with scales of ^oJjdo or greater, the ridges of roofs are represented, and the slopes of the roofs are shaded with India ink, the illumination being oblique. Certain flat tints within the outlines show the special use of buildings. Thus, those belonging to a particular administration are tinted with deep carmine ; those belonging to the State, to parishes or departments, with vermilion or orange red ; to the engineer service, with grayish blue ; and to the artillery, with violet. Grayish blue alone is prescribed when the special use is not designated. For Communications — The outlines in India ink, with the interior left white. With scales greater than -e,-^^^, true breadths are represented ; with -^-^^-^, except in villages, the breadths are increased about -^ of an inch. With ■^-^\^^, each of the edges of a principal route or high- way is represented by a heavy inner and fine outer line : for departmental routes only one of these interior lines is made heavy ; and in each case the breadth between the interior lines is ■^ inch. For a main road one heavy and one fine line are used ; the breadth is ^ inch : fine lines with the same interval are used for its regularly travelled branches. A paved road but little used is represented by a full and a dotted line with an interval of -^>^ in. ; a wood-road not paved, by two dotted lines, with same interval ; and a path, by a single heavy line, full if used for beasts of burden, and dotted if a simple footpath. For parks and gardens, the paths and carriage-ways are in green. These breadths are reduced proportionally for smaller scales. For Enclosures — Fences, by full or dotted lines in India ink, with large dots to represent 78 TOPOGRAPHICAL DRAWING. the posts ; walls, by a heavy red line ; hedges, by a row of small, shaded trees in green, for scales up to Y'm ! f°^ smaller scales, small, rounded forms at short intervals, with dots be- tween. Wells and springs, by red and blue outlines respectively, filled with blue ; boundaries, by square red dots. Rules as to Lines and Tints. — With scales of -^^-^ or greater, full lines are used except in the following cases : For undetermined features, heavy dotted lines ; for subterranean fea- tures, the same with the dots close together — red for dry masonry, blue for aqueducts, and black for mine-galleries. For continuous intermediate contours, fine broken lines, the dashes long ; but when portions only are represented, fine dotted lines are used. For the smaller scales, broken, dotted and full lines are used for communications, as con- ventional signs to indicate their nature, importance and degree of practicability. A general rule for clearness of expression is to have the breadths of lines proportioned to the importance of the object represented, and three grades are thus prescribed for the larger scale, — the finest, for limits of cultivated land and for contours (the latter a trifle heavier than the former) ; the second or medium grade, for paths, common roads, outlines of buildings and shore-lines, and the heaviest for the inner lines of highways. The details must be accurately represented in outlines before tinting, — the outline of masonry in red ; of water, in blue ; of trees, isolated or in masses, hedges and paths of parks and gardens, in green ; and of other features, in black. The tints should be of such relative intensities as to make any of them readily distinguish- able from the others ; therefore surfaces of small extent receive the darker or stronger tints. The green for trees and cleared land is made quite strong and of a bluish tint ; light blue is used for streams and bodies of water of sufficient breadth to be clearly defined by their out- lines. The lighter tints are laid first, as the proper gradations and relative intensities are thus better determined. As a precaution against the fading of any tint, and to avoid any uncertainty as to the ob- ject represented, the initials in India ink of the names of the different kinds of land are placed within the enclosures. Before tinting, the drawing is washed by causing water to flow rapidly over it ; and since red " runs" more easily than the other colors, the red outlines are drawn first to give them a longer time to dry. Plate XVII. ( from Capt. Wheeler's Report upon the Third Int. Geog. Congress and Ex- hibition, published by the U. S. Corps of Engineers) illustrates the French system of coloring as applied to maps of large areas. As may be observed, but six colors are employed, and with very pleasing results. The following systems of coloring are now employed for this purpose by the countries designated (from same report) : 5/«zVz.— Black for arable ground and lettering ; blue for water ; red for buildings, construc- tions, and ordinary routes ; green for forests, horticultural tracts and grazing-lands ; sienna for contours. Switzerland. — Blue for water ; bronze for contours ; brown hachures for surfaces which cannot be easily represented by contours ; black for other details. Prussia. — ReUef is expressed byMuffling's system (par. iii) for slopes of 5" and 10°, and by Lehmann's (par. 106) for more abrupt ground, 34 hachures per inch being used. COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 79 PART IV. COPYING, REDUCTION AND ENLARGEMENT OF MAPS AND MODELLING. Section I. — Copying, Reduction and Enlargement of Maps (Plate XVIIL). 139. In important topographical surveys, and especially those covering a tract with much detail, the first complete plot is usually made to a larger scale than that required for the finished map, and, with the field notes and sketches, is preserved for future reference. This plot is then copied with great care to the required scale, the details and relief are refined as much as possible, and the map is finished ; and, if desired, is ready for reproduction. If, however, it is to be reproduced by any of the photographic processes, either the original plot or an exact copy of it is carefully finished throughout, and then reduced to the required scale by this process ; or a copy of it is made to an intermediate scale, in which those details which would become indistinct in the reduction are "generalized" (see par. 141), and this is used for the purpose. Photographic copies on glass — transparencies — on account of their freedom from changes due to atmospheric influences, are very useful as models for the engraver. This branch of drawing is therefore an important element in map-construction, and the different methods are described in detail. In any of the methods used, some means of testing the accuracy of the work are re- quired, of which the simplest consists in having the original drawing and the sheet for the copy subdivided into sets of squares: — of equal size in both, if the copy is to be to the same scale as the original ; or correspondingly increased or diminished in the copy, for an enlarge- ment or a reduction. The test then consists in ascertaining by measurement if the lines of both drawings cut corresponding points of the sides of the squares. In photographic reductions, the presence of the squares serves to detect any local distor- tion (see also I., par. 140). Meridian lines and parallels of latitude may serve this purpose in maps drawn to a very small scale. If the squares prescribed in par. 128 are on the original drawing, and it is found by measurement that no correction for distortion due to atmospheric influences is necessary, then for a copy to the same scale, the vertices may be transferred to the sheet for the copy as prescribed in II., par. 140 ; otherwise, the squares for the copy are accurately constructed ; and in copying the details, corrections either by eye or by measure- ment are effected proportional to the changes in dimension or position that have taken place in different directions. If it is undesirable to draw the squares upon the original, tracing-paper securely fastened to the latter may be used for the purpose. As in all other topographical drawing, construction-lines are in pencil. 140. Copying to the Original Scale. — I. Geometrically. — The sides of the squares serve as 8o TOPOGRAPHICAL DRAWING. lines of reference, distances are transferred with the dividers or by measurement, and the dif- ferent points of any detail are thus accurately located. Right lines are located by fixing their extremities ; and circles and other regular curves, by fixing the extremities of their radii or axes. Right lines, intersecting details already copied, may be drawn on both sheets, and intermediate details located by means of co-ordinates from corresponding points. Contours are located by their intersections with lines already drawn ; and if a sufificient number of the latter does not exist for their complete determination, others favorably placed can readily be added. This process is very slow, but susceptible of great accuracy. The three-legged dividers (Fig. 68) are very useful in fixing a point by its distances from any two already determined. II. .Sy /VzV/^zwf.— A very accurate means of copying points (extremities of right lines, radii of circles, etc.) consists in fastening the original drawing, face up, upon the sheet for the copy, and then with a fine needle pricking through the required points to the surface beneath. It is particularly useful in copying triangulation- and traverse-lines. Any difificulty in recog- nizing the different points is obviated by raising a corner of the original — the other corners being securely fastened— and marking the points as soon as determined. A fine needle does not injure the surface, and the holes may be closed entirely by placing the drawing on a smooth surface and rubbing the reverse side with an ivory handle. III. By Tracing. — To copy a drawing on tracing-paper or linen (par. 31), this material, glossy side up, is securely fastened along its edges to the original, by drawing-pins or other means ; and as every detail is distinctly visible through it, it is simply necessary to follow the lines with the pen. If the sheet is not large enough to cover the drawing, it may be held in place, and pin- holes thus avoided, by fastening strips of paper along its edges and to those of the drawing. On account of the warping produced by the wetting required if the copy is to be tinted or shaded, it is best to first stretch the sheet, the water being applied with a sponge to white paper laid over the tracing-paper which soon becomes sufficiently limp to attach to the stretcher (par. 38). Tints appear well applied to the back of tracing-linen. In copying from a tracing, the original should lie upon a white surface to make the lines distinct. Vegetable tracing-paper can be made more pliable, will lie closer to the drawing and work better, by soaking it for about*an hour in a solution of 4 parts of water to I of glycerine, and then drying it on a line in the open air. For pencil-work, paper thus prepared requires a light rubbing with talc or sandarach. The " Blue Process" (par. 147) is much used in connection with tracings, for copying to the original scale. IV. Copying on Glass. — A pane of glass of suitable size is disposed within or near a win- dow, and at an angle of 20° to 30° inclining downwards toward the draughtsman, so that by means of white paper placed a short distance beneath, or by an arrangement of mirrors, the light may be reflected upward through it to the eye. The original drawing, face up, is then fastened to its upper surface, the sheet for the copy is superposed, and if the paper is not too thick, the lines will appear sufficiently distinct to trace with pen or pencil. The best result obtains by excluding all light except that which is reflected through the glass. COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 8i If the paper of the original is too thick, a tracing may be substituted for it; and if the fair sheet is so thick that the Hues appear confused, the extremities of the latter may be first pricked through with a fine needle, so that when the fair sheet is superposed, the lines are readily determined by the brilliant points at their extremities. The squares prescribed in par. 139 prove of great assistance in adjusting the sheet when the size of the drawing makes it necessary to copy it by sections. V. By Transfer. — The sheet for the copy is fastened to a flat, smooth surface, transfer- paper (par. 32) is laid over it, and the original, face up, is superposed and secured in place. A steel or agate point, sharp enough to make a fine line, and at the same time not cut the paper, is then drawn over the lines to be copied, which are thus duplicated on the fair sheet. To avoid omissions, squares are drawn upon the original, or it is otherwise divided into sec- tions which are finished separately. Any displacement is readily detected during the progress of the work by retransferring any of the lines, and observing if the two transfers coincide. Reducing and Enlarging. 141. Generalization of Detail. — In making reductions, as indicated in par. 139, certain changes in the representation of details, as compared with the original, are required. Dis- tances, breadths of line and interval, which in the latter are clearly shown, would, if faithfully copied, often become indistinct. Thus, in reducing by photography from ^fl^pg to ginnnr' distances which are considerable quantities according to the former scale disappear, and outlines of buildings distinctly sepa- rated unite and appear as black dots. A certain generalization is therefore necessary; outlines of isolated details are made larger than their proportional scale-dimensions, groups of individual signs are massed into a less number or into a single sign ; communications and enclosures are reduced to single lines, if necessary ; and slopes, narrow in plan, such as banks of canals and ditches, are represented in like manner. In regard to contours, it is evident that in order to preserve the same minuteness of detail as to forms of ground, the contours must all be copied ; but if this would result in bringing them so close as to produce complexity, or obscure other details, the number of con- tours must be diminished ; e.g., for a reduction from y^jVir with an equidistance of 50 feet, to ^o^fto , copy alternate contours and mark the equidistance " 100 feet ;" if to ^ oon o' copy every fourth contour, and mark the equidistance " 200 feet." In reducing to scales much smaller than the original, the general configuration is preserved, while many minor features — the smaller spurs, ravines, cols, etc. — which would become indistinct, are omitted. If an equidistance, not a multiple of that given, is required, the contours to be copied are so interpolated in the original as to preserve the continuity of surface, and at the same time conform to the new scale. In large establishments, these changes are made according to a system of scales, as may be observed in Appendix 20, U. S. Coast Survey Report for 1 860-1. The system prescribed in the text (par. 138) will serve as a general guide ; but as no rigid rules can apply to all cases, much is left in this matter to the judgment of the draughtsman 82 TOPOGRAPHICAL DRAWING. 142. Different Methods.— Q(torc^ttx\zaS. and mechanical methods are used in reducing and enlarging drawings. The former include the " Method by Squares" and other geometrical means, and the latter the use of the proportional dividers, pantograph, eidograph and photography. For the MetJiodby Squares.— To prepare the original, let AB and AC (Fig. 69) be adja- cent inner edges of the border. From A set off the equal distances A — i, i — 2 4 5, and through their extremities draw lines respectively parallel to the edges, as shown. The greater the amount of detail, the greater will be the number of squares required. The fractional parts which remain, as in the upper and right-hand portions of the figure, when an edge is not exactly divisible by A—i, serve the same purpose as the squares. The fair sheet (paper for the copy) is then prepared by constructing first the rectangle of the inner edges of the border, corresponding to the proportions fixed upon for the reduction or enlargement. For a simple lineal reduction, e.g., a dimension of the copy to be i, ^ that of the original, make ab and ac (Fig. 6ga) \,i the length of AB and A C respectively, and correspondingly for the equal distances a—i', i'— 2' 4'— S': and similarly for an enlargement. The squares are then drawn as before. For a reduction by areas, proportional squares or rectangles may be constructed either arithmetically or geometrically. Arithmetically ; since similar figures are to each other as the squares of lines like placed ; let a represent the area of the copy ; b, the area of the original ; c, the required line of the copy; and o the given line of the original ; then a : b :: c' : o' : or c = V t 0'' ; e.g., suppose AB and ^C (Fig. 69) respectively 35 and 24 inches in length ; then for a redu ction to f, ab (Fig. 69a) = ^/f 35' = 28".6, and ac = I9".6. For an enlargement, c =r -ff'. Or one side only of a rectangle need be calculated ; the adjacent side is then obtained as shown in Fig. 70: For a reduction : by setting off from A' on the original A'B'C'D' , the distance yi'^' equal to the calculated side; then erecting the perpendicular at b', and draw- ing through the intersection y of the latter with the diagonal A'D', c'y parallel to A' B' . A'b'c'y is the rectangle required. All rectangles having their diagonals in the same line are similar, and their corresponding sides are therefore proportional. The copying-squares are then located by subdividing adjacent sides of the rectangles into parts like placed, and proportional to those of the original (see par 49). The geometrical reduction depends upon the familiar principles that the square described on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides ; and that if a perpendicular be let fall from the vertex of the right angle to the hypothenuse, the squares of the sides about the right angle will be to each other as the adjacent segments of the hypothenuse. Thus, given the square CO (Fig. 71); to construct another that shall be in any required proportion to it, as 2 : 3. Describe the semi-circumference CED ; subdivide CD into 3 equal parts ; erect the perpendicular FE, and draw CE : then will CE' = f CD\ ED" is evidently ^CE' or ^ CD\ For an enlargement : To construct a square that shall be a multiple of any given square, as KI (Fig. 72). Draw the diagonal HL ; make //i^ equal to HL ; then Hd' = 2KI. Produce IL; COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 83 make IM = Hd, and He = Hm ; then H? = iKI. Similarly, Hf'' — \KI, and so on for any other multiple of KI. If the rectangle of the border is not copied first, it is advisable as a check to first copy the square outline of the greatest number of squares, and then subdivide it into smaller squares corresponding to the original. The squares having been constructed, the points in which lines of detail cut .the sides of the squares of the original are noted, and the required lines are made to cut correspondinrj points marked on the fair sheet. The details are either drawn in by eye, or more accurate means are used— proportional dividers being very handy in this connection for exact work. 143. Proportional Dividers (Fig. 73). — This instrument is very convenient for copying to any scale. When the legs are closed, the slide which bears the pivot can be moved along the slots, and clamped with its index at any desired division of the scale. When opened, the distances between the points at the two extremities bear a ratio to each other, indicated by the number marked on this division. \,\, on the scale of " lines" indicate distances bearing the ratios of i : 2, l : 3 ; if on the scale of " circles," that the areas of circles described with these distances as radii, or of similar plane figures having these distances as homologous lines, are as i : 2, i : 3 Scales of " solids" are found on some instruments. An arc is also sometimes attached, by which the legs may be clamped at any angle. Even with careful handling, the points soon wear off ; and the scales then serve only as a means of approximate adjustment, which, however, is readily perfected by trial — -using a scale of equal parts. 