BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 4.2.3M-A2..0. J 't^/o'f. 35I3-J Cornell University Library TA 590.R32 3 1924 022 865 889 The date shows when^thls volume was taken. To renew this book copy the call No. and give to the librarian. . 80 S'09 HOME USE RULES All Books subject to Recall. ,^-'<^ Books not used for '^ ' "-..a instruction or research J»tJ 1^ '\9a'" are returnable within 4 weeks. Vol,umes of periodi- cals and of pamphlet?' are held in tbe library as much as possible. For special purposes they are given out for a limited time. Borrowers should 2 c Vilu. not. use their library ^ kWT privileges for the beije- fit of other persons. other persons. Books not needed during recess periods should be returned to the library, Or arrange- ments made for their return during borrow- er's absence, if wanted. Books needed by more than one person are held on the reserve list. Books of special value and gift books,- when the giver wishes it, are not allowed to circulate. Readers are asked to report all cases of books marked or muti- lated. Do itot deface books by marks and wrltlne. Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924022865889 Topographical "''"''';:,;'," Surveying and Sketching By THOMAS H. REES, Major, Corps of Engineers, United States Army, Formerly Assistant Professor Department of Engineering, U. S. Military Academy, West Point, N. Y.; Instructor Department of Military Engineering, Engineer School of Application, U. S. Army ; Senior Instructor, Department of Engineering, U.S. Army Service Schools, Fort Leavenworth, Kan. Adopted by Direction of the Oommandant pok Use IN THE IT. S. Abmy Service Schools, AT FoBT Leavenworth, Kansas. 1908 3> In V H h li ?i f ' Ot? COPYBIGHTED BY THOMAS H. BEES, May, 1908. KBTCHBSOJSr PBINTINS OOMPANT, LEAVENWORTH, KANSAS. INTRODUCTION. This text book was, for the most part, originally prepared in the form of lectures and pamphlets for the instruction of student ofBcers of the U. S. Army Service Schools at Fort Leavenworth, Kansas, and it has had the test of several years of use, criticism and correction by a most intelligent body of instructors and students. In "Surveying" no departure from the regular and recog^ nized methods has been attempted. Indeed, at this time, it would be difBcult to add anything new or original to the many excellent text books on this subject. Attention is invited, however, to the treatment herein presented of the adjustments of instruments, particularly with reference to the level tube; to the marking of the stadia rod which automatically applies the correction for the constant of the instrument; to the methods of applying latitudes and departures in reducing and plotting a survey and in computing areas ; and to the methods of resection with the plane table. In the subject of "Sketching," by which is meant the ap- plication of rapid and approximate methods of surveying with hand instruments, the methods adopted and developed in the Department of JEngineering of the U. S. Army Service Schools at Fort Leavenworth are described and explained. By these methods an individual road sketch covers twenty miles of road in a day, and shows topography by 20-foot contours and all es- sential details to a distance of 400 or 500 yards on each side of the trail. By combining the sketches made on a number of parallel and cross roads a fair topographical map covering an area at least 20 miles square (400 square miles) may be made in one day by a group of sketchers. A day's position sketch IV covers from one to two square miles with 10-foot contours, and shows all minor incidents and details of the ground. By com- bining the work of a number of sketchers, the extent of the strip of country that may be covered in a day is limited only by the number of sketchers available, and the resulting map will compare favorably, for all practical purposes, with one made by transit or plane table, and regulring weeks for its com- pletion. These rather surprising claims would be made with hesi- tation were it not for the fact that the above stated results have been attained repeatedly in the work at the Service Schools. The many complete books of tables of logarithms and cir- cular functions, such as Wentworth's, Ludlow's, etc., now avail- able at small cost, make it unnecessary to attach such tables to a text book, and therefore, only a table of reductions of inclined stadia readings to horizontal distance and difference of elevation is appended hereto. Fort Lbavbnworth, Kans., April 30, 1908. CONTENTS. PAQB Introduction 1 CHAPTER 1. Definitions and Principles 1 Kinds of Surveys 4 Systems of Coordinates 6 Map Making 9 Map Reading 17 Relation Between Vertical Interval and Scale 22 CHAPTER II. Scales 43 Tlie Selection of the Scale 45 Scales for Sketching 51 Construction of Scales 55 CHAPTER III. Instruments 69 Measurement of Direction 69 The Transit 71 The Telescope 72 The Stadia 78 The Vernier 89 CHAPTER IV. Adjustments of the Transit 99 The Level Tube 100 First Adjustment 101 Second Adjustment 102 Third Adjustment 105 Fourth Adjustment 107 Fifth Adjustment 112 Remarks on the Adjustments 113 Effects of Faulty Adjustment 115 CHAPTER V. Care and Use of Transit 121 Transit and Stadia Survey 126 The Record 133 To Plot the Traverse 139 Method by Latitude and Departures 143 Determination of Areas 148 vt CHAPTER VI. PAGB The Magnetic Needle 161 The Compass 164 Adjustments of Surveyor's Compass 171 To Set Off the Declination 176 The Prismatic Compass 177 Box Compass 178 Compass Survey 181 CHAPTER VII. To Determine True Azimuth 187 To Find the Declination of the Needle 191 To Determine Latitude 194 To Determine Correct Local Time 196 CHAPTER VIII. The Sextant : 199 Adjustments of the Sextant 202 Use of the Sextant 210 The Clinometer 211 CHAPTER IX. The Plane Table 219 Adjustments 223 Plane Table Survey 226 Traversing With Plane Table 229 CHAPTER X. Measurements of Distance 234 The Wye Level 238 Adjustments 239 Level Bods 244 CHAPTER XI. Use of Level Rod and Level 249 Differential Leveling 261 Profile Leveling 264 Cross-section Leveling 273 Contour Surveying 275 CHAPTER XII. The Chain 291 Chaining With an Incorrect Chain 297 Lining In 298 CHAPTER XIII. The Steel Tape 313 To Measure a Base Line 316 Program of Operations 317 Measuring Distance by Time 329 (JUAi-lMfiK AJ.V. PAGE Trlangulatlon 335 Vertical Angulation 342 Trlangulatlon Survey 345 CHAPTER XV. Sketching 349 The Aneroid Barometer 358 Scales ol Map Distances 358 The Sketching Ruler 359 Methods of Sketching 361 The Notebook Method 361 Sketching Case Method 368 Field Drawing Board Method 371 Sketching With Improvised Instruments 374 Position and Outpost Sketches 378 Remarks on Sketching 381 TABLE. Reduction of Inclined Stadia Readings 401 CHAPTER I. DEFINITIONS AND PRINCIPLES. Topography is the determination by observation, measurement, and record on the ground ; and the representation by projection and convention on a plane surface, of the forms, features, structures, acci- dents of surface, and incidents of rock, earth, water and vegetation, of a considered portion of the earth's surface. Military Topography is the dgtermination and rep- resentation of such forms, features, structures, acci- dents of surface and incidents of land and water as have especial significance in connection with military operations. The term "topography" is also applied in a gen- eral sense to a description of the physical features, collectively, characterizing any region, but in these pages the word will be used in the sense first stated. The eye of an observer can take in only a limited portion of the region surrounding him. Much is hidden by intervening objects, and of that which he Sees he can form only a vague and inaccurate esti- mate as to relative positions, distances and elevations of the points, objects and features observed. When he leaves the ground he has only his memory of the genera] appearance of the ground to rely on, and any report which he might make of his observations would necessarily reflect the vagueness and inaccu- racies of his impressions, magnified by the usual limitations of memory. If, however, he has been able on the ground to measure the distances between the most important objects and features, and the directions of tjje lines joining them, and their relative heights, and has made careful note and record of these measurements, he will take with him information concerning the region covered that is definite and accurate as far as it goes, and that will serve as the basis of a report or description founded on ascertained and recorded facts and not on impressions and memories. But the same information may be expressed more clearly, accurately and comprehensively, in the form of a drawing than in that of a verbal description, no matter how minute and exhaustive the latter may be. The observer would, therefore, make on paper, to scale, a horizontal projection of the points located by his measurements and would represent incidents of sur- face, structures and principal features, by the adopted conventions drawn in their proper relative positions with reference to the located and projected or plotted points. It is evident that in a horizontal projection on a plane surface, elevations cannot be represented graph- ically. They can be shown, however, by writing at a point, or on a horizontal line, or within a horizontal surface, that is represented on the drawing, the num- ber which expresses in feet the height of the point, line, or surface, respectively, above an assumed hori- zontal surface called the datum surface. Such a drawing is called a map, and a complete topographical map will show at a glance the relative positions of all points within its limits, the distance and direction of any two points, one from the other, the actual height above datum of any point, and the relative neignis oi any two or more points, tne siope of the ground at any point, the form and shape of the surface of the ground, the character of the sur- face and of the vegetation covering it, all occurring incidents of water, and all structures erected by man. Like information could not be conveyed in many pages of verbal description. Neither could it be acquired by the mosi careful study of the ground itself. It is a mistake to assume that maps are useful only when the ground itself cannot be seen. No view of the ground from its highest point, no examination of the ground made by riding over all its parts, can give the definite, pre- cise and comprehensive knowledge that may be acquired by the study of a good map. The most complete knowledge is obtained by examination of the ground in connection with study of the map. The first gives general ideas of its appearance, its prominent features, its topographical character, and the nature of soil and vegetation. The second gives precise information of relative positions, distances, directions, elevations, slopes, drainage and topograph- ical features. When both ground and map cannot be studied the map is to be preferred. The man who has made a survey of a piece of ground, including both field work and plotting, will have more accurate and complete knowledge of that ground than others, however carefully they may have studied both map and ground. Practice in surveying and sketching is also the best school for training the eye and mind to estimate distances, slopes, elevations, relative positions, topo- graphical forms, etc., and to appreciate the military functions and limitations of the terrain. In all military operations knowledge of the ground is essential ; therefore all military men should have a thorough knowledge of maps and of map making, that is, of topography. A map is a horizontal projection to scale on a plane surface of a considered portion of the earth's surface. The art of making the observations, measure- ments, records, and computations that determine di- rections, distances, elevations, sizes, areas, »volumes, and movements on the earth's surface, and of ex- pressing the results in the form of a map or drawing is called surveying. KINDS OF SURVEYS. Surveys are made by several different means and methods, and .for many different purposes. The re- sulting kinds of survey receive names characteristic of the means or methods employed or the purpose in view. ' With reference to the means employed, the survey may be a compass and chain, a plane table, a transit, a stadia, a level, a sextant, or a barometric survey, each made with the instrument named. With reference to the purpose or object of the survey, there are Land, City, County, Mining, Geo- detic, Geographic, Boundary, Hydrographic, Topo- graphic, Geological, Engineering, and Coast Surveys, with others of rarer application. These pages will concern themselves principally with Topographic Surveying, which is the basis of all others. Whatever the kind of survey, whether with refer- ence to means employed or object in view, the meas- urements to be made are those necessary to determine the position of an unknown point with reference to a point whose position is known or is assumed to be known. SYSTEMS OF COORDINATES. In order to determine the position of a point with reference to a known point, a system of coSrdinates is necessary. There are three systems of coordinates in general use, all of which are applicable to and used in surveying. 1st. A system of rectangular coordinates, in which distances are measured in three directions that are at right angles to each other. Thus, (Fig. i) from a known point measure north 220 feet, west 300 feet, vertically + 50 feet, and the point at the extremity of the vertical is fully determined with reference to the known point. ' 2d. A system of polar coordinates, in which are measured a horizontal angle, a vertical angle and a distance (radius vector) in the direction determined by the angles. Thus, (Fig. 2 ) at a known point meas- ure from the north toward the right a horizontal angle of 35°; from the horizontal direction thus de- termined measure a vertical angle + 6°, and in the direction so fixed measure a distance of 300 feet to the required point, which thus becomes fully deter- mined with reference to the known point. 3d. A system of triangular coordinates, in which the position of a point is determined with reference to two known points. TJie lines joining the three points (two known and one required) will form a tri- angle in which three parts may be measured and the other three determined by solving the triangle, thus fixing the position of the unknown point with refer- ence to the two that are known. This system is ap- plied as follows (Fig. 3): From a known point A as an origin, determine the direction, the horizontal distance and the difference of elevation to a second point B, which thus becomes known The horizontal line between these two known points is called the base line, and it is the base of a triangle of which a third point C, whose position is required, is the vertex. If the two angles at the base or the two sides be measured, the triangle may be solved and the location of the vertex or unknown point be determined in its horizontal relations to the two known points. The elevation of the unknown point with refer- ence to either of the known points may be deter- mined in like manner. The solution of the horizon- tal triangle has given the horizontal distance of the required point C from the known point A. This is the base of a right-angled triangle, in which the hypothenuse is the inclined line A C joining the points, and the side is the difference of elevation. If , the angle at the base be measured, the triangle may be solved and the difference of elevation be deter- mined. Let A, B and C be points on the surface of the* earth. Assume A to be the origin or first known point, B to be the second known point, determined by measuring its direction from A, its horizontal dis- tance A B', and the difference of elevation B'B. C is a point to be determined by triangular coordinates 1 from A and B. The base line is the horizontal dis- tance ( A B' or A'B) between A and B. Measure the horizontal angles at A and B, and knowing the base and adjacent angles solve the triangle and thus de- termine the position of C or C" which are in the ver- tical line through C. Now, knowing A C, which is the base of the right triangle A C C, measure the ver- tical angle at A, (C A C) and solve the triangle to de- termine the side C C or difference of elevation. Or knowing B C", measure the vertical angle at B, (C B C") and solve for C" C, the difference of elevation of B and C. In either case the point C becomes fully determined wilh reference to A or B. Any two of these three systems may be combined in the same determination by using one system in the horizontal plane, and another in the vertical plane. In the horizontal plane triangular or polar coordinates are generally used, although the rectangular coordi- nates find frequent application in running short off- sets from the main line for the purpose of filling in small details. In the vertical plane, rectangular or triangular coordinates are generally used to deter- mine the difference of elevation. If polar coordinates are used in the vertical plane they are reduced to rec- tangular coordinates before plotting the work. A fourth system, which may be called the tAree point system of coordinates, is sometimes used in the horizontal plane. In this system (Fig. 4) threfe points, A, B and C, must be known. At the unknown or re- quired point D, measure the two angles subtended by the lines A B and B C respectively ; that is, the angles A D B and B D C. The angle A D B determines that the point D is on the circumference of a circle of which A B is a chord subtending an angle at the center equal to twice the angle A D B ; and the angle B D C determines that the point D is also on the circumference of a circle of which B C is a chord sub- tending an angle at the center equal to twice the an- gle B D C. These two intersecting circles being drawn, one point of intersection is found at B and the other will determine the position of D, the required point. Graphical Method. — -Having plotted the known points, A B and C, on paper, use a separate sheet of tracing paper and on it draw through any point three lines making with each other the observed angles A D B and B D C. Place the tracing paper on the plot and move it till its three lines pass through the cor- responding points, A, B and C, of the plot. The apex of the angles will then indicate the position of the point D, which may be pricked through and thus plotted. If the observed angles be set off on a three- arm protractor and the edges of the arms be made to pass through the three plotted points, the center of the protractor will be at the point D. Analytical solution : Let O = angle A D B. D' = angle B D C. B = the sum of the two angles at B. A and C = the angles at A and C and let side A B ^ d, side BC = d'. In the quadrilateral the angles at B and D are known, hence A + C = 36o°— (B + D4-D')^P (i) ' C = P— A -(2) In the two triangles the side B D is common and d sin A d' sin C BD = = (3) sm D sin D' From (2). Sin C = sin P cos A — cos P sin A Substituting in (3). d sin A d' sin P cos A d' cos P sin A sin D sin D' sin D' d' sin P :os A = =1 d + d' cos P ) sin A sin D' sinD sin D' \ whence, cot A = cotF 1 d sin d' sin D +1 cosP ] 1 1 (4) Having found A and C, either of the two triangles may be com- pletely solved and the position of D determined. This system of horizontal coordinates finds its most frequent application in hydrographic surveying, for locating soundings by reading two sextant angles in the boat. In this case the surface of the water is the datum plane and the vertical coordinate is given by the lead line. It is sometimes useful in topographic, surveying, especially with the plane table, as will be explained later. It will be seen that whatever may be the system of coordinates used, at least three measurements must be made in order to fully determine a point and that these measurements involve direction and distance. In many surveys, such as land, county, boundary surveys, etc., elevations are disregarded, and only the horizontal coordinates considered. These give suffi- cient data to project or plot the required points, but the elevations remain unknown and the survey is not a topographical survey. MAP MAKING. From a known point as an origin, any number of unknown points within convenient and reasonable distance may be determined. Any one of the de- 10 termined points may be taken as a new origin of coordinates from which other points may be deter- mined, and so on, until the necessary determinations extend over the area considered. In topographical surveying, the ultimate object, namely, the production of a map drawn to scale and showing in plan or horizontal projection all of the es- sential lines and forms that appear on the ground, must be kept constantly in view. All the lines that actually appear on the ground, such as the lines defining roads, trails, railroads, terraces, embankments, cuts, gullies, bluffs, streams, coasts, fences, buildings, etc.i may be readily determined by measurements of directions and distances, and as readily plotted on paper by laying off these directions and horizontal distances to suitable scale and drawing the lines so determined. The determination and representation of the ir- regular conformation of the surface of the ground in the shape of hills, ridges, knolls, spurs, valleys, ravines, rolling ground, plains, etc., are apparently not so easy, since there are no definite points or lines on the ground by means of which the shape or confor- mation of the surface may be represented. It is necessary, therefore, to assume a system of imaginary lines on the surface of the ground which will, when reproduced on the map be a geometrical representa- tion of the surface. Two different systems of lines have been used for this purpose, viz: I St. The lines of steepest slope. On the map these lines are called hachures. 2d. The lines cut from the surface by horizontal planes at equal vertical intervals. These lines are called contours. 11 The first system is now little used except some times for the purpose of indicating roughly the ex istence of hill forms in any locality without attempt ing to represent their actual shape. Formerly this method was much used and careful rules were fol lowed by which different degrees of slope were indi cated by different widths of hachure lines. This pro duced the effect of hill shading and caused the hil masses to appear to stand out in relief on the map (Fig. 4a.) By the use of contours the shape of the surface o the ground may be geometrically represented wit! any desired degree of accuracy, the method bein^ that of "One Plane DeFcriptive Geometry," in whicl points and lines are projected on a horizontal plane and elevations are shown by writing at the projected points and on the projected horizontal lines, numben expressing in feet (or yards or meters) their height; above an assumed horizontal plane called the datun plane. Numbers thus used are called references, and are usually written in parenthesis to distinguish there from othf r numbers which may be used on the draw ing to express horizontal distances. Any point that has been determined by measure ments on the ground is represented on the map b) making a dot at its plotted position, as at a (Fig. 5) and writing near the' dot its elevation in parenthesis (326.4). Two other points, similarly determined are shown plotted at b (331.7) and at c (324.1). A horizontal line is fully represented by drawing its horizontal projection, as a d, and writing on the line its elevation (326.4). 12 An inclined straight line is fully represented by- drawing its horizontal projection and marking the elevations of two of its points. Thus the line b c through the points b (33 1.7) and c (324.1) represents a fixed inclined line. To find the elevation of any other point on this line, as the point d, proceed as follows: The rise along the line from c (324.1) to b (331.7) is 7 6 feet and the horizontal distance from c to b scales 196 feet. Therefore at d, which scales 62 feet from c, the rise is ^Vt ^ 7.6 = 2.4 feet, and the elevation of d is 324.1 +2 4 = (326.5). To find a point on the line be having any assumed elevation, say (326.5), proceed as follows: The re- quired point is 2.4 feet above c. The rise from c to b is 7.6 feet in the horizontal distance 196 feet. The 2.4 point 2.4 feet above c is therefore — - X 196 = 62 feet 7.6 from c. Scale off 62 feet from c and mark the point d thus determined. It is the point on the line b c whose reference is (326.5). All similar problems may be solved in the same manner by simple proportion, but the graphical solu- tion is much easier and quicker, especially when a number of points on the same line are to be deter- mined. In Fig. 5 the ruled rectangle represents a piece of paper ruled with two sets of equally spaced lines drawn parallel to the sides of the rectangle. A piece of cross-section paper or of profile paper, machine- ruled by the makers, is suitable. One set of divisions is taken as a vertical scale and is so numbered at the sides of the rectangle as shown, using only the range 13 of numbers that includes the references of the plotted points. Lay the edge of the card through the plotted points b and c ; from b follow by eye, using the ruled lines as a guide, a perpendicular to the edge up to the point that reads on the scale (331.7), the reference of b, and dot the point as at b'. From c follow in like manner a perpendicular to the edge up to the point c' where the reading of the vertical scale is (324.1) the referente of c. Lay a straight edge (ruler or folded edge of paper) through the points b' and c'. To find the reference of any point, as d on the line be, follow the perpendicular through d up to the straight edge at d' and read the vertical scale. It is .found to be (326.4), which is the reference of the point d. To find a point on the line b c whose reference is (326.4), find on the straight edge the point d' whose reading on the vertical scale is (326.4) and drop the perpendicular d'd to the edge of the ruled paper. The point d is the required point. From the points of intersection of the straight edge with the ruled horizontals, drop perpendiculars to the edge of the paper and dot the corresponding points on the line b c. They will be points on b c whose references are (321), (322), (323), etc., to (334) as shown. When once in position the ruled paper and the straight edge must not be moved until all required points have been properly projected and marked. The projecting lines need not be drawn on the ruled paper. The ruled lines and the spaces between them can be followed by eye with sufficient accuracy, and the same piece of ruled paper can be used for any number of pairs of points. 14 The dots on the line a (326.4) b (331.7) are pro- jected in the same manner as those on b c and are points on the line ab having references (321), (322), (323), etc. The edge of the ruled paper is, of course, placed on the line a b in determining these points. This is a quick method of "interpolating contour points." Three points, or a line and a point, or two inter- secting lines, or two parallel lines, determine a plane. If the ground surface containing the three points plotted at a, b and c, were a plane surface, that plane is fully determined and represented by the three points a, b and c, or by the line b c and the point a, or by the two intersecting lines b c and b a, or by the two parallel lines b c and a a'. A plane surface is best represented by drawing its horizontals at equal vertical intervals and marking them with their references. To thus represent the plane determined by the" three points a, b and c, and show horizontals at one foot vertical interval, find as already explained points on b c whose references are (321), (322), (323), etc., and on b a points having the same references. Draw straight lines through the pairs of points that have the same references and mark them (321), (322), (323), etc., as shown. These lines will be horizontals or contours of the plane. For any plane the contours will be straight parallel lines equally spaced. The ground services are not generally plane sur- faces, but rather rounded or curved surfaces with varying slopes and curvatures. If numerous points be determined on the ground and if these .points be so selected that they fall along the lines where the slopes and curvature change most abruptly the sur- 15 faces between those lines may be considered as plane surfaces and so represented on the map. Since three points determine a plane, it is best to select the points on the ground in such wise that lines joining them shall divide the surface into triangles in each of which the surface is approximately a plane surface. This is illustrated in Fig. 6. The determined and plotted points are marked with their references. Lines joining these points divide the surface into triangles. On these lines contour points at lo feet vertical intervals are inter- polated by the method just explained. Contour points having the same reference are joined by straight lines crossing the several triangles. In practice, the sides of the triangles are not ac- tually drawn, except by the rows of dots made in interpolating contour points, and each contour is drawn as a continuous curved line passing through contour points of the same reference. This is shown in Fig. 7 which is constructed from the same plotted points as those used in Fig. 6, and which represents practically the same surface. The curved contours furnish a graphical and natural representation of the curved surface of the ground. It will now be seen that in surveying the proper selection of the points to be determined on the ground is an important consideration, and that a careful study and analysis of the shape of the ground is the first requirement. The controlling agents in the formation of the earth's features are rainfall and drainage, and these elements constitute the best basis for a study of the forms which they produce. Where- ever it falls, the water flows along the lines of steepest slope and increases in volume as it flows. From the 16 highest lines of ridges and spurs it diverges and flows away on either side to lower levels. Along the lowest lines in the valleys, between ridges and spurs the water accumulates in streams and cuts channels in the soil, making, gullies, arroyos, ravines, canons or wide river bottoms between bluffs, depending on the volume of water, on the changes in volume and on the character of the soil. Erosion of the surface, caving banks, land slides on undermined bluffs, etc., gradually work the changes that result in the irregu- lar rounded forms called ridges, knolls, shoulders, hills, spurs, cols ,or saddles, ravines, valleys, etc. The highest line that can be traced 6n the groxmd between two adjacent valleys, or the line that parts the water flowing into one valley from that flowing into the adjacent valley, is called a watershed. The lowest valley line along which the water from both sides accumulates and flows in a stream, is called a water course. On any surface the direction of steepest slope at any point is the direction of a line tangent to the surface at that point and perpendicular to a hori- zontal line of the surface through that point. If the directions of steepest slope be followed from point to point in a surface, the line so traced is a line of steepest slope of the surface. At every point it is normal to the horizontal of the surface that passes through the point, and it is the line along which water will flow under the action of gravity. On the surface of the ground the lines of steepest slope, or water-flow lines, are normal to the contours, because contours are horizontals of the surface. Watershed lines and water course lines are normal to the contours and are, therefore, lines of steepest 17 slope, aad all other lines of steepest slope join them tangentially. The forms that have been described as resulting from the erosive action of water are not always ap- parent because many soils and rock resist erosion, and because other agencies are or have been at work. Frosts; winds, waves, glaciers, earthquakes and vol- canoes have done their share, and may have pro- duced the most varied and fantastic forms. As a rule, however, the drainage system of a country is the best guide in studying its topography. The water course lines are usually plainly marked by the channel or gully washed out by the running water. Watershed lines are not so apparent,' but may be traced by following the highest line of the ridge or spur. On the slopes between watersheds and water courses the lines that separate the steeper from the gentler slopes can usually be traced. MAP READING. As careful study of the ground is required in order that its surface may be properly determined and represented on the map, so also is careful study of a map necessary to an understanding of the infor- mation that it conveys. Note first the scale of the' map and keep in mind a map distance that represents a convenient unit of measure, such as lOo feet, looo yards, or i mile. Keep in mind also the vertical interval between con- tours. Note the drainage system as indicated by the water course lines and by the directions of the lines of steepest slope (normal to the contours). Trace 18 the watersheds and note that they always terminate at the junction of two water courses. A contour is a continuous line, and must close on itself. If it does not close within the limits of the map, it must cross the map continuously. Contours will unite and form one line only where they represent a vertical surface, like the face of a vertical cliff. Contours can cross only where they lie in an undercut or overhanging surface. Eroded banks too steep to be represented by con- tours are usually indicated by hachures. A single contour cannot lie between two contours both having a greater or lesser reference than its own. Closely spaced contours indicate steep slopes, and widely spaced contours indicate gentle slopes. A wide space that contains no contours is probably a horizontal plane, or at least a surface within which the differences of elevation are less than the vertical interval between contours. If the spacing of contours increases from the top toward the foot of a slope, the slope is concave ; if the spacing increases from the bottom toward the top, the slope is convex. A change from wide to close spacing and again to wider spacing indicates a point of inflection or of reversal of curvature at the point of closest spacing. A change from close to wide and again to closer- spacing indicates a point of inflection or of reversal of curvature at the point where the spacing is widest. A contour which forms a closed ring or oblong between two continuous contours may indicate either a kno'U or a depression. If its reference is the same 19 as that of the higher of the adjacent contours, it rep- resents a knoll, and if the same as that of the lower, it represents a depression. Depressions usually con- tain water, and wherever water is shown a depression is of course evident. A stream or water course indicates a valley, and valley contours are always convex or salient toward the head of the stream. Ridge or spur contours are convex in the direction of the downward slope. The elevations of all points that lie on contours are given by the references of the contours. The elevation of any point not on a contour may be found by interpolating between the references of adjacent contours. The slope of the surface at any point may be found in terms of the tangent of the angle of slope by dividing the vertical interval between contours by the scaled distance between contours at the point considered. Knowing the tangent, the angle in de- grees may be taken from a table of tangents. The point A (Fig. 8) is one-third the distance from the 300 to the 320 contour, and its elevation is therefore (307). The -sjertical interval is 20 feet and the scaled dis- tahce between contours at A is 540 feet, the tangent of the slope is therefore -g^j"^- =.03 7 = tangent of 2° 7'. In any other direction as long as the line c d through the point A the tangent of the slope is the vertical interval 20 feet divided by the scaled length of the intercept c d of the line between adjacent contours, and is y|^^.oi82 which is the tangent of 1° 3'. To trace on the surface through a given point a line of given slope, reduce the tangent of the slope to a fraction having the vertical interval as its numer- 20 ator. The denominator of that fraction taken from 'the scale of feet will be the intercept which the re- quired line taust have between adjacent contours. Thus, if the given slope be i° 3', its tangent is .0183 = yfg-j- Mark off 1 100 feet from the scale on the edge of a strip of paper, and lay .the edge through the given point A so that one mark shall fall on the 300 contour and the other on the 320 contour, as at c d, or c' d'. The line may be continued with the same slope up to e and down to b by the same method. Is the point C ( 260) visible from the point B (460) over the intervening knoll at D (320)? The point C is 200 feet below B, and the distance is 7400 feet, giv- ing yVw^^V ^s the tangent of the angle of depres- sion from B to C. The intervening point D is 140 feet below B and the distance is 3700 feet, giving tSinr = — as the tangent of the angle of depression from B to D. The point D therefore lies below the line from B to C, and C is visible. Graphical Solution. — Lay a strip of ruled paper (cross-section or profile paper) along the line C B. Assume a convenient vertical scale and mark it on the divisions at the end of the strip, beginning with 260, the reference of the lowest contour. A profile is a line cut from the surface of the ground by a vertical plane. To construct a profile on the line C B, at each contour point, trace by eye a perpendicular on the ruled paper and mark its intersection with the cor- responding horizontal as shown by the vertical scale. A line C D' B' drawn through the points so deter- mined is the profile on the line C B. Draw the 21 straight line B' C. It passes above the knoll at D'and C is therefore visible. To find where a given line pierces the ground, construct a profile on the projection of the given line and on the profile draw the given line. From the point of intersection of profile and given line drop a perpendicular on the projection of the line. This will determine the point where the given line pierces the ground. Thus, a line of sight from B (460) is tangent to the knoll at D. Where does it pierce the ground? Construct the profile B' D' C and draw a line from B' tangent to the knoll at D'. It intersects the profile at E' from which drop a perpendicular to E, which is the point where the given line pierces the ground. The ground from D to E is invisible from B. To determine areas which are invisible from an assumed point on a map, draw lines radiating from the assumed point and, with a strip of profile paper, construct on each line a profile of the ground. From the profile position of the assumed point, draw on each profile lines tangent to the salient parts of the profile, and prolong each tangent to the point where it pierces the profile. Drop perpendiculars from the profile points of tangency and from points where the tangents pierce the profile, to determine the cor- responding points on the map. Draiw a line on the map through the successive points of tangency. It will be a horizon line for a point of view at the assumed point. Draw a line on the map through the points where the tangent lines pierce the ground; the area between this line and the horizon line will be invisible from the assumed point. 22 If the tangent lines do not pierce the ground be- yond the points of tangency, the horizon line deter- mined by them is the most distant horizon, and all of the ground beyond it is invisible from the assumed point. In drawing profiles for the solutions of problems like the foregoing, only such parts of the profile as are necessary in the solution need be drawn. An in- spection of the ridge and valley forms indicated by the contours will show which parts of the profile need be drawn. In drawing profiles the vertical scale is exagger- ated, and the slopes indicated are much steeper than the natural slopes. RELATION BETWEEN VERTICAL INTERVAL AND SCALE. Since the vertical interval (which will be called V. I.) and the scale remain constant for any map, it is evident that the only variable for different slopes on a map is the map distance between contours (called 'M. D.) and that a scale can be constructed showing the M. D.'s that correspond to different de- grees of slope. Let AB (Fig. 9) represent a profile cut from the surface of the ground by a vertical plane, which also cuts A C and D B from adjacent contour planes, B C being the V. I. Then will A C be the horizontal ground distance (G. D.) between contours, to be rep- resented on the map by the corresponding map dis- tance, M. D. Let A be the angle of slope or the angle BAG. Then will 2a A C = B C cotan A or G. D. = V. I. cotan A and if the scale of the map be — n I I M. D. = — X G. D.= — X V. I. X cot A, n n n being the denominator of the R. F. Find the val- ues of M. D. for successive values of A in degrees and make them the ordinates of a curve with the corre- sponding values of A for abscissae. Thus, let n = 2000 and V. I.= 10 ft., then For A = 1°, cot A = 57.3 and M. D. =■ IOX57-3 = 0.286 ft. = 3.44 in. 2000 For A = A = A = A = A = cot A = 28.6, cot A = 19.1, M. D.= M. D.= 5°, cot A= 11.4, M. D.:: f, cot A = 8.^14, M. D.= cot A= 5.67, M. D.= 1.72 1. 15 0.69 0.48 0.34 A = 15°, cot A = 3.73, M. D.= 0.22 A = 20°, cot A = 2.75, M. D.= 0.17 10 A = 30°, cot A = A = 40°, cot A = A = 60°, cot A = A = 90°, cot A = 1.73, M. D.= o.io 1. 19, M. D.= o.o7 0.58, M. D.= 0.035 0., M. D.= o.o in. in. in. in. in. in. in. in. in. in. in. Lay off degrees along an axis as shown in Fig. 10, and at the degree marks erect ordinates equal to the corresponding M. D.'s. Join the points thus deter- 24 mined by a fair curve and it will be the curve of M. D.'s for a scale of ywwit ^^^ V. I. = lo feet. For small angles the cotangent varies inversely as the angle nearly, hence, without appreciable error cot 1° cot A = A I and the formula M. D.= — X V. I. cot A becomes n V.I. cot I" M. D. = n A in which A is the only variable. Remembering that the cot 1° -- 57.3, the scale of map distances may be computed without reference to a table of cotangents. Find the M. D. for 1° then For 2" divide by 2 For 3° divide by 3 etc. etc. The scale or curve thus determined is also plotted in Fig ID. The divergence of the two curves is not apparent up to 20", and at 30° the divergence is still small. The method just given may therefore' be used for determining M. D. up to 20°. In Fig. 8 the scale is ^ ligo ' ^^^ the V. I. is 20 feet. Substituting in the formula V. I.cot 1° M. D. = n A and leaking A ^ i" there results 20 X 57-3 X 12 M. D. = = 0.65 in. 21120 25 as the map distance between contours for a slope of i^ Then for i°, M. D. = \^- = .65 in. for 2"", M. D. =-8/ = .33 m. for 3^ M. D. = -6^s_ = 22 in. for 4°, M. D. = Y- = -i6 in. for 5°, M. D. = Y = -i3 in- fer 6°, M. D. =\s. = .ii in. for 7°, M. D. =4jS- = .09 in. for 8°, M. D. = Y = -08 in. for 9°, M. D. = ^ = .07 in. for 10°, M. D. = fl = .065 in. for 11", M. D. =11 = .06 in. for 12°, M. D. = II = .054 in. etc. etc. Space the degree marks on the axis 0-12° (Fig. 8), and at the degree marks erect the corresponding ordi- nates equal to the M. D.'s just determined. Draw the curve as shown. Intermediate M. D.'s may be found by interpolating at the proper point of the curve. Thus for 4°-3o' take the ordinate midway between 4° and 5°. V.I. cot 1° In the formula M. D. ^= it is evident that n A V. I. and n may be multiplied or divided by the same number without changing the values of M. D. There- fore, a standard system of scales and corresponding V. I.'s may be adopted, having the ^ame scale of M. D.'s or slopes for all maps that conform to the standard. 