UNIVERSITY OF CALIFORNIA AT LOS ANGELES TELESCOPE COMPASS. A MANUAL PLANE SURVEYING; CONFINED TO WORK WITH THE COMPASS. WITH AN APPENDIX. AMPLY ILLUSTRATED. REVISED AND ENLARGED. BY THOMAS BAGOT, SUPERINTENDENT OP RIPLEY COUNTY, INDIANA. INDIANAPOLIS, INDIANA: THE NORMAL PUBLISHING HOUSE. J. E. SHERRILL, PROPRIETOR. 1889. Entered according to Act of Congress in the year 1881, BY J. E. SHEERILL, In the office of the Librarian of Congress at Washington. CARLON A HOLLENBECK, PRINTERS AND BINDERS, INDIANAPOLIS. TA I DEDICATE THIS BOOK TO MY MOTHER THE AUTHOK. 356 PREFACE. EVERY person who studies Surveying from the text-books in gen- eral use, and afterward is called upon to discharge the duties of a surveyor, must, in the course of time, become aware of two things : (1) that he has spent time in learning much that he has never had, and probably never will have, occasion to use, and (2) that a great deal he needs to know, and must know, is not to be found in the books. This is the case particularly under the Rectangular System, and the author's experience has led him to believe that a necessity ex- ists for a book dealing directly with the problems continually coming up before surveyors throughout the country, and that such a book will be cordially received by every person who wishes to understand the subject as it is comprehended in general practice. And this, dear reader, accounts for the existence of this little book. You will find it simply a brief treatise on Compass-Sur- veying, shorn of everything superfluous, and yet embracing all that is necessary to a good understanding of the subject. Very few geometrical or trigonometrical terms are employed, and all the problems may be mastered by any person having a moderately good knowledge of arithmetic. The author does not claim that the book is above criticism, but, on the contrary, he is well aware of the fact that a person disposed to criticise may find in it an ample field in which to exercise his talents. He trusts, however, that a search for its faults will re- sult in disclosing enough merit, even among so much demerit, to excuse him for writing it, and so trusting, he submits it to the public. NEW MARION, INDIANA, May, 1883. (4) CONTENTS. CHAPTER I. INTRODUCTION. No. of Art. ( 1). Surveying defined. ( 2). Branches. 1. Topographical surveying. 2. Geodetic surveying. 3. Plane surveying. 3). Measurements, how 1. Actual area nearly always greater than computed area. 2. Smooth surface conceived underneath. 3. Impossible in many instances to compute the area of real surface. ( 4). Corners defined. ( 5). Surveying instruments used. ( 6). Transit and compass. 1. Remarks on the transit. 2. Remarks on the compass. 7). Chain and pins. ( 8). Chaining over hills. ( 9). Flag-staff, drawing instruments, etc. (10). Assistants needed. CHAPTER II. DESCRIPTION OP THE COMPASS. (11). The compass circle, how divided. (12). Magnetic needle and center pin. (13). Degrees, how marked. and 90 points. (14). Compass box, plate, sights, etc. (5) (15). Description of needle. Delicacy, how determine^. (16). Horizontal angles, how measured. (17). Letters "E" and " W" reversed on compass face. (18). Eeason for this. (19). Kule for reading bearings. (20). Reverse bearing. (21). Northerly and southerly bearings. (22). Running lines east or west. (23). Rules for measuring angles. 1. When both readings are in the same quadrant. 2. When one reading is in each of either the two north quadrants or the two south quadrants. 3. When one reading is in each of either the two east quad- rants or the two west quadrants. 4. When one bearing is in each of two opposite quadrants. (24). Reasons for these rules. (25). Glass cover to compass box. Electricity, how excited in it, and how removed. (26). Needle affected in other ways. (27). Sight compass and telescope compass. Either may be either a plain compass or a vernier compass. (28). Description of the vernier. (29). How used. (30). Other kinds of verniers. (31). Repairing the compass. 1. To re-magnetize the needle. 2. To sharpen the center pin. 3. To replace a spirit level. 4. To adjust a new sight. 5. To straighten the center pin. 6. To straighten the needle. 7. To put in a new glass. 8. To regulate the movement of the ball. (32). The compass, how carried. (33). Needle should be lifted from center pin when the compass is not in use. (34) Compass should be kept level when not in use, and the needle allowed to assume its natural position. (35). The telescope compass gradually growing in favor. CONTENTS. 7 CHAPTER III. THE VARIATION OF THE MAGNETIC NEEDLE. (36). Meridian defined. (37). The true meridian. (38). Methods of determining the true meridian. 1. By a shadow. 