144. The Angle of Reduction is used in connection with ordinary dividers, and is constructed as follows : Make NO (Fig. 74) equal to a side of a square on the original map ; from O, as a centre, with OP =^ to a side of a square of the copy as a radius, describe an arc as shown, and from iVdraw the tangent NP; then PNO is the angle of reduction. To use it; let No be a dimension taken with the dividers from the original map ; with one point at o, extend the dividers until an arc described with the other point would be tangent to NP, then of is the reduced dimension required. Reductions may be made with great accuracy if this angle is traced in fine lines on metal. An allowance may be made for alterations or changes in the squares of the original, due to atmospheric influences, by using the mean length of two adjacent sides for NO; or the squares may be subdivided into 4, 8, others, and the smaller squares then copied separately. 145. The Pantograph. — This instrument, having a variety of forms, consists essentially of four straight bars of wood or metal forming an articulated parallelogram. Fig. 75 repre- sents a common form. The bars A W, AF, and BE are of equal length, and the pivots at A and B are stationary ; the fourth bar, CD, equal in length to AB, is pivoted at its extremities to sliding plates, which can be clamped at any of the divisions marked on AW and BE, and is always made parallel to AB. The entire instrument is pivoted to a piece of metal, W, sufficiently heavy to remain immovable while copying. Projecting points on its lower sur- face are sometimes provided to keep W in place. At T — A T being equal to A W — is a tracing- point with which to follow the lines of the original drawing. At Z' is a pencil-point which may be clamped at any desired division of CD, and which copies the lines traced at T. As may be observed, the instrument is arranged in the figure for reducing. 84 TOPOGRAPHICAL DRAWING. It is in adjustment when (Fig. -jQ) AW : DW :: AT: DP, or as ;« : n, — m and n repre- senting respectively the scales of the original and of the copy. For, let TV be a right line described by the tracer, then will PP , described by the pencil, be a right Hne parallel to TT , and TT : PP' :: m : n. From the two positions of the instrument, AW: DW :: AT: DP, and A'W : D'W :: A'T : D'P ; and since during the movement, D'F remains parallel to A'B', it follows that W, P, and T are in a right line as at first. Then, since TW : PW :: A T : DP, and TW: PW:: A'B' : DP :: AT: DP; therefore TW : PW :: TW: P'W, and the triangles WTT and WPP' are similar ; whence it follows that TT and PP are parallel, and also that they are in the desired ratio; because TT : PP :: TW : PW, or as AW: DW ^^ m : n. Consequently, the figures described by the pencil and tracer are similar. From the foregoing, the instrument is not in adjustment unless ABCD is an exact par- allelogram ; and T, /"and W are in a right line. The test of the former condition is made by measurement, and of the latter by means of a stretched thread. The adjustment for reduction to any required scale needs no further explanation ; the pivot at W and the tracing-point are simply placed at the o divisions of the bars to which they are attached ; and the bar CD and the pencil-point, are so adjusted as to fulfil the con- dition ^ PF : DW:: AT: DP :: m : n. The graduation of the bars assists in making this adjustment, and, with a fine instru- ment, serves to make it directly. For a copy, m = n, W a.nd Pare interchanged, and a copy in reverse results. 71 If — is greater than \, a more rigid system is obtained, as shown in Fig. yj, by inter- changing T and W, moving CD to CD' , and adjusting the pencil-point at P as already described. For an enlargement, the tracer and pencil are interchanged ; but on account of the multi- plication of errors due to inaccurate tracing, the pantograph is not recommended for this purpose, except when rough copies only are required. For accurate work the instrument should be of metal ; to insure freedom of movement, it should be supported on casters at such points as A, B, C and D\ or, as in some of the ex- pensive forms, guyed to an upright fixed in W, and about which it freely turns ; the work should be performed upon a smooth, level table. It is convenient to have the pencil-holder hinged so that the pencil-point maybe raised, to prevent false lines being made in passing over blank portions of the original. The pencil may be weighted to make a heavy line, and a cup is sometimes provided in which shot are placed for this purpose. For right lines, a straight-edge is used to guide the tracer ; and for curved lines, a curve. The open triangle is convenient for copying rectangular outlines. To check the motion of the instrument when its inertia tends to carry the tracing-point beyond a desired limit, or move tangentially to a curve, an ivory frfction-roller which runs along the paper, is in some instruments attached near the tracer, and is manipulated by the left hand, while the right guides the tracing-point. To secure steadiness in working, both elbows should rest upon the table. The only dif^culty arising in properly placing the instrument at first is in making the field swept by it, on both original and fair sheets, as wide as possible. The squares prescribed in par. 139 should be drawn, so that when it becomes necessary to move the instrument, the COPYING, REDUCTION AND ENIARGEMENT OF MAPS. 85 correctness of its new position may be tested by retracing the sides of the squares on the original, and observing that the pencil-point follows the copy. Another form of the pantograph, in which the pivots at all the vertices of the parallelo- gram are fixed in position, is shown in Fig. 78. The tracer T\s fixed to BF, but the pencil P, and the weight W, can be clamped at any desired division of DE and DC. With 7" and P placed as shown, the copy is in reverse ; the proper or " erect " copy being obtained by inter- changing Pand W.. The placing, adjustment and manipulation of this form of the panto- graph are, in general, as described for the other. That the copy is similar to the original is shown in Fig. 79. For let TT' and PP' repre- sent respectively right lines of the original and copy ; from the first position of the instru- ment, shown by the heavy lines, AT: DW :: AP : DP\ from the second position, A'T' : D'WwA'P' : DP; and since ^T' and D'Wars parallel, the second proportion shows that F, W 3.nd T' are in a right Hne ; therefore T W : WPwAD : DP, and T'W: WP-.'.A'D' : DP; consequently TW : T'W:: WP : WP. The triangles TWT' and PWP' have the angles at W equal ; they are therefore similar ; and it results that PP' and TT' are parallel, and that for any position of F, the ratio — — - = 7777,= yrp = ^ is constant. In another form of pantograph called the micrograph (Maes), (Fig. 80), the articulations at A and C only are stationarj^^, the bars are usually of equal length, and the adjustment for any desired ratio of reduction is effected by increasing or diminishing the lengths of the sides of the parallelogram by an arrangement of holes and pins, or by means of sliding-boxes and clamps at B and D. This form is unsuited to accurate work. To adjust the pantograph to any desired ratio of reduction, when the bars are not gradu- ated: First, see that T, /'and Ware in a right line; and with 7" trace a long right line, which denote by L On the line marked by F, set off a distance L', such that yj — — ; then change the alignment of T, F and W, until F describes L' while T traces L. 146. The.Eidograph (Fig. 81) is a very exact and convenient instrument for the reduction of drawings between the limits of \ and full size. It revolves about a vertical pin which pro- jects from the heavyweight fFand enters the box B, through which the centre beam has a transversal motion, and to which it can be clamped. The two pulleys,/, fitted to the ends of the beam are of the same diameter, and have a simultaneous movement by means of the two steel bands, b, which can be adjusted to that degree of tension which will make the bars AP and TD exactly parallel, and at the same time secure steadiness of motion. These bars slide through, and can be clamped to, boxes attached to the under-surfaces of the pulleys. The tracer and pencil are attached to sockets at 7" and P respectively. A cord extending from T'to P serves to raise the pencil-point when necessary in the operation of tracing. The beam and bars are metal tubes graduated into equal parts, reading from o to 100 each way from their middle points, while readings to thou- sandths are effected by means of verniers attached to the boxes. A weight which fits the beam is used to preserve the equilibrium, when W is not at the centre of gravity of the instrument. To adjust the instrument ; make the vernier-indices coincide with the zero-points on 86 TOPOGRAPHICAL DRAWING. beam and bars, and mark points with both tracing- and pencil-points ; revolve the instrument until the pencil-point coincides with mark made by tracer. Tracing-point should then coin- cide with pencil-mark : if not ; mark the new position of tracing-point, bisect the interval between it and the pencil-mark, and by means of the adjusting-screws of the bands, make the tracing-point coincide with the bisection. The bars should now be parallel and the two points and the vertical pin of Win a right line; and if the graduations are correct, these con- ditions should also obtain when the indices of all the verniers are set to like readings. It is evident from inspection of the figure, that when the adjustments are properly made, similar figures must be described by the tracer and pencil. To set the instrument for reduction in any desired proportion, as copy : original ::« : m; — Let X represent the required reading; since there are 200 divisions of each scale, the fulcrums of the beam and of each bar, when the verniers are adjusted to this reading, divide the scale into two parts, one part containing 100 — a: — n divisions, and the other 100 -{- x = 7k divisions; consequently n : m :: 100 — x : ioo-|-;ir, from which x = 100 — -r — . E.g., sup- pose the copy is to be J (lineally) of the original, then x = 100 = 60, and the vernier indices are set on this division of the scales. For reducing, the indices of the beam and of AP are placed on the halves toward P, and those of TD on the half next to Z> ; for an en- largement, this arrangement is reversed. 147. Copying by Photography. — This method has the advantages of accuracy and rapidity, and is generally used when a number of copies of the same map is required. The two processes commonly used are the "wet plate" and dry plate," in each of which a negative on glass is first made, and from this the copies are printed. Aside from the chem- ical preparations and their general application, which can be learned from a photographer or from some of the many manuals of photography, there are certain special rules to be observed in order to obtain a good negative. In the preparation of the subject (the map to be copied), the lines must be very black and on a white surface. Stick India ink of good quality is reliable, and with a magnifying-glass to examine as to variability in intensity, no error need be committed in this respect. The addition of a little red or yellow coloring-matter — carmine, crimson lake, burnt sienna, or bichromate of potassa — to the water used in preparing the ink is advisable, as the lines thus " take" blacker. Since red and yellow "take" nearly black, and blue nearly white, bluish white paper is to be preferred. The paper should also be smooth, like the " Plain Saxe" much used for photographic purposes. The subject should be drawn to a scale at least \ larger than required for the copy, in order that slight irregularities may disappear in the reduction. If a greater reduction is required, the subject must be generalized as described in par. 141, and allowances made for the diminished sizes of scales, letters, etc.; e. g., in reducing from xTrrcrTr ^° ToTinr' the rep- resentative fraction must be changed to ^-^-J-fnr o" the subject, and the letters and lines made respectively larger and broader than are appropriate to the former scale. A very fine line in the subject is liable in any case to become indistinct. The lens used should be of the " rectilinear" type, so that right lines on the extreme boundaries of the sub- COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 87 ject will not be curved or distorted. Any fault in this respect is readily detected by exam- ination, or, if necessary, by measurement of the image on the ground glass. To produce sharply defined lines, very small stops should be used, so that the rays which act upon the plate may be confined to those which enter near the optical centre of the objective or front lens. Rules are prescribed for determining the relative distances of subject and ground glass from the optical centre in order to obtain the desired reduction ; but to ensure an exact copy, the safest plan is to construct the outline of the subject on the ground glass, according to the scale required, and then focus so that the image of the outline of the original shall coincide with it. The subject and ground glass should be parallel, and the optical axis, or axis of the lens tube produced, should intersect the middle point of the subject, at least very nearly. It is more convenient for adjustment to have the subject and ground glass vertical. To fulfil these conditions, rigid and adjustable stands are used for both camera and subject ; but for want of better means, a level table for the camera, with a vertical support to which the subject may be fastened, will serve. 'Y.\\& subject should be strongly and equally lighted ; and, as much as possible, all light except that proceeding from the subject should be excluded from the lens. To obtain a fair negative is by no means a difficult operation ; but, in addition to the information acquired from experts, considerable practice is required to make a perfect one. The following is a brief description of a process successfully used by the writer for obtain- ing a negative suited to map-work : With Wet Plates : For albumenizing ; white of i egg + 2 qts. water, thoroughly shaken with broken glass and flowed on the glass, which should be perfectly clean, and dripping from a fresh applica- tion of clear running water. Dry under cover at least 24 hours before using. For collodionizing; a thin collodion of a sherry color, — made so, if necessary, by the addition of a little iodine (scales dissolved in 95^ alcohol). For sensitizing ; a silver bath, having a strength of 42 gr. to i oz. of water, the latter being first cleansed by adding to it an ounce or so of silver nitrate and placing it in the sun till the impurities are precipitated, and then filtering it through cotton. When perfectly transparent, add a few drops of nitric acid (C. P.), sufficient to make the bath redden blue litmus paper ; then suspend in it a plate, coated with collodion on both sides, for about 24 hours. To sensitize a plate, two or three minutes' suspension in this bath is sufficient. For the exposure, use direct sunlight illumination if practicable ; if not, use reflected sunlight and place black-paper cone on lens-tube. Time, with smallest stop, about 30 seconds, — soon ascertained in practice. For developing: a saturated solution of sulphate of iron 2 parts + double sulphate of irofi and ammonia i part ; to 2 quarts of which add sulphate of copper \ oz. To i oz. of this add 8 oz. water + l oz. of acetic acid. Flow and develop until light parts just begin to turn dark, then wash. For fixing, use a bath of \ oz. cyanide potassium -f- 16 oz. water ; time, about 10 seconds, or until light parts disappear. Wash about half a minute in clear, running water, and dry on edge in rack. If intensification is needed, use sat. sol. bichloride mercury, in which leave plate until film 88 TOPOGRAPHICAL DRAWING. turns white ; then wash, and place it in sol. of ammonia i oz. + water lo oz., and leave till film is blackened. Wash and dry. To produce very dense negatives, use a bath of i oz. sulphate of copper + 2 oz. bromide potassium + 6 oz. water ; wash, and flow with a weak solution (say 20 gr.) of nitrate of silver. Wash and dry in rack. With Dry Plates (" Rapid " brands) : A printed circular giving the process best suited to each of the many kinds of dry plates manufactured usually accompanies each box of plates ; but for map-work the following proc- ess is a simple and good one : Use same illumination of subject as above described for "wet plate." Time of exposure with smallest stop ; begin with 3 seconds. Wash one minute under the tap, then develop with the following solutions : No. I, a filtered sat. sol. of oxalate of potassa -)- a few drops of oxalic acid. No. 2, a filtered " " " sulphate of iron + " " " sulph. " (C. P.) To four parts of No. I add one part of No. 2, first diluting each separately with half its bulk of water. For the bath, use enough to cover the entire plate, and develop until lines are visible on glass side ; wash, and fix in bath of hyposulphite soda i part -j- water 5 parts ; when clear, wash at least I hour in running water. These plates may be intensified as described for wet plates; but will seldom need it, unless for lithographic work. For the printing: I. Albumenized paper is lightly rubbed smooth with a canton-flannel pad, then sensitized by floating on a silver bath of 50 gr. to the oz. 2. Fumed with strong ammonia 30 minutes. 