26 For military maps the system adopted is based upon military requirements, as explained in the chap- ter on scales, and is one in which V.I. I 1056 or is that in which the scale expressed in "inches to I mile" multiplied by the "V. I." is equal to 60, as shown in the following table : Scale. V. I. Feet. REMARKS. Inches to I mi. R. F. = — n % 1 1^ VlTT 120 I -i-%\l.TS 60 Route map. ^y.' ijiw 40 2 TT^TT 30 3 ITT2 20 Road map. 4 ITJIS 15 5 T!!^T¥ 12 6 TT7T50 10 Position map. 1% T?¥3- 8 8 7^'iir 1% 10 ^■jVs 6 , 12 tjVu 5 Fortification map. - 15 1 4 20 fm 3 30 .tV^ 2 60 TTIT? I The scales and corresponding vertical intervals found most useful for military maps are those marked 27 "route map," "road map," "position map," and "fortification map," respectively. The map shown in Fig. 8 conforms to this system as a position map and the scale of slopes there shown is the standard scale for this system. The eye soon learns to estimate the map distances that correspo'nd to different degrees of slope, and by practice great facility is acquired, not only in reading the slopes shown by contours on a map, but also in making a sketch map of the ground where slopes are often estimated and contours spaced by eye. If no system be adopted this facility can never be acquired. 29 31 Tig. 4- iSca/e <^'=//7f,: To^o y/'^/O 33 cb-f96'h oriz on fa/, 7. 6 r/se. cc/ ^ 6Z' /7or/zo/7fa/, Z'.i r/'se. Horizonfa/ Sca/e //n.^/OOft. r.-^. 6. 35 (9<>aj, (Hi) ^ca/e e'^fn?/. Tohs VJ.^10'. S7 ^co/c S'^lm. io^ I//../0' ng. 9. 9p: Cb AtAf> 0/67y}A/C£ 4 /AC 43 CHAPTER II. SCALES. The .scale of a drawing is the ratio obtained by di- viding the distance between two points on the draw- ing by the distance between the corresponding points of the object represented. In topography the scale of a map is the ratio ob- tained by dividing the distance between two points on the map by the horizontal distance between the corresponding points on the ground. This ratio or scale is in practice expressed in three ways. I St. In the form of a fraction which expresses the above described scale ratio. This fraction is always reduced to one in which the numerator is unity and in this form it is called the representative fraction or R. F. Thus R. F. :f^VT5" ^eans that any map distance is i^oa of the corresponding horizontal ground dis- tance, and vice versa, that the distance between two points on the ground is 4800 times the distance be- tween the corresponding points on the map. There- fore if the numerator be taken to represent a map distance, as i inch or i foot, or i centimeter, the de- nominator will represent the corresponding ground distance in the same unit of measure. If the R. F. is ^-5-5-, then I inch on the .map represents 4800 inches on the ground ; i centimeter on the map represents 4800 centimeters on the ground ; i foot on the map represents 4800 feet on the ground, etc. The same 44 scale ratio may be expressed in many different forms, thus I I in. 3 in. i centimeter 4800, 400 ft., 1200 ft., 48 meters are equivalent expressions, and the transformations suggested are useful in solving problems relating to scales. The R, F. is the universal means of comparing scales, no matter in what ceuntry and with what sys- tem of measurement the map was constructed. It is the basis of the construction of all working scales and of the solution of all problems relating to scales. 2d. The scale may be expressed in the form of a statement, in' words and figures, of the relation be- tween map and ground distances. For example, 3 inches = i mile, i inch = 200 feet. 3d. The scale may be expressed graphically by drawing a line on the map and marking its divisions not with their actual lengths, but with the ground distance that they represent. In other words, instead of writing the statement, " 3 inches = i mile," draw a line 3 inches long and mark it i mile. If a photograph could be made from a great height in a camera pointed vertically downward, the result would be a map of the region included in the picture. If it included a straight road with mile posts marked o, I mi., 2 mi., 3 mi., etc., these mile posts would ap- pear in the picture with the same marks, i mi., 2 mi., etc., constituting a scale of miles for the map. If the dimensions of the picture were Try-g-jj-jj- of the dimen- sions of the area covered, the scale of the map would be R. F. = Tj-g-iTir °r ^ ino^a. = 1 mile, and the actual distance between mile posts in the map or picture 45 would be I inch. This is exactly what every graphi- cal scale on a map should be, viz : a picture or repro- duction on the map of distances on the ground. Such a scale furnishes a convenient method of ascertaining at once without calculation the value of any ground dimension, and is generally used in connection with either or both of the other methods. It has the ad- ditional advantage of remaining true under all con- ditions of enlargement or reduction by photography, or shrinking or expansion of the paper due to atmos- pheric influences. Suppose we reduce a map by photography so as to halve the linear dimensions. If the original map bears the legend i inch = 200 feet and this is not erased or covered, it is no longer true. A building 200 feet long would be represented on the original map by i inch. On the reduced map the length of its representation would be but y^ inch. So, the scale would be ^ inch = 200 feet. But the inch of the original graphical scale marked 200 feet would itself be reduced to ^ inch and the scale would remain true. Like the legend, the R. F. would no longer be correct, the numerator being halved while the de- nominator remains the same. THE SELECTION OF THE SCALE. When a drawing is to be made of any object to serve a technical purpose, it must be drawn to scale. Suppose you want to have a desk or table con- structed according to your own idea. The only way in which you could make the carpenter or cabinet maker understand what you want would be by a drawing. It would be inconvenient and unnecessary 46 to make a full sized drawing to natural scale, and you would take any sheet of paper that you might have at hand, say a double sheet of letter paper, which un- folded would be about i ixi6 inches. If you are go- ing to have your desk made about four feet long, you would draw a line say twelve inches long to repre- sent one edge of the desk, and in doing so you have adopted a certain scale, viz : 12 inches ^ 4 feet: or, 12 inches =48 inches: or, I inch =4 inches: or, 3 inches = i foot: or, ifoot = 4 feet: or, R. F. = i< : or, I centimeter ^ 4 centimeters: or, .01 meter = .04 meters. All of which are equivalent expressions and mean exactly the same thing. The expression ordinarily used would be R. F. y,, or, scale, 3 inches = i foot, or the graphical expression, a line six inches long rep- resenting two feet. Let it be required to draw the plan of a building eighty feet long by fifty feet wide, and to show door- ways, window openings, thickness of walls and par- titions, and similar details; The first question that arises is, "What scale shall be adopted?" The gov- erning conditions are that the smallest details required must be shown to scale, and that the sheet or draw- ing must not be so large as to be unwieldy and incon- venient. The smallest detail to be represented is the thickness of partitions, say nine inches. The largest 47 is the length of the building-, eighty feet. For the supposed purpose of the drawing it will suffice to be able to scale off dimensions to the nearest ^ inch, and if rh inch be assumed as the smallest portion of a line clearly appreciable to the eye, we should use a scale not smaller than tU inch = }( inch, or i inch = 25 inches; or R. F. jV. Now what will be the di- mensions of the drawing if this scale is adopted? The length of the drawing will be nV of the length of the building. A X 80 feet. = 38! inches, and the width iV X 50 feet = 24 inches. These are less than the length and width of one of the ordinary sizes of sheets of drawing paper, which are 27 X 40 inches, and the scale adopted would be suitable. If the smallest details required were the widths of doors and windows to the nearest inch, we would take y-J-j- inch = I inch, or i inch = 100 inches, or a scale of R. F. y-J-jj^, or I inch ^ 8^ feet. For convenience take I inch = 8 feet, and the drawing would be only 10 inches long and 6^ inches wide. Now pass to the consideration of a larger subject, and assume that a representation of the Post of Fort Leavenworth is required, including only the portion occupied by buildings. Here again the first con- sideration is the scale to be adopted, as determined by the purpose or object which the map is to subserve. On the one hand the map might be intended to show to the nearest foot the dimensions of all buildings and the distance between them, the widths of roads and sidewalks, and the location of water and gas pipes, sewers, hydrants, fences, small streams, bridges, cul- verts, etc. Oij the other hand, it might only be re- quired to show the general shape of buildings and their relative location, the position and average width 48 of roads and walks, the car lines, if any, and perhaps the water mains and principal sewers, with an allow- able limit of error in plotting or scaling of about four feet. A map must frequently be used and referred to in the field where there is no table to spread it upon, and no dividers available for taking off distances accurately. For this reason tV inch is considered the smallest portion of a line that can readily be esti- mated or scaled on maps in the field . Then in the first case cited above, we should take not less than -sn inch to represent i foot or 12 inches, and we have a scale of tVIu. = 12 in. or, I in. = 600 in. = 50 ft. or, R. F. = ^h. The principal post buildings are included within a space say 3000 ft. long by 2400 ft. wide, and the map dimensions would be ishv of these respectively, or five feet long by four feet wide. This is not an excessive size, and the scale adopted would be suit- able for the purpose. If necessary, however, the drawing could be made on two sheets of a more con- venient size. For the second case cited we would take ^t inch to represent four feet, or i in. ^ 200 ft. = 2400 in., or R. F. = TiW. The size of the map would be i?W X 3000 ft. = 15 in. long and nW X 2400 feet. = 12 in. wide. In the same manner we would determine the scale suitable for a map of the Reservation of Fort Leav- enworth. This reservation is, we will say, four and one-quarter miles long and three miles wide in the extreme dimensions. The smallest details required 49 would be the buildings, and these not in accurate di- mensions, but simply in their relative positions and approximately true in size and general form. A possible erroy in plotting and scaling of ten feet would be allowable, and we will therefore test a scale of tV inch = lo ft., or i in. = 500 ft. = 6000 in., hence R. F. = ■snwa. . The longest dimension on the ground is 4}( miles X 5280 = 22440 ft., and the length of the map will be T^-us X 22440 ft. = 3.74 ft. =44.88 in.,or say 45 in., width 3 1 .68 in. Drawing paper comes in rolls and in sheets. The rolls are from 30 in. to 62 in. wide, and 10 to 40 yds. long. We can, therefore, cut from the rolls any size of sheet that is desired. The sheets vary in size from "Cap" 13 in. x 17 in., to Antiquarian, 31 in. x 53 in. A convenient size is the Double Elephant, 27 in. x 40 in. If this last is the size of sheet available for the map now under consideration, the scale tested is_ found to be too large, and the question arises, what reduction of scale is necessary. The drawing must be reduced from 45 in. to 36 in. in length to provide for a border 2 in. wide at the ends. The ratio of re- duction is, therefore, II = t, and the new scale will be TTTSTT X 4 ='7tW Tsm X 22440 ft. = 3 ft. = 36 lu. loug. rsni! X 15840 ft. = 2.112 ft. = 251^ in. wide, which leaves nearly one inch border at top and bottom. We have now seen that the first consideration in determining the scale for a drawing is the accuracy with which the smaller details are to be delineated, and that the second consideration is the size of the resulting map or drawing. There is a third consideration not yet mentioned, viz : the degree of precision of the instruments or the means used to measure the distances and direc- 50 tions on the ground. If a triangulation survey is to be made with an accuracy of -j-j-BnT in'distances and to the nearest 25" of arc in direction, the probable error of measurement for the average length of lines (say three miles) would be about 30 inches. The scale then should provide for the plotting and scaU ing of distances to the nearest 30 inches. In the extreme care and precision that should be used in plotting the triangulation stations of an accurate survey, ^^ of an inch on the paper would not be overlooked, and the resulting scale is w in. = 30 in. or, I in. ^6000 in. ^ 500 ft. R. F. = TnrVo ' and 10.56 in. = I mi. Now consider, on the other hand, a foot recon- naissance made by. pacing distance and using the hand compass for directions. The average length of courses may be assumed as 2000 feet. An accuraey greater than Th can not be expected, and the probable error in the average course would be jhn X 2000 or 20. feet. It would be absurd to use a scale by which five feet could be clearly represented, when the actual error in the measured distance may be 20 feet. Neither do we wish to increase the chances of error by using too small a scale. To keep on the safe side let us take ^V of an inch to represent 1 8 feet ; then I inch = 900 feet = 10800 inches, R. F. = rSTTTT 5.89 inches =^ a mile, or, for convenience, say 6 inches = i mile, with R. F. = xsho and I inch = 880 feet. 51 This is the largest scale that one would be justified in using under the circumstances, and is suitable for the sketch of a limited area in which all the details and accidents of the surface are to be shown, as in the case of a military position to be occupied for de- fensive purposes, or of a site for the encampment of an army. SCALES FOR SKETCHING. The first subject that presents itself in connection with military field sketching is the question of the scale, and this must be considered with reference to the purpose of the sketch and the amount of detail that the sketch must show in order to fulfill its pur- pose. The two general classes of sketches are Road Sketches and Position Sketches. The first are made with a view to organizing and ordering the marches of any army, and to this end the sketch should show distances by the scale to the nearest minute of marching — say 80 yards, in each hour's march ; it should show halting places, passing places and camping sites ; in towns and villages it should show the principal intersecting streets, per- haps only 100 yards apart; to properly develop the slopes and forms of the ground the sketch should show contours at 20 ft. vertical interval without crowding them or confusing other details ; in prac- ticable country the slopes will not often exceed six degrees, which will place the contour lines about 60 yards apart. On the map the contours should not lie closer than A of an inch apart. As this is the smallest detail to be considered, it should be used as 52 a test for determining the scale and their results TTj in. = 60 yds., or i" = 21600", which is practically 3" = I mi. This scale will show all' other necessary details with the requisite accuracy, and it will place a day's mounted reconnaissance of say fifteen miles on a paper 45 inches long, a convenient length for the sketching case, or a day's foot reconnaissance of ten miles on two lengths of the 15-inch drawing board. This scale, 3"^ i mile, R. F. ^xhis is adopted for all road sketches. For position sketches the primary considerations are the posting of troops in position down to regi- ments or even battalions, the scaling of ranges to the nearest 15 yards, and the definition of slopes and forms of the surface of the ground by -contours at not more than 10 feet vertical interval. For scaling ranges, the scale should be not smaller than iVin. = 15 yards., or tAs-u, or about 6" = i mi. For posting troops, the general should be able to indicate on the map, to his colonels, by a line about one inch long, the front to be occupied by each of his regiments, say 300 yards, which requires a scale of Tuhn or about 6" =: i mi. For contours of 10 ft. V. I. and slopes up to 6° the smallest horizontal distance between contours will be about 30 yds. To keep the contours for these slopes at least tV in. apart on the map requires iV in. = 30 yds. or Tshu or about 6" = 1 mi. For all the purposes of a position sketch, therefore, a scale of 6 in. to the mile will suffice, and this scale is adopted. An outpost sketch should fulfill practically the 53 same conditions as a position sketch, and the same .scale should be used. The purposes of the road sketch may be extended to furnish, by the combination of sketches on several roads, a general map of the country to be used for or- ganizing and ordering the march of an army on sev- eral parallel roads, for locating the cavalry screen, for arranging the details of concentrations and deploy- ments, pursuits and retreats. For the.se and other purposes in grand tactics the scale of the road sketch is too large, giving maps, of unwieldy size and show- ing much unnecessary detail. The combined road sketches will therefore be reduced to give maps of convenient size showing the immediate zone of oper- ations say 30 to 40 miles on the sides ; and for this a scale of I in. to the mile or ^jh-s, with contours at 60 ft. vertical interval is suitable. For strategical maps showing the entire theater of operations, the published civil maps on scales of 10 to 25 miles to the inch will serve, or the tactical maps may be still further combined and reduced. For the defensive organization of a place, which in the eventualities of war becomes important, and which must hastily be fortified by strong field works to re- sist a threatened attack, maps must be prepared in great detail, to show all the minor folds and accidents of the ground, all obstacles and screens, dead spaces, lines of trenches, covered communications, the field works themselves in approximate true dimensions, and accurate ranges to well defined land marks within the zone of defense. In this connection questions of defilade will arise, which gives importance to differ- ences of elevation equal to the height of a man. 54 The above details can be shown on a scale of 12 in. = I mi., or R. F. ^iVir with contour intervals of 5 feet. A lunette 150 yards long will be represented by one inch on the map, and a map error of ?V in. in scaling ranges would be only 7}4 yards. If time were available an instrumental survey would be made, but generally the methods of- field sketching would have to be resorted to for filling in, using especial care to secure accuracy. The resulting series of scales for military field maps is as follows : No. R. F. [ncHes to One Mile. Contour Interval Feet. I TJSir 12 5 2 ■nsiis 6 10 3 JTTIXf 3 20 4 TiTTffff I 60 5] smsi! tV Tsshnm i'T Purposes. Defensive organization of an im- portant place. Position or battle map. Road maps for marches and minor tactics. Route maps, maneuvers in grand tactics. Strategical maps. It must not be understood that the foregoing con- clusions are absolute and definite. On the contrary, for any one of the purposes considered a considerable range in the selection -of a scale is permissible, and the above determinations are based as much on the past experience of many of&cers, as to suitability and convenience of scales, as they are on the discussions given. But it .is of the utmost importance in military work that a uniform system of scales be adopted and 55 adhered to by all persons engaged in field sketching and in preparing and combining the resulting maps. For the purposes stated the foregoing system of scales has been adopted for military maps. For special purposes and under different conditions, the scale should be selected in accordance with the prin- ciples that have been discussed. CONSTRUCTION OF SCALES. In order that a graphical scale may be read quickly and conveniently it must show exact whole numbers of units, tens, hundreds or thousands of the unit of measure, and the primary and secondary divisions of the scale should also read in exact units or multiples of ten of the unit of measure. Graphical scales are of two kinds, reading scales and working scales. A reading scale is a scale drawn on a finished map or drawing and used to read dis- tances thereon in the desired unit of measure, such as feet, yards, meters, miles, kilometers, etc. A working scale is a scale used in m9,king or plotting the map or drawing, and it gives distances in the unit of measure that was used in determining dis- tances on the ground. The units of measure that may be so used are feet, yards, meters, chains, paces; steps or strides of a horse, walking, trotting or can- tering; minutes (meaning the distance passed over each minute) for a horse walking, trotting or can- tering; revolutions of a wagon wheel (meaning the distance passed over by the wagon for each revolu- tion of the measured wheel); the length of a piece of cord or rope that may be used to measure distances, etc. In fact, any fixed length may be taken as a unit 56 of measure for determining distances on the ground, provided that length be known in inches. It would be possible to reduce each measured distance, no matter what the unit used, to feet or yards, and then lay off the distance by a scale of feet or yards, but this- requires a computation for each measurement, and involves time and labor. By using a properly constructed working scale reading in the units of measure used, all measured distances may be plotted directly without reduction. In particular cases the reading scale and the work- ing scale will be the same, as for instance when, ground measurements are made in yards and the map scale is to be read in yards. Generally, however, the working scale will have a unit of measure different from that desired for the reading scale. The leading scale is placed on the map wherever there is space available for it. Usually it appears just below the title, and questions of symmetry deter- mine its length at about 3 or 4 inches on small maps up to about 6 or 8 inches on large maps. Its exact length will of course be determined by the total number of units of measure to be represented. For very accurate work the working scale should be constructed on the paper in order that expansion and contraction will affect paper and scale equally. But for ordinary work it is more convenient to con- struct the working scale on the edge of a ruler and apply it directly to the line to be plotted. Its length should be such that the longest measured distance may be plotted by one application of the scale. When the scale is given as so many inches to the mile, or as one inch equals so many tens, or hundreds of feet, yards, etc., the construction of the scale of 57 miles, feet or yards consists simply in laying off inches on a straight line and marking the points of division with the proper number of miles, feet, yards, etc., as the case may be. In other cases a simple compulation and construction are necessary. The problem may be stated as follows: Given the R. F. and the number of inches in the unit of measure, to construct the scale. When the R. F. and the length in inches of the unit of measure are not stated they must be deduced from given data. Rule : — Transform the R. F. to a fractional form in which the denominator shall be the desired whole number of units, tens, hundreds or thousands of the given unit of measure. The new numerator will be the length of the scale in inches. Lay off this length on a straight line and divide it by construction into a convenient number of equal parts, each represent- ing an entire number of units, tens, or hundreds, etc., of the unit of measure. Mark the points of di- vision with the number of units of measure repre- sented, starting from the assumed zero point. Examples : I. A man takes 238 paces in a distance of 595 feet. Construct a working scale of paces with a scale of 3 inches to i mile. 3 inches 3 inches i = = = R. F. I mile 63360 inches 21 120 595 ft. = 2^ ft. = 30 inches = i pace. 238 paces I in. I in. ,2.84 inches 2 1 120 in. 704 paces 2000 paces 58 Solution: — First, find as shown above the R. F. ^= ITT^TJ Second, find the number of inches in the unit of measure; in this case, i pace ^= 30 inches. Third, the R. F. shows that i inch represents 21,120 inches. Reducing the denominator to paces there results the equivalent ratio, i inch to 704 paces. Assume that the space available or suitable for the scale is about 3 inches long, and it is evident that 2000 paces can be represented within the available space. Using 2000 paces as the denominator of the transformed ratio, the numerator is found to be 2.84 inches. Draw a line 2.84 inches long to represent 2000 paces (see Fig. No. i.) and divide it into four primary- divisions each representing 500 paces. Subdivide the left hand division into 10 equal parts, each rep- resenting 50 paces and mark the divisions as shown. If the space available for the scale had been about 4 inches long the scale could be drawn to represent ^800 paces, thus : 1 I in. 3.98 in. 21 120 704 paces 2800 paces that is, a line 3.98 in. long would represent 2800 paces. If divided into seven parts, each part would represent 400 paces. The left hand division would be subdivided into eight parts, each representing 50 paces. 2. With the same R. F. (irW) construct a work- ing scale to read minutes for a horse that walks a mile in 1 5 minutes. 59 The R. F. being given, first find the number of inches in the unit of measure (i minute). I mile 63360 in. = = 4224 in.= I min. 15 min. 15 min. Next, transform the R. F. thus : I I in. 3 in. 21 120 5 mm. 15 mm. Assuming as before that the space suitable for the scale is about 3 inches long, it is found that 15 min- utes will be represented by 3 inches, and the scale is so constructed (see Fig. No. 2 ). 3. _With the same R. F. (tttitt) construct a reading scale of yards, i yard = 30 inches. I I in. 2.73 in. 2 1 120 586.6 yds. 1600 yds. Finding that i inch represents 586.6 yds., it is evi- dent that within the 3 inch space available, 1600 yds. may readily be represented, and using this as the new denominator, there results 2.73 inches as the new numerator or length of scale. (See Fig. No. 3.) The necessary computation may be slightly sim- plified by the method of construction illustrated in Fig. No. 5, thus: I, I in. 3 in. 21 120 586.66 yds. 1760 yds. Lay off a horizontal line 3 inches long, and on the inclined line, with a convenient scale of equal parts, (in this case the scale of 50ths on the triangular scale) 60 mark the points 500, 1000, 1500 and 1760, or mark each 100 if desired. Join the 1760 point with the end of the 3 inch line and draw parallel lines as shown; erase that part of the scale to the right of the 1500, or extend the scale to 2000 yards if desired. This method is applicable in all cases, but it requires a set of conveniently divided scales of equal parts like those on the triangular scale, and this is not always at hand. The first method may be used with any scale of inches, because the inch may "be readily divided into tenths and distances laid off by estimation to the nearest looth of an inch, which is sufficiently accurate for ordinary purposes. If it is desired to scale off with accuracy distances smaller than those shown by the secondary divisions of a scale, a diagonal scale is used. The construction and use of such a scale are illustrated in Fig. No. 4. The scale on the bottom line is made in the usual manner with secondary divisions reading to 10 feet. By using the diagonal lines the scale may be read accurately to i foot, and by estimation between the horizontals to the nearest tV of a foot. The hori- zontal lines must be equally spaced and the diagonals must be so inclined as to cover one secondary division exactly. In the example given in Fig. No. 4, to scale off any distance, as 273 feet, place the dividers on the horizontal line marked 3 and include the distance A B which is made up as follows : Two primary divisions ==200 ft. Seven secondary divisions == 70 ft. tV of a secondary division = 3 ft. Total, - - 273 ft. 61 The diagonal scale is especially useful for read- ing different units on the same scale ; thus, if the primary divisions read yards, the secondary divisions may be made to read feet, and the diagonal scale with 12 spaces will read inches; or, if the primary divi- sions reads meters, the secondary divisions may read decimeters, and the diagonal scale, centimeters; then by estimation between the horizontals the scale could be read to the nearest millimeter.. 4. Construct a scale to read paces of 32 inches, R. F. Ww tit I I in. 6.22 in. R. F.= 3600 1 12.5 paces 700 paces 5. Construct a scale to plot distances from time, the measurements being made by timing a horse walking a mile in 12 minutes, with R. F.^-^h^. The scale will read minutes, the word in this case signify- ing the distance covered in a minute. 12 minutes ^ I mile = 63360 inches. I minute = 5280 inches. I I in 2.91 in. R.F.= = = 20000 3.78 min. II piin. 6. With R. F. tbW construct a scale to plot meas- urements taken in meters, the meter being 39.37 inches. I I in. 4.92 in. R. F. = = 8000 203.2 met. 1000 met. 62 7- Construct a scale of yards considering that 3 inches ^ i mile. 3 in- 3 in- i = ^ = R. F.- I mi. 63360 in. 21 120 Then with R. F. ^jhn construct scale to read yards. 8. A French map has a graphical scale only. We find that a space marked 100 meters measures 2.23 inches. Construct a scale to read yards. A meter is 39-37 inches. 2.23 in. 2.23 in. i = = R. F. 100 met. 3937 in 1765.46 Then with this R. F. construct scale of yards. 9. A map has no scale. A certain distance which measures on the map 2.5 inches, measures on the ground 523 paces of 30 inches each. Construct scale of feet. 2.5 in. 2.5 in. i = = R. F. 523 paces i5690in. 6276 Then with this R. F. construct scale of feet. . 10. A map has been roughly made in the field as follows: A picket rope of unknown length was used to take the measurements, and a scale was made by assuming a certain line on the paper, dividing it into equal parts and marking each division one rope. After obtaining facilities for measurement it was found that a division on the paper measured .28 inches and that the rope was 84 feet long. Construct scale of feet. .28 in .28 in i = === R. F. 84 ft. 1008 in. 3600 63 With this R. F. construct scale of feet. II. A map has been plotted using a scale of paces, R. F. ijUts. This scale was constructed under the supposition that a pace measured 32 inches, but it was afterward found that the pace was only 30 inches. Find the true R. F. of the map. R. F.= tiUj7. That is, one pace on the map rep- resents 12,000 paces on the ground. A pace on the map has been plotted as equal to 32 inches, while the pace on the ground is really but 30 inches. Hence we have I X 32 I 12000X30 11255 which is the true R. F. of the map as constructed. In place of the rule previously given, the follow- ing formula may sometimes be found useful in the construction of scales. Let X = required length of scale line in inches I - = representative fraction. n m == length of unit of measure in inches. s = total number of units to be represented by the scale. Then m s n Assume for s a convenient round number in tens, hundreds or thousands, etc., and find the value of x to two decimal places. 64 PROBLEMS. 1. Construct a scale of R. F. zs^w to read feet. 2. Construct a scale of R. F. ^sW to read yards. 3. Construct a scale of 15 inches = i mile to read yards. 4. Construct a scale of 6 inches = 80 chains to read meters. One chain ^ 792 inches: i meter = 39.37 inches. 5. Construct a diagonal scale of R. F. rir to read yards, feet and inches. 6. A horse takes 600 steps in walking 500 yards. Construct a scale of i inch = 600 feet to read steps. 7. A horse trots a mile in 6^ minutes. Con- struct a scale of R. F. isTm to read minutes. 8. A man on a bicycle finds that his cyclometer registers 3^ miles when he has been going at uni- form rate for 18 minutes. Construct a scale of R. F. rzhs to read minutes. 9. Construct a scale of R. F jttjti to read revolu- tions of a 20-inch wheel. 10. A map has no scale. It is found that a dis- tance which measures 4 inches on a map measures 1760 yards on the ground. Construct a scale of yards. 11. A French map has only a graphical scale and we have no metric scale. A length of scale marked 200 meters measures .75 inches. Construct a scale of feet. Meter = 39.37 inches. 12. We wish to send a map made in this country to a place where the metric system is used. It has on it only the graphical working scale, and that shows that a line representing 2000 paces of 30 inches equals 2.84 inches. Construct a scale of meters. 65 13- A road sketch was made with a horse that was supposed to trot a rnile in 7J^ minutes, and was plotted to a scale of R. F. stt^tt. It was afterwards found that the horse trotted a mile in 7 minutes. What is the true R. F. of the sketch? Construct scale of yards. 67 //o./ tVo^MoJe/ll^ Of 30^c» P/fce'i- zSoo'ya'e'S- A/oS,. foMtriureS. 1^-f No 5 L I , I , I ■ I ■■!■ ■R.F. Zl,/,6-o/v/n. •icfit-e TO ne/ip to / -^""^ ^<;o Fe-eT. "io 69 CHAPTER III. INSTRUMENTS. Surveying involves measurements of direction and distance, and the instruments used are specially designed and constructed with these objects in view. MEASUREMENT OF DIRECTION. The direction of one point from another is the direction of the straight line joining the two points. That direction must be determined with reference to the direction of some line that is known or that is assumed as the standard of direction. If the direc- tion of one line be known, and the difference of di- rection of this line and another be determined the direction of the other becomes known. The difference of direction of two lines is an angle, and the directions of lines are therefore determined by measuring angles. The unit of angular measure- ment is the degree or jU of a circle. To measure smaller parts of an arc the degree is divided into 60 minutes and the minute into 60 seconds. To determine the direction of a line, that line must be physically established and must be apparent to the senses. We can imagine a straight line join- ing any two points, but we cannot measure an imagi- nary line. Nature's nearest approximation to a phys- ical straight line is a ray of light, and this taken in its opposite sense as a line of sight is almost always 70 used to establish a straight line between two points. It is not perfectly straight, because a ray of light, in passing through a medium like air of varying density, is by refraction deviated from its straight course. This deviation, for distances involved in ordinary surveying, is slight and may generally be neglected. Moreover, it occurs only in the vertical plane and therefore affects only vertical angles. In horizontal angles no appreciable error arises from refraction. The only other straight line that nature provides is the plumb line, and this determines a vertical line only. To determine a horizontal line, the spirit level is used. A horizontal line or a vertical line is taken as the standard direction from which to measure vertical angles. A horizontal line directed toward the north is usually taken as the standard direction from which to measure horizontal angles, although any known direction line may be so used. All instruments that deteirmine direction are so constructed as to measure the angle between the line of sight to the required point and some other known or assumed direction line. The essential parts or elements of an instrument for measuring angles are : 1. Pointers or sights or telescope, establishing a well defined line of the instrument that may be brought, by revolution about a fixed axis, into coinci- dence with any desired line of sight or direction line. 2. A circular arc graduated or divided into de- grees, with its center in the axis of revolution and its plane perpendicular thereto. 71 3- Means of placing the plane of the circular arc in the plane of the angle to be measured or its axis perpendicular to that plane. 4. An index by means of which the pointing of the sights or telescope may be read in degrees on the circular arc. 5. A case or frame or standard to connect and support the several parts of the instrument while permitting the necessary relative movements of the parts. Every instrument used to measure angles will be found to have all of the above mentioned parts or elements. THE TRANSIT. The transit is an instrument designed to measure the angular coordinates, both horizontal and vertical, in any system of coordinates. With the stadia at- tachment it also measures the distance coordinate or radius vector. This instrument appears in different forms, de- pending on the different special purposes for which designed, as the surveyor's transit, engineer's tran- sit, mountain transit, miner's transit, etc., but they are all essentially the same in principle and operation, and differ only in size or in some special attachment. Since the transit is designed to measure both horizontal and vertical angles it must have the parts necessary for both purposes, viz : I. A telescope in which the line of collimation is the particular line of the instrument that is brought into coincidence with the line of sight through the known point or origin and the distant required point. 72 One point of this line of coUimation must remain fixed at the origin while the line itself must revolve about it. 2. A horizontal axis through the fixed point per- pendicular to the line of collimation, for turning off vertical angles. 3. A vertical axis through the same fixed point for turning off horizontal angles. 4. A graduated circular arc centered oh the hori- zontal axis and perpendicular thereto for measuring vertical angles. 5. A graduated circular arc centered on the vertical axis and perpendicular thereto for measuring horizontal angles 6. Indices, which in the transit take the form of verniers for reading the angles on the vertical and horizontal circles. 7. Adjusting screws, by means of which the vertical axis may be made truly vertical; the horizon- tal axis, truly horizontal ; and the line of collimation, perpendicular to the horizontal axis. 8. Level tubes, by means of which the verticality of the vertical axis may be assured. 9. A support for the instrument in the form of a tripod or a trivet. THE TELESCOPE. The parts of the telescope are the object glass, the eye piece, the cross wires, and the tube. The ob- ject glass is a compond achromatic lens. The rays of light, emanating from any point in front of the lens and passing through it, are brought to a focus behind the lens, and thus an image if the point is 73 formed. The ray which passes through the center of the lens is not deviated, therefore the image of the point will lie on the straight line through the point and the center of the lens. Every point of an object in front of the lens will have its corresponding image behind the lens, and thus an image of the object is formed which may be viewed by means of the eye- piece. Each point of the object, the corresponding image of the point, and the center of the lens, lie in the same straight line, therefore the image of the ob- ject is reversed, or appears as if revolved i8o degrees about the axis of the lens. The image formed by the object glass can be seen by the naked eye, but it is very small, and a distant object cannot be distinguished because its image is so minute. The eyepiece is a microscope which mag- nifies the image and causes it to appear larger and therefore nearer than the object itself as seen with the naked eye. It enables the observer to distinguish a signal, staff, or target, or other object, which to the unaided eye would be invisible. An erecting eye- piece is a combination of lenses, usually four in num- ber, which reverses the image and caus'es it to appear erect to the observer. An inverting eyepiece is a combination of lenses, usually two in number, which does not reverse the image, and the latter is therefore seen inverted. The particular line in the telescope which is to be brought into coincidence with the line of sight to the required point 'is defined by the cross wires. These are two fibers of cobweb or of the finest plat- inum wire fixed at right angles to each other across a ring and placed in the telescope within focusing range of the eyepiece and object glass, both of which are 74 mounted in tubular slides operated by rack and pinion movements. The eyepiece is first focused to give a clear and distinct view of the cross wires. Then the object glass slide is moved until the image of the ob- ject observed falls in the plane of the cross wires, when it will also be in the focus of the eyepiece, and will be distinctly seen. If, when the eye is moved slightly from side to side, the wires appear to move on the image, the focusing of the object glass is not correct, and it must be readjusted until no such move- ment is apparent. Now the telescope may be so moved and pointed that the intersection of the cross wires will fall ex- actly on the image of the point observed, and when this is done it is known that the intersection of the cross wires, the center of the object glass, and the point observed, are in the same straight line, viz., in the line of sight to, or ray of light from the point ob- served. In other words the line of coUimation of the telescope has been made to coincide with the line of sight to the point observed. The line of collimation of the telescope is the straight line through the intersection of the cross wires and the center of the object glass. It coincides with the undeviated ray that falls on the intersection of the cross wires. When the intersection of the cross wires is made to coincide with the image of a point, the telescope is said to be pointed at or directed toward that point. The tube of the telescope holds the parts in their proper relative positions, protects them from injury, and shuts out unnecessary light. The Horizontal Axis. — -The telescope is mounted on trunnions the axis of which intersects the line of col- 75 limation perpendicularly, and which rest horizontally in trunnion beds on standards high enough to permit complete revolution of the telescope in the vertical plane. The standards are fastened to a horizontal plate. The Vertical Axis. — The circular plate that carries the telescope has a central stem or spindle whose axis is vertical and, if prolonged, passes through the in- tersection of the horizontal axis and the line of col- limation. The spindle fits and turns in a socket in an outer hollow spindle and thus permits complete revo- lution of the telescope, in a horizontal plane. The Circular Arcs. — A graduated circle or semi- circle is attached centrally and perpendicularly to one of the trunnions of the telescope, and turns with the telescope in its vertical revolution Its index and vernier are fastened to the standard and remain fixed. By means of this circular arc and its vernier, vertical angles may be read. The outer hollow spindle is expanded at the top into a saucer-like plate, on the upper rim of which is the horizontal circular arc with its center -in the ver- tical axis. This lower plate marked with the gradu- ated circle remains fixed, while the upper plate carry- ing the telescope revolves on the vertical axis and thus turns off horizontal angles that may be meas- ured on the arc. The index and the vernier are fastened to the upper movable plate. Most transits have two verniers placed diametrically opposite to each other. By taking the mean of the readings of both verniers, any error due to eccentricity of the circle is eliminated. The outer spindle, which carries the lower plate, fits vertically in a socket, in which it may be turned 76 to set or "orient" the lower plate and circle in any desired position. When set or oriented it must be fastened or clamped in the socket to prevent further motion. This socket supports and centers all the working parts of the instrument and must itself be firmly sup- ported, with its axis vei-tical. The supports provided are leveling screws arranged as follows: The socket is fastened in a horizontal frame or flange which is pierced by three, or four vertical screws having milled beads. These screws rest on a fixed base plate and form adjustable supports for the instrument, by means of which the axis of the instrument may be made vertical. The verticality of this axis is de- termined by two spirit levels fastened horizontally and at right angles to each other above the plates. One level would be sufficient for this purpose, but two are generally used in order that the instrument may be leveled without having to turn it on its ver- tical axis. The leveling screws support but do not fasten the socket of the instrument on its base plate. For this latter purpose there is provided a ball and socket joint when four leveling screws are used, or a spiral spring when three leveling screws are used. The base plate is often made double, so that one part will slide upon the other and permit slight lateral motion of the instrument for exact adjustment over the given point. This arrangement is called a shift- ing center. The base plate is screwed on to the top of the tripod, which is a stand with three legs hinged at the top and arranged to support the instrument at a con- venient height. 