2. By Polaris. (a). Polaris and Alioth. (b). Pole between them. (c ). The plumb-line. (d). Greater accuracy. (e ). When upper culmination occurs during the day. (39). Table showing time of culmination of Polaris. (40). Magnetic meridian. (41). East variation and west variation. (42). Agonic line. (43). Isogonic lines. [variation of the needle. (44). The north magnetic pole; the effect of its movement on the (45). Secular change of variation. (46). Diurnal change and annual change. (47). Table showing diurnal change by hours. (48). Diurnal and annual changes usually disregarded in practice. (49). Electric disturbances. (50). The "dip" or inclination of the needle. (51). Many magnetic phenomena imperfectly understood. CHAPTER IV. EFFECT OF CHANGE OF VARIATION ON OLD LINES, AND METHODS OF CORRECTING BEARINGS. (52). Bearings of lines subject to constant change. (53). Illustration. (54). Effect of re-surveying lines without considering the change. (55). Area of tract not affected. (56). To determine the present bearing of a line. 1. Bearing at time of a previous survey. 2. Date of previous survey. 3. Annual amount of secular change. This may be de- termined in various ways. (1). By comparison of bearings. (2). By establishing a true meridian. (3). By interpolation. O CONTENTS. (57). Table showing the variation of the needle at important sta- tions. (58). Table showing annual amount of secular change in certain localities. (59). Tables to be used in approximation. (60). Determining variation for past times. (61). Advantage of basing bearings on the true meridian. (62). Magnetic meridian bearings subject to constant change. (63). To determine the true bearing of a line from its magnetic bearing. 1. When the variation is west. 2. When the variation is east. (64). Bearings to be changed. (65). When supplement is to be taken. (65). Changing from the true meridian to the magnetic meridian. CHAPTER V. METHOD OF RUNNING LINES. (67). Finding a corner, etc. (68). Determining position of corner from witnesses. (69). Where trees are not available for witnesses, other things are used. (70). Method of describing witness trees. (71). Starting the survey of a line. (72). Course and distance of line must at least be known approxi- mately before the line can be surveyed. (73). Setting the compass and measuring the line. (74). Setting off variation on the vernier. 1. When the variation is east. 2. W T hen the variation is west. (75). Surveying the line. (76). How chained. (77). Pins, stakes, etc. (78). Relative places on line of men engaged in the survey. (79). Continuation of the line to the opposite corner. (80). In case the line does not strike the corner. (81). Illustration. (82). Rules for correcting the stakes. (83). Terminations of lines and starting point regarded as vertices of an isosceles triangle. (84). Examples in moving (correcting) the stakes. (85). Abbreviations used. (86). Errors caused by imperfection of instruments, etc. (87). Correction of assumed bearing. (88). Rules for correcting bearings. 1st method. Derivation of rule. - (1). Trigonometrical lines. (2). Sine and cosine defined. (3). Study of relation of sine and cosine important. (4). Application of these lines. (5). -When the angle is large. 2d method. (1). Modification of first method. (2). Illustration. 3d method. 4th method. (89). Examples of lines whose bearings are to be corrected. (90). Is amount of correction to be added or subtracted? 1. Rule for north-east and south-west courses. 2. Rule for north-west and south-east courses. (91). Examples under these rules. (92). Thus far, all bearings have been based on the true meridian. (93). Bearings of lines based on the magnetic meridian. 1. Rule for correction. 2. Demonstration of the rule. (94). Rule good while the needle moves westward. (95). Bearings to be corrected. (96). Abbreviations, etc. (97). Completion of the survey. (98). Assistants generally sworn. (99). Instruments should be tested frequently. (100). Backsights. CHAPTER VI. UNITED STATES RECTANGULAR SURVEYING. (101). Public lands, how divided. (102). Townships, how divided. (103). These provisions sufficient. (104). Fundamental lines and initial point. (105). Some imperishable mark chosen for the initial point. 10 CONTENTS. (106). Survey of range lines or meridians, etc. (107). Survey of parallels, etc. (108)". Convergence of meridians. Correction lines. (109). Auxiliary meridians, etc. (110). Survey of townships north of the base-line and east of the principal meridian. (111). Survey of townships north of the base-line and west of the principal meridian, etc. (112). Excesses and deficiencies. (113). Congressional township and civil township. (114). Survey of sections. 1. Preliminaries. 2. How to obtain bearings. 3. Sections, how numbered. 4. Survey of section 36. 5. Survey of the rest of eastern tier of sections. 6. Completion of the township. (115). Full sections and fractional sections. The "double frac- tional." (116). Meander corners and meander lines. (117). Making up the field-notes. (118). Monuments adapted to the country surveyed. (119). Work of Government deputy extends only to the division of the township into sections. (120). Advantages of the Rectangular system, etc. CHAPTER VII. THE DIVISION AND SUBDIVISION OF THE SECTION. (121). Subsequent surveys must be made in accordance with the original. (122). The ideal section and the real section. (123). The divisions and principal sub-divisions of the section. (124). Names of these divisions and sub-divisions. (125). Descriptions of land generally qualified by the words "more or less." (126). Corners, how named. 