3. Exposed in printing frames in the usual manner until the lines are slightly bronzed. 4. Washed in running water, two changes, ten minutes altogether. 5. Put in bath of salt one part -|- water 50 parts, until lines turn red. 6. Toned in chloride gold solution which contains one grain to a sheet of paper, and consists of 8 oz. of borax ( i part) and water (80 parts), to i gr. of chloride of gold. Time, until red lines turn black. 7. Fixed in fresh hypo-solution as for plates — time, 10 minutes ; then placed in strong solution of salt and water for 10 minutes, and finally washed in running water for about 5 hours. Clear hydrant-water is used in all of these operations. Transparencies, or positives on glass, are valuable for retaining the lines of an original map in their true dimensions and relative positions, unchanged by atmospheric influences during the slow process of engraving, and also for copying from directly, as described in IV., par. 140. The negative itself may be used for the latter purpose ; but if so intended, the film side of the plate should be placed to the rear in the plate-holder, so that the fair sheet may be placed in contact with the image. The ^^ Blue Process' is a ready means for copying. A tracing of the original is first made (III., par. 140), using very black ink and no very fine lines. Smooth, white, sensitized paper is then exposed underneath the tracing to the action of light, and the result, after washing in water, is a duplicate of the tracing, in white lines on a blue ground. COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 89 Paper already sensitized is usually obtainable, but may be prepared as follows: Dissolve Parts. Potassic ferri cyanide (red Prussiate potash) i No. I. in water 4 , Ammonio ferric citrate i Nc ( An lo. 2. \ K 1 in water 4 and filter separately. In a room dimly lighted, or under gas-light, place the paper flat on a table, mix two parts of No. I and three of No. 2 in a saucer, and with a broad soft brush apply this mixture to one side of the paper, being careful to cover it well, and avoiding streaks by going over it a second time with oblique strokes : hang the paper up to dry, and keep in a dark place till used. The paper, and solutions also, are best freshly prepared. A convenient printing-apparatus consists of a rectangular wooden frame, large enough to hold the tracing, and about three inches deep, in the top of which is fixed a strong plate of clear glass ; a smooth flat board which fits inside the frame ; and two or three pieces of felt or blanket of the size of the board. To use it, place the frame glass down, lay the tracing face down on the glass, and on this the sensitized paper also face down. The pads are then laid evenly on the paper and pressed firmly with the board, which is secured in place. The latter should be hinged in the middle, or in smaller sections, to admit of examining the progress of the printing. In direct sunlight, with fresh material, an exposure of from 5 to 8 minutes is sufificient ; but with a subdued light, an hour or more is sometimes required. Warm weather requires less time than cold. A strong metallic gray color indicates sufficient exposure. When long enough exposed, the print is placed in water for 2 or 3 minutes, or until the drip shows no yellow tinge, and is then hung up or placed between folds of blotting-paper to dry. The following formula is from Lieut. Harris' "Instruction in Photography:" Dissolve Parts. ( Potassic ferri cyanide i No. I. -( . ^ m water o j Pot 1 ir 1 , Ammomo-fernc citrate 1 2 No. 2.\. in water No. 3. Ammonia liquor, concentrated i^ When needed, mix i and 2, and add 3. To write upon a blue print, use a saturated solution of sal soda and a clean pen. To obtain blue lines on a white ground, use a negative of the subject in lieu of the tracing. A negative may be extemporized by placing the subject face down on a copying-glass (IV., par. 140), and, with a sharp point, tracing the lines on the surface of a plate of glass covered with any smoothly laid material not so opaque as to render the subject invisible in 90 TOPOGRAPHICAL DRAWING. copying. Or, to dispense with the latter condition, make a tracing and transfer it by carbon paper, so it will be reversed on the prepared glass ; then cut out the lines with a sharp point. For black lines on a white ground, Colas' process may be used. A hard well-sized paper is coated as described for the blue process, with the following^ preparation : Parts. Persulphate of iron i Perchloride of iron 2 Tartaric acid i Gelatine i Water 30 When dry, expose under the tracing until the greenish-yellow tint of the paper disappears, except where covered by the opaque lines ; then develop in a bath composed of Parts. Gallic acid i Alcohol 10 Water 50 Fix in water as in blue process. The following formula is for red lines on a white ground: j Uranic nitrate 40 to 50 grains [ Water i oz. ( Potassic ferri cyanide i c grains No. 2. -^ ,,^ ^ ^ ( Water i oz. Float Saxe or any other equally smooth paper 2 or 3 minutes on i and dry ; after exposure, develop by floating on 2 until details are brought out (30 to 60 seconds), then wash 5 min- utes in water. 148. Modelling. — A topographical model or relief-representation of a tract, in plaster or other suitable material, certainly presents the different features with greater clearness and force than can be done by any other means. The configuration of the surface, mountains, hills, surface-undulations and the streams and buildings are shown in miniature ; and, with the appropriate colors, so distinctly, that a thorough knowledge of the tract is obtained at a glance. From a military point of view, models are excellent references in the study of cam- paigns and battles, affording a most thorough appreciation of important positions ; and for geological purposes, particular formations may be clearly represented in colors on the faces of different vertical sections. The Details of Construction include the preparation of a topographical map of the tract to be represented, a suitable vertical scale, the model, the mould, the cast, the representation of minor details, and the coloring. The map having been completed, a vertical scale suited to the configuration is selected. For a map scale of 6 inches to i mile, the following exaggerations of height afford pleasing reliefs; COPYING, REDUCTION AND ENLARGEMENT OF MAPS. 91 For a mountainous tract, \; if only hilly, \\ if gently undulating, f. For smaller scales except for very rugged tracts, the exaggeration should be correspondingly increased. For a tract consisting wholly of mountains, no exaggeration is necessary. There are several ways oi forming tite model ; the two generally in use are : I. With Clay. — Fasten a duplicate of the map to a flat, well-seasoned board, or, better, to a slab of plaster ; drive pieces of strong wire vertically into the board and through points of the characteristic surface-lines, such as the highest and lowest points of water partings and courses, and where marked changes in their direction occur; also through the peaks and the lowest points of depressions, the junctions and principal bends of rivers, etc.; nip off the wires, so that their lengths measured by the vertical scale will correspond to the heights, as given by the contours, of the points through which they are driven. Then fill the spaces with fine clay to the tops of the wires, and mould the surface with the fingers, or suitable tools, to conform to the map. In case of any delay in the work, the clay may be kept moist by the application of a wet cloth. II. By Successive Horizontal Layers. — The base of the model should be a fiat, rigid sur- face, representing the horizontal plane through the lowest point of the tract. Select cardboard or other plane-surfaced material that may be easily cut, of a thickness corresponding to the equidistance measured by the vertical scale ; transfer the lowest con- tour of the map to a piece of cardboard (V., par. 140) of sufficient size, cut the latter along this contour, and with glue or otherwise fasten the layer thus formed to the base of the model. The next higher layer is then formed in a similar manner and superposed in its true position on the first, and so on to the highest summit of the tract. Where depressions occur, corresponding cuts are made in the different layers. The layers need not be continuous pieces, particularly in large models, but the annular breadth should be sufficient to enable them to be firmly fastened in place. To aid in properly superposing them, corresponding marks should be made, or holes punched in consecutive layers. The terraced surface thus produced is then rounded out and made to conform to the natural surface, either with wax softened in warm water, putty, or other plastic material ; and when this is firm the model is ready for use. Another plan consists in first transferring the lowest contour to the base, prepared as above described, and levelled ; then a flat strip of soft metal, as lead, is bent so that its inner edge shall coincide with the contour, and the strip resting upon the base thus form a mould for the lowest zone. This is then filled with plaster, as hereafter described for the mould, and to a height corresponding to the equidistance ; when dry, the next higher contour is trans- ferred to it ; the second zone is formed in a similar manner, and so on to the summits. The terraced formation may then be rounded as above described, or by cutting. To make the Mould: construct a strong wooden frame that will exactly enclose the model and extend an inch above it ; place it tightly around the model, and with a camel's hair brush, cover the surface of the latter with sweet oil, leaving the sides of the frame dry. Then pour over the model plaster-of-Faris, mixed to a proper consistency with water, filling the frame. After 15 or 20 minutes the mould is removed, dried in the_sun or by gentle artificial heat, then given two or three coats of drying-oil, and in about two days is ready for use. 92 TOPOGRAPHICAL DRAWING. For large representations, the model is cut into sections not exceeding 3 or 4 feet square, the frames are made very strong, and each section is moulded separately. To make the Cast, oil the inner surfaces of mould and frame, and pour plaster into it, as for the model. When set, the cast is taken out — a few gentle raps with a hammer near the base of the frame expediting the matter, if necessary — and dried, as above described. It is then given one or two coats of a very thin, hot solution of isinglass preparatory to color- ing it. Pieces broken off may be fastened in place with glue, or, when this is impractica- ble, /rt/z>r-/««c^/ may be shaped to fit and glued in. For the Representation of Details, the method of copying by squares (I., par. 140) is re- sorted to. A frame is fitted to the outer edges of the cast as prescribed for moulding ; squares are ruled upon the map, and the points of intersections of the dividing-lines are transferred to the upper edges of the frame. By means of a straight-edge and pencil, the latter being held vertically, the corresponding lines may then be drawn upon the cast, and serve as refer- ences for filling in such of the details as may be suited to the scale employed. Water-colors, thoroughly mixed, and to which a little mucilage is added, are used for the coloring, which should conform to nature as closely as possible. Glass may be used to represent the large bodies of water — a greenish-blue tint serving to heighten the effect. Streams are colored with Prussian blue ; buildings, moulded or cut out of some light mate- rial, are fastened in place and colored to represent the material of construction. For drawing-models, the contours should be represented. This may be effected as above described ; or, before the details are drawn, by placing the cast on the level bottom of a small tank of water, and marking each shore-line as the water is reduced in depth an amount corresponding to the equidistance. PROJECTIONS FOR MAPS OF LARGE AREAS. 93 PART V. PROJECTIONS FOR MAPS OF LARGE AREAS. 149. For a large tract several hundreds of miles in extent, it is necessary to plot the lines, angles and points, so they may have as nearly as possible their relative positions, as if projected upon the spheroidal surface represented by the mean sea-level. For this purpose, in the survey, the tract is divided for topographical representation into zones so small that they do not differ sensibly from their projections upon horizontal planes tangent at their middle points ; and since a meridional arc of 1° in length differs from its projection on such a plane by but 6^ feet, or -^\-^ of an inch to a scale of i inch to i mile, it follows that with the scales ordinarily used any zone may contain right lines considerably greater than 1°, or 69^ miles, in length (par. 52). The separate plots of these partial areas are then assembled on a projection of meridians and parallels. Since a sphere or spheroid is not developable upon a plane surface, in other words cannot be so spread out as to exactly coincide with it, this projection cannot represent the meridians and parallels in their true relative positions ; therefore, although the primary triangulation fixes the geodetic positions {determines their latitude and longitude), of two or more points of each zone, there must be discrepancies between actual distances and their representations upon the assembled plot, and these will vary principally with the extent of territory and the kind of projection employed. For topographical maps, the cylindrical and conical projections are generally used. 150. Cylindrical Projections. — Suppose a cylinder tangent along the equator, and the meridians and parallels projected upon it by their planes produced. If this cylinder be now rolled out upon any of its tangent planes, these projections will be developed into right lines at right angles to each other. Arcs of the equator will be represented in their true length, while arcs of the meridians and parallels will be respectively diminished and increased as their distances from the equator increase, the difference being small near the equator, but rapidly increasing toward the poles. If, instead of this, a right cylinder be passed through a parallel, say of 30° N., a similar result obtains, except that the arcs of the parallels between 30° and 0° are diminished, the arcs of the given parallel only being faithfully represented. Of the modifications of the cylindrical projection commonly used there are : I. The Rectangular Projection. — To construct it, draw P/^ (Fig. 82, Plate XIX.) to represent the middle parallel of the tract to be projected, and graduate it, according to the scale of the map, into degrees, minutes, or parts or multiples thereof (Table VI., Appendix), as may be desired, corresponding to their true lengths on this parallel ; and through the points m , thus determined, draw right hnes perpendicular to PP\ they represent the required merid- ians. The parallels are fixed in a similar manner by setting off from PP, on any meridian, the consecutive distances Mp' , Mp , respectively equal to the reduced length of a de- 94 TOPOGRAPHICAL DRAWING. gree or part thereof (Table IV., Appendix), for the corresponding intervals between the par- allels. II. The Projection with Converging Meridians. — Draw CC (Fig. 83) to represent the cen- tral meridian, and graduate it into degrees or parts thereof (Table IV.). Through the deter- mined points,/ , near the upper and lower edges of the sheet, draw the parallels PP, P'P'; and, beginning at CC, subdivide each into parts corresponding to the length of a de- gree or part thereof (Table VI.) of longitude, measured on this parallel ; the right lines mm' , through the points thus determined, are the required meridians. Intermediate parallels p'p' are drawn as desired. This projection is suited to larger areas than (I.). The adaptability of either projection to the required purpose depends upon the extent of territory to be represented and the scale of the map (par. 52). 