77 When it is desired to set up the instrument on a wall or stump or on a high framed station, use is made of a trivet, which is a brass crowfoot with three points of support. Clamps and Tangent Screws. — The motions of revo- lution of the instrument are controlled by clamps and tangent screws. The clamp is a ring or a segment of a ring which turns freely on one of the two parts to be connected, but which can be fastened to that part by a clamp screw having a milled head. The ring is extended into an arm that is permanently fastened to the other part. In this manner a ring on the trunnion of the telescope is fastened by its arm to the standards, a ring on the spindle of the lower plate is fastened by its arm to the upper plate, and a ring on the same spindle of the lower plate is fastened to the socket. Any motion of the instrument may therefore be clamped or released at will by tightening or loosen- ing the corresponding screw. In each of the arms that join the parts, there is inserted a tangent screw connection which gives a very slow motion to the part affected, and permits accurate pointing of the telescope at the point ob- served or accurate setting of the verniers at any de- sired reading. The Plumb Line. — A known point on or near the ground is usually marked by a stake with a tack driven in its top. To read horizontal angles at this point the transit must be set up with its center exactly over the point. A plumb line is hung from the center of the spindle and the instrument is so placed that the point of the plumb bob is at the center of the tack or mark. The vertical axis of the instrument will then pass through the given point. 78 ATTACHMENTS OF THE TRANSIT. The following named parts are not essential parts of the transit proper, but are attachments that serve some useful or convenient purpose : The Telescope Level is a level tube placed beneath the telescope and attached to it. The axis of the level tube should be parallel to the line of collimation of the telescope and should be in the same^ vertical plane. With this attachment the transit may be used as a surveyor's level. THE STADIA. The stadia wires are two additional horizontal wires that are placed one above and the other below, at equal distances from the horizontal cross wire of the telescope. They are used in connection with the sta- dia rod to measure the distance from the center of the instrument to any unknown point at which the stadia rod may be placed. The principle of the stadia in its simplest form is as follows : With the eye at the peep-hole E (Plate i, Fig. i ), sighting past the fixed wires W and W', there will be intercepted on a rod at A, the space S S', and on a . rod at B, the space T T'. Since the triangles ESS' and E T T' are similar, the distances E A and E B are proportional to the intercepts S S' and T T'. If the in- tercept S S' is one foot at a distance E A of lOO ft., then any other intercept, as 3.53 ft. on the rod would indicate a distance of 353 ft. from E to the rod. When the stadia wires are placed in the telescope 79 the principle of similar triangles still applies, but the intercepts on the rod are proportional to distances measured from a point in front of the lens and not from the eye. This may be explained as follows : A law of lenses is that the reciprocal of the prin- cipal focal length is equal to the sum of the recipro- cals of the- conjugate focal lengths, or I 1 I -=-+ - (0 f fl f^ in which f is the principal focal length and f ^ and f ^ are the conjugate focal lengths, that is, f^ is the dis- tance from lens to focus for rays of light coming from a point at a distance fg from the lens, and f is the distance from lens to focus for parallel rays or rays from a point at an infinite distance. Let I (Plate i. Fig. 2) be the center of the tran- sit, A' and B' be the positions of the stadia wires, L be the position of the lens, and A and B be the points whose images fall at A' and B' respectively, and let s = A B = intercept on rod, i = A' B' = image of intercept, f = F" T ) ^1 p y > conjugate focal lengths, c = I L = distance from center of instru- ment to center of lens, f = L D = principal focal length, d =1 F = required distance from center of instrument to rod. From similar triangles, I s fi : fa :: i : S ,.-.— = — (2) 80 I I I From equation (i), — = (3) I Eliminating — and solving for i^, f f , = f + — s (4) i which shows'that the distan'ce f 3 from the lens to the rod is equal to the constant distance f, plus s multi- f plied by the constant ratio — . Also, that if the dis- i tance f be laid off from L to D, the remaining dis- tance D F from D to the rod is proportional to the intercept s. This is illustrated in Plate la, Fig. 2, where three different positions of the rod A B are shown with the corresponding positions A' B' of the stadia wires when focused on the rod. The triangles be- hind the lens, having equal bases A' B' and different altitudes are dissimilar. Therefore the correspond- ing triangles in front of the lens having the rod inter- cepts A B for their bases and their apices at L are also dissimilar, and distances from the lens to the rod are not proportional to the rod intercepts. But straight lines drawn through the extremities of the rod intercepts will meet at the point D, and distances from this point are proportional to the corresponding rod intercepts. The point D is at a distance from the center of the lens equal to f, the principal focal length of the lens. 81 In most instruments, it is the lens that is moved in focusing the telescope for different distances, but the figure is simplified by supposing that the stadia wires are moved, as is the case in some telescopes. The motion is so slight that it may be disregarded except in explaining the theory. To get the total distance d from the center of the instrument I to the rod, the small distance c from I to L (Plate I, Fig. 2) must be added, and there results — f d = c + f + — s (s) i f is found by measuring the distance from the lens to the cross wires when focused on a star or on a distant horizon, c is found by measuring the dis- tance from the horizontal axis to the lens when focused on an object whose distance is a mean of the expected readings. The slight variation in c due to the movement of the lens in focusing for different distances is neglected, f and i cannot be measured f with sufficient accuracy to determine the ratio — . i Equation (4) may be written in the form f f, -f (6) 1 s f and the ratio — may be found by measuring s for i some particular measured value of fg. On level ground, measure from the plumb bob of the transit the distance (c + f ) and from the apex D, so determined, lay off in the same direction a base 82 (f g — f) of say 400 feet. At this distance a rod will be held vertically and the observer will signal the rod man to set a target or a pencil point, first at the point cut by the lower wire and then at the point cut by the upper wire. The intercept s between these two points is carefully measured in feet; then the base f2-f (fj — f) divided by the intercept (s) gives — which f s is equal to — . If with a base D L of 400 feet the in- i f tercept is 3.965 feet, the ratio — will be ^"" = 100.88. i Now if the rod be held at any other point, the new intercept s may be measured oh the rod and the distance from the apex D to the rod will be 100.88 X s. The total distance from the center of the instrumenl would be d= (c + f) +(100.88 X s). (c+f) is a known constant and s, the intercept, is seen on a graduated rod between the stadia wires of the telescope. The rod is so plainly marked- that the observer can read its divisions through the telescope. Multiply s by the determined ratio and add (c + f). The result is the required distance from the center of the instrument to the rod. If the rod is graduated in feet this multiplication must be performed for every distance read, but if a scale be constructed on the rod with a unit equal to the intercept at 100 feet from the apex D, and with decimal subdivisions then the rod readings increased by (c X f) will give distance directly, and the labor of multiplying or of referring to a table is avoided. 83 To mark a stadia rod in this manner, suppose as before that the intercept on a rod held at a distance of 400 feet from the apex D is 3.965 feet. One-fourth of this or 1 1.89 inches would be the intercept at 100 feet, and is the unit to be laid off on the rod to rep- resent 100 feet. The rod should be a light smooth board about yi inch 'Ca.xo^s., ^y^ inches wide, and 12 to 15 feet long, painted white. The rod may be stif- fened by attaching a longitudinal rib on the middle line of the back. On the face of the rod mark the center point and lay off the unit ( 1 1.89 inches in the case now considered) successively each way from the center on the middle line toward the ends. Subdi- vide each unit into twenty equal parts, each repre- senting 5 feet. Draw cross lines with a try-square and sharp pencil at each point of division and then paint red or black marks or designs, making strong char- acter^ that can be plainly seen through the telescope at a distance. The best marking for a stadia rod is that shown in Plate i, Fig. 3. The spaces representing 100 feet are distinct black masses separated by white, and can be readily counted even at a distance. The 50-ft. points are plainly marked by the large reentrant angles and the 25 and 75-ft. points by the double sali- ents. The lo-ft. points are the small reentrants and the 5 -ft. points are the small salients. The five-foot intervals may be subdivided by eye and read by es- timation to the nearest foot or half foot. To read the rod set one wire at an exact loo-ft. division and count the whole hundreds up to the one that is cut by the other wire and add the tens and units to the exact point cut by the second wire. Thus, if the lower wire is at A and the upper at B the 84 reading of the rod is 231 ft. and if (c + f) is 0.8 ft. the total distance to the rod is 231.8 ft. If a rod is to be marked to read yards, it is con- venient to avoid too large a figure and to take as the unit the intercept that corresponds to a distance of 40 yards. Subdivide and fnark in the same manner as is shown for the loo-ft. division. The smallest di- visions will then read to 2 yards, and, by estimation, to the nearest foot. For meters, proceed in the same manner as for yards. If the rod be held on a point that is higher or lower than the instrument the line of sight will be in- clined, and to get a correct reading the rod should be held perpendicular to the line of sight. To do this is generally impracticable or at least inconvenient. Moreover, it is not the actual distance along the line of sight that is required, but rather the horizontal distance and the difference of elevation between the two points. Therefore the rod is held vertically and its reading and the angle of elevation or depression to the required point are taken. A reduction is then made to get the horizontal distance and difference of elevation between the two points. The formulas for this reduction are deduced as follows: (Plate I, Fig. 4-) Let K be the known point at which the transit is set up. P be the required point at which the stadia rod is placed vertically. I be the center of the transit, and F be a point on the rod such that the height F P is equal to I K. Knowing the rod reading, the vertical angle A, and the angle subtended by the stadia wires 2 B, re- 85 quired the horizotital distance K L or I H, and the dif- ference of elevation, L P or H F, between the points K and P. Let b = K L or I H, the horizontal distance, a = L P or H F, the difference of elevation. s = s' + s", the distance read on the vertical • rod. 2 r = T T' — the rod reading or true distance if the rod were perpendicular to the line of sight at F. This rod reading would be the distance DF or d — (c+f). Then b = (c-|-f-|-2r) cos A. and a = ( c -f f + 2r ) sin A. To find 2r ; in the triangle P T R s': r:: sin (go° + B): sin ( i8o° — A — 90°— B): and in the triangle FT'R' s": r:: sin (90° — B): sin ( 180° — A — 90° + B): Whence cos B cos B s'^r — (i), ands^'^r (2) cos ( A + B ) cos (A — B ) cos B cos B s ■= s' + s" = r \- r .: (3)— cos(A-f B) cos(A — B) 2r = s (cos A — sin A tan A tanaB) (■4) b=(c + f)cosA + s (C0S2A — sin2A tan2B) (5) a ^ (c + f ) sin A + s (sin A cos A — sin2A tan A tan2B) (6) B is a small angle equal to one-half the angle subtended by the stadia wires. Tan^ B is therefore a very small quantity and the second term in the parenthesis in each of equations (5) and (6) may be neglected without appreciable error. In most transits the inter- cept of the stadia wires at 100 ft. is about i ft., hence the tan B = jjj and tan "B =- zuin-^- Neglecting this small quantity the formulae become b = (c + f) COS A + s cos 2 A (7) a = (c + f) sin A + s sin A cos A (8) 86 The values of b and a derived from these formulas are tabulated and the tables are published in works on surveying or in separate volumes. One part of the table gives the values of the first terms for (c + f) equal to the usual values from 0.75- ft. to 1.25 ft., and for angles A from zero to 20 or 30 de- grees Another part of the table gives the values of the second terms for s ^ 100 and for angles A from zero to 20 or 30 degrees. For other values of s mul- tiply the tabular number corresponding to the angle A by ^1^. When great accuracy is not required a method of marking the rod that is only approximately correct is adopted. The small distance, (c + f), is neglected. The base is measured from the vertical axis (or plumb line) of the transit and at the farther end of the base a rod is held on which is marked the intercept of the wires. This intercept divided by the number of hundreds of feet in the base gives the unit to be used in marking the rod. A rod so marked will give a cor- rect reading only for the distance equal to the length of the base. At lesser distances the readings will be too small and at greater distances the readings will be too large. For, let A (Plate II, Fig. i) be the center of the transit and let A B be the measured base. The in- tercept at B is measured and the rod is graduated as already explained. If this rod could be seen through the telescope when held at C, the true apex, the in- tercept would be zero and the distance would be called zero, whereas it is actually (c+fj. The reading is therefore too small by (c+f ). If B E is made equal to B C, the intercept at E would be G H, which corre- sponds to the distance A E', too great by (c+f). The 87 error that is made at any point in using a rod so graduated is, therefore, b— s e = (c+f) , in which e = the error, (c+f) b is the constant of the instrument, b is the measured base and s is the rod reading or observed distance. Notwithstanding the error that is introduced, this method of graduating stadia rods is often used in order to avoid the labor of adding the (c+f) con- stant or its reduction to every reading; but since the stadia will measure distances with an accuracy of about I in looo, the introduction of an error which may be cumulative should be avoided. The following method of graduating stadia rods is correct for horizontal readings and only slightly in error for readings reduced to the horizontal or verti- cal by the usual tables or diagrams. f . The formula d=c+f+ — s may be written in the following form: i f i d = — [(c+f) — + s], i f that is, the distance d is proportional to the intercept, i s, increased by the constant (c+f) — . Having gradu- f ated the rod accurately by the first method lay off. i this constant (c+f) — on either side of each even loo- f ft. division and fill in the space so marked with black at the sides of the rod as shown in Plate i. Fig. 3. In reading the rod, set the lower wire on the upper line of this black mark, but read from the adjacent loo-ft. division. The effect of this is to add to every i intercept a rod distance of (c+f) — which corresponds f to a ground distance of (c+f). The rod reading, therefore, includes the correction (c+f). It is just as easy to set the initial wire at the (c + f) mark as it is to set it at the exact loo ft. mark and no further correction for (c + f) is required. In reducing inclined readings of this rod to the horizontal by the formula b = (c + f) cos A + s cos 'A the first term, (c+f) cos A, disappears because the (c + f) is included in the rod reading, s, and here a small error is introduced by multiplying the (c + f) part of s by cos ^A whereas it should be multiplied by cos A only. But (c + f) seldom exceeds i ft. and cos A does not differ much from unity because the angle A seldom exceeds 15 degrees. For this angle cos A = 0.966 and cos ^A = 0.933. Hence the error, for c + f ^ I ft. and A= 15°, is only 0.033 ft. which is negligible in stadia work. In applying the formula a ^ (c + f ) sin A+ s sin A cos A to obtain differences of elevation the first term disap- pears because (c + f) is included in s, and ai^ error arises from multiplying the (c + f) part of s by sin A cos A, whereas it should be multiplied by sin A only. But the error is very small. If A = 15 degrees, sin A^o.259 and sin A cos A ^0.250; hence for (c + f)=i ft. the error is only 0.009 ft. Generally 89 the vertical angle is less than 1 5 degrees and the errors are correspondingly smaller. With this rod, therefore, horizontal distances are read correctly and inclined readings are reduced to the horizontal and vertical, without appreciable error, by taking the values from the usual tables or dia- grams without reference to the (c + f) corrections. The (c + f) part of the table may in this case be totally disregarded. THE COMPASS. The transit usually has a compass mounted cen- trally on the upper plate. It is similar to the sur- veyor's compass and will be described under that head. THE VERNIER. A measuring scale is a line divided by cross marks into equal parts, each part being a unit of the magni- tude to be measured. In surveying, such scales are used for measuring distances and angles. For distances the scale is marked on a chain or tape or rod, and the parts or divisions are usually feet subdivided into tenths and hundredths, or meters subdivided into centimeters. For angles the scale is a circle divided into 360 equal parts, each part being the measure of one de- gree. Each part may be subdivided into two, or three, or four, or six smaller parts, each measuring 30. or 20, or 15, or id minutes of arc, respectively. The divisions of a scale are usually numbered from an assumed origin or zero mark to facilitate 90 counting the number of divisions included in the dis- tance or angle to be measured. The index of a scale is a mark which indicates the point at which the scale is to be read. Thus, if a tape be stretched with its zero at one mark on the ground and its edge passing through another mark on the ground, this second mark is the index show- ing the exact point at which the scale is to be read. In measuring angles, the index is a mark or arrow head which moves along the circular scale for differ- ent pointings, and for any pointing shows the exact point at which the scale is to be read. When the index happens to fall exactly at one of the marks on the scale, the number of that mark is the reading of the scale. When the index falls be- tween two marks on the scale a fractional part of a division is to be added to the number at the mark -which the index has passed. This fractional part may be estimated by eye with considerable accuracy. A vernier is a device for measuring the fractional part of a division of a scale from the index back to the preceding mark of the scale. Fig. I, Plate la, is a portion of a linear scale showing the 4.0, 4.1, 4.2, 4.3, and 4.4 ft. marks, with each tenth of a foot divided into ten equal parts, which are therefore hundredths of a foot. The zero or origin of the scale is to the left. If the index, or point at which the scale is to be read, be at A, it is seen that it is exactly at one of the division marks of the scale, and the reading is 3.980 feet. If the index lie at the point B, the scale reads 4.12 ft. and a frac- tion over. This fraction may be estimated by eye to be about seven tenths of a division beyond the 4.12 mark, giving a reading of 4.127 ft., but to measure 91 this fractional part of a division a vernier is used. Return to the point A and observe that an auxiliary scale is constructed to the right of the index, that its length covers nine divisions of the main scale, and that it is divided into ten equal parts. Then each of these parts is one tenth of nine divisions or nine tenths of one division, or each part is smaller than a division of the main scale by one tenth of a division. If the index and vernier be moved to the right one tenth of a division, the first mark of the vernier will coincide with a mark on the scale. If the index be moved two tenths of a division, the second mark of the vernier will coincide with a mark on the scale, and so on. With index at B, it is the seventh mark of the vernier that coincides with a mark of the scale and this shows that the index has passed seven tenths of a division beyond the 4.12 mark, giving a reading of 4.127 ft. This vernier reads in the same direction as the scale and is called a direct vernier. At C is shown another form of vernier constructed to the left of the index. Observe that its length covers exactly eleven divisions of the scale, and that it is divided into ten equal parts, each part being equal therefore to eleven tenths of a division of the scale or one tenth longer than a division. The index at C is a little beyond the 4.40 ft. mark, bringing the second mark on the vernier into coincidence with a mark on the scale. This shows that the index has passed two tenths of a division beyond the 4.40 mark, giving a reading of 4.402 ft. This vernier reads to the left or in a direction opposite to the readings of the scale, and is called a retrograde vernier. 92 In general, if a vernier have n divisions, its least I reading is — of a division of the main scale. If it n cover n — i divisions of the scale, it is direct ; if it cover n + I divisions of the scale, it is retrograde. Or if the vernier divisions be smaller than the scale divisions, it is direct ; if larger, it is retrograde. It is convenient to have the zero of the vernier at the index, but this is not necessary provided always that the zero mark of the vernier and the index have simultaneous coincidence with marks on the scale. Any mark on the vernier may be taken as the zero mark. Sometimes the middle mark is so taken, and the vernier reads from the middle to one end, and continues from the opposite end back to the middle. On many transits the limb is graduated or num- bered in both directions, and the vernier must also be numbered in both directions, one set of numbers to be used for reading the limb in one direction and the other set for the opposite direction ; or else two sepa- rate verniers are provided. If a limb of a transit be divided into degrees and half degrees its smallest reading is 30 minutes. A vernier to read minutes would have 30 divisions. If it covered 29 divisions of the limb it would be direct ; if it covered 31 divisions it would be retrograde. Di- rect verniers are commonly used. On a circular arc the degrees are subdivided into three equal parts, each measuring 20 minutes. Its vernier has 40 divisions and covers 39 divisions of the arc. What is the least reading of the vernier? Is it direct or retrograde ? 93 A circular arc is divided into degrees. A vernier is to be constructed having a least reading of five minutes. How many divisions will the vernier have and how many divisions of the arc will it cover if direct? How many if retrograde? A linear scale is marked off in meters and centi- meters. Its vernier has ten divisions. What is the least reading? If direct, how many centimeter di- visions does the vernier cover? PjL/1 / ti j: Yards |^^_,,^_^ feti, vMtters Fig. 4 97 TLhTL U. 99 CHAPTER IV. ADJUSTMENTS OF THE TRANSIT. There are certain parts of the transit, as well as of other instruments, that the makers are unable to fix permanently and rigidly in their proper places with certainty and exactness. Such parts are made movable by means of opposing screws or otherwise, and their adjustment consists in bringing the mov- able parts into their proper positions. The parts of the transit that are adjustable are the plate levels, the cross wires, the horizontal axis, the telescope level, and in some instruments the vernier of the vertical circle, the object glass slide, and the eyepiece. The adjustments are : — 1. To make the axes of the plate levels perpen- dicular to the vertical axis. 2. To cause the line of collimation to revolve in a plane. 3. To cause the line of collimation to revolve in a vertical plane. 4. To make parallel the axis of the telescope level and the line of collimation. 5. .To make the vernier of the vertical circle read "zero" when the line of collimation is horizontal. 6. To center the object glass slide. '7. To center the eyepiece. 100 The adjustments of the attached compass are the same as those of the surveyor's compass and will be considered under that head. THE LEVEL TUBE. A level tube is a tube of glass so bent and ground that a longitudinal vertical section of the upper in- terior surface is an arc of a circle convex upward. This tube is nearly filled with ether and is sealed. The unfilled space forms a bubble of ether vapor which always seeks the highest part of the tube. A plane tangent to the upper siJrface of the bubble at its middle point is therefore a horizontal plane. A line tangent to the longitudinal section of the tipper inside surface of the tube at its middle point is called the axis of the tube. When the middle of the bubble is at this middle point of the tube this axis of the tube is horizontal. If a level tube be revolved about a vertical axis, every point of the tube will revolve in a horizontal plane and that point of the tube that was highest will remain highest throughout the revolution. There- fore the bubble will not move from its original posi- tion in the tube. Conversely, if a level tube be revolved about an axis and the bubble does not move from its original position, the axis of revolution is vertical. If the bubble is at the center of the tube and remains there during the revolution, the axis of the tube is hori- zontal and is perpendicular to the vertical axis of revolution. If a level tube be revolved i8o° about an axis not vertical or be reversed 180° on fixed points of support 101 the motion of the bubble in the tube, measured in degrees of arc, will be twice the angle made by the axis with the vertical or by the line supports with the horizontal respectively. For: Let AC (Plate II, Fig. 2) be the level tube, P O be the axis of revolu- tion, and B O be a vertical line through the center O of the arc of the tube. The bubble will be at the highest point B. After revolution of 180° about PO, the point of the tube that was at B will be found at D, and the point that was at D will be found at B, B D being perpendicular to P O, but the bubble will remain at the highest point B and will therefore have traveled in the tube over the arc D B equal to twice the arc P B that measures the inclination of the axis to the vertical To make the axis vertical, it and the tube must be turned in their own plane till the bubble is at P the middle point of the traveled arc. Having made the axis vertical, the tube may be moved inde- pendently till the bubble is at its middle point, and the axis of the tube will then be perpendicular to the axis of revolution and horizontal. Reversing the level tube 180° on two fixed points of support is equivalent to revolving it 180° about an axis that is perpendicular to the line joining the points of support, and the same demonstration holds good. The' foregoing principles are applied in all adjust- ments of level tubes, and in all cases where a level tube is used to determine a horizontal or vertical line. First Adjustment of the transit : To make the axes of the plate levels perpendicular to the axis of the instrument': First, make the axis of the instrument vertical and then adjust the plate levels so that their bubbles shall be at the middle points of the tubes. 102 For the first part of this adjustment it is best to use the telescope level because it is -more sensitive and accurate than the plate levels. Level the instrument approximately, and turn the head to bring the tele- scope level into the vertical plane containing two op- posite leveling screws. Bring the bubble to the mid- dle of its run and note its exact position. Reverse the instrument 180° on its vertical axis, and when the bubble has come to rest note its new position. With the leveling screws bring the bubble back over one- half of its displacement. This will bring the vertical axis into a vertical plane that is at right angles to the axis of the bubble. Now turn the head of in- strument 90° to bring the level into the plane con- taining the other pair of leveling screws, and repeat the operations described above. This will bring the axis of the instrument into another vertical plane, and since it lies in two vertical planes it must be their line of intersection and be vertical. The fore- going operations should be repeated in both positions as a check. The axis of the instrument being now verticaf, adjust ' the plate levels by means of the capstan headed screws, and bring their bubbles to the middle mark of the tube. The axes of the plate levels will then be perpendicular to the vertical axis, and when the instrument is again set up, and leveled by bring- ing these bubbles to the middle of their tubes, the axis of the instrument will be vertical. Second Adjustment: To cause the line of collimation to revolve about the horizontal axis in a plane. It will do this only when it is perpendicular to the hori- zontal axis. If it is not perpendicular thereto it will in its revolution generate a cone and not a plane. 103 Select a place where the ground is clear and ap- proximately level in opposite directions from the point to be occupied by the transit. Set up and level the transit, (i) Direct the cross wires upon a well defined point in any direction which may for refer- ence be called east. It will be best to drive a stake at a distance of aboiit 200 feet with a half driven nail in its top. Bisect the nail with the cross wires, clamp both plates, and (2) plunge the telescope to the west. About 200 feet from the transit in this direc- tion, an assistant will drive a second stake, being signaled by the observer to place it in the line of sight through the cross wires. The assistant will then hold a pencil point on top of the stake and move it as indicated by the observer till it is bisected by the cross wires. The point so determined is marked by a dot on top of the stake. (3) Reverse the transit on the vertical axis and point again at the eastern point, bisecting the nail accurately. Clamp the plates and (4) plunge the telescope again to the west. Signal the assistant to set the pencil again at the point inter- sected by the cross wires and mark the point on the top of the stake. If this point does not fall on the stake, another stake must be driven and the point marked on it, but usually the Second pencil point will fall near the first, and if a broad stake has been used both points will fall on its top. Draw the line con- necting the two points and mark a point at a distance from the second pencil point equal to one-fourth the length of the line. This last point will lie on a line that intersects the vertical axis of the transit and is per- pendicular to the horizontal axis of the telescope in its last position. If the cross wires be moved by their adjusting screws till their intersection falls upon 104 this last point the line of coUimation will be made perpendicular to the horizontal axis. For, let T (Plate II, Fig. 4) be the position of the transit, and (i) let T E be the direction of the line of collimation when it first bisects the east point E, and let P T P' be the perpendicular to the horizontal axis. (2) When the telescope is plunged the line of collimation will fall on T W the angle W'T P' being equal to PT E. (3) Now the transit is reversed through the angle W'T E and the line of collimation again bisects E. The perpendicular to the horizontal axis will revolve through the same angle and fall in P" T P'", the angle P" T E = W T P" == P T E. (4) Plunging the telescope again to the west lays the line of collimation on T Wg with the angle Wg T P'" = P"T E. Prolong the line E T to W", and the angles P'T W and P"'T W will be equal respectively to P T E and P"T E, therefore equal to each other and to P'TW, and to P'" T W^, which last angle is the angle of deviation of the line of collimation from the perpendicular to the horizontal axis in its last position. The points W and Wg have been marked by pencil dots. Mark P'" at a distance from Wg, equal to one fourth of the line WgW. Then adjust the cross wires so that their intersection shall fall upon P'" and since T P'" is perpendicular to the horizontal axis the line of collimation will also be perpendicular to that axis. To test the adjustment mark the middle point W of the line W W^. Turn the telescope on E and bi- sect accurately, then plunge the telescope, and if the cross wires bisect W the adjustment is correct. If not repeat the operations above described till the ad- justment is completed. 105 Third Adjustment: To cause the line of colliraa- tion to revolve on its horizontal axis in a vertical plane. It will do this only when the horizontal axis is truly horizontal. First Method: Set up the transit about 50 or 60 feet from the wall of a building and point the tele- scope approximately in a plane perpendicular to the wall. Direct the cross wires accurately upon a well- defined point near the top of the building and clamp both plates. Plunge the telescope downward to a point near the ground, and have an assistant mark with pencil or chalk the point on which the cross wires fall. Reverse and plunge the telescope and direct the cross wires again at the upper point. Clamp the plates and again plunge the telescope down to a point near the ground at the same height as the other point. The assistant will mark this second point. If it falls upon the first marked point the adjustment is correct. If not, the horizontal axis is shown to be inclined, and the higher end is toward the side cor- responding to the second marked point. Mark the point that is midway between the two marks. This middle mark and the upper point will lie in the ver- tical plane through the center of the transit. The line of collimation, when it is plunged downward from the upper point should pierce this middle mark. With the adjusting screws of the standard, lower the high end or raise the low end of the hori2;ontal axis. Sight again at the upper point, clamp the plates and plunge downward to the lower points. Note how much of the correction has been made, and adjust till by estimation the remaining part of the correction has been effected. Repeat the test and readjustment until the line of collimation can be m'ade to pierce 106 both the upper point and the lower middle mark, in its revolution on the horizontal axis. ■ To explain this adjustment it is only necessary to show that the lower middle mark is in the vertical plane through the center of the transit and the upper point. Let C (Plate II, Fig. 3) be the intersection of the vertical and horizontal axis of the transit, shown in projection on the plane of the wall, and assume that the right hand standard is the higher, giving A B as the projection of the horizontal axis. Let P be the upper point on the building, lying in the vertical plane through C perpendicular to the wall. When the telescope is pointed at P and then plunged down- ward, the line of collimation will trace on the wall the line P M', perpendicular to the projection A B of the horizontal axis. When the telescope is reversed and plunged, and pointed again at P, the higher standard will be on the left and the horizontal axis will be projected in A' B' making the angle P C B' equal to P C B. When the telescope is again plunged downward, its line of collimation will trace on the wall the line P M" perpendicular to the projection A' B' of the horizontal axis. Since the angles B C P and B' C P are equal, their complements M' P C and M"P C respectively are equal, and if M' and M" lie in the same horizontal, a point M, midway between them, will lie in the vertical plane through P and C. The foregoing demonstration is not mathemati- cally exact, but the errors balance and the result is correct. Second Method: Sight at any well defined high point, as the gable of a roof or the corner of a chimney, and clamp the plates. Place a bucket or pan of water on the ground in front of the transit in such a posi- 107 tion that the observer can see in the water a reflected image of the selected point. Plunge the telescope downward and sight at this image. If the cross wires bisect it the adjustment is correct. If not, make such adjustment of the standard as will cause the cross wires to bisect both the selected point and its image when turned on the horizontal axis from one to the other. The lines of sight from the instrument to the selected point and to its image in the water lie in the same vertical plane, and when the line of collimation is made to revolve in that plane the axis of revolution is horizontal and the adjustment is correct. Fourth Adjustinetit : To make parallel the axis of the telescope level and the line of collimation: First, make the line of collimation horizontal, then adjust the level so that its axis shall be horizontal (bubble in center) and the two lines will be parallel. First Method: An understanding of this method depends upon a knowledge of the principles of "level- ing," and beginners should revert to this method after they have studied the "level" and its use. Drive two stakes about loo feet apart to serve as points of support for a vertical measuring rod. The ground between the stakes should be approximately level and free from obstructions. Call one stake the Bench Mark (B. M.) (Plate III, Fig. i.), and the other the Turning Point (T. P.). Set up the transit on the line joining the two stakes and close to the B. M. stake so that the eye end of the telescope shall be about ^ inch from a measuring rod held vertically on the Rtake. Level the transit and bring the bubble of the telescope level to the middle of the tube. With the eye at the object glass, sight backward 108 through the telescope and mark with a pencil dot or a target the point on the rod seen at the center of the small field of view. The height of this dot as meas- ured on the rod above the B. M. is called a Back-sight (B. S.). Denote this height by "a," and enter it in the record in the column "B. S." opposite B. M. in column "Station.'' Sta. B. S. + H. I. F. S. Elev. B M a a o T P b a — b T P a' a— b + a' BM - W a — b+a' — b' If the elevation of the B. M. be assumed as zero", "a" will be the "height of instrument" (H. I.) or of the line of collimation above the B. M., and is entered in column H. I. Place the rod on the stake T. P., and noting that the bubble is still at the center, direct the telescope upon the rod and mark the point intersected by the cross wires. This height of rod is called a " Fore- sight," and is the vertical distance from the line of collimation down to the stake "T. P." Denote it by "b" and enter it opposite "Sta." T. P. in column "F. S." The observed "elevation" of the stake T. P. with reference to B. M. is (a — b). Enter this in the column " Elevation." Move the transit to T. P. and set it up on line 109 and close to the stake, so that the eyepiece shall be about one-half inch from the rod. held vertically on T. P. Bring the telescope bubble to the middle of the tube and read the rod as befoi"e by looking back- ward through the telescope. The result (a') is a back- sight- on T. P. and is so entered in the record on the third line. Since the elevation of the T. P. is a — b and the line of coUimation is higher by a', the new H. I. is a — b + a', in column "H. I." Sight forward through the telescope and read the rod held again on stake B. M. This reading, b', is a fore-sight and is recorded opposite sta. B. M. in the fourth line and in column "F. S." It indicates that the stake B. M. is lower than the line of coUi- mation by b' and gives a — b + a' — b' as the observed elevation of the stake B. M. If the line of coUima- tion were horizontal this observed elevation of B. M. would be equal to the assumed elevation, zero. Con- versely, if the observed elevation of B. M., a — b + a' — b', is not equal to the assumed elevation, zero, the line of collimation was not horizontal, and the error a — b + a' — b', was due to its inclination. If the value of the error is positive, the back-sights, a and a', were too great and the line of collimation inclined downward; if negative, the back-sights, a and a', were too small and the line of collimation inclined up- ward. The two sights, from B. M. to T. P. and from T. P. to B. M., were equal in length and in inclination. Therefore in the last sight from T. P. to B. M., the error was- one-half of the total error, or y^ (a — b+a' — b') ^= c. Move the last mark on the rod at B. M., upward if c is positive and down if c is negative, over a distance equal to c. Direct the cross wires upon this new mark, and the line of collimation will be horizon- 110 tal. With the adjusting screws of the level, make its axis horizontal, (bring bubble again to the middle of the tube). The axis of the level tube and the line of collimation will then be parallel. To test the adjustment, repeat the operation above described and if the new value of a — b + a' — b' =o the adjustment is complete. In this method a slight error due to curvature of the earth's surface is introduced, but as this error amounts only to o.ooi ft. in 200 feet, and to 0.00025 ft. in 100 ft., it may be disregarded. By analyzing this method it is seen that it con- sists of running a line of levels that closes on the starting point, and thus provides means of determin- ing the error due to the inclination of the line of collimation. Since there are two courses of equal length, one-half the error belongs to each course, and this gives the correction "c," which must be applied to make the line of collimation horizontal. EXAMPLE. Sta. B. S. + H I. F. S. Elev. B. M. 3.256 3.256 T. P. 5.813 — 2.557 T. P. 4.086 1.529 — 2.557 B. M. 2.506 -0.977-^-2 = — 0.488 In this case the value of c is — 0.488 ft. and the last mark on the rod must be moved downward 0.488 ft. If the cross wires be made to bisect this new mark the line of collimation will be horizontal. Ill Second Method: On ground which is fairly level, set up and level the transit. Bring the bubble of the telescope level to the middle of its tube and clamp the telescope. At a distance of loo ft. measured ac- curately from the center of the transit, drive a stake firmly and upon it hold a rod. Mark the point on the rod that is cut by the horizontal wire of the telescope. Turn the telescope i8o degrees on its vertical axis and at the same measured distance (lOo ft.) in the new direction drive a second stake, and testing its height by the rod, continue driving until the mark on the rod is cut by the horizontal wire of the tele- scope. The tops of the stakes will then lie in the same horizontal plane, no matter what may have been the inclination of the line of collimation. Move the transit and set it up close to one of the stakes so that the eye end of the telescope shall be about one- half inch from the rod held on the stake. Looking into the object glass, sight backward through the tel- escope at the rod and mark the point of the rod that is seen at the center of the small field of view. Then place the rod on the other stake and looking forward through the telescope, bisect the same point with the cross wires. This will make the line of collimation horizontal because it has been made parallel to the line joining the tops of the stakes. A second trial should be made to check the first result. Having verified the horizontality of the line of collimation, adjust the telescope level by the screws that control it and bring its bubble to the middle of the tube. The axis of the level will then be parallel to the. line of collimation. 