1. Section corners. 2. Quarter-section corners. 3. Half-quarter corners. 4. Fourth-quarter corners. CONTENTS. 11 (127). Section lines and center lines. (128). Position of corners, how determined. 1. Section corners and exterior quarter-section corners. 2. Center corners, methods of setting. (1). By crossing the center lines. (2). By bisecting east and west center line. 3. Half-quarter corners. 4. Fourth-quarter corners. 5. Other corners. (129). Examples in setting corners. (130). Survey of the divisions and sub-divisions of the section: 1. Quarter section. 2. Half-quarter section. 3. Fourth-quarter section. (131). General rule. (132). When some of the boundary lines are known. (133). Tracts to be surveyed. (134). Independent corners, lines, and tracts. (135). Illustration of independent corners, etc. (136). Dependent corners, lines, and tracts. (137). Dependent lines, etc., how surveyed. (138). Examples of dependent tracts. (139). Description by "metes and bounds." (140). Rules for setting corners, etc., in full sections, generally apply to fractional sections. (141). When a tract of land lies partly in one section and partly in another. CHAPTER VIII. FIELD-NOTES. (142). Field-notes of sectional survey by the Government deputy. (143). Contents of full section, supposition regarding it, etc. (144). Plot of township. (145). Explanation of the plot. (146). List of witnesses to corners. 1. Exterior corners. 2. Interior corners. (147). Other particulars. 1. Area of fractional quarters. 2. Creeks, etc. 3. Offsets. 12 CONTENTS. (148). Surveyor needs copy of original field-notes. (149). What surveyors' records should contain. (150). Pocket record. (151). Description of pocket record. 1. Left-hand page. 2. Right-hand page. (152). Instructions regarding bearings. (153). Representation of surveys on the plot. (154). Approximating the bearing of a line. (155). Illustration. (156). Principles underlying the method. (157). The field-book. (158). Method of keeping the field-book in independent surveys. (159). When new witnesses should be taken. (160). Method of keeping the field-book in dependent surveys. (161). Names of stations, witnesses, etc. (162). Records by authorized surveyors taken as prima facie evi- dence in favor of surveys recorded. (163). Other methods of keeping field-notes. CHAPTER IX. RE-LOCATION OF CORNERS. (164). Trouble caused by lost corners. (165). Nature of this trouble. Means of re-locating corners. 1. Remains of missing corner or witnesses. 2. By course and distance from some other corner. 3. By retracing old line by marks on trees, etc. 4. By projecting lines. (1). Illustration of this method. (2). The reverse of the method by which the corners were established. (3). Another illustration. (4). Examples for practice. 5. A last resort. (1). Setting S. } corner. (2). 'May not agree in position with corner lost. (S). A case illustrated. (4). Lost section corner. (5). Case of disagreement illustrated. (6). When a quarter corner can not be found. CONTENTS. 13 (166). Re-locating original corners to the variable quarters of fractional sections. (167). To set a quarter corner between two fractional sections. (168). To set an exterior corner to a fractional section, or to any exterior section. 1. When there is an offset. 2. When there is no offset. (169). Examples. (170). He-location of subsequent corners, etc. CHAPTER X. DESCRIPTIONS OF LAND. (171). Necessity of a good description. (172). Length of lines and area of tracts should be given in sur- veyor's measure. (173). Tables of equivalents. (174). Examples for reduction. (175). Fractions of a chain expressed in links. (176). General rule for reduction. (177). Area of tracts, how expressed. (178). Description of independent tract. (179). Examples of errors in descriptions. (180). Examples for correction. (181). Erroneous descriptions. (182). Use the words " more or less." (183). In describing dependent tracts the course and distance of each boundary should generally be given. (184). The description should state whether the bearings are based on the true meridian or on the magnetic meridian. (185). Lines running north, etc. (186). Surveys generally made in accordance with the description. CHAPTER XL OBSTACLES TO ALIGNMENT AND MEASUREMENT. (187). Obstacles met on line. (188). Two classes of obstacles. (189). Methods of spanning obstacles of first class. 1. By perpendiculars. 2. By an equilateral triangle. 3. By a right-angled triangle. [urement. (1). When an obstacle both to alignment and meas- (2). When an obstacle to measurement alone. 14 COXTENTS. 4. By symmetrical triangles. 5. When a fence is built on or near the line. (1). When offset line terminates on the opposite side. (2). When the offset line terminates on the same side. (a). When the distance missed is greater than the offset. (b). When the offset is greater than the distance missed. 