151. Conical Projections. — Suppose a cone tangent to the earth's surface along a parallel of latitude, and the planes of the meridians and parallels produced to intersect it ; the meridians will be projected into right lines passing through the vertex — elements of the cone — and the parallels into circumferences of circles intersecting the projected meridians at right angles. If the cone be now cut along an element, and rolled out upon any of its tangent planes, the projections will appear as in Fig. 84, — the meridians all converging to the vertex, V, and the parallels as arcs of circles described from F as a centre. The parallels still intersect the meridians perpendicularly, but arcs of meridians are diminished toward the poles and increased toward the equator, while arcs of the parallels are all increased, except those of the parallel of tangency, which are represented in their true dimensions. Arcs of the meridians may be correctly represented by laying off from the parallel of tangency and along the central meridian (the element of contact of the plane of develop- ment) consecutive distances corresponding to the length of a degree of latitude, or part thereof, and with Fas a centre describing arcs of circles through the points of divisions. Of the conical projections, those most in use are : I. The Simple Conic Projection, in which the parallel of tangency is the central parallel of the tract to be represented, and is made the central parallel of the map sheet. Its radius is r = Rcotl, (i.) in which / is the latitude of this parallel, and R the mean radius of the earth's surface. To construct it ; draw the right line VC (Fig. 84) to represent the central meridian, and let M be assumed as its intersection with the central parallel. On VC set off Mp, , each equal to the length, reduced to the scale of the map, of a degree of latitude or part thereof, (Table IV.) ; also MV ^ r (Eq. i) : then from Fas a centre, describe the arcs/'/', PP, through M,p, , and they will be the required parallels. On PP set off Mm, mP, each equal to a degree, or part thereof, of this parallel (Table VI.), and the right lines Vm, VP, will be the required meridians. When the value of r places Fat an inconvenient distance, as is usually the case with small areas, points of the central parallel may be determined by rectangular coordinates as follows : Let VC (Fig. 85) represent the central meridian ; through M, its assumed intersection with the required parallel, draw AB perpendicular to VC, and use VC and AB as the axes of X and F respectively (par. 57), J/ being the origin. PROJECTIONS FOR MAPS OF LARGE AREAS. 95 The coordinates of any required point, as P, are x = mP and y =^ pP. Denote by v the angle MVP made with VC by the meridian through P\ then x is the sin V ; and y is the ver sin v ; and since the radius of the required parallel is r, the correspond- ing values of the coordinates are jr = r sin V, and j/ = r ver sin v. (2.) To find V measured on MP, let d represent the difference of longitude of VC and the meridian through P. Referring to a and b (Fig. 86), in which R' is evidently equal to R cos /, 90° -.dR':: d:M'p'\ and similarly, 90° : dr :: v : M'p' ; therefore d''R = drv ; substituting for R' and r their values and reducing, R cos/ , ■ , V— d ^ — --. = d sm /; R cot / and by substituting this value for v in (Eq. 2), X— r sin {d sin /), and y ^= r ver sin {d sin /). (3.) For a point 1° distant from M, d ^= i ; For a point i' distant from M, d= -^•, and similarly for other differences of longitude. Three or more points having been fixed, a curve drawn through them will be the parallel required. The meridians will be represented by normals to this parallel at the desired points of subdivision ; and additional parallels, concentric with PP, may be described through cor- responding points of subdivision of the normals. II. Bonne s Projection. — In this, the line representing the central meridian is first sub- divided into the required number of intervals, and the parallels are then constructed through the points of subdivision. Each parallel is then subdivided into degrees, or parts thereof, cor- responding to the intervals required between the meridians, which are then drawn through the points of subdivision. The latter are thus curved lines intersecting the central parallel only at right angles. Distances measured on the central meridian and on any parallel are faithfully represented, and the zones between parallels are in their true relations to each other and to the central meridian. Distances along other meridians are increased in the same ratio that the cosines of the angles between the radius of a parallel and the tangent to a meridian, at their point of intersection, are diminished ; and the result is, that while the areas of the different quadrilat- erals are unchanged, their diagonals become unequal, and this inequality increases toward the polar corners of the map. The construction of the parallels by points is as described for the central parallel in (I.) III. The Polyconic Projection. — In this projection, each parallel is the line of contact of a tangent cone ; therefore, in the development, the parallels are arcs of circles described from different points of the central meridian as centres, and with radii equal to the cotangents of their latitudes. The meridians are curved lines intersecting the parallels perpendicularly, the arcs of the meridians being exaggerated in proportion to the increase of longitude from the central meridian. This is termed the rectangular polyconic projection : and is constructed by setting off, upon the central meridian, distances corresponding to rectified arcs (Table IV.) of 96 TOPOGRAPHICAL DRAWING. one degree, or parts thereof, and with the different radii R cot /, the centres evidently being on the central meridian, describing the parallels through the points of division : the merid- ians are then curved lines drawn normal to the different parallels. For narrow belts of country the ordinary polyconic projection is used. The central meridian alone is normal to the parallels, and is developed in its true length. The parallels are con- structed as above described, and the meridians are curved hnes drawn through the extremities of the arcs of the different parallels, corresponding to equal differences of longitude measured upon them. For the smaller areas usually represented in topographical maps a graphic construction termed the equidistant polyconic projection is used. The limiting meridians and parallels, and the scale being given, draw a meridian line VV, (Fig. 87), as nearly through the middle point of the sheet as the proper spacing of the merid- ians and parallels will permit. Beginning at A, near the top of the sheet, set off on VV, AM, and MN respectively equal to the lengths of the degrees, or parts thereof (Table IV., Ap- pendix) of the central meridian, corresponding to the given latitudes ; and draw BC, DE, and FL exactly perpendicular to VV, and extending across the sheet. Set off on these per- pendiculars, from A, J/ and N, the distances ^^, Ab, Me, iV/" equal to the corresponding values of x (Table V., Appendix) for the extreme meridians of the survey ; and at the ex- tremities b, c, d /of these distances, set off, parallel to VV, bb' , cc' , ff, equal to the corresponding values oi y in the same table. Through b' , d' and /', and through c', e' and g' draw the meridians HI and KL, which for areas not exceeding about 200 miles square will be sensibly straight lines. Other meridians kk' , IM are drawn as required. Then subdivide the sections b'd' , c'e' into the desired number of equal parts b'b" , b"b"', of the same value as the intervals between the meridians (i.e., if the latter are minutes, make the former minutes, etc.) ; and the curves // through the corresponding points of division will be the required parallels. For very small areas, the points 3', c', g' may be joined io A,M d^nd iV by right lines. The accuracy of the construction is verified in the usual manner by comparing any plotted interval, as s, with its actual length reduced to the scale of the map, and by the measurement of diagonals as tt, t't', symmetrically disposed as to VV, and which should be of equal length. 152. Plotting the Triangulation. — The meridians and parallels plotted, and the latitudes and longitudes of the triangulation-points given, any intersection as of the former lines is an origin of co-ordinates for any point as r, — x and y being the number of seconds of longitude and latitude respectively from 0. To verify the accuracy of the plot, compare the actual distances, reduced to the scale of the map, between triangulation-points, with their plotted distances. To plot meridians and parallels upon a map already filled in with the topographical details, needs no further explanation ; the latitude and longitude of any point and the direction of the meridian through it being known, the construction is simply the reverse of that just described. Tables for the values in minutes and seconds of x and y for differences of longitude, to include 30° on each side of the central meridian, are given in the U. S. Coast and Geodetic Survey Report for 1884. TOPOGRAPHICAL SKETCHING. PART VI. SKETCHING-INSTRUMENTS, METHODS AND EXAMPLES. Section I. — General Considerations, Instruments, Locating Points, Measurement OF Distances and Heights, the Skeleton. 153. General Considerations. — The aim in the following pages is to show how with simple means to represent a route and the important features bordering it, a military position, or a tract of small or large extent, in an inteUigible manner and with a degree of accuracy suited to the special purpose for which the sketch is made. The mode adopted in treating this subject is to first describe a method of working with the presumption that suitable hand-instruments and plenty of time are at command, and then vary the details to suit less favorable conditions. The operations performed are based upon the general principles of topographical survey- ing and drawing, and their degree of accuracy depends upon the kind of instruments used, the method of working, and the time devoted to it. Facility in plotting is a prime requisite, and a theoretical knowledge of the manipulation and use of the more exact topographical instruments is of great assistance ; but it is only after much practice in the measurement of angles and distances, both instrumentally and by eye-estimation, that the ability is finally acquired of intelligibly representing a tract of country, or a route, while passing rapidly over it. Whether done in haste or at leisure, it embraces the following operations : I. Constructing the skeleton or outline of the sketch ; II. Filling in the details ; III. Representing the configuration of the surface, or hill-sketching. These are performed either in the order given, or the first two together and the last then added, or all three are carried on at once as the work progresses ; the method adopted depending mainly upon facility in sketching and the time at command. A beginner is obliged to perform them in the order given, until enough experience is gained to combine them, and thereafter the advantage obtains that no part of the ground need be revisited. 154. Sketching-Instruments (Plate XX.). — The instruments ordinarily used are a hand- compass, hand-level, clinometer, protractor, lead-pencil, rubber eraser and note-book ; or in place of the last, the required number of sheets of paper, secured to a light, smooth, rectangu- lar board or cardboard of convenient size. A red pencil for contours and a blue one for 13 98 TOPOGRAPHICAL SKETCHING. streams are particularly useful in preventing confusion of lines when much detail is required. With water-colors, some of the features, such as woods, cleared and cultivated land, bodies of water, etc., can be represented with great rapidity. The above are the essential instruments ; others of importance, suited to special pur- poses, are described in their appropriate places. a. Hand-compasses. — Two kinds are in general use, the " rectangular box-compass" and the " prismatic compass." Each consists of a cylindrical compass-box, usually of brass, three or four inches in diameter and from one to one-half an inch in depth ; containing a magnetic needle, with its N. end marked, and a compass-card having its edge graduated into 360 degrees, numbered toward the right, or in the same direction as the hour-divisions of a dial. The needle is free to move horizontally upon a pivot attached to the centre of the bottom ; but to prevent wear of the pivot when the compass is not in use, a lever is so ar- ranged that one end lifts the needle off the pivot when the other is pressed down. A glass cover protects the needle and card. These two kinds differ essentially in the attachment of the card and devices for sighting. In the rectangular box-compass (Fig. 88), the compass-box is sunken its depth into a square piece or case of hard wood provided with a hinged lid, and the 0° and 180° line is par- allel to the hinged edge. To take a Bearing; open the lid to the right at right angles to the case ; hold the case hori- zontally, and in such a position near the eye that the distant object can be seen along one of the faces of the lid, and the vibrations of the needle watched at the same time ; check the vibrations with the lever ; and when the needle is at rest, hold it in place and read and record the number of the division next to its N. end. With a little practice the mean of two or more successive delicate vibrations can be observed, and more accurate work ensured. A form of this instrument has the lines of sight at right angles to the lid, a vertical slit in the middle of the latter, and a sight-vane diametrically opposite which can be turned down when not in use. In the prismatic compass (Fig. 89) the card is attached to the needle, and the 180° divi- sion is usually at the N. end. The four "cardinal points" are also marked on the card. The adjustable sights, s and /, with slits in them, are hinged to the upper edge of the box, at points diametrically opposite each other. The sight s, held next to the observer's eye, contains a lens, and a glass prism which reflects the number immediately below on the card to the eye, at the same time an observation is taken through the slits at the distant object. The front sight, when folded down for the purpose of placing the box in its case, lifts the needle off the pivot, in the manner already described (see Fig. 88). By pressing a knob which projects from the side of the box, usually under the front sight, a spring bears on the rim of the card and the needle-vibrations are checked. In the instrument shown, the cam c serves to retain the card free of the pivot when the box is closed. To take a Beari^ig; adjust the prism so that the numbers on the card can be easily read ; hold the box steadily in a horizontal position so that the distant object, of which the bearino- is required, may be clearly seen through the slits in both sights ; check the needle, and record the number which appears to be cut by the hair in the slit of the front sight. If the card is graduated from 0° to 180° on each semicircumference, care must be taken to record the semicircumference on which the number is read, as E. 90°, or W. 90°, etc. SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. 99 A mirror, shown attached to /, is used for taking bearings of the sun or of very elevated points, the colored glasses attached to s serving in the former case as a protection to the eye. It is not, however, of much particular value, since by use of a plumb-line a point vertically beneath that of which the bearing is desired may readily be found. The common pocket-compass, graduated with the " points of the compass" only, may be made serviceable by notching its rim directly over the N. and S. points and using these notches for sighting. By carefully tipping the box in the direction of the needle, the latter will be held in place for reading (the S. notch should be held next to the eye, and the "point" under N. end of needle recorded) ; or the card may be easily graduated into degrees, the box encased, and the compass used as already described. The protractor, and its use in plotting bearings taken with differently graduated com- passes, are described in Topographical Drawing, pars. 22 et seq. b. Hand-levels and Clinometers. — A Hand-level is shown in Fig. 90. It consists of a metal tube 5 to 7 inches in length, and closed at each end with plane glass. When held hori- zontally, the rays from the bubble b pass through an opening beneath to the mirror m, which usually occupies the left half of the tube and is inclined at an angle of 45° with the axis of the latter, and are reflected to the eye at ^; in this position of the instrument a horizontal cross-wire within the tube bisects the image of the bubble and any distant point on a level with the eye. In the improved form, the eye-piece is telescopic and contains a semicircular lens which magnifies the image of the bubble. To measure differences of level; locate a point of the slope on a level with the eye ; proceed to that point and locate another, and so on, to the height desired. The number of observations, multiplied by the height of the eye above the ground, will be the required difference. Its most important use is to ascertain from the sketcher's position the points of inter- section of a contour with distant objects. There is also a great variety of clinometers, — instruments for measuring angles of eleva- tion or depression. One of the best is " Abney's Reflecting Level" (Fig. 91), which is a combi- nation of the hand-level and clinometer. A rectangular metal tube, T, about 4^ inches long, is furnished with mirror, eye-aperture and cross-hair, as in the hand-level; the limb. A, fixed to the tube, is graduated to degrees, while the index-arm, i, which carries the level, L, is provided with a vernier which reads to 10 minutes. To make L horizontal, the milled head, h, is turned with the left hand while the instru- ment is held with the right. To use it as a hand-level, set the index at 0°. To measure the angle of elevation or depression of a point, sight the point, make L horizontal as above described, and read the arc passed over by the index of the vernier. A scale of slopes from \ to ^-^ is also marked on the limb ; but in using it, the front edge of the arm is the index. To test the adjustment, set the index at 0° ; sight at a distant object and move the entire instrument until a point of the object and the bubble are bisected by the lower edge of the mirror ; then revolve the instrument 180° about its axis and bisect the bubble as before. If 100 TOPOGRAPHICAL SKETCHING. in adjustment, the same point will again be bisected : if not, correct half the error by means of the screws which fasten the level to its support. For lack of better means, a protractor and plumb line may be used for both hand-level and clinometer;— the plumb-line being suspended from the "centre" of the former; and the diameter edge, held uppermost, used as the line of sight. In measuring angles of inclination, the line of sight is directed parallel to the general surface of the slope ; but if the slope is seen in profile, the clinometer is held at arm's-length, with its axis or longer edge parallel to the profile line. The lead-pencil should be of medium grade, and the note-book or paper ruled in accord- ance with the special method adopted in making the sketch (par. i6o). 