112 Fifth Adjustment. — To cause the vernier of the vertical circle to read zero when the line of coUima- tion is horizontal. This adjustment should be made in connection with the fourth adjustment. 1st. When the line of collimation has been made horizontal by .either of the methods of the fourth ad- justment, then, without moving the telescope, loosen the screws that hold the vernier of the vertical circle and move the vernier until it reads zero on the limb. Fasten the vernier in this position and the adjust- ment is completed. 2d. When the vernier on the vertical circle is not adjustable this adjustment must be made by- moving the line of collimation, that is, the cross wires, instead of the vernier. Therefore, when mak- ing the fourth adjustment, set the vertical circle to read zero and then make the line of collimation horizontal by adjusting the cross wires with the top and bottom adjusting screws. This may disturb the second ad- justment and a test should be made to correct a pos- sible lateral displacement of the cross wires. 3d. When the vertical arc is a full circle the fol- lowing method may be used. Level the transit care- fully and direct the telescope at a well defined point or mark, 300 to 500 feet distant and read the vernier of the vertical circle. Reverse and plunge the tele- scope and point it at the same mark. Read the ver- nier again and find the mean (one-half the sum) of the two readings. Adjust the vernier so that its reading shall be the mean of the two readings. It will then indicate the correct vertical angle and will read zero when the line of collimation is horizontal. If the vernier is not adjustable, turn the telescope till the 113 reading of the vertical circle is the mean of the two readings, and then with the top and bottom adjusting screws bring the intersection of the cross wires again upon the selected point or mark. When this last method is used for the fifth adjustment it should precede the fourth adjustment, which is then made as follows: Set the vertical circle at zero. This makes the line of collimation horizontal. Bring the bubble to the middle of its tube by the adjusting screw. This makes the axis of the bubble tube hori- zontal and therefore parallel to the line of collima- tion. REMARKS ON THE ADJUSTMENTS. It cannot be assumed that the adjustments of the transit or of any instrument can be made with perfect accuracy, but when the adjustments are carefully made and tested, the errors that remain uncorrected are so small that they do not appreciably affect the results in ordinary surveying. When greater accur- acy is required each observation should be repeated (with the instrument reversed if possible) and the mean of the two readings adopted. For the greatest attainable accuracy, the best of instruments must be used and the observations must be repeated, five, ten, or even twenty times, with alternate reversals of the instrument whenever reversing is applicable. In the adjustments given no attention has been paid to making the line of collimation pass through the point of intersection of the horizontal and verti- cal axis, because this condition cannot be fulfilled at the same time with the others. If it does not pass exactly through this point the errors that arise are 114 those of parallax only, and are so small as to be neg- ligible in ordinary surveying. When the second adjusttnent has been made it is impossible to make any other lateral adjustment of the cross wires without disturbing the second adjust- ment. Therefore the line of collimation cannot be made to intersect the vertical axis without sacrificing the second adjustment. It may, however, be made to intersect the horizontal axis without interfering with other adjustments. If it is desired to make this ad- justment, proceed as follows: Set up the transit about 50 feet from a building or wall. Rest the object end of the telescope in a notch cut in the top of a board that has been set firmly in the ground. Sight through the telescope at the wall and signal an assistant to mark a pencil dot at the point intersected by the cross wires. Reverse and plunge the telescope and rest the object end again in the notch. Mark the point that is now intersected by the cross wires and draw a horizontal line midway between the two dots. With the top and bottom adjusting screws move the cross wires till their intersection falls upon the horizontal line. The line of collimation will then intersect' the horizontal axis, as will be evident from an examina- tion of the figure (Plate III, Fig. 2). O is the projec- tion of the horizontal axis. C L D is the first posi- tion of the line of collimation. C'L D' is its second position after reversing and plunging. PLO is its adjusted position passing through the point P mid- way in elevation between D and D' and therefore passing also through O, the horizontal axis, midway between C and C. The notch in the board holds the point L at the same place in both positions of the telescope. 115 If this adjustment is made the fifth adjustment must be made by adjusting the vernier of the vertical circle and not by moving the cross wires. If the vernier is not adjustable its error must be treated as an index error to be applied to all readings. The object glass slide is not always made adjust- able. When it is adjustable, the adjustment is .care- fully made by the manufacturer and the adjusting screws are covered by a ring to discourage ill-advised attempts at readjustment. The adjustment of the eyepiece consists simply in moving it by its adjusting screws so that the inter- section of the cross wires shall be in the center of the field of view. Whenever the cross wires are adjusted, care must be taken to preserve the verticality of the vertical wire. To test this, sight at a plumb line. EFFECTS OF FAULTY ADJUSTMENT. f I.) If the vertical axis is not truly vertical the limb of the instrument, which is .perpendicular to the axis, will not lie in a horizontal plane and the ob- served angles will not be the horizontal angles be- tween the points observed. Neither will the vertical circle lie in a vertical plane, nor will a zero reading of this circle always indicate a horizontal position of the line of collimation. Observed vertical angles will therefore be in error. (2.) If the line of collimation is not perpendicu- lar to the horizontal axis it will not follow a vertical line through a given point when revolved on the hor- izontal axis. The horizontal angle between any two points on a plumb line is, of course, zero ; but if the 116 line of collimation be directed first at a high point, and then be plunged downward to a lower point, some change in azimuth will be found necessary to bring the cross wires again to the plumb line. This change is the error due to the inclination of the line of collimation. This same error would affect the ob- served horizontal angle between any two points hav- ing respectively the same angles of elevation as the two points considered. If two points have equal angles of elevation there would be no error from this cause in the observed horizontal angle between them. The reading of vertical angles is only slightly af- fected by a' small obliquity of the line of collimation. (3 ) If the horizontal axis is not truly horizontal the error in horizontal angles is similar to that caused by obliquity of the line of collimation. The cross wires would not follow a plumb line, and their devia- tion from the plumb line in plunging from a high to a low point would be the error in the horizontal angle between those points, or between any two points having respectively the same angles of elevation as the points considered. If two points have equal angles of elevation there would be no error from this cause in the observed horizontal angle. In reading vertical angles a slight error would be introduced because the plane in which the angle is- measured would not be a vertical plane. (4.) If the fourth adjustment is faulty, errors will be made when using the transit as a level. These errors are discussed in connection with the descrip- tion of the level. (5.) If the vernier (or index) of the vertical cir- cle is not properly adjusted, the error of adjustment 117 will' appear in all single measurements of angles of elevation or depression. If not adjusted, the index error should be determined and applied to all read- ings of this circle, or the mean of a direct and re- versed reading should be taken. p^AT^. jn f--^^ Fig. 3 ^•-4'*'* 121 CHAPTER V. CARE AND USE OF THE TRANSIT. When first opening the case containing a transit or other instrument note the position of the telescope and of the clamp screws, and also the points of contact on the bearings that support the instrument in its proper position. Always thereafter return the instrument to its case in the same position. Observe also whether the instrument is provided with plumb line, reading glass, adjusting pins, screw driver, oil can, and dust brush, each in its proper receptacle. To assemble the transit, set up the tripod with the legs equally spread and screw on the head of the instrument, taking care not to cross the threads and not to use excessive force. Attach the plumb line. In carrying the transit, unclamp the telescope and lower plate, but clamp the upper plate, and stop the compass needle. The undamped parts permit the transit to adjust itself to shocks and knocks with- out undue strain ; the clamping of the upper plate maintains the orientation that is often to be trans- ferred from one station to another ; and the stopping of the needle prevents wear of the pivot. To set up the transit means to place it in position for use with its center over the station point and with its axis made vertical by means of the leveling screws. The station is usually marked by a tack in the top of a stake that is driven into the ground. 122 Any well defined point on or near the ground may be used. Spread the legs of the tripod equally around the station point and adjust them so that the plates shall ■be nearly level and the plumb bob nearly touching the tack that marks the station. With the shifting center bring the point of the plumb bob over the center of the tack or other station mark, and, with the leveling screws, make the axis of the transit ver- tical. To accomplish this last object proceed as fol- lows : Turn the head of the transit so that the plate levels shall be parallel respectively to the-lines joining opposite leveling screws. With one pair of screws bring the bubble of the parallel level to its center by turning the screws simultaneously in opposite direc- tions, that is, by moving the thumbs toward each other or away from each other. The bubble will fol- low the motion of the left thumb. With the other pair of leveling screws bring the other bubble to its central position. Repeat these operations alternately till both bubbles stand at the center of their tubes. The axis of the transit will then be vertical. Setting up or to set up the transit includes all of the operations just described, and it will be so under- stood in the following pages. To reverse is to turn the transit i8o degrees, or approximately i8o degrees, on its vertical axis. To plunge is to turn the telescope through any in- dicated angle on its horizontal axis or trunnions. When sighting through the telescope for the final pointing at any definite point, be careful not to touch the instrument at any part. The lightest touch of the fingers on the clamps or the contact of the cloth- 123 ing against the tripod will cause an appreciable error in the reading of an angle. Transits are built mainly of brass, bronze and wood. These are comparatively soft materials and will not stand hard usage. The application of any considerable force will wear or strip the threads of the screws, bend the screws or spindles, or otherwise dis- arrange the delicate mechanism, and ruin the instru- ment. The handicraft in the use of an instrument consists in applying the strength and skill of the fin- gers only. Never apply the strength or weight of the wrist, arm, or body. The transit should be kept perfectly clean and dry. Dust and grit are the greatest enemies of the transit. Next to these comes water. Therefore, brush it with the camel's hair brush or wipe it dry with a soft cloth whenever it has been exposed to dust or moisture. The transit should be kept in its case when not in use, if only over night, or else the head of the instrument should be covered with the gossamer hood. The care of the graduated circles deserves special attention. The lines are engraved on silver and filled with black wax. Through neglect the circles may become tarnished and dirty. An occasional rub- bing with a chamois skin will keep them in good order. Dirt is easily removed by a cloth moistened with alcohol. If the graduations are worn and scratched by dust, grit, and neglect, the instrument should be sent to the makers for repairs. The use of emery, chalk, pumice, or other polishing powder or composition, on a graduated circle is an evidence of gross ignorance, and an end to the usefulness of the instrument. 124 The working parts of the transit should move smoothly and freely without catch or jerk, and with- out chafing or grating. A grinding sound indicates the presence of dust or grit in the bearings. When this occurs the instrument should be taken apart, cleaned with alcohol and oiled by rubbing with a well oiled rag, not by a squirt can. Only the best clock or typewriter oil should be used. The transit should always be handled as if it were made of thin glass. The principles and general methods involved in surveying are the same for all instruments and must be stated before proceeding with a description of the use of the transit. Surveying is a time-consuming and expensive operation, and satisfactory progress cannot be made if the comfort and ease of the members of the party are considered: A surveying party should be com- posed of men who are willing to rough it, to work early and late in all kinds of weather, and to lend a hand at any labor that will expedite the work. In clearing out long lines in wooded country every man in the party becomes an axeman and does his share of the work. In building high stations every man becomes a carpenter. Whatever the work, each man bears his share, and there is no excuse for any mem- ber of the party to be idle at any time. The plotting of the field sheets should keep pace with the field work. The evenings and the stormy days are utilized in reducing the notes, plotting the work, and in overhauling and cleaning the equip- ment; When a draughtsman is employed, he plots the final map as fast as the notes of the surveyor are furnished him, and when the field work is completed 125 he should require only two or three additional days to finish the map, while the surveyor or chief of party prepares his report. ' When the beginner first sets up an instrument and looks around over the country which he is re- quired to survey, his task seems hopeless. There ap- pears to be no definite "point" to which he can make a measurement, because all points seem alike indefi- nite. It is in the selection of the points to be deter- mined that the skill, judgment and experience of the topographer are tested. The manipulation of the transit or other instrument is a mechanical operation readily performed after a little practice. But this op- eration is valueless unless the points observed are critical points. For example, in surveying a quadrilateral field inclosed by straight fences on the four sides, let four points, at the middle of the four sides respectively, be determined as a, b, c and d. (Plate III, Fig. 3.) These points would give no information as to the shape of the field. But, let the four corners A, B, C and D be located by observations, and the shape and size of the field are fully determined. On ground with varying slopes, as shown in profile in Plate III, Fig. 4, the location of the points a, b, c, d and e would give no idea of the shape of the profile, but if the points A, B, C, D.and E, where the slope changes, be located, the different slopes are fully determined. Again, let the points a, b, c, d and e, (Plate IV, Fig. i) on the 680 ft. contour be located ; the line thus determined would be the dotted line a, b, c, etc.; but if the critical points. A, B, C, D, etc., be located and plotted, the curved line drawn through these 126 points will represent with considerable accuracy the actual run of the contour. A critical point is a point that is common to two or more intersecting lines or surfaces. On curved lines or surfaces, it is a point where the curvature is most pronounced or where the curvature begins or ends. Points at the corners of a house, at fence cor- ners, at the junction of streams, at a bend in a road or stream or shore line, on the crest line of a ridge or bottom line of a valley, at the crest and foot of a slope, terrace, embankment or cut, at the salients and reentrants of the border of a forest, at the be- ginning and end of curves on a railroad or road, etc., are critical points. If all such points be determined the lines and surfaces that join them are determined, and the map made by plotting these points and filling in the details by the usual conventional signs and symbols will be a correct representation of the fea- tures and incidents of the area considered. If the critical points are not determined, the shape, features and incidents of the ground will not be fully deter- mined, and cannot be correctly represented on the map, no matter how many random points may be, de- termined. Note the water course, watershed, and change- of-slope lines and on these lines select the critical points. These and other critical points already de- scribed are the points to be determined in surveying. TRANSIT AND STADIA SURVEY. The transit with stadia attachment is a complete and universal surveying instrument. When set up at a known point and directed at a required point, the 127 reading of the horizontal circle, vertical circle and stadia rod, give respectively the horizontal angle from the assumed standard direction, the vertical angle from the horizon, and the distance; that is, the three polar coordinates that fully determine the unknown point. The surveyor simply .points the telescope and then reads and records what the instrument tells him. PARTY. EQUIPMENT. One surveyor Transit, note book, pencil, etc. One rodman Stadia rod hatchet One axeman Axe and stakes. This is the minimum, and is suitable only for a small survey. For an extended survey the party should consist of one surveyor, one recorder, one draughtsman, two rodmen and two axemen each having the necessary equipment. If the party is in camp there must be added one cook and one teamster with camp equipage, team, and wagon. Such a party will cost from $20 to $25 per day, and will cover daily from one-half square mile to one square mile, de- pending on the character of the country and on the •accuracy and amount of detail required. The cost per square mile will therefore vary between $20 and $50 for an accurate topographical survey with the tran- sit and stadia. It requires an experienced surveyor to run such a party and keep every man employed all the time. The beginner will find it diificult to keep one rodman and one axeman busy, and is doing well if he covers one square mile in a week. It does not take long, however, to become an experienced surveyor, if the principles are well understood. The fussing, the false moves and the hesitancy of the first week will largely disappear in the second, and a month of steady 128 field work will make a good surveyor of a man who tries. On the other hand, after years of experience there will still be new things to learn. Long practice and a knowledge of the higher mathematics are nec- essary for very accurate work. In beginning the survey, if an established bench mark is accessible it should be used as the origin or starting point of the survey. (A bench mark is a definite point on a fixed and permanent object the elevation of which point with reference to a fixed or assumed datum plane has been determined. Refer to the subject of "Bench Marks.") If no existing bench mark is available, select a suitable point and make it a bench mark by assuming its elevation. Set up the transit near the bench mark, selecting as the initial station (Q 0) a point from which a good view may be obtained to surrounding critical points including the bench mark. Stations are usually designated by the following symbols: Q represents a transit or plane table sta- tion, O a compass station, A a triangulation station. A dot is usually placed in the center of each station symbol. To orient the transit at the initial station unclamp the upper plate and turn it so thatvernier Ashall read zero. Clamp in this position and set accurately by the tan- gent screw, using a magnifying glass to read the ver- nier, and place its index exactly at the zero mark of the limb. Unstop the needle and turn the head of the in- strument so that the north end of the needle shall read zero at the N point of the compass circle. Clamp the lower plate and insure an accurate zero reading of the needle by means of the lower tangent screw. 129 The horizontal circle is now oriented and fixed in po- sition by its clamp. If the upper clamp be loosened and the alidade turned so that the telescope shall point at any desired object, the vernier (A) and the needle will each indicate the horizontal angle between the north point and the point observed. The compass circle should be graduated to the left from zero to 360 degrees, and the lower limb to the right from zero to 360 degrees. The readings of the compass and of the index to the lower limb will then be the same for all angles, and the compass will serve as a check on the readings of the index. Beginners should always note the compass reading and see that it veri- fies the reading of vernier A. This verification ex- tends only to the degrees and quarter degrees because the compass cannot be read as accurately as the ver- nier, but it is in the degrees that errors, or rather mistakes, are often made. , The reading of the vertical circle gives the angle of elevation or depression to the point observed, and the reading of the stadia rod held at the point ob- served gives a distance which by means of the for- mulas (page 85) or by the table or diagram referred to on page 86 may be reduced to the horizontal distance and difference of elevation or vertical distance. When the transit is set up, the rodman holds the stadia rod on the ground vertically and close to the telescope, and places a rubber band around the rod at height of the telescope trunnions. This marks on the rod the point to be observed when reading verti- cal angles. The surveyor notes the height of this mark on the rod and enters it in the record as Height of Instrument (H. I.), which here means height above the ground, not above datum plane. This record 130 enables the surveyor to see that the rubber band is not displaced during subsequent observations from this station. The rodman holds the rod first at the bench mark and then, in succession, at other points to which he is directed by the transitman. He mtist always hold the rod vertical with its center over the selected point. When the rod is properly placed at a selected point, the surveyor directs the telescope at the rod and bisects the rubber band with the cross wires, using only the clamp and tangent of the vernier plate and the clamp and tangent of the telescope. The lower clamp must not be touched after orientation. He reads vernier A, and the vernier of the vertical circle, and the compass, and records the, readings. Then sighting again at the rod he brings the lower stadia wire to the nearest lOO-foot division, (or to the "c + f" mark) and counts the hundreds, tens and units of the rod divisions up to the point cut by the upper stadia wire. This is entered in the record as the stadia reading. When all the critical points that are accessible from station ( □ 0) have been observed a new station (D I) is established. A point is selected for this sta- tion that will advance the survey in the desired di- rection and will afford good sights to the new points that must be determined. The readings on the rod to this station are the same as those made to deter- mine other points, but they are made with greater care and deliberation. Any error made in the loca- tion of a station affects all subsequent determinations. When the readings on the new station (DO are completed, unclamp the lower plate (leaving the ver- nier plate clamped), unclamp the telescope, and stop 131 the needle. Carry the transit to station i and set it up ; free the needle, plunge the telescope on the hor- izontal axis and cause the rodman to set the rubber band at the new height of instrument. Send the rod back to station 0, and with the telescope plunged sight at the rod and bisect the rubber band, using only the lower clamp and tangent and the telescope clamp and tangent. Do not touch the clamp or tan- gent of the vernier plate. Since vernier A has re- mained set at the forward pointing from D^^ to Qi and the telescope now points back from D i to Q 0, the line of coUimation is parallel to its former posi- tion, and so also is the zero line of the horizontal cir- cle parallel to its former position. This backsight from D I to no therefore orients the transit at D i with every line on. the lower plate parallel to its for- mer position at n and the azimuth readings will have their origin or zero at the north point as before. Check the orientation by reading the needle. The compass reading should coincide with the reading of vernier A. Plunge the telescope back to its normal position, unclamp the vernier plate and proceed with the ob- servations on the critical points that can be located from n I- The .last observation at D i is the one which locates 02, and the transit is then carried to n 2 and oriented by a backsight on Q i with telescope plunged. The accessible critical points are then located from Q 2 as described for D and D i • The foregoing operations are repeated at successive sta- tions until the required area or line has been covered. Orientation by a backsight with telescope plunged is affected by an error in the adjustment of the line of collimation. It is therefore considered better 132 to point the telescope backward by reversing it i8o° instead of plunging it. To do this, set vernier A at a reading that differs by i8o° from the last forward reading and point at the last preceding station. The instrument is now oriented and the work may pro- ceed as described. The foresights and backsights between stations along the main traverse must be made with special care and accuracy since they serve to refer all de- terminations back to the origin, and any error is transmitted through all subsequent observations. The accuracy of the traverse as a whole may be checked by running the traverse in a circuit and closing on the initial station, or by running the traverse between two points that have been accu- rately determined by some other method, as by trian- gulation. If no such check is applied, there can be no certainty as to the accuracy of the survey, and large errors, or even mistakes, may be made without detec- tion. If the traverse closes on a known point, whether it be the initial point or a triangulation point, any error or mistake will be detected. Legitimate errors within the limits of error imposed by the re- quirements of the work may be adjusted, but mis- takes can be corrected only by running the traverse over again to the point where the mistake was made. If the compass is constantly used as a check on the azimuth readings, there should be no mistakes in the determination of azimuths, but there is no constant check on the stadia readings except care and atten- tion on the part of the surveyor. A mistake is a wrong reading of a vernier or of a rod, a pointing made at the wrong point, a faulty 133 orientation of the instrument, or a false entry or omission in the record. Mistakes are avoidable. Errors are small deviations from accuracy due to the imperfections of the instrument and of the human faculties. Errors are unavoidable. SUMMARY OF OPERATIONS. 1. Set up the transit at D 0, and orient by the needle or by pointing in a known direction. 2. Backsight on bench mark. 3. Sideshots on the accessible critical points. 4. Foresight on Q i- 5. Move to D I and set up transit. 6. Orient by backsight on D 0. 7. Side shots on required points. 8. Foresight on D 2. 9. Repeat 5, 6, 7 and 8 at successive stations. 10. Close on the initial station or on a triangula- tion station. THE RECORD. The notes taken at each station constitute a sep- arate section of the Record." The heading of each section (see example) gives the numeral of the station occupied, the height of instrument (H. I.) at the station, and a blank space for the elevation of the station point. Below this heading enter, ist, the read- ings of the backsight to the previous station ; 2d, the readings of all side shots taken at this station; 3d, the readings of the foresight to the next station. 134 The readings for each sight or sideshot include the azimuth; the bearing (used as a check only); the vertical angle, + for angles above the horizontal and — for angles below the horizontal ; and the stadia distance. 135 Example : ( First page or title page of eaeh note book.) SURVEY OF FORT LEAVENWORTH, KANSAS AND VICINITY Made under the direction of by Surveyor, in charge of party. Recorder. Draughtsman. [ Rodmen. Axemen. Azimuth and bearings are from the magnetic north to the right, zero to 360 degrees.* Book No ( books in all), March 3, 1906, to 190 *The record should show whether azimuth and bearings are magnetic or true and in what direction they have been read. 136 (LEFT HAND PAGES OF RECORD. ) (1) (8) (3) (4) (5) (6) (1) (8) (9) Point Observed. Azimith Tern. A. Bearing Compass Vertical Angle Stadia Horiz. Distance. Vertical Distance. Eleva- tion Remarks. (?f c&fe. 0, ^ ^ = ^SS. (g^. o / / BM I 231-46 231^ —3-21 489 732.41 BM No. 65 US Sur'y M Riv C. P. etc. etc. etc. Sta. I 35-24 35K + 5-52 728 Top of Turtle Hill ^ S^c.. /, 3IC jr= ■i'-i'S. S'^. Sta. 215-24 215X —5-20 726 Backsight C. P. Fence Corner C. P. S. E. Corner House C. P. .S. W. Corner House etc. etc. etc. Sta. 2 Near Forks of Stream 137 The note books used in, a survey should also be named and numbered on the cover. These books are not the personal property of the surveyor, but per- tain to the office or person for whom the survey is made.. The note book, map and report constitute the complete record of the survey. The field record begins with a statement describ- ing the method of reading angles, and the orienta- tion adopted. The notes taken at the initial station have the heading, "At Sta. 0, H. I. =453, Elevation •," and under this heading there are entered all the readings taken at this station. The first sight is taken on a bench mark, supposed in this case to have been estab- lished by a government survey. In column (i) write the designation of the point observed (B. M. i); in column (2) the azimuth, or reading of ver. A (231" — 46'); in column (3) the bearing, or reading of the needle (2313^°); in column (4) the vertical angle, or reading of the vertical circle ( — 3° 21'); in column (5) the stadia distance, or reading of the stadia rod (489). Columns (6), (7) and (8) are the reductions to hori- zontal distance, vertical distance, and elevations above datum, respectively, deduced from columns (4) and (5) and filled in after the day's field work is com- pleted. Column (9) contains definite descriptions of' points observed,' and any other needed remarks of an explanatory nature. Similar record is made of readings on other points, and the last observation from Sta. 0, is made on Sta. i to determine a new point from which further obser- vations may be taken. The right hand page of the record should contain a free-hand sketch made in the field as the work pro- 138 gresses, and showing all points located by the obser- vations and the objects and features that are thus located. The horizontal angles may be plotted direct from the records, but the vertical angles and stadia read- ings cannot be plotted until they have been reduced to horizontal distance and vertical distance. The horizontal distance can then be plotted or laid off to scale, from the station point, in the direction given by the azimuth, to the required position of the point observed ; and the elevation of this point may be shown by writing, at its plotted position, the number that indicates its vertical distance in feet above the datum plane. The map position of the point is thus fully expressed. The methods of reducing inclined stadia readings to horizontal distance and vertical distance by form- ulae or tables or diagrams have been fully explained under the head of "The Stadia " Reduction by means of the formulae is the most accurate method, but is a slow and tedious operation seldom used. The tables are sufficiently accurate, and their use involves only a simple multiplication of the stadia reading by the tabular numbers that cor- respond to the vertical angle. The diagram gives the ' horizontal and vertical distances without computation and with an accuracy equal to that attainable in reading the stadia or in plotting the results. It may be used, therefore, for all sideshot reductions. For reductions of sights on the main traverse, if greater accuracy is required, the table should be used. In plotting the survey the main' traverse and the sideshots are considered separately. The traverse is j)lotted first, and when all the traverse stations are 139 correctly located on the paper the sideshots taken from each station are plotted. The points, so deter- mined serve as the basis for drawing the lines and conventional signs that represent in proper relative positions, the objects, features and incidents that have been located. TO PLOT THE TRAVERSE. A protractor is a thin flat piece of metal, paper, celluloid, wood, or other material, near the edge of which are radiating lines spaced at degree or half- degree intervals. The center point from which these lines radiate is also marked, and the degree marks are numbered usually at lo" intervals to facilitate counting. To plot any angle at a given point on a given line, place the center m'ark of the protractor at the given point and the zero line of the protractor on the given line. Count the degree marks in the desired direction to include the number of degrees in the given angle and make a dot on the paper at the point so determined. A line drawn through the center point and the dot so marked will make with the given line the given angle. In plotting azimuths or bearings, the given point is the station point on the paper and the given line is the meridian line or other assumed standard direction line drawn through the station point If not already plotted, assume a point on the. paper to represent the initial point of the traverse, or station zero. Write its elevation and draw through it a meridian line. With the protractor lay off the azimuth of station i, and draw the line representing 140 the first course. With the working scale lay off the horizontal distance to station i. At station i so de- termined write the elevation and draw a meridian. Proceed in the same manner from station i to station 2, from 2 to 3, and so on to the end of the traverse. To plot the sideshots, set the protractor again at station 0, and lay off all the azimuths and distances taken as sideshots from that station. Mark the de- termined points, write their elevations, and draw the objects thus located. Proceed in the same manner at all the other stations. On the lines that join the plotted critical points, interpolate contour points and draw contours through points having the same ref- erence. Plot in the same manner all the traverses of the survey. To finish the map, ink the drawing and fill in, with conventional signs, the grass land, woods, marshes, cultivated ground, etc.; draw title, legend, scale, true and magnetic meridians and border. Erase all pencil lines and clean with sponge-rubber or bread crumb. When the traverse begins and ends at known points or accurately determined points, as in the case of a closed traverse or of one that runs from one tri- angulation point to another, a check is furnished by means of which the errors made in traversing may be adjusted. If the traverse is correct, the plotted, or the computed position of the final station should coincide with the known position of that station. If these two points do not coincide, the line joining them is the error of closure and, in the contrary sense, is the correction that must be made in the plotted or computed position of the final station. This correc- tion is used as a basis of comparison for deterniining 141 the corrected positions of all other stations of the traverse under the assumption that the error is accu- mulative and proportional to the distances of the sta- tions along the traverse from the origin. This as- sumption is not supposed to be true, but in the absence of any positive evidence to the contrary it is supposed to give positions that are probably more nearly cor- rect than any other positions that can be assigned. If a traverse be run between two known points A and B, of which a {z^j) and b (spr) are the plotted positions, and if the line a, i, 2,j, b' (Plate V, Fig. i,) represents the plotted position of the traverse, \b\28y') representing the point B as determined by the tra- verse] the error bb' {-^) in the position of b' is due to the accumulated errors made in running the traverse. The position of b' , {28'j) may be corrected since its true position, b {sgi) is known, and the positions of the other stations may be adjusted to conform to the correction bb' (+^) applied at b' . Each plotted station must be moved in a direction parallel to the line b'b, over a distance, /-/', 2-2' or j-j' respectively, that is proportional to the distance of the station from the origin a. The known ratio is the error of closure divided by the total length of traverse, and this multiplied by the distance of each station from the origin gives the corresponding cor- rection. 142 The operations of adjustment may be tabulated as follows: (I) (2) (3) (4) (5) (6) (7) LENGTH DISTANCE STA- OF FROM HOR. VERT. OBS. COR. TION COURSE ORIGIN CORR. CORR. ELEV. ELEV. a. O 253 253.0 I 428 428 1.7 + I.I 264 265.1 2 410 838 3-3 + 2.1 278 280.1 3 540 1378 5-5 + 3-4 282 285.4 V ig6 1574 6.2 + 4.0 287 2gl.0 6.2 4 Hor. Ratio = = .004. Vert. Ratio =; = .0025. 1574 1574 Through the stations draw lines parallel to l>'6 and on them lay off (column 5) 1.7 ft. at G i, 3-3 ft. at D 2, 5.4 ft. at □ 3, giving the corrected stations, i', 2' and 3'. At these new station points write the corrected elevations (column 7). When the adjust- ment is completed, erase the original traverse and adopt the new one. When a traverse forms a circuit and closes on the initial station, the same method of adjustment is ap- plicable. When a survey is made up of a number of inde- pendent traverses each closing on a known point, the errors of each traverse may be adjusted and will not be carried to the next ; but if the entire survey be made by a continuous traverse which does not close on any known point, there is no opportunity for adjustment, and no check on the accuracy of the 143 work. A traverse should not run more thai;! three miles without closing on a known point, and it is bet- ter to have check points at shorter intervals when practicable. In the foregoing method of adjustment the hori- zontal error of closure may be due as much to errors in plotting as to errors in measuring the angles and distances. A better method of adjusting and plotting a traverse is the following : METHOD BY LATITUDES AND DEPARTURES. The polar coordinates given by the transit and stadia for each course are first reduced to the rectan- gular coordinates. The courses are considered consec- utively in a forward direction along the traverse. The vertical coordinate or difference of elevation for each course is given by the reduction of the stadia and vertical angle readings. The result is positive if upward and negative if downward. This reduction gives also the horizontal length of the course as already explained. The latitude of a course is the component of its horizontal length in a north or south direction, and is equal to the horizontal length of the course multi- plied by the cosine of the azimuth. The sign of the cosine determines the sign of the latitude. The departure of a course is the component of its length in an east or west direction, and is equal to the horizontal distance multiplied by the sine of the azimuth. The sign of the sine determines the sign of the departure. The azimuth is assumed to be measured from the north point to the right, zero to 360 degrees. 144 The signs of latitudes and departures are indicated in Plate V, Fig. 2. Let O be the origin or station occupied for any course and let the radius of the. circle be the horizontal length of the course. If the course O a lies in. the first quadrant the cosine and sine of the azimuth are positive and there- fore the latitude (Oaj) and the departure (Oag) are positive. In the second quadrant the cosine is negative and the sine is positive; therefore the latitude (Ob^ ) is negative and the departure (Obg) is positive. In the third quadrant the cosine and the sine are both negative, therefore the latitude (Ocj) and the departure (O c^) are both negative. In the fourth quadrant the cosine is positive and the sine is negative, therefore the latitude (O dj) is positive and the departure (O dj) is negative. A positive latitude is also called a northing and a negative latitude, a southing. A positive departure is called an easting and a negative departure, a westing. The algebraic sum of all the successive differ- ences of elevation on all the courses of a traverse is the difference of elevation from the initial to the final station of the traverse. Thus if the traverse be shown in profile as in Plate V, Fig. 3, the algebraic sum of the differences of elevation is +20 — 9 — 16+8 +22 = -j-25, indicating that the point b is 25 feet higher than the point a. But if the point b is known to be 27 ft. higher than «, the total vertical error in the traverse is — 2 feet, and the correction for each course is assumed to be proportional to the length of the course, with the total correction, +2 ft., divided by the total length of traverse, 1094 ft., as the known ratio, Y-o-*5^ = .0018. Multiplying this by the several 145 lengths of courses gives +047, +0.53, +0.39, +0.35, +0.23, as the corrections to be applied to +20, — 9, — 16, +8, and +22, respectively, to give the corrected differences of elevation, +20.47, — 8.47, — 15 -61, + 8.35, +22.23, the algebraic sum of which numbers is 27.00 as it should be. In like manner the latitudes and departures are adjusted, for — The algebraic sura of the latitudes of all the courses should be equal to the latitude of the traverse, and — The algebraic sum of the departures of all the cohrses should be equal to the departure of the trav- erse. If the traverse runs from one known point to an- other its total latitude and total departure are known and are used in adjusting the latitudes and departures of the several courses of the traverse. In the particular case of a traverse that closes on the initial station, the algebraic sums of the differences of elevation, and of the latitudes, and of the depar- tures, should each be equal to zero. Let the line a, i, 2, 3, b, Plate VI, Fig. i, repre- sent a traverse running from A to B. The depar- tures of the several courses are respectively +a i, + 1 2, — 2 3, +3 b, and their algebraic sum is the de- parture of the traverse, +ab. The latitudes of the several courses are respec- tively — a' i', + 1' 2', +2' 3', +3' b', and their algebraic sum is the latitude of the traverse. +a' b'. If the sum of the departures is not equal to the known departure of the traverse, the difference is the error of closure of the departures and is distributed 146 among the sevaral departures in proportion to the lengths of the corresponding courses. Likewise, if the sum of the latitudes is not equal to the known latitude of the traverse, the difference is the error of closure in latitude and is distributed among the several latitudes in proportion to the lengths of the corresponding courses. Thus: Course. Horizontal Length Departure. Corrections. Corrected Departures Departures from A. a, I I> 2 2, 3 3 b 360 412 196 310 + 321 + 346 - 87 + 291 — 1.40 — I.6l — 0.76 1. 21 + 319-59 + 344 39 - 87.76 + 289.79 + 319-59 + 663.98 + 576.22 + 866.00 a, b 1278 + 871 — 5.00 + 866.00 + 866.00 Known departure = + 866 Closing correction = — 5 Ratio = — ^^V? = — .0039 This ratio multiplied by the several horizontal lengths gives the corrections which, applied to the departures, give the corrected departures, whose con- tinued sum gives the departures from A. In the same manner adjust the latitudes and com- pute distances from an origin a' with attention to signs. In the same manner adjust the differences of ele- vation, and compute the elevations above datum. To plot the traverse, draw on the paper an east and west line O Y (Plate VI, Fig. i), a,nd as.sume any point as a^ as the origin of departures. From this point lay off the computed departures to i j , 2 j , 3 ^ , and bj, and at these points erect perpendiculars. Draw a north and south line O X and assume any point as a' as an origin of latitudes. From this point lay off the computed latitu'des to i', 2', 3' and b' and at these points erect perpendiculars. The correspond- 147 ing perpendiculars intersect in the points a, i, 2, 3, and b, and these points are the plotted positions of the traverse stations. At each station point write the computed elevation of the station. This method of plotting is facilitated by ruling the paper in squares with sides representing con- venient units of measure. This furnishes plotting scales in both directions by means of which the lati- tudes and departures are quickly plotted. The advantages of this method of adjusting and plotting a survey are the following : 1st. The adjustment of errors is not affected by errors made in plotting with a protractor. 2d. The latitudes and departures may be com- puted and summed up in a short time, and errors or mistakes may be detected while there is a chance to correct them. If mistakes are not detected until the draughtsman plots the notes it will usually be im- practicable to send the field party back to locate and correct the mistakes. 3d. Plotting by rectangular coordinates is more accurate than plotting with the protractor. 4th. The area included within a closed traverse or within any boundary that has been determined by sideshots from a traverse may be readily and accu- rately computed without plotting the survey, as will now be explained. 14^ DETERMINATION OF AREAS BY MEANS OF DOUBLE MERIDIAN DISTANCES OR BY DOUBLE LATITUDES (plate VI, FIG. 2). The meridian distance of a point is its distance from an assumed meridian measured on a parallel of latitude, as i' i (Plate VI, Fig. 2). The double meridian distance of a straight line is the sum of the meridian distances of its extremities ; thus i' 1+2' 2 (Plate VI, Fig. 2) is the double merid- ian distance of the line 1,2. The double meridian distance of a course multi- plied by the latitude of the course is twice the area included between the course and the meridian, and between the parallels of latitude drawn through the ends of the course ; thus (o' 0+ 1 ' i) X o' i ' is twice the area o, i, i' o'. The meridian is so assumed that all meridian distances are positive. Therefore, this double area is positive when the latitude of the course is positive, and negative when the latitude is negative. Rule : Find the double area for each course of the closed traverse. Find the algebraic sum of these double areas and divide by two. The result will be the area inclosed by the traverse. The sign of the result is immaterial since only its numerical value is required. Explanation : In (Plate VI, Fig. 2) take the sum of the positive areas i, 2, 2' i'; 2, 3, 3', 2'; 3, 4, 4', 3'; .4. 5> 5'. 4'; ^^^ subtract the sum of the negative areas o, I, i', o'; 5, 6, 6', 5'; 6, 7, 7', 6'. The result is the area o, i, 2, 3, 4, 5, 6, 7. Double areas are used only for convenience, as it is easier to divide the sum of all the areas by two than to divide each double area by two and then find the sum: 149 The result will be positive when the traverse runs contra-clockwise and negative when the traverse runs in clockwise direction, but as before stated the sign of the result is immaterial. The same result is obtained by multiplying the double latitude distance of each course by the depar- ture of the course to get the double areas, the signs being determined by the signs of the departures. One-half of the algebraic sum of the double areas gives the required area enclosed by the traverse. It is often impracticable, on account of fences and other obstructions, to run a traverse on the boundary line of an estate or farm, but the traverse can be run near the boundary with sideshots to the corner points. The rectangular coordinates of the corner points may be computed as already explained and the area may then be determined by the method of double meridian or double latitude distances. 150 CO HI Doub. Area. • r^ Doub. Area. + M Doub. Mer. Dist. fX) Mer. Dist. M UP ^ o s3| O M |3 c^ CO ►J r~- o Corr. Vert. Dist. in L/iT£: jar. • n ;:d JB.Af. Fi^ 1 T.n rig. 2 Fig. 3 Fig. 4 155 P/./fT-£: is: Fig.l /^l/^TE jz:. C^6s;tj ^^5i) a. f£6^j Fig.i. o CZ98) 270' tV. £90' i«o Fig. 2, 3 /9S' l^ig- 3 h'>t. Fi0-.i 0,7, s;i,6. n g-2 ^r^t ir* 161 CHAPTER VI. THE COMPASS. THE MAGNETIC NEEDLE. If a Straight bar of steel be magnetized and bal- anced horizontally upon a sharp-pointed pivot, with the least possible friction it will at any place assume a fixed position in the magnetic meridian through the place and will therefore determine a direction line, which may be taken as the standard for measuring other directions. A thin bar of steel so magnetized and balanced is called a magnetic needle. It furnishes the only avail- able means of determining the magnetic meridian at any place, and the magnetic meridian must therefore be defined as a line that coincides in direction with the magnetic needle at the place considered. The directions indicated by the needle at differ- ent places on the earth's surface converge approxi- mately toward the magnetic poles of the earth. These poles do not coincide with the poles of the earth's axis, and therefore the magnetic and true meridians do not generally coincide. The angle between the magnetic and true merid- ians at any place is called the declination of the needle for that place. There is a certain line on the earth's surface at all points of which the magnetic and true meridians 162 coincide, and the declination is zero. This line is called the agonic line (without angle). In America it now passes through Michigan and South Carolina and intervening States, and is very slowly moving west- ward. Conceive the agonic line to divide the earth roughly into two hemispheres. In the eastern hem- isphere, thus formed, the north end of the needle in- clines to the west of the true meridians and the declinations are therefore west, varying from o" to 1 80°. In the western hemisphere, so formed, the north end, of the needle inclines to the east of the true meridians and the declinations are east, varying from 0° to 180°. The declination becomes 180° only in the regions between the true poles and the mag- netic poles. The lines of magnetic force do not lie parallel to the earth's surface, but dip downward, to the north in the northern hemisphere, and to the south in the southern hemisphere. A needle hung at its center of gravity would therefore assume a position inclined to the horizontal. To counteract this tendency the needle is counterpoised, usually by a light coil of fine wire on the end that tends to rise. Lines on the earth's surface at all points of which the declinations of the needle are equal are called isogonic lines. These lines, like the agonic line, are in America moving slowly westward. Judging by the results of observations extending over several cen- turies, it is predicted that in time this westward mo- tion will cease and the isogonic lines will move east again. These motions are attributed to a secular (long period) change in the positions of the magnetic poles, 163 and the north magnetic pole is supposed to be mov- ing toward the west in the region north of America. Consequently in America the north end of the needle is moving slowly toward the west following the mo- tion of the magnetic pole; that is, west declinations are increasing and east declinations are decreasing. On the other side of the earth opposite conditions obtain, and there is an intermediate region, presum- ably that toward or from which the magnetic pole is moving, in which there is no change in declination. This slow variation of the needle as it follows the moving magnetic pole is called the secular variation of the needle. The rate of this variation has been determined in different localities in minutes of arc per year, so that, if the declination be determined at any place* and time, the declination at other times may be deduced. It is best, however, not to rely on computed declinations, but rather to determine it by actual observation at the place where and time when it is wanted. Besides the secular variation there is also a daily variation of the needle. In the early morn- ing the north end of the needle moves slowly to the east reaching its extreme eastern position about 8 A. M. Then it moves to the west and reaches its ex- treme western position at about 1:30 p. M. The angle between these extreme positions varies from 5' to 15' of arc, depending on the season of the year. The needle has its mean position at about 10:30 A. M., and 8 P. M. The daily variation is so small that it is usually disregarded in compass surveying, but in de- termining the declination of the needle at any place it should be accounted for. The compass needle is disturbed by the near presence of iron or steel or magnetic iron ores. The 164 steel bows of eye-glasses, an adjusting pin, a knife, a watch, or a steel button will afifect the needle if brought within a few inches of it. If the compass be set up near a railroad track, near iron pipes in the ground, near electric wires carrying heavy currents, near any iron or steel structure, or near a deposit of magnetic ore, its readings will be erroneous. A disturbance of the needle due to any of the fore- going causes is called "Local Attraction." THE COMPASS. The compass is one of the most simple and con- venient of instruments for measuring horizontal angles. Its principal advantage over other angle measuring instruments lies in the fact that it is al- ways oriented. Its principal defect is that the orien- tation is inaccurate and varying, so that generally certain allowances or corrections must be made. Referring to the parts which have been mentioned as essential in angle measuring instruments, there are found in the compass: 1. . The pointers or sights which fix the particu- lar line of the instrument that is brought into coinci- dence with the line of sight in the required direction. 2. A graduated circle upon which the direction of the sighting line is given in degrees of arc from the standard direction which is indicated by — 3. The magnetized needle which is balanced on a pivot at the center of the graduated circle, and which always lies in the magnetic meridian. 4. Various means are provided for making the graduated circle .horizontal. Some compasses have leveling screws and plate levels like the transit, 165 others have a ball and socket joint and plate levels, and others are simply held horizontally in the hand. 5. The several parts of the compass are assem- bled in a case or box which is held in the hand if it is a pocket, or box, or prismatic compass, and is mounted on a tripod or Jacob's Staff if it is a survey- or's compass. There are two general classes of compasses known as needle compasses and car , compasses. In the needle compass the graduated circle is at- tached to the box or plate and turns with the sights or pointers. In this case the index is the north end of the needle which remains fixed while the gradua- tions of the circle pass by it. In the card compass the graduated circle is marked on a circular card or on a light ring of metal which is atta'ched to the needle, and which therefore remains fixed while the index which is marked on the box turns with the sights or pointers. These two methods of reading angles may be compared with the two methods applied in reading the vertical and horizontal circles respectively of th6 transit. The vertical circle of the transit turns with the telescope while its index and vernier remain fixed. So, in the needle compass, the circle turns with the sights while the index or needle point re- mains fixeS. The horizontal circle of the transit re- mains fixed while the index and vernier on the upper plate turn with the telescope. So, in the card com- pass, the graduated circle remains fixed with the needle, while the index on the box or case turns with, the sights. From these two methods of reading angles there result two corresponding methods of graduating the 166 circular arc. All compasses should read zero (or 360°) when the sights point north, 90° when the sights point east, 180° when the sights point south, and 270° when the sights point west. In the needle compass remember that the index or north end of the needle remains fixed while the graduations of the circle pass by,it. Then in turning from north, zero, to east 90°, the numbers of the graduation must run toward the left, contra-clocfewise, and so on all around the circle; that is, the graduations must be numbered in a direc- tion (left) opposite to the direction in which the angle is turned or measured (right). In the card compass the circle, attached to the needle, remains fixed while the index with the sights or pointers turns past the graduations. Therefore in turning from the north, zero, to east 90°, the num- bers of the graduations must run to the right or in clockwise direction. That is, the graduations must be numbered in the same direction (right) as that toward which the angle is turned or measured (right). Unfortunately, compasses are not all graduated in the manner indicated above. Some are numbered from zero at the north and zero at the south, both ways to 90° at the east and 90° at the west. -The same numbers are therefore found in each of the four quadrants, and in reading a bearing it is neces- sary to add the letters NE, SE, SW or NW, in order to designate the quadrant; thus, N2S°E, 847° 30' E, etc. Others are graduated from zero at north, east, south and west to 45" at northeast, southeast, south- west and northwest, and in this case the letters indi- cating the octant must be added thus W 38° N, S 14° E, E 44° N, etc. 167 Some are graduated from zero at the north in both directions to i8o° at the south, and the letter E or W must be added to indicate the semi-circle in which the angle is read. There will also be found compasses numbered to read zero for a north pointing, 90° for a west point- ing, 180" for a south pointing and 270° for an east pointing. In the mariner's compass the divisions are not degrees, but points, arranged as shown in Plate A, Fig. I. A "point" is one- eighth of a quadrant, and is equal to 1154^ degrees. Smaller divisions are ex- pressed as half or quarter points, thus, E ^4^ N, read "east quarter north," or SW y^ S, read "southwest half south." This method of marking compasses is gradually giving way on modern vessels to the method of marking in degrees numbered to read zero when pointing north, 90° east, 180° south, 270° west, so that a bearing or course may be expressed by a num- ber only and not by a complicated system of letters and fractions. When a choice can be made, a compass numbered contintiously from zero to 360 degrees, and reading N 0°, E 90°, S 180°, W 270°, should be selected. The surveyor, may, however, find that the only compass available is marked by any one of the systems men- tioned, and should be able to use it and to record and plot its bearings. A needle compass should have its graduated cir- cle raised to the plane in which the needle swings, and the point of the needle should almost touch the inside of the circle. In some the circle is placed in the bot- tom of the box and the needle swings above it. These 168 should be discarded when the other kind can be ob tained. The Surveyor s Compass has two sight leaves mounted at the ends of a plate. The compass is mounted in a circular box at the center of the plate. It has a raised circle, which is usually numbered as follows : The two zero marks are placed in a line with the sights and the numbers run both ways from each zero to 90" at the right and left. K fletir-de lis is en- graved in the bottom of the box to indicate the for- ward or object end of the compass. The other is the eye end, and is indicated by S. The letter E is near the left hand 90° mark and the letter W near the right hand 90° mark. When the " fleur-de-lis," or N zero, is at the north end of the needle the sights point north. Turning the sights 90" to the right brings the E 90^ to the north end of the needle, and the sights point east. Another 90° turn brings the S zero to the north end of the needle, and the reading is south. Ninety de- grees more brings the W 90° to the north end of the needle, and the pointing is west. Intermediate point- ings are expressed thus, N 23° E, S 80° 30' E, S 50° W, N 44"^ 15' W, as previously explained. The small- est divisions of the circle are half degrees. With a reading glass the angle can be estimated on a large compass to the nearest 5 minutes. To read the bear- ing, note the degree mark that the north end of the needle has passed and add the fractional part. Re- cord it by writing the letters of the quadrant with the number of degrees between them. If the circle is numbered continuously from zero to 360°, no letters are needed, and this method of marking the circle is therefore to be preferred. In 169 no system of measurement is it desirable to have more than one origin. The graduated circle may, by means of a rack and pinion movement, be turned on the plate about its own axis. This motion of the circle makes it pos- sible to u.se some direction other than the magnetic north as the standard direction from which to meas- ure angles. Compare this arrangement of the com- pass with the similar feature of the transit. In the transit the horizontal circle may be turned at will and oriented with its zero line in any desired position. It may be set in the true meridian and give true asi- muths, or it may be set in the magnetic meridian and give magnetic azimuths. So with the surveyor's compass, may the circle be turned and set to read zero when the sights lie in the magnetic meridian and thus give magnetic bearings, or to read zero when the sights lie in the true meridian and thus give true bearings. The difference between the two instru- ments is that in the transit the circle when oriented remains fixed, while the index and vernier turn with the telescope ; in the surveyor's compass the index or north end of the needle remains fixed, while the cir- cle when set by its vernier, turns with the sights. To set the circle of the compass in the desired position, there is provided an auxiliary circular arc with a ver- nier reading to minutes. Since the arc is generally used to set off the declination of the needle so that it shall read true bearings, it is called the decimation arc, and its range and the corresponding motion of the circle are limited to include only the extreme declina- tions that may obtain in the localities where the com- pass will be used. 170 Incidentally the declination arc and its vernier may be used to read the fractional part or minutes of any bearing. Having pointed the sights in the de- sired direction, count the degrees to the mark that has been passed by the needle and read the declina- tion vernier. Then turn the circle back by its rack and pinion movement till the needle is exactly oppo- site the degree" mark, and read the vernier again. The difference of the two vernier readings will be the number of minutes to be added to the degrees. This refinement is not justified except when it is desired to determine accurately the angle between two lines by reading the bearing of each. In this case, local attraction and the variation of the needle do not affect the result. The vernier should also be used when an, accurate determination of the declination of the needle is made. The plate of the surveyor's compass is fastened centrally and perpendicularly to a socket which fits and turns on a spindle, the axis of which should be vertical. The spindle is mounted on the tripod or on a Jacob's staff by means of a leveling head or a ball and socket joint. Two level tubes are placed at right angles to each other on the plate to assure the verti- cality of the vertical axis. A clamp and tangent screw are provided to control the motion of revolu- tion of the plate. A needle stop actuated by a milled headed screw lifts the needle, from its pivot and holds it against the glass cover. The needle should be lifted from the pivot whenever the compass is not in use, or when it is carried. For reading angles of elevation there is provided a peep sight near the bottom of the rear sight leaf and a tangent scale marked in degrees and half de- 171 grees on the front sight leaf reading upward. For reading' angles of depression there is provided a peep sight near the top of the rear sight leaf and a tangent scale marked in degrees and half degrees on the front sight leaf reading downward. Sight through the peep hole at the distant point and note where the line of sight cuts the scale; the reading of the scale is the angle .of elevation or depression as the case may be. ADJUSTMENTS OF THE SURVEYOR S COMPASS. I. To make the plate levels perpendicular to the vertical axis. This adjustment is the same as the first adjustment of the transit, except that the com- pass has no telescope with attached level tube, and therefore the plate levels must themselves be used to' make the axis vertical. Set up the compass and if it has leveling screws turn the plate till the plate level tubes are respectively parallel to the lines joining op- posite screws. Bring the bubbles to the center of the tubes. Turn the plate i8o". If the bubbles remain at the center of the tubes the levels are in adjust- ment and the axis is vertical. If they do not remain at the center, turn the leveling screws, or adjust the socket joint, so that the bubbles will move back over one-half of the displacement. This will make the axis of the compass vertical. Adjust the levels by means of their adjusting screws so that the bubbles shall return to the middle of the tubes. This makes the axis of the level tubes perpendicular to the ver- tical axis. Repeat the test to check the accuracy of the adjustment, and make further corrections if nec- essary. When the compass is again set up and lev- 172 eied so that the bubbles are at the middle of the tubes, the axis of the compass will be vertical. 2. To make the sight slits vertical. Having leveled the compass, sight through the slits at a plumb line. If either slit appears to be inclined to the plumb line the sight leaf is not vertical. The error may be corrected by filing the base of the sight leaf, or by gluing one or more pieces of paper under the side toward which the leaf inclines. 3. _To straighten the needle. The ends of the needle and the pivot should lie in the same straight line. The following test and adjustment are inde- pendent of the position of the pivot, and are applica- ble whether the pivot is or is not at the center of the graduated circle. Read both ends of the needle. Re- volve the compass so that the point on the circle that was at the north end shall come to the south end of the needle, and read the north end again. If its read- ing is the same as the original reading of the south end the needle is straight. If these two readings are not the same, bend the needle in the hands so that when it is replaced on the pivot, one-half of the ap- parent error shall have been corrected. The needle will then be straight. Repeat the test to check the result. The test for this adjustment might be made by turning the needle by hand so that the south end would come around to the point previously occupied by the north end and then comparing the new read- ing of the north end with the original reading of the south end. The figure (Plate A, Fig. 2) illustrates the two positions of the needle and shows that whether the pivot be at the center of the circle or not, the test will make apparent the double error due to a bent 173 needle, and hence will indicate the correction to be applied. C is the center of the circle and O is the pivot. N O S is the first position of the needle and N'O S' is its reversed position, the points S' and N coinciding. If the points S and N' also coincide the needle is straight. If not, bend the needle till it can be made to occupy the position B O A, A being the midway between S and N. The hand is not steady enough to turn the needle and hold it accurately in its reversed position, but ex- actly the same result is accomplished by turning the compass box as first explained. It makes no differ- ence in this test whether the needle be turned on its pivot through the required angle, or the compass box be turned through the same angle while the needle remains at rest. It is easier in practice to turn the compass box, but the explanation is simplified by as- suming that it is the needle that has been turned. 4. To straighten the pivot so that its point shall be at the center of the circle. After straightening the needle it may still be found that the readings of the opposite ends of the needle do not differ by 180°. This will be due to an eccentric position of the pivot point, and can only be corrected by bending the pivot so that its point shall be at the center ofthe circle. When the compass is turned on the vertical axis, if the pivot point is eccentric this point will describe a small circle about the true center and will carry the needle with it first to one side and then to the other side of the true center, causing a varying difference of end readings. (Plate A, Fig. 3.) The discrep- ancy is zero when the line joining the true center and the pivot is parallel to the needle, and the discrep- ancy is a rnaximum when the line joining the true 174 center and the pivot is perpendicular to the needle. To correct the error turn the compass to the position that gives the maximum discrepancy of end readings and then bend the pivot in a direction at right angles to the needle till the discrepancy disappears. The pivot point will then be at the center of the circle. If there is a discrepancy of end readings of the needle that cannot be corrected by straightening the needle and the pivot, it is due to inaccuracies of grad- uation and cannot be corrected.' 5. To adjust the vernier of the declination arc. When the vernier of the declination arc reads zero the 0° — 180° line (or 0° — o" line) of the compass circle should lie in the plane of the sight slits. Stretch a fine thread from the bottom of one slit to the bottom of the other slit and see that it bisects the openings. Sight through the top of either slit down- ward toward the thread and turn the compass circle by its rack and pinion till the zero line coincides with the thread. If the vernier now reads zero it is cor- rectly adjusted. If not, adjust the vernier without turning the circle, so that it shall read zero. If the •vernier is not adjustable, its reading must be treated as an index error, to be applied in setting off declina- tions. 6. To remagnetize the needle. A sluggish needle is a fatal defect in a compass and no reliance can be placed on its readings. If a compass be set away, when not in use, with its needle stopped in the reverse position with respect to the inagnetic meridian, the needle will in time lose part of its magnetism. When again released the needle will move sluggishly and will not settle exactly in the magnetic meridian, being checked by the slight 175, friction of the pivot. When a compass is put away, therefore, the needle should be allowed to settle in the magnetic meridian and then be lifted by the stop. In this position the earth's magnetism will tend to maintain that of the needle. If the needle has become sluggish through loss of 'magnetism it must be remagnetized. This is best accomplished by placing the needle, with proper at- tention to polarity, in the field of a strong electro- magnet. A few minutes will serve to revive its magnetism. If an electro-magnet is not available a permanent bar magnet may be used as follows : Stroke each half of the needle from the pivot toward the end with that end of the bar magnet that attracts the part stroked. In other words, stroke the north end of the needle with the south seeking end of the magnet and vice versa. 7. A sluggish needle may be due in part to a blunted pivot. 'When this is the case, unscrew the pivot and sharpen it on an oil stone, taking care to turn it constantly and preserve its conical form. 8. To test the scale of vertical angles. A zero reading on this scale should indicate a horizontal line. Level the compass carefully and sight through the peep hole and the zero mark of the scale at a rod held vertically, and three or four feet from the compass. Mark the point pierced by the line of sight. Reverse the compass 180° and sight from the zero mark through the peep hole at the rod and mark the point now pierced by the line of sight. If the two points coincide in height, the zero line is horizontal. If the second point is below the first, the last line of sight inclined downward and the zero 176 mark was too high. If the second point is above the first the zero mark was too low. The error may be corrected by glueing slips of paper under the lower sight leaf and thus correcting one half of the ap- parent error. Whenever glue is used a coat of var- nish should be applied to protect the glue from damp- ness. Another way to make this correction is to' bend the plate slowly and carefully in a vice, but this should not be attempted by a novice. The compass attached to a transit is similar to the surveyor's compass, the telescopj taking the place of the sights. To make the fifth adjustment (vernier of declina- tion arc) of the transit compass, proceed as follows : Set the declination vernier at zero. Lay a straight- edged ruler across the compass box and adjust it so that its edge shall coincide with the zero line of the com- pass graduations. Clamp the plates. Sight along the straight edge and set a mark in the line of sight so determined at a distance loo feet -or more. Sight through the telescope at the same mark. If the cross wires bisect the mark the adjustment is correct. If not, turn th'e telescope and bisect the mark. Turn the straight edge and again bisect the mark with it. Turn the compass circle by the rack and pinion move- ment and bring the zero line to coincide with the straight edge. Adjust the vernier so that it shall read zero. If not adjustable the error must be treated as an index error. TO SET OFF THE DECLINATION. Set up the compass and bring the needle and the declination vernier to a zero reading. If the declina- 177 tion be west D degrees, turn D degrees to the east. If the declination be east D degrees, turn D degrees to the west. The sights or telescope will now be pointing' in the true meridian. Clamp the plates and with the rack movement turn the compass circle till the needle again reads zero, setting the declination off accurately by means of the declination vernier. When the declination has been set off the compass will give true bearings. THE PRISMATIC COMPASS. The. prismatic compass is a card compass with front and rear sights, the rear sight having an at- tached magnifying prism which reflects an image of the graduated edge of the card into the eye when a sight at the distant point is taken. With any other form of the compass, the sighting line must first be pointed at the distant object and then the compass reading be made, but with the prismatic compass the graduated circle is read simul- taneously with- the pointing of the sights. The index of this compass is the rear sight, and since it is desired that the compass read zero when pointing north, 90" when pointing east, etc., the zero should be placed at the south of the card, 90° at the west of the card, 180° at the north, and 270° at the east. The mirror in the prism reverses the image of the graduation marks, and the numbers on the card must therefore be printed in reversed form like the face of the printer's type, in order that the rnirrored image may be direct. 178 The prismatic compass is the most accurate of the hand compasses and is a valuable instrument for rough surveying or careful sketching. BOX OR MILITARY COMPASS. This is a simple needle compass mounted in a rectangular block of wood with a hinged cover. In some, the edge of the cover when raised to a vertical position is the sighting line, and in this case, holding the cover toward the right, the zero should be at the forward side, the go° mark at the left, the 180" mark at the near side and the 270° mark at the right. In another form a white line is marked on the inside of the lid. When the lid is wide open this line is in the prolongation of the 0° — 180" line of the compass. To point the compass, open the cover about 120° and turn it to the front. Hold the com- pass in a level position and a little below the eye. The white line will be seen reflected in the glass top of the compass. Make this reflected line coincide with the 0° — -180° line of the compass and while main- taining this coincidence sight over the top of the white line at the distant object. When the needle comes to reet, read its north end. Many forms of pocket compasses are manufactured, but they are generally unsuitable for surveying or sketching. For these purposes a needle compass should have the following features: 1. A good sighting line. 2. A light sensitive needle with a stop. 3. A raised circle lying in the same plane with the ends of the needle. 179 4. Plainly marked graduations dividing the cir- cle into single degrees and numbered continuously from zero to 360" in counter-clockwise direction. The 0° — 180° line should be parallel to the sighting line, with the zero at the forward side. 5. A strong, water-tight case. For the card compass the requirements are the same except that the graduated circle is marked on the card, the numbering should be in clockwise direc- tion, and the index shoyld be so placed as to read zero when the sights point north. To plot the bearings given by a compass. It has been explained that compasses are grad- uated and marked in several different ways, and that the same direction with reference to the meridian may be expressed by a number of different compass readings, depending on the kind of compass and the manner of its marking. In order, therefore, to plot the bearings given by any particular compass the pro- tractor should be marked or numbered to correspond with the readings of the compass. This does not mean that the protractor should be marked like the circle of the compass used, but rather that it should be marked to give the same readings as the compass. For example, a needle compass, like the box com- pass, that reads zero when pointing north, qo° when pointing east, 180° when pointing south and 270* when pointing west, is graduated to the left or coun- ter- clockwise and has the zero mark at the forward side, the 90° at the left, 180'' at the near side and 270^^ at the right, whereas a prismatic compass that gives the same readings would be graduated to the right or clockwise, and would have the zero at the south point 180 of the card, 90° at the west point, 180° at the north, and 270'' at the east. For both of the compasses just described the mark- ings of the protractor should be the same and should be numbered in clockwise direction from zero to 360", and the zero mark should be placed toward the north end of the meridian when the center mark is at the station point. ^ The rule to be followed for determining the proper marking of a protractor to correspond with any com- pass is as follows: Point the sighting line of the compass to the north and read the compass. Mark this reading on the pro- tractor at the north end of the diameter that is to be placed in coincidence with the meridian line on the plot. Next point the compass to the east and read it. Mark this reading on the protractor at a point 90° to the right of the north reading. Then point the compass to the south and read it. Mark this reading at the south end of the meridian diameter of the pro- tractor. Lastly, point the compass to the west and read it. Mark this reading on the protractor at a point 90° to the left of the meridian diameter. Mark also the intermediate points of the protractor at 10" (or 5°) intervals to correspond with intermediate com- pass readings. The protractor will then plot, with- out reduction, the recorded readings of the compass. If the protractor has been marked by the maker with numbers that do not correspond with the com- pass used, it is onl)' necessary to disregard those numbers and mark the proper numbers with pencil or ink. Conversely, if the compass is not marked in the desired manner, new numbers may be added and the old ones disregarded. 181 A plotting diagram for any compass consists of two lines crossing at right angles, and marked at their ends with the N, S, E and W readings of the compass. It indicates the proper markings of the protractor to be used with that compass. The figures on Plate A, Fig. 4, give the compass graduations and correspond- ing plotting diagrams for four compasses, the first two being box or needle compasses, and the other two being prismatic or card compasses. For any- other compass the plotting diagram would be con- structed in a 'Similar manner. In military topographic sketching it may be neces- sary for compass notes taken by one person to be plotted by another. Even when the sketcher plots his own notes he may be working with a compass furnished him for the particular occasion and differ- ing in its marking from the one he is accustomed to using. To remove all chance of error a plotting dia- gram of the compass used should be placed at the be- ginning of every set of compass notes. COMPASS SURVEY. PARTY EQUIPMENT Surveyor Compass, note book, etc. Front chainman Crayon and marking pins Rear chainman - Chain and note book Rod and axeman Rod and axe The principles and methods involved in survey- ing with compass and chain are the same as those of a transit survey except that distances cannot be measured with the stadia, and must be chained. The surveyor directs the work and uses the com- pass. He sets up the compass at the selected initial 182 station and sees that the declination vernier reads zero, if magnetic bearings are desired, or that it reads the proper declination, if true bearings are desired. He reads bearings and vertical angles to selected critical points and directs the chainmen to measure the hori- zontal distance from his station to these points. The last observation at station zero is made to determine station i, which is so selected as to give a good view of surrounding critical points and to advance the sur- vey in the desired direction. He records bearings, vertical angles, distances and points observed, on the left hand page of the note book, and on the right hand page makes a free hand sketch of the ground covered, showing all points observed and the objects and features so determined. FORM OF RECORD. ( Title page, description of instrument, names of party, etc., same as for transit survey : ) Bearing. Vert. Ang. Hor. I?ist. Di£E. Elev. Elev. Point Observed. At Station 0. Ele V. 568. N. 56° E. -4° 332 — 23.2 544-8 Fence cor. et S. S% E. c. + 3X 428 -f-26.I etc. 541-9 Sta. I At N. 5X W. station -3^ I. 428 Ele — 27.8 V. 541.0 540.2 B. S. Sta. 183 If only a boundary or a line is to be run without reference to topography, the columns, Vert. Aug., Diflf. Elev., and Elev. are omitted. Having taken all necessary observations at station zero the surveyor stops the needle and carries the compass to station i. The first sight from station i is a backsight at station zero, taken to verify the for- ward readings from that station of bearing and Vert. Ang. The operations already described are repeated at successive stati ms to the end of the traverse, which should, if possible, close on the initial station or on a known point. The chainmen measure all required distances as directed by the surveyor. The operation of chaining will be described under the head of "Measurements of Distance." The rodman is provided with a staff on which is a mark whose height above the ground is equal to the average of the heights of the two peep holes on the rear sight leaf of the compass. The rodman holds this staff at the "point observed and the surveyor sights at the mark when reading vertical angles. Great accuracy in determining elevations cannot be expected by this crude method of measuring vertical angles, but with care fair results can be obtained in a small survey. Since horizontal distances are measured, the dif- ference of elevation is obtained by multiplying the horizontal distance by the tangent of the vertical angle. The elevation of a " point observed " is found by adding algebraically the "difference of elevation" to the " elevation" of the station occupied. 184 After the notes are reduced, the adjustment of the traverse, the plotting of the traverse, the plotting of the side shots, and the drawing of the map, are the same as the like operations in a transit survey. When the foresight and backsight bearings be- tween two stations do not agree it is probable that there is "local attraction" at one of the stations. Select a third station at as great a distance as is con- venient from the other two and read foresights and backsights on each side of the triangle. The station at which all foresights and backsights disagree is the one probably affected by local attraction. If the .foresight and backsight agree between the two other stations they are probably not affected. A station affected by local attraction should be aban- doned if one not so affected can be selected. In regions containing deposits of magnetic ores, local attraction is so general that compass bearings cannot be relied on. In this case the compass may be oriented^ at each station in a manner analogous to that applied in orienting the transit ; thus, backsight at the previous station and turn the circle with the pinion movement so that the backsight shall agree with the foresight. 