155. As to scales and plotting, these subjects are fully described in Topographical Drawing, Part I., Section III. The particular scale to be used depends upon the purpose of the sketch and the area to be covered by it. For a route-sketch, it is ordinarily 4 or 6 inches to a mile. The conventional signs are used in sketching : they are necessarily "oughly and rapidly drawn, but no ambiguity should exist as to their exact meaning. In cases where a doubt might arise, it is best to describe the particular feature by its name written upon it, or by a marginal reference. 156. Locating Points, and Measurement of Distances and Heights. — Points are located by offsets from a given line, by intersections of compass-bearings, or by estimating their posi- tions. As in surveying, offsets are measured perpendicularly to a course. If by compass-bearings, these should be taken from such known points that the intersec- tions will not be at a very acute angle. A check-bearing from a third point serves as a verification. A point is also located by interpolation, i.e., by taking bearings from it of two or more given points, and then plotting these bearings at the latter points, and producing the lines until they intersect, which necessarily locates the required point. To promote accuracy, when time permits, both direct and reverse bearings of a course are taken and the mean is used in plotting. . , , J, , sum of the bearings-]- 180° If the reverse bearmg is less than 180 , then = mean bearing. xr , ... , „ a , sumof the bearings— 180° If the reverse bearing is greater than 180 , then = mean bea-ring. 157. Distances are measured by pacing, by the odometer, by the time required to pass over them, or they are estimated. a. By Pacing. — The ordinary gait, or customary step, and rate of walking are best suited to this purpose, and a regular swing of the arms tends to make the steps of equal length. The steps are counted singly, or each double step only is noted. For plotting, the average pace is the unit of measurement. This is found by pacing a known distance with an easy, even step, and dividing the distance by the number of steps. By repeating the operation several times, a final average is obtained which will be more accurate. SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. loi 1 The Pedometer (Fig. 92) is of great assistance, since by recording the number of paces automatically it permits the entire attention to be devoted to other parts of the work. It is of metal, of the size and shape of a watch, and is carried suspended by the hook h from a button-hole, or pocket-edge of the coat or vest. Its interior mechanism (Fig. 92^) consists of a weight, W, attached to an arm pivoted on the same arbor as the ratchet-wheel, C. When at rest, W\s kept at the highest point of the slot in which it plays (indicated by the dotted line) by the tension of the spring s" . In the motion due to walking, the spring s advances one notch for each descent of W, — s' retaining C in place during this downward movement. The screw, s'", serves to limit the play of Wto any desired extent. The large hand (Fig. 92) advances i division at each step, and registers loo paces each entire revolution. The hand on the right advances i division for each 100 paces, and the one on the left I division for each looo paces. The hands may be set at O, care being taken to turn them in a backward direction. d. The Odometer is an instrument for registering automatically the number of revolutions of the carriage-wheel to which it is attached, the distance passed over being the product of number of revolutions X circumference of the wheel. Its usual form is shown in Fig. 93. The bob, B, is suspended by its lugs, /, to the arbor, a, fixed to the stirrup, S. On the front face of B are two dials of equal diameter which revolve upon the same pivot, b, the teeth of each engaging in the thread of a, as shown. It is contained in a leather case, and so attached to the carriage-wheel that the dials are to the front. It is seen from inspection of the figure that the dials are advanced one tooth for each revolution of the carriage-wheel; and since the outer or front dial contains 100 teeth, and the other 99, the latter gains I tooth, or dial division, for each of its own entire revolutions. The manner of registering is seen in the figure, the motion of the dials being in the opposite direction of the hands of a watch, the hook index, i, (attached to the back of B^ serving for the outer dial, and the o of the latter for the inner one — the former dial evidently recording the number of single revolutions up to lOO, and the latter the number of hundreds of revolutions up to 99. The zeros may be made to coincide if desired, by first removing the screw at b. The present reading is 3483. The distance between any two points is the difference of the readings at these points X the circumference of the carriage-wheel ; but owing to the slip of the latter and other irregu- larities of motion, a special wheel, either trundled by hand or separately harnessed, is some- times used for the more precise measurements. In the ordinary application of it, a little practice will serve to determine the corrections, or the percentage of reduction, necessary to be made for the particular vehicle to which it is attached. c. In sketching from the saddle, the length of the horse's pace, ascertained for the differ- ent gaits, may be taken as the unit ; or, in very rapid work, the distance passed over in one minute (see also par 48). d. The correct estimation of distances by eye is of great importance, and especially so. when, as is often the case in the military service, but little time for making the sketch is available. 102 TOPOGRAPHICAL SKETCHING. The values ascribed to both distances and angles vary with the locahty, atmospheric conditions and time of day. Distances appear too small or too great according as the sun is respectively in front or in rear of the observer. They are usually underestimated in a fog or in rainy weather, after a storm, and when the ground is covered with snow. Distances observed down a slope appear shorter than in a contrary direction, and sharp undulations of the surface make the different ridges appear nearer. Angles of depression of slopes appear too great when observed from the bases, or in pro- file; and in estimating differences of elevation of points, not far removed, of an irregular surface, the tendency is to diminish them, frequently as much as one-half their actual value ; also in trying to find a point of equal elevation on the opposite side of a valley, it almost invariably happens that a higher one is selected. It is only by much practice in observing known angles and distances, and under the con- ditions above enumerated, that the eye is trained to avoid these illusions. The following general indications of distances from Lehagre and the " Feld-Taschenbuch" may be of use. Ordinary powers of vision and a clear atmosphere are assumed. Church spires are visible from 7 to 9 miles ; windmills and large buildings, from 5 to 6 miles ; ordinary buildings, if isolated and white, from i\ to 4^ miles ; windows, from if to 2\ miles ; large trunks of trees, from i^ to i^ miles ; and small trunks and telegraph-poles, from 1000 to 1200 yards. As regards military observations; — At a distance of 1600 yards, a column of infantry pre- sents the appearance of a heavy line of uniform thickness; in the case of cavalry, the line is thicker, and indented along its upper edges. The rifles are distinctly visible if the day is bright. At 1300 yards, infantry files are distinguishable; also whether cavalry is mounted or dis- mounted; and guns are distinguished from their carriages. At 750 yards, the movements of arms and legs, white parts of the uniform, and the horses' heads are observable. At 600 yards, the movements of horses' legs; and the front of a squadron may be esti- mated by the number of files. At 450 yards, men's heads, kind of head-covering ; mounted men are readily distinguished from their horses, and dark colors become visible. At 300 yards, cap ornaments and light-colored facings. At 200 yards, the faces, brass belt-plates, and the intervals between the limbs and the body at rest. At 150 yards, the hands, buttons, and hair of a light color. At 100 yards, the position of the eyes may be seen. e. A distance measured on a slope is reduced to its horizontal value ; therefore the angle of inclination is required, and the first method, prescribed in par. 82, is applicable. Or, roughly, to obtain the horizontal distance : on a slope of 5°, the average pace being 30 inches, an actual distance of 120 paces is diminished by very nearly half a pace ; on a slope of 10°, by 2 paces ; on a slope of 15", by 4 paces ; on a slope of 20°, by 7 paces ; on a slope of 25", by II paces, etc.: these values being obtained by rnultiplying 120 by the ver. sin (or I — cos) of the different angles of inclination. SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. 103 A similar table, on a basis of 100 paces, may be constructed, suited to any other average pace, and to the different degrees of declivity. As a rule, intersections of compass-bearings are used to determine points, the distances to which are in any way difficult to measure directly. On slopes exceeding 15° of inclination, paced distances are in general unreliable. 158. Measurement of Heights. — a. Heights may be measured as described in the use of the hand-level [b, par. 1 54). The clinometer in conjunction with the table (par. 81) serves to find the difference of level between points already plotted ; e.g., the distance between the plotted points, measured by the scale of distances, is 11 34 ft.; the vertical angle, as determined at one of the points by the clinometer, is 10°: from the table, 5.67 ft. horizontal distance corresponds to i ft. of eleva- tion, therefore the difference of level is — ^ 5-67 200 ft. 5 ' \ \ \ \ A Graphical Method is illustrated in the accompanying figure. The summit S having been located by compass-intersections and plotted, and the vertical angle i measured from any plotted point as A ; join A and 5 by a right line, and draw AB, making BAS equal ^^ —^-^ to i\ then .S^, perpendicular to ^5", will repre- 4 0*^ — "^ sent the required height, and may be measured with a scale of equal parts. Or measure AS by the scale, then Sp = AS tang. i. The following table is convenient in sketching in mountainous tracts. The first column gives the angle of elevation or depression ; and the second, the corre- sponding differences of level for distances of i mile. ° Feet. Feet. ■> Feet. • Feet. \ 46 4* 415 8i 789 I2i I170 I 92 5 462 9 836 13 1218 li 138 5i 508 qi 883 I3i 1267 2 184 6 554 10 931 14 1316 2i 230 6i 601 10+ 978 I4i 1365 3 276 7 648 II 1026 15 1414 3i 322 7i 695 Hi 1074 I5i 1464 4 369 8 742 12 II22 16 1514 Differences of level for other distances are found by multiplying the tabular difference by the given distance expressed in miles. b. The Aneroid Barometer (Fig. 94) is very useful for this purpose in hilly or mountainous districts. It is of metal, and, like the pedometer, is usually of the shape and size of a watch. Its interior mechanism is shown in Fig. ga,a. The cylindrical box, B, is of very thin metal, has a corrugated upper surface, and is nearly exhausted of air. It is connected with the spring S, (of which the tension is adjustable,) by means of the pillar / ; and, the rise and fall of its upper surface, due to changes of atmospheric pressure, are thus confined within narrow limits. The motion of .S due to these changes is 104 TOPOGRAPHICAL SKETCHING. communicated to the index hand h, by the levers / and /', shaft b, and chain c, — the latter being kept tense by the hair-spring t. For an increase of atmospheric pressure, the lever-arms are evidently drawn towards S, the chain is wound about the arbor by the hair-spring, and the index moves to the right ; the reverse obtains with a decrease of pressure. Aneroids are usually compensated for temperature by making the lever / of strips of brass and steel soldered together ; and if properly effected, the same reading would exist inside and outside of a room, at the same level, with differences of 30° or 40° F. — time being allowed in each case for the instrument to acquire the temperature of the surrounding air. A common form of graduation is shown in Fig. 94: the outer or altitude scale being movable, so that its o point may be set at the barometric reading of any desired point of the tract, and the height of any other point, as referred to the latter, be read directly from it. In another handy form of the instrument, a pin projecting from the inner edge of the lid can be adjusted to indicate the bar. reading at the starting-point, and the index-hand also can be set at any desired reading. The usual method is to set the o of the altitude-scale at the bar. reading of 31 inches; then if the temperature is about 5o°F., the difference of the readings for any two points is the difference of altitude. At other temperatures the following correction is applied : If the sum of the temperatures at the two stations is greater than 100° F., add xTnro of the height for every degree in excess of 100° ; if the sum is less than 100° F., subtract y^jVir of the height for every degree less than 100° F. The formula is, t^t'— 100- Difference of level = {h- h') fi + ^ "^/ppo ^"H h and h' being the readings of the altitude-scale at the respective stations, and t and t' the corresponding temperatures. Table VII. (Appendix) is a copy of Airy's table, to which the correction for tempera- ture is applied as above described. The following method is prescribed in the U. S. Coast and Geodetic Survey Reports : The index and scale errors, if any, of the aneroid should be determined and entered in the record. If of small value, they may be neglected, as differences and not absolute heights are required. The following formula will give the difference in height between two stations at which the aneroid and thermometer have been read within a few hours' interval of time, — the shorter the interval the better. Difference = 60345 feet (log B — log b) T-\-t , Mean temperature = This factor varies with the mean temperature. For every desfree above 32" F. add 134 to 60345. SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. 105 ^ = 28.72 b = 27.14 Example. log B= 1.4581844 log b = 1.4336098 0.0245746 T-\-t = 62^.5, and, hence, 64432 logs. 8.3904865 4.8091016 Difference = 1583.4 feet 3.1995881 Or the difference in height may be taken from the following table, by interpolation of tenths ; e.g. : 28.72 at 62°.5 for 1220.0 feet 27.14 at 62°.5 for 2803.6 feet Difference.. 1583.6 feet TABLE GIVING THE DIFFERENCE IN HEIGHT, IN FEET, BETWEEN TWO STATIONS, AT THE MEAN OF THE TWO OBSERVED TEMPERATURES. MEAN OF OBSERVED TEMPERATURES IN DEGREES FAHRENHEIT. Barometer. 32° 42° 52° 62° 72° 82° 92° 30.0 29.9 87.'5"" 89;4"" 91.4 93 3 95-3 97.2 99-2 29.8 175-3 179.2 183. 1 187.0 190.9 194.8 198 7 29.7 263.4 269.3 275-1 280.9 286.8 292.7 298.5 29.6 351-8 359-6 367.4 375-2 383-0 390.9 398-7 29-5 440.5 450.3 460.0 469.8 479.6 489-4 499-2 29.4 529.5 541-3 553-0 564.7 576.5 588.2 600.1 29-3 618.8 632.6 646.3 659-9 673-7 687.4 701.3 29.2 708.4 724.2 739-9 755-4 771-3 787.0 802.8 29.1 798-3 8i6.l 833-8 851-3 86g.2 886.9 904 7 29.0 888.5 go8.2 927-9 947.6 967-4 987-2 1007.0 28.9 979-0 1000.7 1022.4 1044.2 1065.9 1087.8 1109.6 28.8 io6g . 9 1093.5 1117.3 1141.1 1164.8 1188.8 1212.6 28.7 ii6i.i 1186.7 1212.5 1238.3 1264. I 1290. 1315.9 28.6 1252.5 1280.3 1308. I 1335-9 1363.8 1391.6 1419-5 28.5 1344-3 1374-2 1404.0 1433-8 1463.7 1493.6 1523.5 28.4 1436.4 1468.4 1500.2 1532-1 1563.9 1595-9 1627.9 28.3 1528.5 1562.9 1596.8 1630.7 1664.5 1698.6 1732.7 28.2 1621.5 1657.7 1693.7 1729.6 1765.6 1801.7 1837 9 28.1 -1714.6 1752.8 1790.9 1828.9 1867.0 1905-2 1943-4 28.0 1808. 1 1848.3 1888.5 1928.6 1968. 8 2009.0 2049.3 27.9 1901.9 1944.2 1986.4 2028.6 2071.0 2113.2 2155.6 27.8 1996.0 2040 . 