9£. COMPASS GK/^DUA TtON- /7 M IH w ^rn c^r^ ^ s:2. S*ti0- PhdSS? Ln ■ * P-" M M 10 M M M '^ *J (Jl B s ^ I I ni S O en <-n. O O O tr ^ M "= W c o Jk 5' H ^ ^ o M M t. W ^i" t-" to O Ut to o O -pi M to p. < ? > w o B *. I-I pa to to O M r.^ l/X c to O w +■» w 0^ oo 00 ^ OO w u\ c^ M vO • o w o t^ Pi to *^ Q 1 < 73 M ° r 00 CO o H H 4i. *. o M Ul c^ O bo -.^ vO SO ^ O O 4^ OT (ji (H r+ LH O W o w o p. 5S m 3 n o a o c O O o D 278 The party should consist of observer, level rod- man, stadia rodman and axeman. Set up the transit at the initial station (i) and orient by the compass or by sighting at a triangulati-on station whose azimuth is known. Level the telescope by its level tube and backsight ( + 6.316) on the B. M. whose known eleva- tion is 880.374, finding H. I. =886.690. If contours are to be run at five feet V. I., the contour next below this H. I. will be 885, and the target will be set on the rod at 1.690. The stadia rod is now placed at B. M. and and the observer reads azimuth, bearing and stadia distance to locate the B. M. The level rod is then carried up or down the slope as signaled by the observer until, when resting on the ground, its target is bisected by the horizontal wire of the telescope. Since the H. I. is 886.690, and the target is set at 1.690, the foot of the rod will rest at a point of the 885 contour. The stadia rod is placed at this point and its azimuth (225-00) and stadia distance (451) are read and recorded. Other points on the 885 contour are in like manner found by placing the level rod and located by readings on the stadia rod. When a new station is required a T. P. is estab- lished by a F. S. ( — 2.000) giving its elevation 884.690, and the selected new station (2) is located by azimuth 230-15, stadia 420 and vertical angle — 0.13. This angle is so small that no reduction to the horizontal is necessary, and the horizontal distance is 420. Setting up at station 2, the transit is oriented by a backsight (50-15) on Sta. i, and the new H. I. is determined by a backsight (+5.903) on T. P. i, giving H, I. =890.593 and a new setting of the rod (5-593) for the 885 contour. 279 When 885 has been run out to the limits of the survey, a new station and T. P. are selected and located for beginning the next contour, say 880. A F. S. (—9.725) gives elevation 880.868 for T. P. 2, and azimuth 220-00, stadia 224, vert. ang. — 5.30 (giving horizontal distance 222), locates station 3. At station 3, a. backsight 40-00 on Sta. 2, orients the transit and a backsight (+ 2.000) on T. P. 2 gives H. I. 882.868, and a rod setting of 2.868 for the 880 contour. If the instrument stations be properly selected with reference to H. I., two contours may be run at the same time by using two targets or marks on the level rod. Thus, in the foregoing example, at sta- tion I set one target at 1.690 and another at 6.690. The first will locate points on the 885 contour, and the second, or upper target, will locate points on the 880 contour. Vertical angles are read on traverse courses only, and are used to reduce stadia readings to the hori- zontal. Elevations are carried along by the level observations. The telescope must be re-leveled for each sight on the level rod. If a rod be made to show stadia readings on the face and level rod readings on the back, only one rodman will be required. He turns the level rod markings toward the observer for the level readings, and the stadia face toward the observer for stadia readings. On a long straight stretch of contour only a few points need be determined, but on broken ground, where the contours curve irregularly, numerous points are required, and especially those at salients and reentrants. 280 In plotting this survey the traverses are first ad- justed and plotted by methods previously described. Then side shots are plotted and contours are drawn through the points having the same elevations. No interpolation of the contour points is required in this case. The notes should, of course, include readings necessary to locate roads, buildings, streams, etc., and these are plotted and drawn in the usual manner. P^.AT£ ZST 283 7=*L/}T£ 7Z • <^ rig, z r.F.J'i- i^4'4' T£ i". ty (4 riATE:xi: 1 to 1<0 c4 289 PLAT a 'SJT' >i 291 CHAPTER XII. THE CHAIN, The chain is made of short pieces of steel wire joined end to end by small rings of the same material. A link is the distance included between the middle points of adjacent joints. The handles at the ends of the chain are included in the end links. The length of the chain is the distance from outside to outside of the handles when the chain is stretched taut. The Gunter's or Surveyor's Chain has loo links, each 7.92 inches long. One chain = 4 rods = 22 yards = 66 feet ^100 links = 792 inches. One mile =^ 8 furlongs = 80 chains. One acre = 10 square chains. One section = i square mile = 640 acres ^ 6,400 square chains. The Surveyor's Chain is convenient for measuring land in order to determine the acreage. The links are recorded as hundredths of a chain ; thus, 8 chains and 35 links is written 8.35 chains. When the area is computed in square chains, the acreage is found by moving the decimal point one place to the left. Thus a rectangle 8.35 chains wide and 10 chains long con- tains 83.5 square chains or 8.35 acres. Ten chains = i furlong == yi mile, therefore the surveyor's chain is convenient in laying out a race track where the quarters and eighths of a mile are important units. 292 The Engineer's Chain has loo links, each 12 inches, long, and it is therefore a scale of feet, 100 feet long. Half chains of fifty links are also furnished by the makers. These are 33 feet long for the surveyor's half chain and 50 feet long for the engineer's half chain. The metric chain has 100 links each 0.2 of a meter or 7.874 inches long. It is therefore 20 meters in length and has 5 links to the meter. It is easily m'is- taken for a Gunter's chain, as it is only 4.6 inches shorter. Chains are usually compared with a standard and adjusted to standard length by the makers, who will furnish also a statement of the pull or tension required to stretch the chain to standard length at a stated temperature. New chains may therefore be depended upon to give fairly accurate measurements of dis- tance. With use, the rings and loops wear away at the points of contact, and may become elongated under tension, or may be partly opened if they are not brazed together. These causes tend to lengthen the chain, and an adjustment to standard length be- comes necessary. This is done, if a standard is avail- able, by comparing the chain with the standard and adjusting one of the handles by means of its adjust- ing screw and lock nuts so that the chain under nor- mal tension and temperature shall be equal to the standard in length. When a chain is used to measure a line with the greatest attainable accuracy a correction for tempera- ture must be applied. This correction is the same for the chain as for the steel tape and will be ex- plained in connection with the latter. 293 Steel pins are used to mark the successive chain lengths on the ground. They are called "marking pins," and are about 14 inches long, pointed at one end and bent into a loop at the other in order that they may be strung on a ring of strong wire like keys on a key-ring. Eleven pins form a set. One pin marks the origin or initial point of the line to be measured. When the other ten pins have been used, a record of ten chains is entered in the note book, and the eleventh pin becomes the initial point for further measurements on the same line. Two pins of the set should be weighted each with a bob of lead near the point. These pins may be attached to a string to provide plumb lines for use when chaining over slop- ing ground. The chain should be stretched horizon- tally, and if the ground is uneven one end, or perhaps both ends, of the chain will be held above the ground. A plumb line is then needed to place the rear end of the chain over the last marking pin or to set a new pin under the forward end of the chain. 'Yo fold the chain begin at the middle and fold the links together pair by pair, holding the bunch of links in one hand and folding with the other. Bind with strap or cord. To open the chain remove the strap or cord and cast the links forward along the line while retaining both handles in the grasp. TO MEASURE A LINE. PARTY EQUIPMENT _ , . I Chain, range pole, one weighted Rear chamman. - ] pin. note book, etc. Fore chainman, '^'"hSt^ ^'"'' '^°^' ^°^'' 294 Mark the initial point with a pin, or better, with a range pole, and mark the distant objective point with a range pole if it is not already marked by some definite selected object. The fore chainman takes one handle of the chain and moves off along the line till the chain is stretched taut. He then faces to the right, kneels on the left knee and braces the right arm against the inside of the right knee. He holds a pin and the handle of the chain in the right hand and moves to the right or left as directed by the rear chainman till the pin is on line. At the signal " Down " he forces the pin vertically into the ground with the left hand and an- swers " Down." He then proceeds along the line dragging the chain and clearing away brush, weeds or other obstructions that would foul the chain. He halts at the signal of the rear chainman and marks the end of the second chain in the same manner as the first. When the pin is set the forward sides of pin and handle must be even. The fore chainman can place himself nearly on line by sighting back over the last pin at the rear range pole, but the ac- curate alignment is maintained by the rear chainman who sights toward the forward range pole and lines in each new pin. Rags of red or white cloth tied to -the pins make them clearly visible. The rear chainman directs the work. When the chain is first stretched along the line he examines its whole length to see that there is no kink or twist at any point. At the initial point he holds the end of the chain (outside of handle) against the front of the pole or over the center mark on the stake, lines in the pin in the hand of the fore chainman by sighting at the forward range pole, calls out "Down" when 295 the alignment is correct, and upon hearing the an- swer, moves forward carrying the rear handle. When he reaches the first pin he halts the fore chairman, holds his handle against the front side of the pin, lines in the next pin, and then pulls the first pin and again advances. When the fore chainman has used his last pin there is one pin in the ground and ten in the hands of the rear chainman, and a distance of ten chains has been covered. This distance is recorded in the note book or clicked on a tally register, and the ten pins are handed to the fore chainman. The chaining is then continued, starting at the pin that remained in the ground. When the objective point is reached the fore chainman halts and holds the end of the chain at this point. The rear chainman lays down the chain, counts and records the number of pins that he has pulled, advances tO' the pin last set, stretches the chain, and reads the number of links between the pin and the objective point, which number is re- corded as hundredths of a chain. The number of record marks in the note book gives tens of chains. The number of pins in hand when the objective point is reached gives units. The number of tags between the last pin and the ob- jective point gives tenths of a chain, and the number of links between the last pin and the nearest forward tag gives hundredths of a chain. A part of a link next to the pin may be estimated to the nearest tenth and recorded as thousandths of a chain. For example. Six record marks in the note book. Four pins in hand. 296 Seven tags between last pin and objective point. Three whole links between pin and next tag, Five-tenths of a link next to the pin, gives a total distance of 64.735 chains if it is a Gunter's chain, or 6473.5 feet if it is an engineer's chain, or 1294.70 meters if it is a 20-meter chain. The Gunter's chain and the engineer's chain give decimal readings, while the readings of the 20 meter chain must be multiplied by twenty to give the dis- tance in meters. A more convenient metric chain would be a chain 10 meters long with fifty links, each two decimeters in length, and with a tag at every fifth link. Each lo-chain record would be 100 meters, each pulled pin would be tens of meters, each tag would be one meter, each half-link would be one decimeter, and each tenth of a half -link would be one centimeter, making the decimal system complete. In chaining up-hill the rear chainman must raise his end of the chain above the ground to make the chain horizontal and hold it in a vertical line through the marking pin. To do this he uses the plumb line with the weighted pin for a bob, and brings the point of the bob against the marking pin while holding the handle against the line at the proper height. In chaining down hill the fore chainman must in like manner raise his end of the chain to the hori- zontal and with the plumb line find the point at which to set the marking pin. Otherwise, he may while stretching the chain, hold the weighted pin under and against the handle, and when aligned drop the pin. He pulls the weighted pin, sticks another in the same hole, and advances to the next point. By using a staff to support the hand holding the 297 raised end of the chain greater steadiness and accu- racy are secured. On steep slopes the end of the chain cannot be raised high enough to make the chain horizontal. In this case parts of the chain (halves or quarters) may be used in succession to keep the raised point within reach. Otherwise, stretch the chain on the ground and measure the slope with a clinometer or a slope board. Record each chain so measured, not as one chain, but as links, by a number equal to lOO X cosine of slope. This reduces the inclined measure- ment to horizontal distance. In practice, having measured the angle of slope of the chain length, take the cosine of this angle from a table and record it with the decimal point moved two places to the right. (Plate 13, Fig. i.) When part of the chain is thus used, the number of links in the partial chain must be multiplied by the cosine of the slope, and the result entered in the record. The sum of the recorded numbers will be the total horizontal distance in links. With the engi- neer's chain the distance will be given in feet. CHAINING WITH AN INCORRECT CHAIN. A chain, rod, wire or tape of any length may be used to measure a line if the length of the chain, or rod, etc., be known. If a chain has been elongated by use, or if it has been shortened by the loss pf some of the links, it may still be used as a measuring unit, if its new length be ascertained. It is only neces- sary to multiply the number of chains applied in any 298 measurement by the length of the chain. The result is the required distance. Thus, if an engineer's chain has worn at the joints so that its length, as ascertained by comparison with a standard, is 100.065 feet, a record of 6.834 chains gives a distance of 6.834 X 100.065 = 683 844 feet. But if some of the links, say four, liave been lost, and the remaining links, 96, are correct, a record of 6 chains and 83.4 links, gives a distance of 6 X 96 links + 83.4 links = 659.4 links. It is often required to lay off on the ground from a given point a given distance, and thus fix a required point. With a correct chain this is readily done since it is only necessary to apply the chain and links along the ground in the given direction the required number of times. But with an incorrect chain such application would not give the required point cor- rectly. To lay off the given distance, divide that dis- tance by the actual length of the chain and laj' off the resulting number of chains and links. Thus, let it be required to lay off 683.844 feet with an engineer's chain that is known to be 100.065 feet long. 683.844 -h 100.065 ^= 6.834 of these chains. Therefore lay off 6 chains and 83.4 links. The point thus determined will be 683.844 feet from the given point as required. LINING IN. To line in is to cause a pin, pole, stake or other mark to be set in the ground on a certain line. The line may be determined by the pointing of an instru- ment, such as a transit, compass, sextant, or alidade of a plane table, or it may be determined by sighting 299 through two points that have been set or selected on the ground. When an instrument is used the observer signals to the rodman to move the rod to a point that is bisected by the vertical wire or sight of the instru- ment. If the required point is also to be fixed in elevation, the rodman is signaled to fix a mark or target on the rod at a point that is bisected by the horizontal wire or sight of the instrument. When the line is determined by two points that have been marked by stakes or poles, a third point may be lined in as follows: First. When the two given points are mutually visible: (a) To line in a third point between the two given points, the surveyor sights from one marked point to the other and signals an assistant to set a mark in the line of sight so determined. If the surveyor is alone he sights from one marked point through the other and selects a third point that is in this line of sight; then moves forward to the desired point and sets a stake in line with either mark and the selected point. {^b) To line in a third point beyond the two marked points, the surveyor places himself on line at the de- sired point by sighting through the two marks, and sets a stake in the line of sight so determined. When a third mark has been lined in, any two of the three marks may be used to fix a fourth point, and thus a- line may be prolonged indefinitely by marking new points in line with any two points pre- viously marked. The greater the distance between the two marked points that determine a line, the more accurate will 300 be the lining in of a new point on that line. There- fore in prolonging a line, the first mark or origin of the line should be used as one sighted point when- ever it is visible from the newly determined point. In prolonging a line over undulating or wooded land the initial point is soon lost to sight and generally- only the last two points will remain visible. In sur- mounting hills or crossing valleys, the range poles that determine the line, and the stakes that mark the line, must be placed at short intervals to be vis- ible from the new point, but on open ground of uniform slope three marked points will keep the chainmen always on line, no matter what the included distance. In prolonging and measuring a line in a known direc- tion two classes of obstacles may be encountered, viz : those that obstruct the view, and those that obstruct progress along the line. When the view is obstructed by an obstacle that cannot be cleared away, the line must be carried around the obstacle by offsets and then continued as before. When progress along the line is obstructed by a pond, a marsh, or by private property to which access is denied, the line may still be prolonged by sighting across the obstacle, but the chaining must be carried around the obstacle by offsets, or across the obstacle by triangulation. Many methods may be devised for passing either class of obstacle. The following simple cases are given as examples: I. To pass an obstacle that obstructs the view, such as a building or a thicket or tall growing crops. (Plate 13, Fig. 2.) Line in and chain to C. At A and C, 301 on the line, 200 or 300 feet apart, lay off perpendicular offsets of equal length, A B and C D, and prolong the line B D, parallel to A C, to some point as G be- yond the obstacle. Chain D E. At E and G lay off perpendicular offsets, E F and G H, equal in length to A B and C D. The line F H will be in prolonga- tion of A C, and the lining in and chaining may be continued from F. Otherwise, thus (Plate 13, Fig. 3) : Having lined in and chained to B, construct on A B (one chain long) the equilateral triangle ABC, and prolong C B to E. Chain BE. On D E construct the equi- lateral triangle D E F and prolong E F to G, making E G equal B E. On H G construct the equilateral triangle H G K. The side K G will be in prolonga- tion of A B, and the triangle BEG will be equilateral. The distance B G is therefore known because it is equal to B E or E G. The line K G may now be pro- longed and chained. 2. To pass an obstacle that obstructs progress along the line, but which does not obstruct the view. (Plate 13, Fig. 4.) Prolong the line by sighting across the obstacle and lining in C E. Having chained to A, lay off perpendicular offsets A B and C D of equal length, and chain B D. Then continue chaining on the line C E. If the obstacle is a stream too wide to be spanned by the chain it may be passed by the method indi- cated in Plate 14, Fig. i. Make the angles at A and B each equal to 60° by constructing equilateral tri- angles, and chain A B. Find the point C beyond the stream where A D and B E prolonged intersect. Then A C = A B, and the chaining may be continued from C. 302 Otherwise, (Plate 14, Fig. 2) construct the 3-4-5 triangle ABC, right angled at A, and at any point D, beyond the stream erect a perpendicular and find the point E where it interests C B prolonged. Meas- ure D E. Then DE: D B:: AC: AB: : 3:4. Hence DB==f DE. Second. When the two given points are not visi- ble each from the other. In this case the line joining the two points can- not be determined by sighting direct from one point to the other, and the surveyor cannot begin at one point and chain toward the other because he cannot see the other. He must first determine and mark on the ground, by marks that are mutually visible, the line that passes through the two known points. The line thus established may be prolonged, by the methods already explained, from one of the given points to the other, and the line may then be chained. To establish and mark the line between two points not mutually visible different methods may be used, depending on the distance between the points, the kind of obstacle that obstructs the view and the accuracy necessary to fulfill the purpose of the survey. If the given points are several miles apart it will be best to make a survey by running an instrumental traverse from one point to the other along the most convenient route. By plotting this traverse the di- rection and length of the straight line joining the two points may be determined. The direction being known, the line may now be ranged out and marked on the ground if desired. Thus (Plate 14, Fig. 3), the direction and length of the line joining the two points A and B are required. The ground between 303 them is undulating and partially wooded, so that the view is obstructed, and standing at either point, the exact direction of the other is not known. Run the traverse A, i, 2, 3, 4, B, with transit, compass or plane table and plot the survey. (Plate 14, Fig. 4.) Draw the line ab. The scale of the plot gives the distance {2yi miles), and the protractor gives the bearing (N 50° E). If the line is to be staked out on ground, set up the instrument at A and point it in the known direction (N5o°E). Prolong this direction by clearing out the line and setting stakes at conven- ient intervals. If the determination of the direction was not accurate the prolonged line will not pass exactly through the point B, but when a point oppo- site B is reached the error can be determined and the line readjusted by moving the stakes to their correct positions on the line A B. Having cleared out the line and marked it at points that are mutually visible, the line may be chained and the distance determined with greater acciiracy than was attained by travers- ing. For shorter distances the following method may be used : Let A and B (Plate 15, Fig. i) represent the two given points which by reason of intervening hills, or trees, or buildings, are not miitually visible. Starting at either point, as A, estimate as nearly as possible the direction toward the point B and range out a trial line, leaving stakes d, e, f, etc., at measured intervals. Arriving at point C where B is seen at right angles to the trial line, erect the perpendicular C B and measure it. • Then correct the position of each stake by moving it on a perpendicular to the trial line over 504 a distance that is proportional to its distance from A k A. Thus, k is moved to k' making k k' = B C, A C etc. This method may be used when an instrument is not available. If the two points are on opposite sides of a hill or ridge, from the top of which .both points can be seen, two rodmen may range each other in by suc- cessive approximations till both are on the desired line. Thus (Plate 15, Fig. 2), A and B being two given points separated by a ridge, the first rodman at C ranges in the second rodman at D by sighting at B. Then the second rodman ranges in the first at C and so on till both are on the line A B. In any case, after the line has been marked at points that are mutually visible, the chaining may be done as already described. With ordinary care in chaining an error of i in 1000 should not be exceeded. With greater care and by using a spring balance to regulate the pull and applying the correction for mean temperature, the error in chaining should be reduced to i in 5000 at least. For ordinary purposes the chain is the most satis- factory and cdnvenient of distance measuring in- struments. It is durable, and may be dragged over rocky ground or through brush without injury ; it is easy to use and rapid in operation. Its decimal di- visions make the record simple and facilitate compu- tations. The only care that it requires is to wipe it dry after use and oil it to prevent rust. It should be 305 tested from time to time and adjusted to standard length for normal pull and temperature. For many purposes, and especially for use over broken ground, the half chain of 50 links is prefer- able to the full chain. In selecting a chain, get one of strong steel wire, not smaller than No. 1 5 gauge, and be sure that the rings are brazed into closed cir- cles. If not brazed they will open and the chain will part. If the rings are oblong in form they will wear always at the same point and the chain will be length- ened with but little use. A bent link, when straight- ened, is permanently elongated. Frequent bending and straightening links will therefore lengthen the chain. A novice in surveying is inclined to ignore the precise methods prescribed in text books and the pre- cautions recommended therein as being trivial and unnecessary. A single attempt at closing a traverse, or measuring a long line whose length is accurately known, will convince him of his error. Haphazard methods improvised by the inexperienced, will result only in absurdities. Nothing but a carefully devised system, thoroughly carried out in all its details, will yield even fair results. For good results, infinite care, attention and forethought are necessary. /oo X C os ^ .^........^^^^ Fig Z no I _^ Fjo Z ■^- .+ i-^^^i ^r^:^ <^'J — N-^ F/g4 ■ zi M/ic^ r --\ Ja/e Jm-.Jmi PLATE: 14 \b r/G 1 •^ — ^ /^""^ -i'\ c 5^ -J!sf=' i = y^^ 2 Pi.AT^ IS. 313 CHAPTER XIII. THE STEEL TAPE. The Steel Tape is a narrow ribbon of spring tem- pered steel. Some tapes are marked by etched lines and numbers, others by brass sleeves. Tapes are made in any desired length up to looo feet, and are graduated in any divisions that may be ordered. The Surveyor's chain-tape is 66 feet long and is divided into lOo links. The Engineer's tape is usually lOo feet long and is graduated into feet, tenths and hundredths. The Architect's tape is 50 or 100 feet long and is graduated into feet, inches and eighths of inches. The longer tapes, 400 to 1,000 feet in length, are usually marked only at the 10 ft. or 100 ft. intervals, with smaller divisions into feet and tenths of a foot near the ends, and sometimes also near each 100 ft. mark. Tapes graduated to meters, yards, varas, or any desired unit may be obtained. For convenience in carrying and handling, tapes are wound upon a reel either in a leather case or in an open frame. The latter is to be preferred because it permits the evaporation of moisture and thus pre- vents rusting. 314 The steel tape is the most accurate and convenient of instruments for measuring distances. The makers will furnish a record of the test applied in standard- izing the tape, giving the pull in pounds necessary to stretch the tape to standard length at a stated stand- ard temperature (generally 62' Fah.), when supported throughout its whole length or when supported only at its ends, or when supported at stated intervals. In making measurements in the field, the same conditions of pull and supports as were used in the test should be applied. The stretch or elongation due to pull may be made to balance the shortening due to sag, and there will remain only the tempera- ture and slope corrections to be applied. The temperature correction. Steel expands or con- tracts 0.000065 of its length for a change in tempera- ture of I ° Fahrenheit. -With a rise in temperature it is lengthened, and with a fall in temperature it is shortened. The standard temperature of tests is usually 62'' Fah. At other temperatures the tape, when properly stretched and supported, will not have the length that is marked on it, and its true length must be determined. It may be accepted as an axiom that a 100 ft. tape is not 100 feet long. Let Ls ^ tested length of tape at standard temper- ature. This is the length that is marked on the tape. Tg = standard temperature = 62° Fah. Tt = temperature of tape when used, and Lt = length of tape at temperature Tt when prop- erly stretched and supported. Then Lt = L, + 0.0000065 ( Tt — T, ) L, Or Lt = Ls [ I + 0.0000065 ( Tt — T, ) ] 315 Rule: — Subtract algebraically the standard tem- perature from tbe temperature of the tape when used. Multiply the result by 0.0000065. Add unity alge- braically, and multiply the result by the standard or marked length of the tape. The result will be its true length, that is, the length of a straight line join- ing the two points at which the end marks of the tape were held, provided that the tape is pulled and sup- ported as it was in the original test. When the tape is applied a number of times suc- cessively in measuring a long line it is not necessary to apply the reduction for temperature to each tape length, but only to apply the reduction for mean tem- perature to the sum of all the tape lengths. Thus, if six lengths of a 100 ft. tape have been applied in measuring a straight line, with temperatures of 68°, 68^°, 6g}4°, 70°, 71° and 71^" respectively, the mean temperature is 69^", and the length of the line is L = 600 (i + 0.0000065 [69^, — 62 ]) = 600.030 ft. When a partial tape is used to complete the meas- urement, and to close on the objective point it must be reduced separately, since each temperature must be given a weight equal to the length of tape to which it applies. Thus, if a length of tape marked 46.35 ft. were applied with a temperature of 73° in addition to the six full tapes, the corrected partial tape would be — L = 46.35 (I +o.oooo"o65 [73— 62 ]) = 46-353 ft- giving a total distance of 646.383 ft. Some tapes are marked by the maker with a tem- perature correction ; that is, there are different 100 ft. marks corresponding to different temperatures. The 100 ft.^mark moves 0.0065 of a foot for a change in 316 temperature of io° F. In using such a tape the tem- perature is read and then the mark which is num- bered with the corresponding temperature is used as the loo ft. mark (Plate 15, Fig. 3). With such a tape it is not necessary to make the temperature reduction except for partial tape lengths. The refinements of using a tested tape with a pull, measured by a spring balance, that will compen- sate for the sag, and of applying the correction for temperature, are adopted only when a distance is to be measured with extreme accuracy, as in measuring a base line for an important triangulation survey. To measure a base line with an error less than i in 50,000. Such accuracy is attainable only by exercising the greatest care and eliminating every possible source of error, especially such as are accumulative. The work should be done in calm cloudy weather or at night — never in bright sunlight. When the sun is shining the mean temperature of the tape cannot be determined. The base line party should consist of one chief who also acts as recorder, one transitman, one rear rodman, one front rodman, one rear marker, one front marker, one axeman, one man to read thermom- eters and one teamster ; total, nine men. The most convenient tape to use is a steel tape, 100 ft. long, graduated to feet, and tenths and hun- dreds of feet, with the zero and 100 ft. marks on the tape several inches from the ends, and with a loop or handle at each end. The tape must have been care- fully tested by the makers, or by the Bureau of Weights and Measures at Washington, or by the Miss- issippi River Commission, St. Louis; or by the U. S. 317 Lake Survey Office, Detroit ; or at some college or university where standards of measures are available for comparison. The certificate of the test should give the pull in pounds necessary to stretch the tape to standard length ( lOO ft.) at standard temperature (62° Fah.) when supported at its ends, and similar tests should be reported for supports at the o, 25, 50, 75 and 100 ft. marks respectively. These last are needed when partial tape lengths are used. The full tape should be used as much as possible. The equipment consists of the tape described, a transit, posts 4"x4"x3' to 5', one for every 100 ft. of base line ; two range poles, one level rod, one stand- ard spring balance, two standard Fah. thermometers; one axe, one maul, one shovel, one scythe, zinc plates for tops of posts, one hammer, one hand saw, one try square ; tacks, note books, pencils and one wagon. If the party is in camp the usual camp equipage will be required. PROGRAM OF OPERATIONS. The base line stations, that is, the points at the ends of the base line, should be marked by permanent stone monuments, sunk below the frost line. The geometric point at each station should be marked by the intersection of two fine lines cut in the top of a copper bolt set in the stone. The horizontal dis- tance between these two points is required. It cannot be expected that the ground between the two base stations will be level and open. Gener- ally it will be sloping or undulating, and obstructed by trees, weeds, brush, fences, etc. It is only on a straight line of sea beach, or on a long tangent of a 318 railroad, or on a level barren plain, or on the ice of a Northern lake or sound, that the ideal conditions of a straight and unobstructed level base line can be at- ♦tained. The first requisite for a base line is that when the line is cleared a signal pole erected over each station shall be visible from the other and from adjacent triangulation stations. The second requi- site is that the ground between the stations shall be fairly smooth, open and unobstructed. These condi- tions must be considered when selecting the base line. Select one station, A, as the origin, clear out the line, and erect a signal pole or target at the other station, B, centered and plumbed over the geometric point. Set up the transit at station A and point it at station B. The transit is then used to line in the posts at IDG ft, intervals along the base, and to estab- lish the grade- in which the tops of the posts are cut. On undulating ground the positions for the posts will soon be lost to view from station A, and the grade in which the tops of the posts should lie, will change. In either case the transit must be moved forward to a point on the line where a further view can be had to continue the line and grade. At station A, orient the transit by sighting at station B, and clamp the plates. Line in the posts as far as they can be seen and drive them vertically and firmly, with their centers on line, and loo feet apart, measuring with care, but not at this stage with the greatest accuracy. Turn the telescope up or down to a position par- allel to the slope of the ground, and clamp it. In Plate 1 6, Fig. i, it is assumed that the first slope is 2°. Measure the height of telescope above the mark 319 on the monument at station A, and set the target on the level rod at the corresponding reading. The rod- man will then hold the rod against the face of each post in succession at a height indicated by the tran- sitman, that will place the target in the line of sight. A line at the bottom of the rod is then marked on the post as the cutting line. When sawed off at these lines the tops of the posts will be on a line of uniform slope. When the slope of the ground changes, the tops of the posts can no longer be maintained in the origi- nal slope and a new grade line must be established, as at station 5 (Plate 16, Fig. i ). The transit is moved to this station and the new grade is estab- lished as described for station A. The setting and sawing of posts to line and grade respectively is con- tinued in the same manner over the different slopes, till station B is reached. The last grade line must pass through the mark on the monument at station B. Each portion of the line that has a new slope or grade must be considered as a separate section, to be treated independently in the reduction to the hori- zontal. Each portion of the line that is measured with a partial tape length must be considered" as a separate section, to be treated independently in the reduction to standard temperature. It is best, there- fore, to make the "slope section" as long as possible, and to make the "partial tape sections'' as few in number as possible in order that the reductions may be simplified. If the same grade and the full tape could be used throughout the line, there would be but one section, and only one reduction to the hori- zontal and to standard temperature would be neces- sary. The grade line should not be more than three 320 and one-half feet, nor less than one foot above the ground at any post. When these limits are exceeded a change of grade becomes necessary. While the transit is still set for a grade the posts of that grade are capped by a plate of zinc tacked on, and a fin^ mark is scratched on the zinc, exactly "on line," as determined by the telescope. Measuring the Line.- — ^When the posts of the first section of the line have been set on line and cut to grade at trial intervals of lOO feet the measurement may begin. The rear rodman fastens the zero end of the tape to his rod or staff by means of a loop of strong cord and uses the staff as a lever to steady the tape and oppose the pull exerted by the front rodman. By raising or lowering the loop on the staff, and by properly inclining the staff, the edge of the tape at the zero mark may be made to coincide exactly with the cross mark on the monument or station post. The rear rodman devotes his whole attention to holding the staff steady and bringing the zero mark on the tape exactly to the mark on the post. The rear marker watches the mark on the tape and assists the rear rodman to bring it exactly to the station mark. When he sees that the coincidence is exact he calls out " Mark" to the front marker. The rear marker and rodman must take care that the post is not touched except by the tape, and that the tape barely touches the zinc when drawn across its top. The post must not be disturbed in the slightest degree. Failure to observe this precaution has caused many errors in base-line measuremfents. The front rodman attaches the ring of the spring balance to his staff by a loop of strong cord and then 321 hooks the balance into the loop of the tape. He uses the staff as a lever to stretch the tape. By raising or lowering the loop and properly inclining the staff, he causes the edge of the tape to coincide with the scratch on the zinc, and at the same time pulls out the spring balance to the reading required by the certified test of the tape. His whole attention is de- voted to keeping the tape on line and maintaining the required pull on the balance. When the rear marker calls out " Mark," he repeats the word if at that instant the balance indicates the proper pull. If not he says "No," and calls for a new trial. The front marker fixes his attention on the lOO foot mark of the tape and holds a sharp steel point, as a knife point or sharp scratch awl at that mark. When the signal, "Mark" of the rear marker is re- peated by the front rodman, he sets the steel point in the zinc exactly at the lOO foot mark of the tape. The tape is moved a little to one side, and he places a straight edge crosswise against the point and scratches a fine cross line on the zinc, taking care not to disturb the post. The tape is then stretched again in the same man- ner, and at the signal "Mark" the position of the cross line is verified. Although a post may appear to be solidly and firmly driven in the ground, still a slight knock or pressure will displace it and vitiate the measurement. Hence, the precaution prescribed against touching the post after it has been marked. When a post has been marked and verified, the tape is carried forward and a new post is marked in the same manner. 322 The axeman examines the tape throughout its length as soon as it is stretched, and sees that it is without twist and that it hangs clear of all obstruc- tions. If it is fouled by weeds, grass, or brush, he clears the obstruction away. If it touches the ground, the earth must be dug away to clear it. When the tape is carried forward, he carries it at the middle point to keep it from dragging on the ground. The thermometer man provides himself with two light rods on which to support his two thermometers. When the tape is stretched he plants one rod at the 25 ft. point, and the other at the 75 ft. point and hangs the thermometers on the rods at the height of the tape, but not touching it. While the marking is being made, he takes and records two or three read- ings of each thermometer, finds the mean reading and reports the result to the chief of party. The chief of party supervises the work and sees that all the details are carefully attended to. He should watch the rear mark during the first reading, and the forward mark and the spring balance during the verification of each tape length ; he should see that all obstructions have been cleared away so that the tape hangs clear, and he should read the ther- mometers to check the report of the thermometer man. It is only by constant watchfulness and veri- fication that he can secure the results for which he is responsible. If he is careless and inattentive, it is certain that the members of his party will be more so. The record is best kept in diagrammatic form, each separate "slope section" or "partial tape sec- tion" of the line being represented by a sketch at the top of a separate page of the note book. 323 Each slope section is represented by a profile line (Plate 1 6, Fig. 2) reading from left to right, and marked with the number of the section in Roman characters, the included stations in Arabic characters, the slope in degrees ( + ) for rising and ( — ) for fall- ing, and with the mean temperature Fah., for each tape length, and mean temperature of section. Mean Temp. = 40)^° F. cos 3° = 0.99863 40.25° — 62° = — 21.75° F. Reduction for slope and temperature : Lt =400 X 0.99863 [i — (0.0000065 X 21.75)]. — (0.0000065 X 21.75 = — 0.000141375 + I. 0.999858625 400 X 0.99863 X 0.99986 = 399.3961 = Lt The method of reduction is indicated in the fore- going example. To reduce to the horizontal multiply the measured length by the cosine of the slope, and to reduce to standard temperature multiply also by I + [0.0000065 (Tt—Ts)] The result is the horizontal length of the section, and if five decimal places be retained throughout the computation, the error introduced in computation will not exceed i in 100,000. For a partial tape section the record and reduc- tion are made in a similar manner (Plate 16, Fig. 3) ; thus : 324 cos 4° ^ 0.99756 76° — 62°== 14° F. Lt = 58 X 0.99756 X [1+ (0.0000065 X 14)] 0.0000065 X 14 = 0.000091 + I. 1. 