4 2084.7 2128.9 2173-5 2217.8 2262.3 27.7 2090.5 2136 9 2183.4 2229.6 2276.3 2322.7 2369.3 27.6 2185.2 2233.8 2282.4 2330.7 2379-4 2428.0 2476.7 27-5 2280.3 2331.1 2381.7 2432.2 2482.9 2533.6 2584-5 27.4 2375-8 2428.7 2481.4 2534.1 2586.8 2639.6 2692.7 27-3 2471.6 2526.7 2581-3 2636.2 2691. I 2746.0 2801.3 27.2 2567.8 2625.0 2681.9 2738.9 2795-9 2852.9 2910.3 27.1 2664.3 2723.6 2782.6 2841.8 2901.0 2960.2 3019.7 27.0 2761.2 2822.6 2883.9 2945-1 3006.5 3067.9 3129-5 A fair approximation suited to rough work is obtained from the formula Z* = 9 (/« - A'). If k is less than 26, or the temperature exceeds 70° F., use 10 instead of 9 for the multiplier 14 io6 TOPOGRAPHICAL SKETCHING. The readings of an aneroid should correspond to those of a standard mercurial barometer reduced to 32" F. ; and frequent comparisons on several successive days will determine the difference, if any, which should be constant, at least for the time occupied in any single sketch, and is applied as an index error to the readings. In a series of observations, the aneroid should be held in the same position for reading; slightly tapped, or swung back and forth before reading, to overcome any friction in the parts; and sufficient time allowed for it to acquire the temperature of the surrounding air. The usual plan is to suspend it by the ring in such a position that, in observing it, the index shall be projected at right angles to the dial-face ; and, if a thermometer is attached, as in Fig. 94, to read the latter first, before it becomes affected by the heat of the hand. The readings of a good aneroid may be relied upon when taken at intervals so short that they are not affected by changes of pressure due to other causes than changes of level or temperature. If the reading at the first of any two stations is subsequently found to differ from that first observed, a mean of these may be taken as the true reading. After being subjected to sudden changes of pressure, the position of the o point changes and a new comparison with a standard is required. A large aneroid is in general more accurate than a small one. In ascending, good results are obtained until the readings decrease 6 inches. In descend- ing, the limit is 8 inches. Its usefulness in obtaining extended profiles of mountainous tracts is apparent. The following form of record is suited to this purpose (see also par. 166) : Station. Distance by Reading. Percentage of Reduction. Bearing. Aneroid No. Inches. Feet. 159. Constructing tJie Skeleton. — I. The general principle of working from the whole to a part requires that reference lines and points should first be established, and the loss of time due to hesitancy in working, on account of uncertainty as to position, and spent in making corrections, thus avoided. With the scale usually employed, 4 or 6 inches to a mile, but a comparatively few refer- ence lines and points are required, and these may be obtainable from a general map containing the principal features of the tract ; if not, the skeleton is constructed by one of the followino- methods. In this as in other parts of the work, the plotting may be done in the field, or the measurements recorded and afterwards plotted. II. By Triangulation.—Y\g. 95, Plate XXI., represents a tract in which the principal points C,D,E, were established by a system of triangulation based on AB, as shown. Most of the principal points were selected on account of their favorable location for hill- sketching. The traversing, for filling in the details, began at E, and, following the roads, was frequently checked upon the As. The traverse stations are marked ©. Fig. 98 is a record of part of the triangulation, and Figs. 96 and 97 show different forms of keeping the traverse notes as applied to the first few courses. SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. 107 For clearness of illustration, a small tract is chosen ; but it is evident that for one of great extent the operations would be similar. The conditions governing the selection of a position for the base are that it should be centrally located, free from marked irregularities of ground, in open ground, its extremities intervisible, and that several of the principal points shall be visible from both extremities. Its length vi'iW depend upon the extent of the area to be represented. The present ten- dency in surveying, to save time, is to use comparatively short bases measured with great accuracy ; and, applying this to sketching, a base of from 500 to looo yards would serve for a tract of from 5 to 10 miles square. Its bearing is taken from each extremity, and its length is carefully paced, — a mean of at least two measurements being required for this most important line. It is also advisable to use a rigid support for the compass here and at all the principal points. The base is then plotted, approximately in its relative position on the paper with refer- ence to other lines of the tract ; and, if practicable, so that the N. shall be toward the top of the sheet. In extending the triangulation from the base AB (F\g. 99, Plate XXI.), the principal points C, D, should be selected so that the angles ACB, ADB shall not be much less than 30°. If the location is unfavorable to this condition, a point as E (Fig. 100) is first fixed, then a point as D' can be located by bearings from B' and E; ox D' may be subse- quently fixed by intersections from other points. At least one check-bearing is required for each of the principal points ; and if the latter are not conspicuously marked upon the ground by natural or artificial objects, a signal of some description should be set up. Generally in taking a bearing, such a point of the distant object should be selected as may be visible and easily recognized at points from which it is to be subsequently observed. As a test of the accuracy of the skeleton, in addition to that given by check-bearings ; measure in the field the distance between any two stations independently plotted, and com- pare with the plotted distance. Instead of writing on the plot the names of objects sighted upon, the form shown in Fig. 98 may be used. Each station is given a distinguishing letter or numeral, which should be marked upon the plot. It is advisable to observe a larger number of points than is really needed to form the skeleton, so as to admit of a selection of those best suited to this purpose, and also to furnish others that might prove useful in filling in the details. III. By Traversing. — A polygonal figure, enclosing the main portion of the tract, is deter- mined by following as much as possible the lines of roads and streams. While traversing and following this route, the intersections of the courses by roads, paths, edges of woods and fields and other prominent features, are determined, as are also the positions of commanding summits. As a rule, the polygon thus obtained will not close. If its angles are all salient, the error due to angular measurement is the difference between their sum, and 180° X number of courses less two; if there are n re-entrant angles, subtract their sum from n X 360°, add the remainder to the sum of the salient angles, and proceed as above, as if the angles were all salient. The difference thus obtained is distributed equally among all the angles. If io8 TOPOGRAPHICAL SKETCHING. the polygon still remains unclosed on account of errofs of distance, the total error in distance is distributed among the different sides in proportion to their lengths. (See also par. 63.) The skeleton is then completed by determining the interior lines, of which the points of intersections with the different courses have been fixed, as above prescribed. IV. By Traversing Combined with Radiation and Intersection. — A route, as G, H, I, (Fig. loi), is chosen, from which the commanding points a, b,c, . . . . ,to the right and left, are visible ; it is then traversed and these points are determined by any of the various methods heretofore described. The lines Ha, Hb, .... form so many traverse routes for obtaining the interior details. a,b, .... should be so situated and numerous as to conjointly command a view of the entire tract. If a height is available from which a greater portion of tlie tract is visible, a commanding point of it is marked, and I'adial lines are determined, in direction, to all the principal points. These lines are then traversed, and suitable points are established from which to begin the minor traverses. If the distance between any two of the principal points is known, all the others visible from them may be located by intersections, and this part of the skeleton determined at once. V. By Parallels and Perpendiculars. — The principal points are first determined by any of the methods already described. If the tract is open and not much intersected by streams or other obstacles, it is then divided into four parts by right lines intersecting at right angles at a central point. One of these lines is then traversed and suitable points of it determined, through which, perpendiculars being established, the tract is thus subdivided into parallel zones of convenient width, of which the details to the right and left of alternate perpen- diculars are sketched by traversing the latter. VI. Figure 102 illustrates the use of alignments and prolongations in determining the out- line of a fortification, h, e, e', f, i and g are points already determined. At h, it is ob- served that h and i are aligned on the face MN; at e, that e and / are aligned on MO; at e' , that e' and ^ are in the prolongation of NP: and it is evident that by joining these points, as shown in the figure, their intersections will fix J/ and N\ and that the entire outline can be determined in a similar manner. This method, in its general application, has a very wide range, and in eye-sketching is often of great value. In experienced hands it serves not only for determining the skeleton, but also, by reference to the principal lines, for filling in the details. (See also pars. 163 and 167.) 160. Filling in the Details. — The skeleton having been determined and plotted, the details are obtained by traversing in such directions as may be best adapted to representin"- the char- acter of the surface, and also afford opportunities for checking the measurements by closino- upon, or referring to, the principal or other points of lines already established. As previously stated, this part of the sketch may be completed in the field, in which case errors are immediately detected and corrected ; or it may be more convenient, as in un- favorable weather, or when secrecy is required as is sometimes the case in the military ser- vice, to record the measurements and plot them afterwards. The entire skeleton, or, if too large, a tracing of part of it of convenient size, might be taken into the field, and the details plotted directly upon it ; otherwise, a note-book of suit- able size, say 6x8 inches, ruled according to one of the following forms, is generally used: I. Form of Note-book for Plotting the Sketch in the Field. — This is illustrated by Fi"-. 96. Three vertical columns are ruled on each left-liand page, as shown ; the central column is for SKETCHING INSTRUMENTS, METHODS AND EXAMPLES. 109 the stations, bearings of and distances between the different points of the traversed route ; the side columns are for tlie lengths of offsets to the right and left respectively ; and the side spaces are for descriptive remarks. Each right-hand page is prepared for the plotting by ruling it into inch-squares, a side representing I mile, looo yards or feet, etc., which may be subdivided at pleasure, and which assist in setting off distances. A protracting circle is described in the central part of the page, as shown, and its divisions are numbered to correspond to those of the particular com- pass in use. (See par. 60.) Instead of using a circle, the edges of the sheet may be graduated into degrees and parts thereof, thus forming a rectangular protractor sheet. It is used as follows : Begin at the bottom of the central column, note the date and minute of starting ; and immediately above it, represent the first station, or starting-point, by its proper sign. Use the letters of the alphabet, or numerals, in regular order, to designate the stations. Select some well-defined object in advance — the farther off the better, provided the distance to it can be conveniently measured — take its bearing and record it immediately above the station. Mark the station and plot the bearing (in direction only) on the right-hand page, assuming that set of lines for meridians, and starting from such a point on the page as will ensure getting as much of the route on it as possible ; e.g., if the route runs northeasterly, mark the top of the page N and begin near the lower left-hand corner. Then proceed to sketch the route to the second station. In locating objects from intermediate points of a course, record at the instant of halting the total distance from the preceding station. Sketch an object as soon as located, holding the note-book in its true position with reference to lines upon the ground. On leaving a course to measure an offset, mark the point left, so that the course may be resumed at its proper place. Descriptive remarks are made in their appropriate column and opposite the sketch of the objects to which they apply. For obvious reasons, avoid using the compass in the immediate vicinity of iron, such as a bridle-bit or wagon-tire. II. Form of Note-book used when the Plotting is not done in the Field. — As illustrated in Fig. 97, each right-hand page is ruled with two vertical lines, forming a central column for the stations, bearings and distances of the route traversed. The side spaces are for rough sketches of objects on the right and left of the courses, and which are shown approximately in their relative positions. The left-hand pages are for descriptive remarks. In its use, the date, time of starting, stations, bearings and distances of the route are noted as prescribed for Form I. The central column, however, represents the successive courses ; and, to prevent the confusion of lines which would otherwise occur in changing direc- tion, is interrupted at the extremities of each course by double lines drawn across the page. Intermediate bearings and offsets are noted by drawing lines free-hand, from the point of observation, in the direction of the object observed, and writing the bearing or distance, and if necessary, " to house," " to wood," etc., upon them. The central column is regarded as with- out breadth in representing objects which cross it; e.g., a road is terminated at one side of the column and resumed in its proper direction at the point horizontally opposite. no TOPOGRAPHICAL SKETCHING. A point from which a cross-traverse is run is indicated by enclosing its distance, marked on the course, by a line. In any mode of sketching that may be adopted, the direction of the meridian and the unit of measurement must be recorded ; and if the sketch itself conveys but a part of the required information, the verbal description must furnish the rest. i6i. Features to be sketched or otherwise described. — The purpose of the sketch of course determines the nature and amount of detail to be included ; but for practice, it is best to note all details of any prominence. For military purposes those features only are noted which affect the movements, disposi- tions and accommodation of troops, either in march or battle. The following are some of the most important : Roads. — 'W\). There were no other points that could be fixed between D and the skirt of the wood {R). R. " I then galloped to the top of the hill, and placing myself in a line with the two farms (/and K), that line was assumed to be correct ; and then observing the angles between AT and Sta. Martha, between isTand C, and ^and P, R was fixed. " At R, I could see (over the trees) the village of Calvarosa Ariba, and also a chapel (called an Hermita) on this side of it, the directions to which were taken ; also to the remarkable hill (5), and the abrupt slopes of the ground to the rear (C^and V). "A line was drawn to the fall, or gap, in the ground {T), taking great care that this, as well as those to 5, Calvarosa, and the chapel, were as correct as possible in regard to the line from P, because the connection of the right of the position rested on this SKETCHING WITHOUT INSTRUMENTS. 117 point, and the accuracy of the winding up of the sketch would depend on the cor- rectness with which those angles were taken. T. " Next to T ; and as, on reaching it, it was clear that none of the points on the left of the position could be seen, except R, it became necessary that the distance from R to T should be judged as accurately as possible — which distance became a fresh base. At T, thus fixed, all the right could be seen, and the Hermita could be inter- sected, as well as the ground to the rear (U, V, and E). The direction {X) of the smaller hill was taken, and the line over its summit, it was observed, passed to the abrupt right-hand slope of the ground {W^, to the rear of the position. A farm also, in a hollow of some wood to the front, was also noted. X. " I then went to the smaller hill, intending to go to the top, but the rocks were so rugged I could not ride up ; so, standing on a line between it and T, at X, that sta- tion was fixed by observing the direction to E and to the Hermita. " The line to Calvarosa from R was next intersected, which fixed that place. The direction to the houses {Z) was also laid down, and this place turned out to be the village of Arapiles ; and the two remarkable hills were the celebrated hills of the same name. " The line W being intersected, gave the boundary of the ground (I'): the farm in front, observed from T, could no longer be seen. " Passing, then, down by the right and along the hollow between the two great hills, I went to the Hermita, and this point having been before fixed, from thence the direc- tion of the further fall of the great hill (S) and two slopes of the hill on the further side of the Calvarosa valley were secured, as well as the direction of the water- course above and below. I then passed down the valley, and wound up the sketch at 0. " Going back from thence to C, I proceeded along the main road to D and E, putting in, on judgment, the village of Carvajosa, as well as the point E, where was a house, and where the great Salamanca road passed. " I returned to Cabrerizos, finding the Duke where I had left him, and handed him the sketch, having been absent about two hours and a half. I made a verbal report to his Grace, pointing out the high hill (S), which we could plainly see from the spot where we then stood, observing that it was doubtful how far guns could be brought there, not having had time to ride thither. The Duke gave me back the sketch, to put it in ink, which I did, sitting down on the ground, and returned it to his Grace. " In the afternoon of that day the position just sketched was occupied by part of our army; and the enemy having, by signal, communicated with the forts of Salamanca, re-crossed the Tormes at Huerta, and retired on the Douro. "Some weeks after (July 21, 1812), this position was again occupied, but being too strong to be attacked in front, the French marched round it, and the battle of Sala- manca took place next day, to the right of the ground here sketched — viz., to the right of the village (Z) of Arapiles. In making this, the sort of sketch-book was used given in the note below. — P. B." The " sketch-book" referred to consisted simply of a pasteboard 6x9 inches, with black parchment cover and flap ; and furnished with a loop on each side to which a string was at- tached, and by means of which it could be suspended from the neck and in front of the waist. ii8 TOPOGRAPHICAL SKETCHING. i68. Landscape-sketching (Figs, in and 112, Plate XXIII.). — For topographical pur- poses, a landscape-sketch is a free-hand drawing of the outlines of important features. It affords a good general idea of a tract, particularly if the ground is much diversified and the sketch is made from a favorable position, and is at times the only available method of repre- sentation. True sizes and relative positions should be given as nearly as possible ; the outlines of nearer objects made heavier than those more remote, a regular gradation according to dis- tance being preserved if possible ; and the point of the compass from which the sketch is made and the estimated distance of the sketcher's position from a central object are noted. Facihty is acquired by practice, but to those who find it " impossible " to sketch, the following suggestions may be of value. Draw a horizontal line to represent the horizon of the point of view, placing it at its proper height on the sheet relative to the field of view, and through its middle point draw a vertical line. Observe a prominent object centrally located, and placing the sheet suddenly in a vertical position in front of the eye so that the horizontal line shall be projected in the horizon, and the vertical pass through the object, mark the position of the latter on the ver- tical. The retention of an image upon the retina makes this an easy matter. Points near the extreme right and left of the field may then be fixed in a similar manner, taking care each time to hold the sheet in its original position. These with a few other prominent points being thus fixed, it is a simple matter to fill in intermediate lines and objects by eye. The usual method is to measure the distances between objects and differences of level by means of the pencil, held at arm's-length and perpendicularly to the line of sight. 169. Itineraries, or verbal descriptions of features, with a record of the measurements necessary to plot them, are at times the only data available for making the sketch. The fol- lowing form is a convenient one, and fully explains itself : Names of Consecu- places, tive and their distances distances between from the important start! ng-- points of point. the route. Designations of impor- tant points. — These are determined by changes in the direction or construc- tion of the roads traversed, in the nature of the soil ; the en- trance to a defHe, ter- minals of steep slopes, obstacles, a bridge, ford or building ; forks of the road, paths, etc. Lengths, measured on the road traversed, of its different features ; declivities, levels, etc. Breadth Sketches of the road or profiles traversed. of defiles. where bridges, changes in fords or this occur. other im- portant objects. Detailed descriptions of the condition and practicability of the road traversed, of the terrene it passes through ; of villages, dwell- ings, streams ; of important positions on the right and left; nature and dimensions of bridges ; fords, periods of overflow ; capacities of ferries as to number of men, horses or carriages, and time required for round trip. Available material and means for the repair of road and bridges. Remarks. The notes would of course conform to the purpose for which the itinerary is made. PHOTOGRAPHY APPLIED TO TOPOGRAPHICAL SKETCHING. 119 PART VII. PHOTOGRAPHY APPLIED TO TOPOGRAPHICAL SKETCHING. (Plate XXIV.) 170. Photography has long been employed in Europe for surveying-purposes. With good working-apparatus, the results are accurate, the necessary material for a map is obtained in a much shorter period and with less labor than by other means, all the details of each view are given and selections for representation may be made at pleasure, the computations are sim- plified, the plotting may be performed by those not engaged in the survey, and the different views serve to verify the accuracy of the plot. The obstacle to its general application is that it is unsuited to the survey of thickly wooded tracts, and those containing no commanding points ; and when the ground is much diversified, it is sometimes necessary to supply details hidden by intervening objects. Two methods are employed : in the first the views are taken upon a vertical plate, as in ordinary landscape-photography, and in the second upon a plate horizontally disposed. 171. First Method. — a. The Camera used is of the ordinary pattern ; but is provided with means for levelling it accurately upon the tripod. For this purpose, it may be placed upon a stout flat board attached at its middle point to a spindle projecting from the tripod-head, and which rests upon three levelling-screws, a nut at the head of the spindle serving to clamp the base of the camera firmly to the board. The latter may be made circular and its edge roughly graduated into degrees, so that the camera may be turned through any desired angle. A detached level, either circular or formed of two tube-levels at right angles, is used in the adjustment. The objective, fitted to a vertical slide, is of the rectilinear class which produces no sensible distortion. During the operations, the ground glass, and the sensitive plate which replaces it, must be vertical, and therefore perpendicular to the optical axis. It is best to focus for distant views once for all, and to mark upon the instrument the corresponding positions of its parts. No special size of camera can be prescribed ; but it is evident that the larger the photo- graph (or print, as it is generally termed in the following description), the more easily can the details be observed. The objective should be of sufificient size to give clear definition on the outer edges of the largest plate suited to the camera. b. The principles of this method rest upon the fact that photographs are true perspectives. As shown in Fig. 113, the camera being adjusted and focussed as above described, O, the optical centre of the lens or combination of lenses, is the " point of sight ;" P, the point in which the optical axis produced pierces the ground glass, is its projection on the plane of the picture; a horizontal line through P on the ground glass is the "horizon." 120 TOPOGRAPHICAL SKETCHING. A and B representing any two distant points, and a and b their images, the angles AOB and aOb are equal. (For clearness of illustration, the image is assumed as erect.) OPi% the focal distance; or, if a combination lens is used, it is the equivalent focal dis- tance. From the foregoing, the measurement of horizontal angles is readily effected by means of the print (Fig. 114) ; e.g., required the horizontal angle between two points E and F,—e' and f on the print ; place the latter on a flat surface ; draw a horizontal line, as HH; project by verticals, as shown, the points e' and/ in e" and /"; from P' , the middle point of HH, set off the perpendicular P'0' = PO\ then e"0'f" is the required angle, and can be measured with a protractor. To ascertain vertical angles; e.g., the angle of elevation of E: set off perpendicularly to O'e", e"e"' = e"e' , and draw O'e'" ; e"0'e"' is the required vertical angle, which also may be measured with a protractor. Or either angle may be determined trigonometrically and with great nicety ; e.g., to measure the horizontal angle e" O'f": with a finely graduated scale of equal parts measure Pf" P'e" Ff" and P'e" ; then tang. P'O'f" — -~; ; tang. FO'e" = j^, ; the angles are then ob- tained from the table (III. Appendix) and their sum is the angle required. The vertical angle may be similarly ascertained from the expression tang. e'O'e' = -rr^,- c. The difficulties that may arise in the application of these principles are the determina- tion of the horizon and of the distance O'F' = OP. In the following description, unless other- wise stated, the instrument is presumed to be levelled and focussed as already described. In a carefully constructed camera, when the objective slide is in its normal position, the optical axis produced pierces the ground glass at its middle point, or at the intersection of its diagonals; a right line through this point parallel to the base of the ground glass is hori- zontal. Distant points of which the images are observed to be coincident with this line may be recognized in the print, and a right line drawn through them is the required horizon. Similarly with a negative on a plate which accurately fits the holder, or with a print not dis- torted in its preparation, the middle point is the intersection of its diagonals, and a line through it parallel to the base is the required horizon. In practice, however (see also h, par. 172), this line is fixed by the interposition of a fine horizontal cross wire immediately in front of the exposed plate. To secure sharp defini- tion of this line on the print, means should be devised for pressing the wire into contact, or very nearly so, with the plate during exposure, or the wire may be fastened within the plate- holder itself. It may be made adjustable by attaching its ends to slides, or in vertical grooves. Its exact position may be verified with the aid of a surveyor's level. The images of distant points, ascertained by the latter to be on a level with the camera, should lie in the horizon of the print ; if not, and the line joining them is parallel to the horizon, correct by the ob- jective slide; if oblique to the horizon, the wire is out of adjustment. Any truly horizontal line of the print serves for the measurement of horizontal angles ; but the horizon is more con- venient for determining differences of level. (See also^, par. 172.) To find OF. — If the position of the optical centre is known, it may be measured directly PHOTOGRAPHY APPLIED TO TOPOGRAPHICAL SKETCHING. 121 on the instrument — P being the intersection of the diagonals of the ground glass. Otherwise it may be determined as follows: On the ground glass draw a vertical line through P\ select two distant points whose angular distance @, measured horizontally with a transit or theodolite, is about \ the field of view of the camera ; make the image of one of these points coincide with the vertical ; then expose a plate, and make a print from the negative. Project these images by verticals into the horizon ; then OP = OF = d cot. @, in which @ is known, and d is the distance between the projections of the points measured on the print. Measurements made upon the negative usually afford the more accurate results. The following method (Abney's) may be used for determining the focal distance directly with the camera : As illustrated in plan in Fig. 115, place two rods R and R' 10 yards apart, so that R shall be 75 yards from the ground glass G, and RR' perpendicular to GR. Draw a vertical line through the middle point of G\ focus on R, making its image coincide with this vertical ; measure the distance between the images of .^and R' and denote it by d; then, from the simi- lar triangles thus formed, OG=OP== 75^ ■ 10 + d' 2700 ^ I !> e.g., if ^ = 1.5 in.; OP— ^ ^ ^ > - = 1 1.2 in., practically. The measurements are made horizontally. d. Points are plotted as follows: Suppose a base measured, and two views embracing the same objects are taken from its extremities; it is apparent that if these views are arran^ea upon the plotting-sheet so that their horizons occupy their true positions relative to the plotted base, the intersections of right lines drawn from the extremities of the base to the projections of corresponding points on the horizons will fix the required points on the plot ; the operation would thus correspond to plane-table work. A view taken from a third known point, and embracing some of the points already fixed, would, if plotted, evidently serve as a check upon the work. 172. Application of the First Method . — e. The simplest way known to the writer of apply- ing the foregoing principles to topographical sketching is that recommended by M. Javary, which is essentially as follows: A hand-compass is used in conjunction with the camera; a base is measured (II., par. 159). and a skeleton formed in which the principal points are stations which command favorable views of the surrounding country. This triangulation may be per- formed first, or simultaneously with the photographing, — the hand-compass being used for this purpose. As may be already inferred, points to be determined must appear on at least two prints. In addition to this, the bearings of a few prominent points, say 3 or 4 of each view, near the right and left extremities and the middle point of the field, are taken with the compass. These 16 122 TOPOGRAPHICAL SKETCHING. latter points must be readily recognizable in the prints, and, for convenience of expla- nation, will be called orientation-, or simply orient-points. f. The field-work completed, the skeleton is plotted first, and the details are then filled in from the prints as follows: Let A and B (Fig. 1 16) represent the plotted stations from which views V and V respectively, of the same objects, have been taken. Project the orient-points upon the hori- zons of these views and mark the projections of each view on the edge of a slip of paper placed coincident with the horizon. At the point A plot the bearings Aa, Aa' , Aa" of the orient points of V\ apply the corresponding slip to these bearings so that the latter will intersect the projected orient-points (as in the graphical solution of the 3-point problem) ; a line drawn along the edge of the slip — represented by the heavy line in the figure — will then be the required position of the horizon of V. It should be tangent to the circle described from .(4 as a centre, with the focal distance OP' as a radius ; and the point of tangency is the middle point of the horizon. Four bearings serve to fix the horizon with greater accuracy. The horizon of V is fixed in a similar manner; and it is apparent that all required points of FandF' can now be plotted, by first projecting them on their respective horizons, then transferring them by means of the slip as above, and fixing them by intersecting lines from A and B, as indicated for the orient-points O in the figure. To facilitate the operation, the prints should be carefully and equally dampened by placing them between sheets of blotting-paper, and then pasted, without rubbing them, to cardboard. The same points or objects on different prints are marked with the same numbers ; their pro- jections are similarly marked on the slips, and the latter are then pasted at their extremities in their proper position on the plotting-sheet. Needles, to which silk threads are fastened, are placed in the principal points, the threads are then stretched over points correspondingly numbered, and their intersections are marked and numbered accordingly on the sheet. g. To Ascertain Differences of Level. — Let Z? (Fig. 117, a vertical projection) represent the plotted distance from A to B,d the focal distance {O'P', Fig. 116), and h the apparent height of B above A as measured from the horizon of A on the print ; then the difference of level of A and B is : e. g., D — 800 feet ; ^ = 1.5 Inches, and d — 12 inches ; then, H= 800 — = 100 feet. 12 To refer all vertical measurements to a common datum — as in this case, the heights of points ascertained on view from B, to the horizontal plane through A — it is only necessary to add to them the difference of level of A and B. h. In the method here recommended, a horizontal wire is unnecessary for determining the horizon of a print, and the operation is as follows : In the field-work, the vertical angles of the orient-points are measured with a hand-clinom- eter, at the same time the bearings of these points are taken. The horizon of the print is then determined, as follows (Fig. 118): C, an orient-point PHOTOGRAPHY APPLIED TO TOPOGRAPHICAL SKETCHING. 123 projected at c, being established by intersections as already described, and the vertical angle /3 measured with the cHnometer ; A'c'xs measured on the print; then cc' — A'c tang /J; and this distance is laid off vertically from C. Similarly for another orient-point, D, da ^^A'dt^ingfi'; da' is set off vertically from D\ and a right line through the extremities of cc' and dd' is the horizon of the view taken from A' . Evidently the direction in which cc' and dd' are laid off depends upon the vertical angle being one of elevation or of depression, and the image being erect or inverted. A horizontal wire attached anywhere across the plate is needed, however, when but one vertical angle is measured ; since one point only of the horizon being then determined as above described, the horizon is a right line drawn through it parallel to the line marked by the wire. i. For the plotting of levelling-measureinents, the calculations should be recorded. The fol- lowing form is a convenient one (see^, this par, and Fig. 1 17) : • Number of h. d. D. H. Reference of Remarks, Point. Station. Print. Inches, Inches. Feet. Feet. Station. Point. The refei'ences are then noted on the plot, near the numbers of the points; and the contours are interpolated as described in pars. 86 et seq., the prints showing nearly all the details of the configuration. Any detail not represented by photographic means is supplied by other methods of sketching. k. The most convenient scale for plotting surveys is -j-sVo; ^""^ ^^ ^ smaller one is desired it is advisable to plot to this scale, generalizing the detail (par. 141), and then reduce the map. For sketching, since the field-work is much less exact, a much smaller scale is used. TYis. degree of accuracy attainable by this method is stated by M. Javary to be a maxi- mum error of j^jVir for both the plane-surveying and leveUing. This is the result as deduced from a number of cases; and with a focal distance of 20 inches, a scale of ^-j^, and the use of a microscope to examine the prints. As to the choice of stations, on which a successful result entirely depends, no rigid rules can be given. Poinis in open ground, commanding a wide view, are the most suitable ; and to avoid an unnecessary increase in their number, it is best, if practicable, to arrange them in sets of three, so that those ofeach set will face the same part of the tract. 