00009 1 58 X 0.99756 X 1.000091 = 57.8637 = Lt When all the sections of the base line have been thus measured, recorded and reduced, the reduced lengths of sections are added together to get the hor- izontal length of base line. When partial tape lengths must be used, it is best to select a length of 25, 50, or 75 feet, if the tape has been tested at these points. In closing on station B, use first a tested length of o, 25, 50 or 75 feet, whichever comes nearest to the station mark, and then measure carefully the small remaining distance, with a pull estimated by com- parison with the pull required for 25 feet. This small section is to be recorded and reduced separately like the other partial tape sections. There are other methods of measuring base lines with the tape, but the method above described takes account of the principal sources of error, and if care- fully followed will give the proposed result, viz., the length of base line with an error less than i in 50,000. Measuring Lines With the Tape. — It is only in meas- uring the base-line of an important survey that such precise methods as have been described are necessary. For ordinary measurements, a method similar to that described for the chain will suffice 325 The tape is stretched by hand with an estimated pull; tape lengths are marked on the ground by marking- pins or small stakes; on sloping ground the tape is held horizontally as well as the eye can determine, and the end points are referred to points on the ground by means of a plumb line. Careful measure- ments thus made on a main traverse should not err by more than i in 5000. Otherwise, the measure- ments may be made on the slope; and if the slope exceeds i" 00', the usual reduction to the horizontal should be made by determining the slope and multi- plying by its cosine. For hasty measurements made to "fill in" a sur- vey by side shots, an accuracy of i in 1000 will suffice. Measuring Distances With the Stadia. — -This is the most rapid, convenient and accurate method available for running traverses and "filling in" when the tran- sit or plane table is used. The principle of the stadia and its use are explained in connection with the transit. The error of a single measurement may be as great as i in 600, but in many measurements the compensation of errors will probably reduce the total error to about i in 3000. Pacing. — The paces of a man who walks steadily at a marching gait are quite uniform, and if the length of a pace be known it forms a convenient unit of measure for rough and hasty surveys or sketches. It is not the length of any single pace that is re- quired, but rather, the average horizontal length of a pace when walking over varied ground, up hill and down hill, on roads and on grass. To determine this average length of pace, measure a course of 1000 to 2000 feet over varied open ground that is fairly representative of the country to be covered in subse- 326 quent work. Mark the ends of the course and pace the distance several times in both directions, counting the number of steps each time. The length of the course in inches divided by the number of paces will give the length of the pace in inches. If the results of the several trials differ by more than two per cent., repeated trials should be made with greater attention to uniformity of step and cadence, until by practice the desired regularity is attained. The gait should be a natural rapid walk that can be maintained for a day's march. A following wind lengthens the pace and a head wind shortens it. Mud and loose sand shorten the pace. On slopes less than four degrees the pace is not materially affected. On steeper slopes the pace is shortened whether moving up or down, but if the pace has been determined on a course of varying slopes such as are met with in the field, no further correction for slope will be necessary. On steep slopes pacing cannot be relied upon and some other method of determining distances should be used, such as triangulation, or the range finder. The paces of different men vary from about 27 inches to 37 inches as the extremes, but for most men the length of pace is from 30 to 33 inches. Many prefer to count strides instead of paces. A stride is the distance between two successive posi- tions of the same foot, and its length is therefore twice the length of a pace. In counting paces or strides mistakes sometimes occur through leaving out or adding entire tens or hundreds. To avoid such errors it is necessary to fix the attention on the count, and this prevents ob- servation and study of the ground passed over. It is 327 well, therefore, to use some form of mechanical counter for recording the count. The passometer is a small tally register that is held in the hand. A lever is pressed by the thumb or finger at ever}' step or stride, and the number is indicated on a dial. The motion of the hand soon becomes a sub-conscious operation that requires no attention. The pedometer registers the vibrations of a care- fully poised weight that is balanced by a spring. The shock of planting the foot at each step produces a vibration of the weight, which is recorded on the dial. The spring should be adjusted to suit the cadence, by tightening for a rapid pace and loosen- ing for a slow pace. The instrument has the shape of a watch and may be carried in the pocket. It is not as reliable as the passometer. Pacing is the method generally used for measur- ing distances in making a rough survey or "sketch," such as a road or position sketch or reconnaissance on foot. In this work the hand compass is generally used for determining directions, with an error that should not be greater than one degree for a single observation. The tangent of one degree is, roughly, ■jJj^. The error in pacing should not exceed i in 60. Therefore, the determination of distances by pacing is equally as accurate as the determination of direc- tions by the hand compass. When the hand compass is used for directions, the pace is just as good as a steel tape for distances. This same principle applies in all methods of sur- veying, viz : When a certain limit of error obtains in the determination of directions, the same limit of error is permissible in the measurement of distances, 328 and vice versa. Thus the transit and the steel tape are used together, the surveyor's compass and the chain are used together, and the hand compass and the pace are used together. The step or stride of a horse at a steady walk or trot is more uniform than that of a man. It is also less affected by wind, mud, sand or slope, and is a good unit of measure, when its average horizontal length has been determined. This length is deter- mined by riding over a measured course on varied ground at the regular gaits and counting the steps or strides. A number of trials are necessary in order to gait the horse to a steady walk and trot that can be maintained in subsequent work. When the average length of the step or stride at the walk and trot has been determined any distance may be measured by riding over it at the adopted gait and counting the steps. A passometer should be used to keep the count in order that the attention may be devoted to careful observation of the country traversed. When a wheeled vehicle is used as a means of transportation, the revolutions of a wheel afford a means of measuring distances. The unit of measure is the distance passed over by the wagon during one revolution of the wheel. To determine its length the wagon is driven over a measured course on a road similar to those that will be traversed in subsequent work, and the revolutions of the wheel are counted. The length of the course divided by the number of revolutions gives the distance covered in one revolu- tion. When this distance has been determined, any distance traversed by the wagon may be measured by counting the revolutions of the wheel. To obtain the horizontal distance, the measured distance on ■329 each slope must be multiplied by the cosine of the slope. A direct measurement of the circumference of the wheel is not a good determination of the unit of measure in this case, because the wheels slip more or less, and on rocky, sandy, or muddy roads the slip is considerable. The front wheel should be used be- cause the hind wheels are sometimes locked by the brakes in going down hill. A rag of white cloth tied to a spoke is a conven- ient mark to indicate successive revolutions. TJie Odometer is an instrument that automatically registers the revolutions of a wheel to which it is at- tached. In one form the mechanism is contained in a leather case and is strapped to the spokes of the wheel. A weighted frame hangs freely on a central axis in the case and remains vertical while the wheel and case turn. Worm gearing communicates a slow motion to the numbered dials in the frame and thus registers the number of revolutions. To read the dial the case must be opened and the frame with- drawn. In another and better form the recording appa- ratus is attached to the front axle and is operated by a lever which is raised at each revolution by a cam or eccentric on the hub of the wheel. The cover of the dial may be left open and readings taken at any time without dismounting. MEASURING DISTANCE BY TIME. If a man moves on foot or horse back, or by wagon or boat, at a uniform rate, the time during which he so moves may be taken as a measure of the 330 distance traversed, provided that the distance cov- ered in a unit of time is known. On land this method of measuring distance is used principally in. mounted reconnaissance, and for this purpose the rate of the horse must be determined. Ride several times at a walk and at a trot over a measured course in both directions and note the time for each trip. Find the average time for each gait. The length of the course divided by the number of minutes taken to traverse it gives the distance corresponding to one minute, and the minute becomes a known unit of measure when the same horse is used at the rated gaits in subsequent work. The horse should be rated by the rider who is to use him in the reconnaissance, since a change of riders changes the rate of the horse. After careful gaiting and rating, the error in measuring distances by this method should not exceed i in 50, or 2 per cent., an ordinary roads. For convenience and accuracy in timing dis- tances a stop watch is essential. The pattern that is alternately started and stopped by one lever and is set back to zero by another is best suited for this work, since it may be stopped and read at every halt and started when the advance is resumed. At the station halts it is set back to zero, and it therefore always indicates the distance in minutes from the last station. To attain equal accuracy in the determination of directions the hand compass or the sketching case is used, and for a like approximation to differences of elevation the clinometer or the slope board will suf- fice. At sea a similar method, called "dead reckoning," is applied for tracing the course of a vessel. The 331 speed is determined by the "log" in miles per hour, and time thus becomes the measure of distance. Di- rections are given by the compass. There are no differences of elevation to be considered until the harbor entrance is reached, and then the stage of the tide is the vertical coordinate that shows whether the vessel is high enough to clear the bar. Check points corresponding to triangulation stations are furnished by the daily sextant and chronometer ob- servations for latitude and longitude. These methods have become familiar to many of our officers in their voyages to the Philippines. The principles involved are identical with those applied in surveying. Dead reckoning is traversing, and the determination of latitude and longitude is spherical triangulation. It is the sextant and chronometer, more than the com- pass and log, that brings a vessel surely to her haven. tx b '0 *)/ J* // -7 /^/97^ /<5 335 CHAPTER XIV. TRIANGULATION. Triangulation is an application of the triangular system of coordinates in the accurate determination of the relative positions of selected points, well dis- tributed over the area considered. Points so deter- mined are used as control points to check and correct the results of the less accurate but more rapid and economical methods ordinarily used in surveying. Let the point A, Plate D, be selected as the initial point of the survey. A second point, B, is first deter- mined by measuring its azimuth, horizontal distance, and difference of elevation from A. The line A B is the base line, and the horizontal distance from A to B is the length of the base line. The operation of measuring a base line is described in Chapter XIII. A third point, as C, may now be determined by measuring two other parts of the triangle ABC, either the two sides a and b, or two of the angles as A and B. Since it is easier to measure angles than to measure distances, angles are usually measured. Measure the angles A and B with a transit, and solve the triangle as follows : C = i8o° — (A + B) c a = • sin A ( i ) sin C ■ 336 c b = sin B (2) sin C in which A and B are the measured angles and c is the measured length of base line, C is the required angle, and a and b are the required lengths of the sides opposite the angles A and B respectively. All of the parts of the triangle being thus found, the point C is fully determined in plan and may be plotted. To check the accuracy of the angle measurements, the third angle at C should also be measured with the transit. The sum of the angles of a triangle is 1 80°. If the sum of the three measured angles is not 180°, the discrepancy is due to errors of measurement, and suitable corrections are applied to give angles whose sum shall be 180°. If the angles have been measured with equal care, it is assumed that equal corrections should be applied to all, and each angle is changed by one-third of the total error so that their sum shall be 180°. For example, suppose that the angle measure- ments give the following results: A = 41° 26' 30" + 10" = 41° 26' 40" B = 93° 58' 15" + 10" = 93° 58' 25" C = 44° 34' 45" + 10" = 44° 34'' 55" Sum = 179° 59' 30" + 30" = 180° 00' 00" The sum of the measured angles is too small by 30", therefore 10" must be added to each of the ob- served angles to give the corrected values which appear in the last column, and whose sum is 180°. It cannot be claimed that these corrected values are correct, but they comply with a known condition, which is better than any observation. 337 If the angles have not been measured with equal care, weights are assigned to the observed angles expressing the estimated accuracy of the. several measurements, and the total error is divided into parts that are inversely proportional to the weights. Thus, in the foregoing example, suppose the weights in order are respectively 3, 2 and 6. Then, the total error, 30", must be divided into parts pro- portional to ){,.}4 and i^, or as 2 : 3 : I, giving + 10", + 15" and + 5" as the corrections, and A = 41^ 26' 40" B = 93° 58' 30" C = 44° 34' 50' Sum = 180° 00' 00" as the corrected angles. The corrected angles are used in solving the tri- angle to find the sides a and b. In like manner some other point, as D, may be determined by measuring the three angles, correct- ing them to distribute the error, and solving the •triangle A B D. Any two of the determined points may be used as the ends of a new base for determining a new point. Thus, from A and C the point F may be determined by the triangle A C F. From A and D the point G may be determined, and the triangle FAG may be completed by measuring its three angles. Whenever, as in this case, a point, such as A, is surrounded by measured triangles, a further check is introduced by the condition that the sum of all the angles around this point must be 360°, and all the angles of the five triangles may be readjusted so that this condition shall be satisfied without disturbing the first condition. 338 If the points C and D be used to determine a point, E, the side C D is first found by solving the triangle C A D, of which two sides and the included angle are known. If A' represent the included angle, a' the required side, c' and d' the known sides, then a' = t/c'» + d'2— 2 c'd'cosA' ( 3 ) Having thus found the side C D, the triangle CDE may be solved by ( i ) and ( 2 ) to determine the point E. The system of triangles may be extended to cover any desired area. The method of adjustment above indicated is the simplest and most obvious, and involves only the con- ditions that the sum of the angles of a triangle is 180°, and that the sum of the angles about any point is 360°. As each triangle, or group of triangles, is adjusted, the result is considered final and each new point is assumed to be accurately determined. This assump- tion is approximately correct for a small plane sur- vey, say 100 square miles or less in extent, made with care and with a good transit. For important and ex- tensive surveys, covering thousands of square miles, geodetic conditions are involved, taking cognizance of the ellipsoidal figure of the earth, and requiring pri- mary triangulation with precise instruments and rig- orous adjustment by the method of least squares. Primary geodetic triangulation is an undertaking of such magnitude that only governments can in- dulge in it. No private or local interest would justify the expense. But in regions that have been covered by primary triangulation, all important local surveys 339 should, when 'practicable, be based upon and con- trolled by the government surveys. In primary triangulation the stations may be from 10 to I GO miles apart and do not serve the purpose of control and adjustment for a small survey. The stations of the primary system are, therefore, con- nected by chains of smaller triangles corrected and adjusted to close on the primary stations. This con- stitutes "secondary triangulation." For the survey of a limited area or for " filling in " an extended survey, the stations of the secondary system may be connected by "tertiary triangulation," giving stations only one to three miles apart. Tra- verses may then be run with transit, or plane table, or surveyor's compass, and by beginning and closing every traverse on triangulation stations each short traverse of from one to three miles may be adjusted and corrected. Errors and mistakes in traversing will not be propagated throughout the survey, but will be detected and eliminated at every closing station. When primary and secondary systems of triangu- lation are not available, as is usually the case, the control of a plane survey of a limited area is secured by a system of small triangles corresponding to ter- tiary triangulation and adjusted by the simple method that has been described. In measuring the angles each observation is re- peated a number of times and a mean of the results is taken as the observed angle. One methed of re- peating the measurements is as follows : 340 To measure at station A the angle 'C A B. Record : At station A, 5 observations. Vernier A. Vernier B. On Sta. C 23° 14' 20' On Sta. B 230° 26' 40' 203 14 20 410° 27' 00" + 207° 12' 40" 10 X angle = 207° 12' 20' = 414° 25' 00' Angle CAB = 41° 26' 30' Set up the transit at station A, clamp the lower plate, point at the left hand station, C, clamp the upper plate, read both verniers and enter the read- imgs in the record as shown above. Unclamp the upper plate, point at B and clamp. This ends the first observation. Unclamp the lower plate, point at C and clamp. Unclamp the upper plate, point at B and clamp. This ends the second observation. Re- peat these observations to include the fifth pointing at station B. Read both verniers. If a vernier has passed the 360° point once, add 360° to its final read- ing; if twice, add 720° to its reading. Enter the readings in the record under the first readings and find the difference of the readings for each vernier. The sum of these differences divided by twice the number of observations gives the required angle. For greater accuracy a second set of observations may be made as follows : Shift the verniers so that a different part of the circle will be used, and make the initial pointing on the right hand station, B. Then turn with the upper clamp; to the left and with the lower clamp to the right, to include the fifth pointing at station C. The remainder of the opera- 341 tion is the same as for the first set. The mean of the two results is taken as the required angle. In practice, all of the triangulation stations are established and marked, including those of the base line. The stations must be selected with a view to covering the ground with well conditioned triangles and preserving mutual visibility between adjacent stations. A good station mark for temporary pur- poses is a piece of 2 -inch pipe sunk about two feet in the ground and projecting one foot. Rods bearing different combinations of flags are set in the pipes as targets and distinguishing marks. When a station is occupied the rod is removed and a plug having a center mark in the top is inserted. Always replace the flag rod before leaving a station. Measure the base line twice in opposite directions and take the mean of the two results. Determine the true azimuth of the base line by an observation on Polaris. Determine the difference of elevation of the base stations by leveling. Measure all the angles of the selected triangles with five or ten repetitions of each angle, adjust the angles and compute the lengths of all the sides. Compute also the true azimuths of all the sides, the latitude and departure of each side, and the latitude distance and meridian distance of each station from an assumed origin. To plot the triangulation stations, rule a sheet of drawing paper with fine pencil lines into accurate squares, such that the side of each square shall repre- sent 100 feet, or 500 feet, or 1000 feet, etc., depending on the scale of the map. Assume a convenient origin and from it lay off the rectangular coordinates given by the latitude distances and meridian distances that 342 have been computed. Mark the points thus plotted with the names of the corresponding stations. When traverses have been run for filling in, and have been adjusted, by the method of latitudes and departures, to close on the triangulation stations, the same system of squares may be used in plotting the traverses. VERTICAL ANGULATION. For vertical control in a triangulation survey, the elevations of all the triangulation stations or of bench marks near them should be -determined. This may be done by running lines of levels between stations, but it is easier and quicker, though not so accurate, to measure the vertical angles between stations and compute the differences of elevation. This latter method is called vertical, angulation, or trigonometri- cal leveling, and is applied as follows : Let A and B be any two stations, and Let H =^ height of instrument at station A, that is, the height of the horizontal axis above the top of the stake or iron pipe. A =: the vertical angle or reading of the ver- tical circle when pointing at the top of the flag rod at station B. R = height of flag rod at station B from top of rod to top of iron pipe, b ^= horizontal distance from A to B. d = difference of elevation or vertical dis- tance from top of iron pipe at A to top of iron pipe at B. then d = H + b tan A — R. 343 When the distance b is greater than one-half mile, the correction for curvature and refraction should be applied. If K represent the distance in miles, then 0.57 K* is the correction in feet, and is always positive. The above formula becomes d = H + b tan A — R+ .57K8. A positive value of d indicates that station B is above station A, and a negative value of d indicates that station B is below station A. When the angle A is an angle of depression it is negative and the second term in the second member of the above formula becomes negative. The difference ot elevation from B to A is also determined by reading at B the vertical angle to the top of the flag rod A. The two results should be numerically equal with contrary signs. Their nu- merical mean is taken as the observed difference of elevation between the two stations. In practice the vertical angles are read when the stations are occupied for reading the horizontal angles. Several readings should be made of each angle with telescope alternately direct and reversed. The record and reductions may be shown as follows : D.— 2° 24' R.— 2° 30' D.— 2° 22' R.— 2° 28' At Station A, Vert. Ang. to B. A = Tan A: 2° 26' —.0425 b = 7495 b tan A = — 318.54 — R=:— 15.72 A to B = — 334.26 7495 K = = 1.42 5280 .57K2= 1. 15 H= 3.78 + 4-93 = — 329-33 Similar record and reduction of observations at sta- tion B on A will give the difference of elevation from 344 B to A, and the mean of the two results is the ob- served difference of elevation. Similar results are obtained for each side of all triangles, and may be adjusted by a method analogous to that used in adjusting the triangles. For, the algebraic sum of the differences of eleva- tion, taken in order around the sides of a triangle or other polygon, is zero. Therefore, the observed dif- ferences of elevation may be adjusted by corrections which shall make their algebraic sum, taken in order around the sides of each triangle, equal to zero. Thus, in triangle ABC, Plate D, let the observed differences of elevation be A to B, — 329.33 — 0.49 = - 329.82 B to C, — 48.60 — 0.49 — — 49.09 C to A, + 379.41 — 0-49 = + 378-92 A to A, + 1.48 — 1.48 = 0.00 The total error is + 1.48, therefore the total cor- rection is — 1.48, one-third of which, or — 0.49, is applied to each side, giving in the last column the corrected differences of elevation, whose algebraic sum is zero. If the elevation of station A be known or assumed, as, say, 823.50, then B, 329.82 feet lower, has an ele- vation of 493.68, and C, which is 378.92 lower than A, has an elevation of 444.58. In adjusting an adjacent triangle, as A C F, the corrected value, — ■ 378.92, instead of its observed value, is used for the side A C and is not further cor- rected, one-half of the resulting total error being ap- plied as a correction to each of the other two sides. 346 C F and F A. When two sides of any triangle have been adjusted as parts of adjacent triangles no further adjustment is possible, as the difference of elevation on the third side becomes known. In any system of triangles or polygons, that poly- gon in which the total error is greatest is first ad- justed and the others follow in the order of the mag- nitudes of their total errors. The corrected differences of elevation are used to determine the elevations of stations, beginning at one whose elevation is known or assumed, and the eleva- tions thus determined are used to check and adjust the elevations in traverses that close on the triangula- tion stations. TRIANGULATION SURVEY. A complete topographical survey controlled by triangulation involves the following operations : 1 . Select and mark suitable triangulation stations well distributed over the area to be surveyed, and forming fair shaped triangles with no angle less than 30 degrees. In some cases lines must be cleared of trees and other obstructions or high framed sta- tions must be erected to give a view of adjacent sta- tions. Two of the stations will be those of the se- lected base line. 2. Measure the base line twice in opposite direc- tions and take the mean of the reduced results. 3. Determine the true azimuth of the base line or of any side of a triangle by an observation on Po- laris. 4. Measure the horizontal angles in all the tri- angles, repeating five or ten times for each angle, and 346 measure vertical angles at each station to all adjacent stations. 5. Adjust the triangles, compute azimuths, lengths of sides, latitudes and departures of sides, and meridian and latitude distances of stations. With the rectangular coordinates thus determined plot the stations on coordinate paper (ruled in squares to scale). 6. Adjust differences of elevation on sides of tri- angles and determine elevations of stations. Mark these elevations at the plotted station points. 7. Run a system of traverses, each traverse be- ginning and closing at triangulation stations, and cover the area considered with a network of traverses such that all desired critical points shall be included. 8. Reduce the traverses to latitudes and depar- tures and'differences of elevation and adjust them to close on triangulation stations, finding the meridian and latitude distances and the elevations referred to the same origin as that used in the triangulation system. 9. Plot the traverses by rectangular coordinates and mark the plotted traverse stations with their ele- vations. 10. At each traverse station plot the sideshots with protractor and scale and sketch in the deter- mined features. Mark the elevations of all deter- mined critical points and interpolate and draw the contours. 11. Ink the drawing and finish the map by ad- ding scale, m.agnetic and true meridians, title, date, legend, and border. 347 'Plate: 2). 349 CHAPTER XV. SKETCHING. The term sketching is applied to hasty methods of surveying that are used when considerable areas or great distances must be covered in a limited time. Necessity for the employment of such methods arises with imperative force in connection with military operations in a region which has not been mapped with sufficient detail for military purposes, or of which suitable maps are not obtainable. Like methods are employed in explorations in new coun- tries and in preliminary reconnaissances made to select lines for new roads or railroads. In sketching, the same principles are applied as in topographical surveying, but the simplest of means are used to assist the eye and mind in estimat- ing directions, distances and elevations, and the hand in representing on paper the features thus deter- mined. A knowledge of the principles of surveying is essential to a thorough understanding of the methods of sketching, and practice in surveying is the best preliminary school of instruction for the sketcher. The surveyor should, by careful and conscious effort, constantly compare the appearance of distances, angles, slopes and heights as they are presented to him on the ground, with their determined values, and with their appearance as represented or re- 350 presented on the map, keeping always in mind the scale of the map and the relative sizes and forms of corresponding features on ground and map. He will thus quickly acquire facility in estimating distances, angles and slopes with considerable accuracy, and will gradually attain a power of representing on paper, with almost unconscious effort, the forms and features which he sees on the ground. By repeated verification or adjustment of his estimates by the check of accurately determined values, he acquires correct standards of comparison that are attainable in no other way. An experienced topographer will need only a few determined critical points for control in sketching in all the intervening and surrounding features, forms and incidents of surface that lie within his unobstructed view. This implies that the map is drawn in the field and that the ground is studied while its shape and features are transferred to the map by the contours and other lines that rep- resent them. Plane table surveying is therefore the best practice for the sketcher. The surveyor should make a men- tal estimate of each distance, angle, slope, and differ- ence of elevation before it is determined, and should, with light pencil touches, sketch on the map his esti- mate of the positions of points and of the shape of contours before they are actually plotted. He will soon find that his estimates have become quite accu- rate, and that he can dispense more and more with determined locations excepting for a few important control points around which he may fill in, by esti- mation and free-hand sketching, the representation of the intervening features. 351 Surveying also teaches the amount of detail that can be shown on maps of different scales. On a large scale map of say 12 inches =; i mile (i in ^ 440 ft.) with contours at 5 ft. V. I., all the minor inequalities of the ground can be shown by the contours, and buildings can be drawn to scale. Therefore when a map of this or larger scale is required, all of these minor details must be determined by measurement or by estimation. On a small scale map of say 3 inches = i mile, with 20 ft. contours, the smaller fea- tures of the surface cannot be shown, and buildings will be reduced on the map to mere dots. The sur- veyor or sketcher must learn to waste no time in de- termining small details that cannot be shown on the map and to confine his attention to the larger masses and general forms that will be apparent at the adopted scale. To cover large areas or great distances rapidly this method of mapping by estimation, and by free- hand drawing must be applied to its fullest extent, with only such assistance in maintaining general con- trol as is afforded by the use of hand instruments. For military purposes, as stated in Chapter II, sketches may be classed, in general, as position sketches and road sketches. To these may be added a third class called route sketches made when hun- dreds of miles are to be covered in exploratory expe- ditions. These differ only in the amount of detail required and in the resulting scales: Position sketch, 6 in. — i mi. V. I. — 10 ft. Road sketch, 3 in. = i mi. V. I. = 20 ft. Route sketch, i in. = i mi. V. I. = 60 ft. 352 Sketches are made on foot, on horseback, and in wheeled vehicles. Position sketches cover relatively small areas, or are divided up into small areas for individual sketch- ers, and require considerable detail. They are there- fore usually made on foot. Road sketches are made at the ordinary rates of marching and should cover from 15 to 20 miles of road in a day. They are therefore usually made on horseback. Route sketches may cover many successive days' marches, and are made either on horseback or in wheeled vehicles. The following instruments are used : For determining angles and directions, the hand compass, either box or prismatic, the pocket sextant, the oriented plane table, or sketching case. Distances are determined by pacing, or by timing a rated horse, or by counting the revolutions of a wheel. A "pace tally" operated by hand is conven- ient for recording paces either of man or horse ; a stop watch is essential in measuring distances by time ; and the odometer is used to record the revolu- tions of a wheel. Slopes are measured with the clinometer or the slope board. Slope and distance give difference of elevation. Elevations are also determined by means of the aneroid barometer. In plotting the work the paper is fastened on a small drawing board or on the sketching case. Direc- tions are plotted by means of a small rectangular protractor, or, when the plane table method is used, by oriented board and pointed ruler. Distances are plotted by means of a working scale marked on the 353 edge of the ruler or protractor. Contour points are spaced by means of scales of map distances marked on the edges of a card or ruler. The above named instruments have been de- scribed in the chapters on surveying, with the excep- tion of the sketching case, the aneroid barometer and the scales of map distances. The Sketching Case is a small drawing board or hand plane table with attached compass and ruler. Rollers on opposite edges of the board stretch the paper and permit it to be shifted as the sketch pro- gresses. The compass is used to orient the board. The ruler is attached to the board by means of a pivoted arm or frame so arranged that the ruler may be shifted to any part of the paper, pointed in any direction and clamped in a fixed position. For read- ing slopes, a slope board or clinometer arrangement is provided. A strap is attached to the back by a swivel connection and the board is held on the left hand or wrist and manipulated with the right hand. Many different designs of sketching cases have been devised in an effort to provide an instrument that will do all of the work and replace skill on the part of the sketcher. In this latter purpose they generally fail, and the more elaborate designs are dis- carded by the sketcher as he acquires skill. The ex- perienced sketcher prefers the simplest of means to aid him in his work. The Aneroid Barometer is an instrument in which a light needle or pointer is actuated by change of at- mospheric pressure. A thin metallic disc with cir- cular corrugations forms a flexible cover for a sealed box from which the air is partially exhausted. The central point of the flexible cover is depressed more 354 or less as the atmospheric pressure increases or de- creases, and this slight motion is multiplied many times and transmitted to the pointer through a train of levers. Atmospheric pressure decreases with ele- vation above sea level, and the pointer may there- fore be made to indicate, on a properly constructed scale, elevations above sea level. The pressure varies also with changing atmos- pheric conditions, and the barometer will not, there- fore, indicate absolute elevation unless the effect of these changes be eliminated. Differences of elevation may, however, be determined by the use of two barom- eters. One barometer remains stationary at a point whose elevation is known or assumed, and its observer records time and barometer readings at half hour intervals throughout the day. The other barometer, while at this initial station, is set at the same reading as the first, and then is carried by the surveyor or sketcher and read at the stations occupied during the day. The time of each reading is also recorded. At the end of the day's work the two records are compared and the readings of the stationary barome- ter are subtracted from those of the surveyor's barom- eter taken at the same times. The results are dif- ferences of elevation, which, applied with proper sign to the elevation of the initial station, give the eleva- tions of the corresponding stations occupied. The foregoing method is applicable in position sketches and in rough surveys that cover small areas each day, but in road sketches, which may cover fif- teen or twenty miles each day and which advance always into new territory, it is impracticable to com- pare the barometer readings of the sketcher with those of an observer at a fixed station. Moreover, 355 elevations of stations are needed at once in order to finish the sketch tip to the station occupied, and sub- sequent adjustment by comparison with a standard is impossible. Therefore, the differences of elevation shown by the readings of the sketchers aneroid must be adopted, usually without correction. In settled weather, barometric changes take place very slowly, and the error from this source in passing from station to station will be small. The accumulated error dur- ing a day's work may be loo feet or more, and this seems large, but if twenty miles have been covered, an error of loo feet in elevation is only i in looo. The hand level and the clinometer are subject to errors greater than this, and the aneroid need not be discarded as a sketching instrument on the charge of inaccuracy. When unsettled weather causes rapid barometric changes, an effort may be made to apply corrections as follows : At 6 A. M. set the barometer to read the known or assumed elevation at the point occupied and record this reading, say 850 feet. At 7 o'clock read the barometer again, and start out on the day's work. Suppose this reading to be 862 feet, indicat- ing a rise of 12 feet in one hour. Then a correction of — 12 feet must be applied to the 7 a. m. reading to get the original reading, 850. Assuming that this rate of change continues for two hours, the correction at 8 A. M. will be — 24 feet, and at 9 a. m. — 36 feet. Intermediate corrections may be interpolated. At 9 o'clock halt for half an hour, and take readings at 9 and 9:30, say 824 and 828 respectively, showing a rise of 4 feet in thirty minutes, or 8 feet per hour. Then at 10 o clock the correction will be • — 44 feet, and at 1 1 o'clock — 52 feet. During a halt from 1 1 to 1 1 130 356 the barometer reads 8io feet without change, and the correction, — 52 feet, is applied for the next two hours. The readings during the next halt at i:oo and 1 130 p. M. are 866 and 864, showing a fall of 2 feet in 30 minutes, and giving corrections of — 48 feet at 2 p. M., and — 44 feet at 3 P. M. At the 3:00 to 3:30 p. M. halt, the readings might be 916 and 910, giving corrections of — 32 ft. at 4 P. M., and — 20 ft. at 5:00 P. M. The resulting eleva- tions of the stations occupied at the odd numbered hours would be- — ■ 7:00 A.M. 862 — 12=850. 9:00 A.M. 824 — 36 = 788. ii:ooA. M. 810 — 52^758. i:oo p. M. 866^ — 52=814. 3:00 P.M. 916 — 44^872. 5:00 p. M. 933 — 20 = 913. For readings at intermediate stations, the time is noted and an interpolated correction is applied. The record may be kept conveniently by plotting times, barometer readings, corrections, and corrected elevations on profile paper as shown in Plate 1 7. The horizontal scale is i inch = 2 hours and the vertical scale, I inch = 40 ft. The 6:00 and 7:00 A. M. read- ings are plotted, and the line through them is pro- longed two hours to 9:00 A. M. The ordinate, for any time, from this line to the "level line," is the cor- rection to be applied to a barometer reading at that time. Thus at 7:45 A. M. the barometer reads 847, which is plotted at a ; but the correction at this time is a' b', and this, transferred by the dividers to a b, gives the point b at 826, which is the corrected eleva- tion of the station. 357 At the first halt the 9:00 and 9:30 readings are plotted at c and d and this line is prolonged to 1 1 :oo A. M. at e. Draw the parallel line c'e' prolonging the correction line to 1 1 :oo A. M. In like manner the rate of change is determined at each halt, prolonged two hours, and transferred to the correction line by a parallel. In Plate 17 the corrections have been ap- plied to the plotted readings at the odd numbered hours, and the line joining the corrected points and marked "-profile" is a rough profile of the road traveled during the day, distance being expressed in time. If progress has been made at a uniform rate of say two miles per hour, a scale of miles may be added along the bottom line as shown. The hori- zontal portions representing the half-hour halts must be disregarded. If corrections were not determined and applied, the elevations shown on the line marked "uncor- rected profile" would have to be accepted. The half-hour halts need not be time wasted, since they may be utilized in finishing the sketch or in lo- cating distant points by triangulation. The aneroid barometer is an expensive and delicate instrument, and must be handled and used with great care. The small pocket aneroid, about the size of a watch, can be read only to the nearest 10 feet, and is not satisfactory as a sketching or surveying instru- ment. The larger size, inclosed in a padded case and reading by careful estimation to the nearest 4 feet, is much to be preferred. Before reading an aneroid at any station, it should remain undisturbed for a few mo- ments and then be tapped lightly to settle the needle. A magnifying glass should be used to get close read- 358 ings. The aneroid must be shielded from the direct rays of the sun and must be protected from shocks. Only the simplest and most obvious use of the aneroid as a sketching or reconnaissance instrument has been indicated, and the corrections that are necessary for obtaining good results in surveying have been disregarded. In mountain surveys the large aneroid or the mercurial barometer is a most valuable instrument. By simultaneous hourly read- ings for a day or more of two barometers at two stations, and by applying corrections for temperature and moisture, the difference of elevation of the two stations may be determined with considerable accu- racy. The use of the barometer in hypsometric sur- veys is a special study that cannot be considered here. SCALES OF MAP DISTANCES. The standard system of scales and vertical inter- vals adopted for military maps, as explained in Chap- ters I and II, gives for each degree of slope a con- stant map distance between contours for all maps that conform to the system, no matter what the scale. For a slope of i" the map distance is 0.65 of an inch. For any other slope up to 20° divide 0.65 by the de- grees of slope, and the result is the map distance for that slope expressed in inches. The instruments used in sketching will read ver- tical angles or slopes only to the nearest half or quarter degree, and scales may be prepared on card- board giving map distances for each quarter degree up to 3°, for each half degree up to 6", for each degree up to 12°, and for each two degrees up to 16°. All of these scales may be placed on the edges of one card 359 as shown in Plate i8. The scales for 9°, 10°, 11° and 12° may be subdivided by eye to give map distances for 18°, 20°, 22" and 24° respectively, although these steeper slopes are seldom encountered. When the slope of the ground along any line is read with the clinometer or slope board, the scale for that slope is applied along the corresponding line on the sketch, and, beginning at the station occupied, contour points are dotted along the line at the marks of the scale. By counting one vertical interval for each space, the elevation of any point on the line may be determined. THE SKETCHING RULER. The ruler used in sketching serves several pur- poses. If the plane table method be used, as with the sketching case or oriented field drawing board, the ruler serves as an alidade, to be pivoted at the point representing the station occupied, and sighted at the "point observed." It serves next as a straight edge for drawing courses and side shots, and finally, with divisions marked on its edge representing to proper scale the units of measure used in sketching, it serves to lay off distances. A convenient form of ruler for these purposes is shown in Plate 19. It is triangular in cross section and is five inches long. The three faces of the ruler give six edges on which scales may be marked. By placing different sets of scales on each half of the ruler, twelve scales are provided, as follows : 1. Yards. 2. Man's pace or stride. 3. Horse's step or stride at walk. 360 4- Horse's step or stride at trot. 5. Horse's time at walk. 6. Horse's time at trot. And scales of map distances in the following sets: 7. %°, i\ 2°, 4°, 8°, 16°. 8. i^°, iy2\ 7°, 14°. 9- %\ ^%\ 3°, 6°, 12°. 10. 2^°, ^%°, II". 11. x%\ 2%°, s\ 10°. 12. 2%°, ^%\ 9°. These are the same map distance scales as those shown in Plate 18, and when this ruler is used the card scales will not be necessary. The ends of the ruler are capped with pieces of sheet rubber, trimmed to project slightly beyond the faces. These edges support the ruler at the ends only, and cling to the paper with adhesion sufficient to prevent displacement when drawing a pencil line at the edge. To assist in this purpose and to prevent displacement by wind, the ruler is weighted with lead in a hole bored lengthwise through the axis. A small ring at one end may be used to attach the ruler by a string to a button-hole. The scales of paces, steps and times must be con- structed for the individual sketcher and his horse. They may be marked on slips of paper and glued to the edges of the ruler. The face of the ruler seen in Plate 19 shows two trotting scales for a horse that trots a mile in 7^ minutes with 45 -inch steps, at a scale of 3 inches to the mile, and the map distance scales correspond to 20 feet V. I. If sketching at a scale of six inches to the mile, divide the numbers on the pace, step and time scales by two and use a V. I. 361 of lo feet. At one inch to the mile, multiply the numbers by three and use a V. I of 60 feet. If strides are counted instead of paces or steps, divide the num- bers on the pace or step scales by two. With this ruler properly marked, the sketcher will be prepared to work mounted or on foot, using paces, steps, strides or time, walking or trotting, at scales of I inch, 3 inches or 6 inches to the mile ; to space con- tour points for any slope at vertical intervals of 60, 20 or ID feet respectively, and to change from one method or one scale to another by turning the corre- sponding edge of the ruler to the paper. Scales for different lengths of paces, steps or strides, and for different rates of travel at walk or trot are given on Plates 20, 21 and 22. Any desired scale may be copied from these plates by placing the edge of a strip of paper on the proper line or by interpolating between the lines. METHODS OF SKETCHING. Different methods of sketching depend on the dif- ferent sets bf instruments that may be used to main- tain general control in directions, distances and ele- vations. In all methods of sketching the chief reli- ance is placed on the ability of the sketcher to draw fr.ee- hand on the paper what he sees on the ground. THE NOTEBOOK METHOD. In this method a record is made in a notebook of all the observations made with the hand instruments in the field, and on the right hand pages of the note- book free-hand sketches are made, showing the ground covered by the notes on the corresponding 362 left hand pages. After the day's field work is fin- ished, the notes are plotted on a sheet of drawing paper, and the "filling in" is done by copying and adjusting the free-hand sketches of the notebook. The outfit required for the field work consists of a box or prismatic compass, a clinometer, a stop watch, an aneroid barometer, notebook, pencils, etc. The sketching may be done on foot, on horseback, or in a wheeled vehicle. With the latter an odometer is usually attached to one of the front wheels to measr ure distances along the trail. The program of operations is as follows: At the initial station, read the aneroid, and if its rate of change has been determined by previous read- ings apply the correction that corresponds with the time of this reading, or plot aneroid readings, correc- tions, and corrected elevations on a piece of profile paper, as previously explained. With the compass, read bearings along the trail to the selected station in advance and to the most important critical points in the vicinity. With the clinometer, read vertical angles to the selected station and critical points, and read also the slope of the ground in the same direc- tions, estimating the distances to these critical points and also the distances over which the observed ground slopes remain uniform. Record these readings on the left hand page of the note book and make a fre'e- hand sketch on the right hand page showing roughly, but approximately to scale, the "points observed" and the features located by them. Observe only the points and slopes that are absolutely necessary for general control, and rely on the free-hand sketch for all details. 363 The record is made in form similar to that of a compass survey, with the following headings and columns : Date Road Sketch from to Made by at trot, i mi. in 8 minutes. PRISMATIC COMPASS. (^Plotting Diagram.) I 2 3 456789 At Sta. Point Obsr'd. Dist. Bear'g. Vert. Ang. Time. Aner- oid. Elev. Remarks. I 7:45 847 826 At Cross Roads. Top of Hill. 360 y 225° + 3X° Slope. 140 y 225° + 1?^° etc. etc. Sta. 2. 2 m 20 s 13° + X° im25S House. Soy R im48s Down Stream. 97° -%° At Bridge. Stream. 260° + ^° At Bridge. etc. etc. 2 8:05 914 8go Water Shed. 250 y 278° + 2° etc. etc. In the first column are the numbers of the stations occupied. Intermediate halting points are recorded in this column by distances along the trail expressed in stop watch time from the last station. At every 364, halt the stop watch must be stopped at the instant of halting and started at the instant of resuming the march. At the station halts, after its reading is recorded, it is set back to zero. It will then indicate always the distance in time from the last station. The third column records distances, expressed in yards for estimated side shots, and in time for the courses between stations. In the fourth column the bearings of lines are entered, and the letters R and L are used to denote rectangular offsets to the right and left respectively of the trail. Thus at i min. 25 sec. from Sta. 1, a house stands 80 yards to the right of the trail. Column 5 contains vertical angles to points observed and slope of ground. When the slope of ground is recorded it is so noted in column 2. Columns 6, 7 and 8 contain the record for the aneroid. When the aneroid is not used, or when its record is plotted on profile paper, these columns are omitted. The column of "Remarks" contains such descrip- tions of stations or of points observed as may be necessary. The sketching on the right hand pages is facili- tated by having a protractor circle printed on these pages as in the Engineer Field Notebook. Know- ing the general direction of the road or trail to be covered, mark that direction at the top of the pro- tractor in terms of its compass bearing and number the other 10 degree marks with their corresponding compass bearings for the compass used. Then, when facing forward along the trail, directions transferred by eye from the protractor to the station points on the, sketch will be approximately oriented with the corresponding directions on the ground. The scale 365 of the sketch must be carried in the mind so that dis- tances may be plotted by estimation. Beginners should refer frequently to constructed scales until a correct mental standard is fixed. For spacing con- tours, the standard map distances for different slopes soon become fixed in the mind and estimates will be fairly accurate. Alternate pairs of pages between those for record and sketch are used for notes describing roadway, bridges, crops, forests, villages, mills, shops, camping places, water supply, defiles, etc., with references by letter or number to points correspondingly lettered or numbered on the sketch. The record and sketch should include all details that can be drawn to scale to a distance of 300 or 400 3'ards on both sides of the trail. Villages and promi- nent natural features should be included to a distance of 800 or 900 yards, their location being determined by intersections from at least two points. A method often used for recording notes of a sketch is to begin at the bottom of the page and work toward the top, placing bearings and distances of the main traverse in a center column, and offsets right and left in columns at the right and left respectively of the center column. This method has no advan- tages over the method given above, and is simply an attempt to keep the notes as well as the sketch ori- ented with the forward progress of the work. The notes are still nothing but notes, and must be plotted after the field work is finished. The customary method of writing from the top of the page down is more natural and less confusing. Plotting the Sketch. — When the day's field work is completed, the sketch is plotted and filled in like an 366 ordinary compass survey. The main traverse along the trail is first plotted and marked with the eleva- tions of its stations. Then the side shots are plotted to locate important control or critical points, and these are marked with their elevations. Contour points are dotted on all lines whose slopes have been determined. If these lines have been well selected, they will in- clude all the principal control lines, especially water- sheds and water courses. Then, by copying the free- hand sketches of the notebook, fill in the details and make them conform to the control points determined by the main traverse, side shots and slope lines. For these purposes there will be required a small protractor marked to correspond with the compass readings, a ruler marked on its edges with a scale of yards and a scale of steps or time, whichever was used in measuring distances, and a set of map dis- tance scales on card or ruler. The use of these in- struments has been explained in previous pages. If the aneroid barometer has not been used, eleva- tions must be carried forward from station to station by means of the recorded slopes and distances, using the map distance scales to determine differences of elevation. Thus, Plate 23 shows three courses of a traverse made at a trot of 8 minutes to the mile, and the re- corded slopes are + 2 ^ °, + 6° and + 3 ^ °, respectively. Use the protractor and the 8 minute scale, i :2 1 120, to plot the courses, and then apply to the first course the map distance scale for 2^°. The known or assumed elevation at the first station is 925, the V. I. is 20 feet, and the next contour point will be at 940. Therefore, place the quarter point of the first division at this station and dot in the 940, 960, 980 and 1000 367 contour points, the last falling at the second station and giving its elevation. On the second course apply the 6° scale and there are found to be seven and one- half of its divisions, giving 1150 as the elevation of the third station. On the third course apply the 3^° scale, beginning at the middle point of the first divi- sion, and point off the map distances. There will be six points, and, estimating by eye, four-fifths of an- other division, which adds 16 feet, and gives 1276 as the elevation of the last station. In like manner, elevations may be carried from station to station throughout the day's traverse. By plotting the slopes on side shots from the sta- tion points, additional contour points are determined and contours may be drawn, as shown in the figure. When the aneroid barometer is used to determine the elevations of stations, no attempt is made to carry a continuous chain of slopes throughout the traverse, but at each station slopes are read on the principal control lines, and in plotting the work the station point is used as an origin for locating and sketching surrounding features, which are made to join those determined from adjacent stations. If the sketch is to be combined with other sketches of parallel roads to form a map of the region covered, it may be finished in black pencil, but if it is to be used in its original form, the following conventional colors are added with crayon pencils to bring out the different details distinctly. Trace the roads with yellow; stream lines, ponds and lakes with blue ; contours with red ; and lay on flat tints of green for forest masses, and of brown for cultivated fields. The fences, houses, bridges, cuts and embankments, rail- roads and other artificial features, remain in black. 368 Every sketch must contain a title, a scale, a mag- netic meridian, the date and the name of the sketcher. The descriptive notes kept on alternate pairs of pages in the notebook are used as the basis of a report which is submitted with the sketch. The reference letters and numbers should appear on the finished sketch. SKETCHING CASE METHOD. The sketching case is simply a small plane table. It is oriented by means of the attached compass, and its ruler serves as the alidade. For reading vertical angles or slopes it becomes a slope board and is held in a vertical plane with the ruler swinging freely on its pivot. Sight along the top line of the board at the "point observed" and read the angular scale on the lower edge at the point cut by the edge of the ruler. The clinometer is to be preferred for reading vertical angles, and may be used in connection with the sketching case. The aneroid barometer is also useful for obtaining elevations of principal stations. In this description it will be assumed that the sketching case alone is used, but it should be under- stood that the clinometer and aneroid are useful ad- juncts. At the initial station hold the board squarely in front of the body and face in the general direction which the sketch is to follow. Loosen the needle and turn the bars or lines of the compass cover to a position parallel with the needle. Draw on the paper parallel with the needle a magnetic meridian. As- sume a point on the paper and mark it as the initial station. Select the critical points that must be de- 369 termined, and for each of them face squarely toward it, orient the board by bringing the bars or lines parallel with the needle, place the edge of the ruler on the station point and sight it at the " point ob- served," draw a line along the edge of the ruler, estimate the distance in yards and plot it with the scale of yards. Having thus plotted the selected points, use the slope board (or clinometer) and read the vertical angles or ground slopes toward the same points, ap- ply the proper map distance scales and point off the contour points. Draw the contours thus determined, shaping them by constant observation of the ground surface, and sketch in all the surrounding details that can be drawn to scale. Lastly, face along the trail toward the next selected station and plot its di- •rection and slope. Walk or ride to the next station, halting when necessary to sketch in adjacent features, and at the new station plot its position by the measured dis- tance in paces or time along the trail. Determine its elevation by map distances from the first station (or by the aneroid), and proceed here, in the same manner as at the first station, to locate and sketch sur- rounding features and details, and continue in like manner from station to station throughout the day's march, finishing the sketch as it progresses. With practice and experience many of the fore- going operations may be curtailed or omitted, and more reliance be placed on the skill of eye and hand in sketching what is seen. The traverse along the trail should, however, always be determined by actual measurement of directions, distances and slopes, and not by estimation. 370 A fifteen-mile road sketch will, at 3 inches to the mile, cover 45 inches of paper, and if the road varies in direction, or if the original orientation has been wrong, the sketch will run off the edge of the 6 inch strip, which is rolled back as the sketch advances. When the main traverse approaches the edge of the paper with future promise of running off, terminate the completed portion of the sketch by drawing a line across the paper, and begin anew, assuming a central position on the paper for the station occupied. Correct the orientation so that it shall conform to the new direction of the trail, and draw the corresponding magnetic meridian. Several such changes may be necessary in the course of the day's work, dividing the sketch into as many separate sections. When the sketch is ended, cut the sections apart and lap the ends of adjacent sections, making the common station points coincide by means of a pin stuck through both. Turn the forward section so that its meridian line shall be parallel with that of the pre- vious section, and cut through both sections on a line that will save as much as possible of the sketch. Fasten the sections together in this position by gum- ming a strip of paper across the joint on the back. The several sections will then be properly oriented with respect to each other and will form a connected map of the strip of country traversed by the trail. If the route followed makes a circuit and returns to the starting point, the sketch must be made to close. Pin the sections of the sketch to a board or table with pins stuck through the common station points of successive sections and make a closed cir- cuit. Then adjust and shift the pin joints so that the meridian lines of all the sections shall be as 371 nearly as possible parallel. Cut through at the joints and fasten the sections together by pasting them on a single large sheet of paper. If portions of the sketch must be trimmed off in making the joints, they must be copied in the corresponding blank spaces on the adjacent section. The sketch is finished and colored in the manner described for the "notebook sketch," not omitting meridian line, scale, title, date, and name of sketcher. While making the sketch a notebook is used for notes that will be needed in preparing the usual re- port that is submitted with the sketch. FIELD DRAWING BOARD METHOD. This method is well adapted to sketching on foot, and is suitable for either road or position sketching, the drawing board is i2"xi5" and ^ inch thick, made of white pine and reinforced at the ends. On the back of the board is a slope board arrangement, consisting of a small plumb line and bob attached at the middle of the upper edge, and a scale of degrees near the bottom edge, with its zero at the middle point. Sight the top edge at the point observed and when the plumb line comes to rest hold it in place with the finger and read the scale at the point cut by the line. This is a rough method at best and can- not be used in a high wind. The clinometer should be used when available. A cord attached to the left front and right rear corners of the board and passing over the left shoul- der and under the right arm supports the board hori- zontally in front of the body when sketching, slings 372 it under the right arm while pacing, and does not in- terfere with its use as a slope board. A sheet of ruled paper is attached to the face of the board by thumb tacks. The outfit consists of the drawing board, a box or prismatic compass, a small rectangular protractor, scales of paces and of map distances, notebook, pen- cils, etc. The clinometer and aneroid barometer are useful adjuncts in place of the slope board attach- ment, and it is understood that they will be used when available and when the best results are desired. A passometer or pace tally is convenient for keeping the count of paces. Ruler, protractor and scales may be combined in one instrument. The best combina- tion is a stiff card, marked on one side with a pro- tractor covering three of its edges and a scale of yards and scale of paces on the two halves of the fourth edge. The other side contains the scales of map dis- tances as shown in Plate i8. Otherwise, the ruler shown in Plate 19 may be used in addition to the pro- tractor. At the initial station sight with the compass at selected critical points and at the next forward station. At the point assumed on the paper for "station occu- pied," plot, with the protractor, these compass read- ings as each is determined, and lay off the estimated distances with the scale of yards. Sight with the slope board (or clinometer) at the same points and read the vertical angle to each. Apply the proper scale of map distances on each line and dot the con- tour points. Draw the contours and fill in the sur- rounding details by free-hand sketching, locating them with reference to the plotted points and lines. While sketching, the board may be roughly oriented 373 by pointing one of the plotted lines at the correspond- ing point on the ground. Pace to the next station, counting paces and halt- ing when necessary to sketch in adjacent features. The halting point is plotted by laying off its distance in paces from the station, and details, right or left, are located by estimated rectangular offsets. Arriv- ing at the new station, plot the paced distance and determine elevation by map distances (or by the an- eroid), and here proceed as at the first station. The lines on the ruled paper are assumed to be meridian lines, and are so used in setting the pro- tractor with its center at the station point and its meridian edge parallel to the adjacent meridian line. Without the ruled lines it would be necessary to draw a meridian line through every station. The letters N. and S. are marked at the ends of one meridian line in order that no mistake shall be made in setting the protractor. Two or three strips or sections of the sketch may be run across the sheet of paper, each section begin- ning anew at the near edge when the sketch has run off at the far edge. When the first sheet is filled a new one is tacked on. The sections are cut apart and joined end to end with parallel meridians and matched station points in the manner described for the sketching case method, and the sketch and report are finished iij like manner thereto. 374 SKETCHING WITH IMPROVISED INSTRUMENTS. An experienced sketcher who has acquired correct mental standards for estimating directions, distances and slopes, and for representing on paper to proper scale what he sees on the ground, needs only a pencil and a piece of paper to produce a fair road or posi- tion sketch. The paper, preferably in the form of a pad or block, is oriented by backsighting on a plotted line, and so held while new directions are added by pencil strokes toward the new points, a glance toward each "point observed" being sufficient to determine its direction. If the sketch runs generally towards a well defined point on the distant horizon, such as a mountain peak, the sketching board or pad may be oriented by sighting one of its edges toward the dis- tant point, and whenever that point is visible the same orientation may be repeated. Estimated dis- tances in yards and measured distances in paces or time are laid off, and surrounding features and de- tails are sketched in by means of the "memory scales" that have been acquired by previous use of the standard scales of distances and contours. If a standard set of scales has not been used, these men- tal standards cannot be acquired with any degree of accuracy. If greater accuracy be desired, as in a position sketch, or if the sketcher has not sufficient experience to rely on estimated distances and mental scales, a simple plane table outfit may be devised and pre- pared with such material as can usually be found. The lid of a cracker box or other small packing case will serve as a drawing board. Cut three sticks, from straight brushwood or branches, about four feet 37U long, lay them side by side, and about six inches from one end lash them together with over and under turns of cord or twine tightened with frap- ping turns in the intervals. Stand this up and spread the legs and it becomes a tripod. The board rests on the projecting top ends and may be leveled by adjusting the legs and sighting in the plane of the board at a distant horizon line. For constructing a set of scales, a scale of inches is the first requisite. The sketcher should know some dimension on the hand, as the length of a finger or the distance between certain lines of the palm in inches. This distance may then be marked on a strip of paper which is folded into a number of equal folds, equal to the known number of inches, and a scale of inches is the result. The size of the glove is the number of inches around the hand at the knuckles. A strip of paper may be wrappedr around the hand as if measuring for gloves ; if the size of the glove is No. 8, the strip that encircles the hand is 8 inches long ; folded, it is 4 inches; another fold gives 2 inches, and a third fold, I inch ; and a scale of inches is provided. One of the inch divisions may be subdivided by eye or by construction into ten equal parts to give tenths of inches, and these may be subdivided by estimation to the nearest hundredth of an inch. With this scale of inches and tenths, any desired working scales of paces, time, or yards may be constructed in the usual manner. Remembering that 0.65, or 6^ tenths of an inch, is the map distance between contours for i degree, lay off this distance with the scale of tenths succes- sively on the edge of a card for the 1° scale, halve 376 these divisions for 2°, halve again for 4°, and so on for 8° and 16". Dividing 0.65 inches by i}4 gives 4^ tenths as the map distance for 1%°; and this may be succes- sively halved for the 3°, 6° and 12° scales. In like manner all of the scales shown in Plate 18 may be constructed and marked on the edges of a visiting card, or other rectangle of stiff paper. Vertical angles or slopes may be read by using the board as a slope board. Select one edge as a sighting edge, and near the middle of this edge on the back of the board attach a thread by driving a small wooden wedge into the board. A bullet or key will do for a plumb bob. Through the point of at- tachment draw a line perpendicular to the sighting edge. The zero of the angular scale must be on this line, and this angular scale must be constructed on both sides of the zero line. With the thread as a radius, strike an arc with a radius of 5.73 (or 5^) inches. Beginning at the zero line, lay off consecutive one-inch chords on this arc in both directions, and divide each chord into tenths. The divisions so marked will be degrees, be- cause the tangent of i degree is i -h- 57.3, and in this construction tenths of an inch have been laid off ap- proximately perpendicular to a radius of 5.73 inches. The degree lines may be prolonged to the bottom edge of the board and there marked by stretching the thread through each of the points. A ruler may be whittled out of any small piece of straight grained wood. It may be smoothed and straightened by rubbing its faces and edges on a board covered with fine sand. The triangular form is best, as this gives a good sighting edge when it is 377 used as an alidade. The scales that have been con- structed may be marked on its edges. A piece of paper is fastened to the board by driving small wooden wedges at the corners through the paper into the board. Pencil and eraser complete the outfit. At the initial station set up the tripod, level the board, and assume a point on the paper to represent "station occupied." Pivot the ruler's edge at this point and sight it at surrounding objects and critical points, drawing the corresponding direction lines, in- cluding one to a point selected for the next station. Estimate distances on side shots and lay them off •with the scale of yards, or pace these distances, if estimates cannot be relied upon, and lay off with scale of paces. Then pick up the board, and using it as a slope board, read the vertical angles or slopes on the same lines, and with the map distance scales dot the contour points from an assumed elevation for the station occupied. Draw the contours, shaping them by observation of the ground forms, and sketch in surrounding details. Place a mark at this station, such as a stake, stone, branch or bunch of grass and weeds — anything that can be seen from the next station — and pace the dis- tance to the next station. Here set up the tripod, level the board and plot the distance in paces from the first station. Find the elevation by map distances and fraction thereof. Orient the board by backsight- ing at the previous station with the ruler's edge on the line that represents the course. Then proceed at this station, as at the first station, to locate and sketch surrounding features and details. Continue these operations at successive stations throughout the day's traverse. At intermediate halts between stations, the 378 board may be held in the hand and approximately oriented by sighting at the back or forward station. Sometimes a point on the distant horizon may be se- lected which will assist in orienting the board when a previous station is lost sight of. An approximate meridian line may be drawn as follows. Place the watch on the oriented board and point the hour hand toward the sun. A line which bisects the angle between the hour hand and the XII mark will then point approximately to the south, and such a line may be drawn on the paper as a meridian line. For this purpose the watch should show local time. Otherwise, if camp is made near the end of the day's traverse and if the sketch is not needed at once, leave the board oriented at the last station, and at dark sight the ruler at the North Star and draw the meridian. POSITION AND OUTPOST SKETCHES. Any of the outfits and methods heretofore de- scribed may be used for any kind of sketching — route, road, or position, on foot or on horseback — but cer- tain outfits and methods are better suited than others for certain purposes. A position sketch is usually made at a scale of 6 inches = i mile with V. I. = lo feet, and includes much more detail than does either the route or road sketch. The area to be covered is correspondingly smaller, or for an ex- tended position, it is divided into areas of from one to two square miles each, and these are assigned to the members of a party of sketchers, whose sketches are subsequently combined into a single, connected 379 map. The sketch does not necessarily or usually follow a road, but runs across country on lines that give the best control and best view. It is made on foot, and the field drawing board or a light plane table outfit are most suitable for the purpose, better than the sketching case or the notebook. A "position" usually extends along high ground that commands the slopes and valleys to the front, and the sketch should include the position and the ground in front to a distance of at least two miles. Run a careful base line traverse for general con- trol along the position, leaving the stations plainly marked on the ground, and determining their eleva- tions with the aneroid barometer. Then run sec- ondary traverses in loops to the front, beginning at one of the main traverse stations and closing on another station. Each loop should inclose a strip about half a mile wide and extending about two miles to the front. The inclosed space not reached from the stations of the loop traverse may readily be filled in by eye. Each strip must be adjusted and con- nected with adjacent strips, so that the whole shall form a connected and consistent contoured map covering the position, and showing all essential de- tails. By assigning such strips to the individual mem- bers of a party of skilled sketchers, an extended po- sition may be covered in a day, and the sketches may be combined and printed on blue print or bromide paper and be ready for issue by dark of the same day. Outpost Sketch. — The purposes of an outpost sketch are similar to those of a position sketch, since it is intended to furnish a detailed map which will serve for the posting of troops, locating field works 380 and batteries and determining ranges, in case an engagement is expected at the position occupied. The method of sketching must, however, some- times be modified on account of the presence of an enemy which will prevent the running of traverses to the front and will confine the operations of the sketcher to the line of outposts. If the ground to the front is open it may be covered by triangulation, that is, by intersections and vertical angles from selected stations. For this method a light plane table outfit is most suitable, and the "improvised out- fit" described under that head will serve. The an- eroid barometer and clinometer will, of course, give better vertical control and should be used when available. Select stations on the "line of observation" which will give good views over all the foreground without unnecessarily exposing the observer. Connect these stations by a careful traverse, which may be run under cover in rear of the line of stations, and reach them by offsets or short branches. Occupy these principal stations in succession, and at each station orient the board and draw direction lines by sighting the ruler at prominent objects and important critical points in the foreground. For the nearer points, ad- jacent stations will give good intersections, but for distant points, widely separated stations will give a longer base and better intersections. When a distant point is located by intersection from two stations, read the vertical angle to that point with slope board or clinometer and determine its elevation by an application of the proper scale of map distances, counting from elevation of "station occupied" as determined by the main traverse. 381 "Write the elevation at each determined point. Having located a sufficient number of control points, interpolate contour points on the lines joining them and sketch in the contours. If the critical points have been well selected, the sketching of intermediate features and details should be easily accomplished. When the country is generally broken and wooded, the foregoing method is not applicable, and it will be practicable only to run short traverses from the main traverse to the front, on roads or paths under cover of patrols. For this purpose the field drawing board or the sketching case is suitable, and the sketches thus made may be combined to produce a fair map of the position and the foreground. Since the outpost sketch is intended to serve the same general purpose as a position sketch, it is usually made to the same scale, namely, 6 inches to the mile with V. I. of lo feet. For an extended position, cover- ing perhaps lo to 20 miles of front, a scale of 3 inches to the mile with V. I. = 20 feet, may be used, and the work would necessarily be divided among the mem- bers of a party of sketchers, to be subsequently com- bined into a continuous map. The same accuracy and attention to detail is not expected in an outpost sketch, since the ground cannot usually be traversed and thoroughly examined as in the position sketch. REMARKS ON SKETCHING. For many purposes a carefully made sketch map is just as good as one made by an accurate instru- mental survey, and since a road sketch will cover in a day the ground that it would require a week to sur- vey, the value of sketching may be appreciated, 382 especially in connection with military operations, where a map, to be of use, must be immediately available. For some civil purposes, such as the preliminary examination of a route for a road or railroad, or in exploratory expeditions in new countries, the meth- ods of sketching are equally applicable and valuable. It need not be assumed that the rough and hasty methods adopted in sketching will give only rough and approximate outlines of the ground covered. The degree of accuracy that may be attained after long practice and careful training of eye and hand is surprising to one not familiar with the results of such training. Maps produced by these methods and by experienced sketchers need no apology on the ground of haste or inaccurate instruments, since they will serve every purpose that an ordinary map is expected to fulfill, and for military purposes will include many details and features not usually shown on the civil maps of the region. Sketching parties, advancing on parallel roads under cover of the cavalry screen, and sending indi- vidual sketchers in turn to cover the cross roads, will ... in one day turn in sketches which, being combined, adjusted, traced and printed, will furnish for issue be- fore midnight of the same day, a map covering the entire front of the army, perhaps twenty or thirty miles from right to left, and extending a day's march, say fifteen miles, to the front. This map should show all of the roads in the region covered, with details of topography on a strip one-half mile wide along each road, and the most prominent features in the inter- vening areas. The accompanying reports will give all the additional information which may be useful 383 in ordering the day's march, selecting camping places, and in making proper tactical disposition of troops and trains. Similar maps are turned out for each day's advance of the cavalry screen. When neces- sary, the intervening areas not covered from the main roads may be filled in by other parties of sketchers, and a new and complete edition of the maps may be issued. Upon contact with the enemy, the sketchers are set to work making detailed position sketches, which should cover not only selected positions, but as much of the ground towards the enemy as time will per- mit. These are also printed and issued for each day's work, but may subsequently be combined and printed as one map (perhaps in several sheets) of the front of operations. For the service which has been outlined above, it is evident that a well trained body of sketchers is necessary and that they must be organized and con- trolled under one chief, who will see that all of the ground is covered, that none of it is covered more than once, and that all the sketches are turned in at the appointed place and time, to be combined, printed, and issued at the earliest possible moment. Every individual road sketch or position sketch should be made with the idea that it is to be com- bined with adjacent sketches. To this end, the stand- ard scale and vertical interval must be adhered to ; the terminal stations must be well defined points, easily identified, and carefully described in the re- port ; cross and branch roads should be marked with the name of, and distance to, the nearest town or vil- lage ; local names of villages, streams, heights, rail- way stations, postoffices, etc., should be ascertained 384 and printed at appropriate places on the sketch, and every piece of paper that contains a portion of sketch should contain also a magnetic meridian, a scale, the date, and the name of the sketcher. If these points are carefully attended to by each sketcher, the chief of the party will have no difficulty in combining the several sketches that are turned in to him. Only the regular methods and operations of sketching have been described in foregoing pages. There are numerous checks, tricks, shortcuts and aids that are applicable under various circumstances and that will occur to the sketcher as he acquires ex- perience. A few may be mentioned to indicate their general character. Horizontal sights with clinometer or slope board from a given station will locate a number of points of equal elevation which may be used as check points in subsequent work. By sweeping a hillside with a horizontal line of sight, a contour may be traced on the ground to serve as a guide in tracing the contour on the sketch. A line of sight to a distant horizon is very nearly horizontal. When two or more prominent objects come into range from a point on the trail, halt and plot the line through those objects. Subsequent intersecting sights on the same objects will determine their posi- tions. When coming on line with a distant stretch of road or river, plot the line and use it as a check on subsequent location of the same stretch in case the trail runs to it. If adjacent reaches of river are thus plotted and joined by curves, that portion of the 385 river will be fully determined, although the trail may not run near it. In road sketching it is not necessary always to follow the road. A road often runs between high banks or hedges that obstruct the view, and it may be better to traverse a line on high ground near the road, which will give better control and more ex- tended view. The road may be readily located by offsets from the traverse. Three or more plotted sights to the same object from different points of the trail should intersect in a common point. A parallel road may sometimes be discovered by noting the passage of a wagon or horseman along it. If several positions of the wagon be estimated and plotted, a portion of the parallel road may be sketched. Similarly, the passing of a locomotive in the distance will disclose the location of a railroad that might not otherwise be discovered. In some regions an outcrop of ledge rock main- tains the same elevation for many miles, and will serve as a check on elevations. Similar checks and aids will occur to the sketcher as his work progresses, and he should make use of every device that will expedite his work and increase its accuracy. Nothing, however, can take the place of the skill that is acquired only by practice. -s? sr Or ^ Cs o^i o 2? O O Cj . © on *^ <^ ^ <£> 1 S ^ § ^ -"!/!$ ^. <- 7 ,'r-, Sx<>r 7 1 -^^ 'o^ 1 '\ > K \- I I 4^ ^ t "-/-- ~"A' ^' / ~ '7~— \ I" \ ^., _ _ — a T "^, ^ ■- T ^5 '^sl i ._4 WJ ^v. -)- t'^; i s " ^i — |--^ ^— -^ ^ i. ^--, . 1 X ^ 2,//2. V.J 7BC 60 ^0 30 2^ SO /^ /S. /O i> ^ Z 2^ 2. /^ >'"~