173. Second Method. — By this method, horizontal views are taken from different stations of the tract ; these are then oriented on the plotting-sheet, and the intersections of radial lines through corresponding points of the views determine the required points of the plan. The contouring is effected as hereinafter described. 124 TOPOGRAPHICAL SKETCHING. Fig. 119 is a vertical section of the kind of camera used for this purpose. AB is the cylindrical dark chamber, with its axis vertical, mounted on a tripod-head fur- nished with levelling-screws. The sensitive plate is placed upon the bottom of it. f is a circular plate resting upon AB, and movable about its centre C in the axis of the cylinder. This plate carries the lens-tube /, containing either a single- or combination-lens, of which the optical axis O is parallel to the axis of AB. ^ is a triangular prism, with its edges placed horizontally, and its incHned face silvered, so that the rays from exterior objects are reflected to the lens and thus focused upon the sensitive plate, — the lens-tube being raised or lowered in its carrier for this purpose, in the usual manner, by means of a milled head, and an index and a scale being attached for adjusting. The images are prevented from extending beyond the centre of rotation by the horizontal disk, D and D' , which contains a very narrow radial open- ing, and which revolves witla /"by means of the shaft S. This opening is in the plane of O and S\ and a fine wire, k;, is stretched across it where O pierces it. If the instrument is levelled and focused, and an exposure is made by revolving P 360° with a rapidity corresponding to the sensitiveness of the plate used, the print made from the plate will be a circular view, or a " tour of the horizon," in which the circumference of the circle, produced by the interposition of the wire, will mark all points on a level with R, and the angles between radials drawn from its middle point will be the true horizontal angles between points which these radials intersect. Any radial also represents the vertical of the points intersected by it. It is apparent that if views are taken from a sufificient number of points to embrace the entire tract to be represented, and are properly oriented upon the plotting-sheet, all the ele- ments for a complete map are presented. The two prints taken from the extremity of the measured base are placed so that their centres shall be at a distance apart equal to the length of the base, according to the scale of the map; and, being secured by pins through their centres, are turned so that the images of the points between these stations, on each print, shall lie in the right line joining the centres. Points are then fixed on the plot by the intersections of right lines through correspond- ing images. A third print is then placed and oriented in a similar manner, its centre being fixed from the other two prints, and so on. The configuration is determined in a manner similar to that described for the first method ; and by the use of the hand-compass and clinometer, the cross-wire may be dis- pensed with. APPENDIX TABLE I. MEASURES OF DIFFERENT COUNTRIES IN METRES AND ENGLISH UNITS. MEASURES OF LENGTH. ITINERARY MEASURES. Country. Designation. Equivalent in Country. Designation. Equivalent in Metres. English Feet. Metres. Efier. Stat. Miles. Foot 0.3048 0.3248 0.3048 0-3139 0.3139 0.2919 0.2832 0.3000 0.3000 O.3161 0.2827 0.2827 3-2809 1.0658 I.OOOO 1.0297 1.0297 0.9576 0.9291 0.9843 0.9843 1. 0371 0.9274 0.9274 Statute mile.. . Myriametre. . . Verst 1609.3 loooo.o 1066.8 7532-5 7532-5 10699.0 7420.2 7586.7 4239-8 4239.8 6181.4 8333-0 1652.8 France Metre Paris foot. . . . Foot France 6.2138 0.6629 4.6806 4.6806 6.6480 4.6108 4.7142 2 . 6346 2 . 6346 3-8410 5.1780 1 . 0270 Mile Denmark Sweden Geog. mile Mile Judicial league League Mile Bavaria Baden Spain Austria Vienna foot.. . Foot Portugal Spain Tuscany TABLE M. FOR MUTUAL CONVERSION OF METRES AND ENGLISH UNITS. Metres to Yards. Metres to Miles. Yards to Metres. Miles to Metres. Metres. Yards. Metres. Statute Miles. Nautical Miles. Yards. Metres. Miles. Metres in Statute Miles. Metres in Nautical Miles. I 1.094 10 0.006 0.005 I 0.914 I 1609 33 1853-25 2 2.187 20 0.012 , o.ori 2 1.829 2 3218.66 3706.50 3 3-281 30 O.Oig 0.016 3 2-743 3 4827.99 5559-74 4 4-374 40 0.025 0.022 4 3-658 4 6437.32 7412.99 5 5.468 50 0.031 0.027 5 4-572 5 8046.65 9266.24 6 6.562 60 0.037 0.032 6 5-486 6 9655-98 11119.49 7 7-655 70 0.043 0.038 7 6.401 7 11265.31 12972.74 8 8.749 80 0.050 0.043 8 7-315 8 12874.64 14825.98 9 9-843 90 0.056 0.049 9 8.230 9 14483.97 16679.23 10 10.936 100 0.062 0.054 10 9.144 10 16093.3 18532.50 I metre = 1.093623 yards. = 0.00062138 statute mile. = 0.00053959 nautical mile. I yard = 0.914392 metre. I statute mile = 1609.330 metres. I nautical " =1853.248 Note. — Tables II., IV., V. and VI. are based upon Clarke's Determinations, the present accepted standards. 126 APPENDIX. TABLE 111. NATURAL SINES, COSINES, TANGENTS AND COTANGENTS. o , Sine D. Tang. D. Cotang. D. Cosine D. 1 0.0000 0.0000 00 1 . 0000 90 30 0.0087 0.0087 114-5887 1. 0000 30 89 I 0.0175 0.0175 57-2900 0.9998 0. 1 30 0.0262 0.0262 38.1885 0.9997 30 2 0.0349 0.0349 2.9 28.6363 0.9994 88 30 0.0436 0.0523 0.0437 0.0524 22.9038 19.0811 0.9990 0.9986 30 87 3 30 0.0610 0.0612 16.3499 0.9981 30 4 o.o6g8 0.0699 14.3007 0.9976 0.2 86 30 0.0785 0.0872 2.9 0.0787 0.0875 12.7062 11.4301 0.9969 0.9962 30 85 5 30 0.0958 0.0963 10.3854 0.9954 30 6 0.1045 0.1051 9-5144 0.9945 0.3 84 30 0.1 132 0.1139 8-7769 8 . 1443 0.9936 30 83 7 0.1219 0.1228 0.9925 30 U.1305 0.1317 7-5958 0.9914 30 8 0.1392 0.1405 7-11:4 m fc 0.9903 0.4 82 30 0.1478 0.1495 3.0 6.6912 — y 1 M 0.9890 30 0.1564 0.1584 6.3138 0.9877 81 9 30 0.1650 0.1736 0.1673 0.1763 5-7958 5-6713 0.9863 0.9848 0.5 30 10 80 30 0.1822 0.1908 0.1853 0.1944 5-3955 5.1446 "bo 3 0.9833 0.9816 30 II 79 30 0.1994 0.2079 0.2035 0.2126 4.9152 4.7046 II 0.9799 0.9781 0.6 30 12 78 30 0.2164 0.2250 0.2217 0.2309 4.5107 4-3315 _3 0.9763 0.9744 30 13 77 30 0.2334 0.2401 3-1 4-1653 > 0.9724 0.7 30 14 0.2419 0.2493 4.0108 •u 0.9703 76 30 0.2504 0.2588 2.8 0.2586 0.2679 3.8667 3-7321 .2 0.9681 0.9659 30 15 75 0.2672 0.2773 3.6059 c 0.9636 30 30 16 0.2756 0.2867 3-4874 0.9613 0.8 74 30 0.2840 0.2962 3-3759 (2 0.9588 30 17 0.2924 . 3007 0.3057 0.3153 3-2 3.2709 3.1716 0.9563 0.9537 73 30 30 18 . 3090 0.3173 0.3249 0.3346 3-0777 2.9887 0.9511 0.9483 0.9 72 30 30 19 30 0.3256 0.3338 0.3443 0.3541 2.9042 2.8239 0.9455 0.9426 71 30 3-3 I.O 0.3420 0.3640 2.7475 0.9397 70 20 30 0.3502 0.3584 0.3739 U.3839 2.6746 2.6051 0.9367 U.9336 30 69 21 30 0.3665 0.3939 2.5386 0.9304 1 . 1 30 22 0.3746 0.4040 3-4 2.4751 0.9272 68 30 0.3827 0.3907 2.7 0.4142 0.4245 2.4142 2-3559 0.9239 0.9205 30 23 67 30 24 0.3987 0.4067 0.4348 2.2998 2 . 2460 0.9171 0.9135 30 66 0.4452 3-5 1.2 30 0.4147 0.4557 2-1943 0.9100 30 Cosine D. Cotang. D. Tang. D. Sine D. APPENDIX. 127 TABLE III.— Continued. NATURAL SINES, COSINES, TANGENTS AND COTANGENTS. » , Sine D. Tang. D. Cotang. D. Cosine D. 1 25 0.4226 0.4663 2.1445 0.9063 65 30 0.4305 0.4770 3-6 2 . 0965 0.9026 30 26 0.4384 0.4877 2.0503 0.8988 1.3 64 30 0.4462 2.6 0.49B6 2.0057 0.89-19 30 27 0.4540 0.5095 1.9626 0.8910 63 30 0.4617 0.5206 3-.7 I. 9210 0.8870 30 28 0.4695 0.5317 1.8807 0.8829 62 30 29 0.4772 0.4848 0.5430 0-5543 3.8 1.8418 I . 8040 0.8788 0.8746 1.4 30 61 0.4924 0.5658 1.7675 0.8704 30 30 30 0.5000 0.5774 3.9 I. 7321 0.8660 60 30 31 0.5075 0.5890 1.6977 0.8616 1-5 30 0.5150 0.6000 4.0 1 . 6643 Sf.H u 1 vi 0.8572 59 30 0.5225 0.5299 2.5 0.6128 0.6249 I. 6319 I . 6003 u 0.8526 0.8480 30 32 4.1 58 30 0.5373 0.5446 0.6371 . 6494 1.5697 1.5399 0.8434 0.8387 1.6 30 33 57 30 0.5519 0.6619 4.2 I. 5108 'bo 0.8339 30 34 30 35 0.5592 0.5664 0-5736 0.6742 0.6873 1.4826 1.4550 I. 4281 at 0.8290 0.8241 0.8192 56 30 55 4-3 II 2.4 0.7002 4.3 30 0.5807 0-7133 4.4 I. 4019 3 O.8141 1-7 30 36 0.5878 0.7265 4.4 i.3764 1 . 8090 54 30 0.5948 0.6018 0.7400 0.7536 4-5 4.5 1-3514 1.3270 (U ■-5 4J . 8039 0.7986 30 37 S3 30 0.6088 U.7673 4.6 1.3032 0.7934 30 38 0.6157 2-3 0.7813 4.7 1.2799 0) G 0.7880 52 30 0.6225 .0.7954 4-7 1.2572 )-• 0.7826 1.8 30 39 0.6293 0.8098 4.8 1.2349 (2 0.7771 , 51 30 0.6361 0.8243 4.9 1.2131 0.7716 30 40 0.6428 0.8391 5.0 1.1918 0.7660 50 30 0.6494 0.8541 5.0 I. 1708 . 7604 30 41 0.6561 2 . 2 0.8693 5-1 I. 1504 0.7547 1.9 49 30 0.6626 0.6691 0.8847 0.9004 5.2 5-3 I. 1303 1.1106 0.7490 0.7431 30 42 48 30 0.6756 0.6820 0.9136 0.9325 5-4 5-5 I. 0913 1.0724 0.7373 0.7314 30 43 47 30 0.6884 0.9490 5.5 1.0538 0.7254 2.0 30 44 0.6947 . 7009 2.1 0.9657 0.9872 5.6 5.7 1.0355 I. 0176 0.7193 0.7133 46 30 30 45 0.7071 I. 0000 5.8 I. 0000 0.7071 2. 1 45 Cosine D. Cotang. D. Tang. D. Sine D. Columns "i? " give the differences corresponding to arcs of i'; e.g., the sin 23°2o' for I' (or 2.7) X 20 = 0.3907 + 54 = 0.3961 sin 23° + the difference 128 APPENDIX. TABLE IV. MERIDIONAL ARCS. From Lat. To Lat. Length in Metres. From Lat. To Lat. Length in Metres. From Lat. To Lat. Length in Metres. 20 21 110705.1 • 30 31 II0857.O 40 41 I I 1042. 4 21 22 IIO718.2 31 32 I 10874. 4 41 42 III061.9 22 23 IIO731.8 32 33 I 10892. I 42 43 II1081.6 23 24 I 10746.0 33 34 Iiogio.l 43 44 IIIIOI.3 24 25 I 10760. 6 34 35 IIO928.3 44 45 IIII2I.O 25 26 I 10775. 6 35 36 110946.9 45 46 III140.8 26 27 II0791.I 36 37 1 10965. 6 46 47 I11160.5 27 28 H0807.O 37 38 I 10984. 5 47 48 III180.2 28 29 II0823.3 I i!o840 . 38 39 I I 1003. 7 48 49 111199.9 39 30 39 40 IIIO23.O 49 50 II1219.5 For arcs of i', divide by 60 ; for arcs of i", by 3600. TABLE V. POLYCONIC PROJECTION. COORDINATES IN METRES, FOR DIFFERENCES OF LONGITUDE OF the Lati- I minute. 5 minutes. 10 minutes. 30 minutes. 1 degree. '■ y- jr. y- ... y- X. y- ... y- 20° 1744.1 0.1 8720.7 2.2 17441.4 8.7 52324.2 78.1 104648.0 312.3 21 1732.9 0.1 8664.3 2 2 17328.6 9 51985.9 81.3 103971.3 325.2 22 1721.I 0.1 8605.4 2 3 17210.7 9 4 51631-8 84.4 103263.1 337-6 23 1708.7 O.I 8543-7 2 4 17087.4 9 7 51262.0 87.4 102523.4 349-6 24 1695.9 0.1 8479-5 2 5 16958.9 10 50S76.6 90.3 101752.7 361.2 25 1682.5 0.1 8412.7 2 6 16825.3 10 3 50475.8 93-1 100950.9 372.3 26 1668.7 0.1 8343-3 2 7 16686.6 10 6 50059.6 95.8 100118.5 3S3.0 27 1654-3 0.1 8271.4 2 7 16542.8 10 9 49628.2 98.3 99255.7 393-2 28 1639.4 0.1 8197.0 2 8 16393.9 11 2 49181.7 100.7 98362.6 403.0 29 1624.0 0.1 8120. I 2 9 16240.1 11 5 48720.3 103.1 97439.6 412.2 30 1608 . 1 0.1 8040 . 7 2 9 16081.4 11 7 4S244.O 105.3 96487.0 421.0 31 1591-8 0.1 7958.9 3 15917-7 II 9 47753-0 107.3 95505.0 429-3 32 1574-9 0.1 7874-6 3 15749-2 12 1 47247.4 109.3 94493.8 437-0 33 1557-6 0.1 7787-9 3 I 15575-9 12 3 46727.4 III.O 93453.8 444.2 34 1539-8 U.I 7698.9 3 1 15397-9 12 5 46193.2 112.7 92385.4 450.8 35 1521.5 0.1 7607.5 3 2 15215 12 7 45645-0 114.2 91288.8 456.9 36 1502.8 O.I 7513-8 3 2 15027.6 12 8 45082.7 115.6 90164.3 462.5 37 1483.6 O.I 7417-8 3 3 14835-6 13 44506.7 116.9 89012.2 467-5 38 1463.9 0.1 7319-6 3 3 14639- I 13 1 43917.1 118.0 87833.0 471-9 39 1443 . 8 O.I 7219.0 3 3 14438. I 13 2 43314.1 118.9 86626.9 475-8 40 1423-3 O.I 7116.3 3 3 14232.6 13 3 42697.8 119. 8 85394.3 479.0 41 1402.3 0. 1 7011.5 3 3 14022.9 13 4 42068 . 5 120.4 84135.6 481.7 42 1380.9 0.1 6904.4 3 4 13S08.8 13 4 41426.3 120.9 82851.2 4S3.8 43 1359-I 0. 1 6795-3 3 4 13590-5 13 5 5 40771.4 121.3 81541.3 485.3 44 1336.8 O.I 6684 3 4 13368. I 13 40104.0 121.5 80206.5 486.2 45 1314-I 0. 1 6570.8 3 4 1314I-5 13 5 39424-3 121. 6 78847.1 486.5 46 1291.1 O.I 6455-5 3 4 12910.9 13 5 38732.6 121.6 77463.6 486.3 47 1267.6 0. I 6338-2 3 4 12676.4 13 5 38028.9 121.4 76056.3 485.4 48 1243.8 0.1 6219.0 3 3 12437-9 13 4 37313-6 121.0 74625.6 484.0 49 1219.6 O.I 6097.9 3 3 i2]95.8 13 4 365S6.8 120.5 73172.0 4S1.9 50 1195.0 O.I 5974.8 3- 3 11949.7 13 3 35848.8 119.8 71696.0 479.3 APPENDIX. 129 TABLE VI. LENGTHS OF DEGREES OF THE PARALLEL. Lat. Metres. Lat. Metres. Lat. Metres. Lat. Metres. Lat. Metres. Lat. Metres. / 20 00 104 649 / 25 00 100 952 o / 30 00 96 488 35 00 91 290 • / 40 00 85 396 45 00 78 849 30 4 314 30 539 30 6 001 30 731 30 4 770 30 8 160 21 00 3 972 26 00 119 31 00 5 506 36 00 166 41 00 4 137 46 00 7 466 30 3 622 30 99 692 30 5 004 30 89 593 30 3 498 30 6 765 22 00 3 264 27 00 9 257 32 00 4 495 37 00 9 014 42 00 2 853 47 00 6 058 30 102 898 30 98 814 30 93 979 30 88 428 30 82 201 30 75 346 23 00 2 524 28 00 8 364 33 00 3 455 38 00 7 835 43 00 I 543 48 00 4 628 30 2 143 30 7 906 30 2 925 30 7 235 30 879 30 3 904 24 00 I 754 29 00 7 441 34 00 2 387 39 00 6 629 44 00 208 49 00 3 174 30 I 357 30 6 968 30 I 842 30 6 016 30 79 532 30 50 00 2 439 71 698 For arcs of i', divide by 60 ; for arcs of i", by 3600. TABLE VII. ANEROID MEASUREMENT OF HEIGHTS. Computed for Temperature of 50" F. Height. Barometer Reading. Height. Barometer Reading. Height.- Barometer Reading. Height. Barometer Reading. Height. Barometer Reading. Height. Barometer Reading. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. 31.000 2000 28.807 4000 26.769 6000 24.875 8000 23-115 lOOOO 21.479 50 30.943 2050 28.754 4050 26.720 6050 24.829 8050 23.072 10050 2 I . 440 TOO 30.886 2100 28.701 4100 26.671 6100 24.784 8100 23.030 1 0100 21.401 150 30.830 2150 28.649 4150 26.622 6150 24-738 8150 22.988 10150 21.361 200 30.773 2200 28.596 4200 26.573 6200 24.693 8200 22.946 10200 21.322 250 30.717 2250 28.544 4250 26.524 6250 24.648 8250 22.904 10250 21.283 300 30.661 2300 28.491 4300 26.476 6300 24 . 602 8300 22.862 10300 21.244 350 30.604 2350 28.439 4350 26.427 6350 24-557 S350 22.820 10350 21.205 400 30.548 2400 28.387 4400 26.379 6400 24.512 8400 22.778 10400 21.166 450 30.492 2450 ' 28.335 4450 26.330 6450 24.467 8450 22.736 10450 21.128 500 30.436 2500 28.283 4500 26.282 6500 24-423 8500 22.695 10500 21.089 550 30.381 2550 28.231 4550 26.234 6550 24-378 8550 22.653 10550 21.050 600 30.325 2600 28.180 4600 26.186 6600 24.333 8600 22.611 10600 21.012 650 30.269 2650 28.128 4650 26.138 6650 24.288 8650 22.570 10650 20.973 700 30.214 2700 28.076 4700 26 . 090 6700 24.244 8700 22.529 10700 20.935 750 30.159 2750 28.025 4750 26.042 6750 24.200 8750 22.487 10750 20.896 800 30.103 2800 27-973 4S00 25.994 6800 24.155 8800 22.446 10800 20.858 850 30 . 048 2850 27.922 4850 25-947 6850 24. Ill 8850 22.405 10850 20.820 900 29.993 2900 27.871 4900 25.899 6900 24.067 8900 22.364 10900 20.782 950 29.938 2950 27.820 4950 25.852 6950 24.023 8950 22.323 10950 20.744 1000 29.883 3000 27.769 5000 25 . 804 7000 23.979 9000 22.282 1 1000 20.706 1050 29.828 3050 27.718 5050 25-757 7050 23-935 9050 22.241 I1050 20.668 1 100 29-774 3100 27.667 5100 25.710 7100 23.891 9100 22 . 200 moo 20 . 630 II50 29.719 3150 27.616 5150 25.663 7150 23.847 9150 22.160 11150 20.592 1200 29.665 3200 27.566 5200 25.616 7200 23.803 9200 22.lig 1 1 200 20.554 1250 29.610 3250 27-515 5250 25.569 7250 23.760 9250 22.079 11250 20.517 1300 29.556 3300 27.465 5300 25.522 7300 23.716 9300 22.038 11300 20.479 1350 29.502 3350 27.415 5350 25.475 7350 23.673 9350 21. 998 11350 20.441 1400 29.448 3400 27.364 5400 25.428 7400 23.629 9400 21.957 1 1 400 20.404 1450 29-394 3450 27.314 5450 25.382 7450 23.586 9450 21.917 11450 20.367 1500 29.340 3500 27.264 5500 25-335 7500 23-543 9500 21.877 11500 20.329 1550 29.286 3550 27.214 5550 25.289 7550 23 . 500 9550 21.837 11550 20.292 1600 29.233 3600 27.164 5600 25.242 7600 23-457 9600 21.797 11600 20.255 1650 29.179 3650 27 115 5650 25.196 7650 23-414 9650 21.757 11650 20.218 1700 29.126 3700 27.065 5700 25.150 7700 23.371 9700 21.717 11700 20.l8l 1750 29.072 3750 27.015 5750 25 . 104 7750 23.328 9750 21.677 11750 20.144 1800 29-019 3800 26 . 966 5800 25.058 7S00 23.285 9800 21.638 1 1 800 20.107 1850 28.966 3850 26.916 5850 25.012 7850 23.242 9850 21.508 11850 20.070 1900 28.913 3900 26.867 5900 24.966 7900 23.200 9900 21.558 1 1900 20.033 1950 28.860 3950 26.818 5950 24.920 7950 23-157 9950 21.519 11950 19.996 2000 28.807 4000 26.769 6000 24-875 8000 23.115 1 0000 21.479 12000 19.959 For other temperatures, use formula D = h — IiVi. two stations, and t and t' their respective temperatures. t+i' h and h' being the tabular heights at any Missing Page PLATE >:V ^AL TINTS ■LAND BRUSHWOOD •OCKY SURFACES ■ Q ASSES i MISCELLANEOUS Wooden Budtciing i Brick do^ ^ Stone do- ^ Road \ / Bridoe .Wooden / r— do , Masonry —1 1 Hedge -m m -Bt^ «*•• ' Cayadry ArtUJjpry InfoJXtry ..■.«S& Orneral Slope Foi^ms. .■:::■- ^ t^( 0^>' If, !• Cn /iM ^^ PLATE XVI. EXAMPLES IN TH A P IPAIRTT' OF M01BIIT§(D)n TO^^y^rr.^Mii-F ■jroiYJE s '12 'ID o Mas l^^ , Sui-vr^veil l>V .John Saiitli , (' E. Scalp I 1(11)00 irU3G l-'.^iii ilist.uM cL'O fl r^^^,^ r r ^, c r J"" . Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page PLATE XXIII. EYE AND LANDSC /VI+Yi}layiof Pelo irau _.>f->: ^». -^ '^^ct^^- ■ v<' ^--•/>^ iV: .' r ^ ! :ape sketching Mvai-osaAi-flicc ,J Mile 'liBUL; .f^^-A^^^^^prAc. . r-- It'"' aT-^ .«^: ^ "f:" : '^; y u /I \ Julius BieriSC- Vh-ti. LiUi Missing Page