6. Surveying over hills. (190). Methods of spanning obstacles of the second class. 1. By a right-angled triangle. 2. By symmetrical triangles. 3. By similar triangles. (191). Other methods, etc. CHAPTER XII. COMPUTATION OP AREA. (192). Advantage of expressing dimensions of tracts in chains and links. (193). Special rules deduced. (194). Examples in computation. (195). Every tract of land a polygon in shape. (196). Rectangles. (197). Parallelograms. (198). Trapezoids. (199). Triangles. 1. With base and altitude given. 2. With no altitude given. (200). Trapeziums. (201). Any figure. (202). Computation of area by latitudes and departures. (203). Latitude and departure, as applied to courses, defined. (204). North and south latitudes, and east and west longitudes. (205). Signs of latitudes and departures. (206). Relation of sine and cosine to departure and latitude. (207). Table of natural sines and cosines explained. (208). Examples in computing latitudes and departures. (209). Columns marked differently at top and bottom. (210). Traverse tables. CONTENTS. 16 (211). Sum of latitudes and sum of departures in every cor- rect survey. (212). A trial survey. (213). Explanation. (214). When a discrepancy exists. (215). Initial line or meridian. (216). Difference between longitudes and departures. (217). How to determine the longitude of a course. (218). Algebraic sum must be used. (219). Simplification of the rule. (220). Computation of area by longitudes. (221). North products and south products. (222). Double longitudes. (223). Method of keeping the data. (224). General rule for computing areas by double longitudes. (225). To determine the most westerly corner. (226). Examples in computation. (227). Courses without departures, etc. (228). Not absolutely necessary that the meridian be drawn through the most westerly station. CHAPTER XIII. LAYING OUT AND DIVIDING UP LAND. (229). No general rule can be given. (230). What are known in problems to be considered. (231). To lay out a square. (232). To lay out a rectangle. (233). To lay out a parallelogram. (234). To lay out a right-angled triangle. (235). To lay out a trapezoid. (236). To lay off any figure. (237). Things to be considered in making partition. (238). Nature of problems chosen. (239). To divide a rectangle into equal parts. (240). To divide a rectangle into unequal parts. (241). Problems. CHAPTER XIV. SURVEYING TOWN LOTS. (242). Description of town lots, blocks, etc. (243). Survey of town, how based. 16 CONTENTS. (244). Usual shape of lots, etc. (245). What the plot of a town should show. (246). Particulars respecting Figure 58. (247). Illustration of a town in which the lots vary in size. (248). Method of surveying lot number 11 in the figure. (249). Examples for practice. (250). Chain or tape used in surveying should be tested frequently. CHAPTER XV. PLOTTING. (251). Plotting defined. (252). Instruments used. 1. Drawing board. 2. T-square. 3. Euler. 4. Drawing pen. 5. Dividers or compasses. 6. Protractor. 7. Diagonal scale of equal parts. (253). Units of the scale may have various equivalents. (254). Plotting bearings. (2551. Examples for practice. (256). Plotting rectangular tracts. (257). Plotting tracts in general. (258). A particular case. When a survey does not "close." (259). The pantograph. (260). Locating objects on the plot. (261). Coloring plots and maps. CHAPTER XVI. SURVEYING WITHOUT A COMPASS. (262). The compass nearly always necessary. (263). Setting corners. (264). Establishing lines. (265). Setting out perpendiculars. (266). Survey of rectangular tracts. (267). Measurements. APPENDIX. Land Decisions. Table of natural sines and cosines. MANUAL PLANE SURVEYING. CHAPTER I. INTRODUCTION. ART. ( 1 ). SURVEYING is that branch of applied mathematics which embraces operations for finding, (1) the relative positions of points on the earth's surace, (2) the area of any portion of its surface, and (3) the contour or shape of any part of its surface, so that it may be represented in maps and plots. ( 2 ). It is divided into three branches : 1. Topographical Surveying, or Topography, includes operations for determining the contour of portions of the earth's surface and re- presenting it on paper. 2. Geodetic Surveying, or Geodesy, takes into consideration the curvature of the earth's surface and is employed in extensive sur- veys. 3. Plane Surveying does not regard the curvature" of the earth's surface and all lines are measured as on a plane. It is used in lo- cal work. (3). All measurements in surveying are made as nearly hori- zontally as possible, and the area of a tract of land is not its ac- tual surface measure, unless the tract be perfectly level, but the amount of land enclosed by its boundaries measured horizontally, instead of with the inclinations of the surface over which they run. 2 (17) 18 MANUAL OF PLANE SURVEYING. 1. The actual area, therefore, is nearly always greater than the computed area, and increases in proportion to the inequality of the surface. 2. We may conceive a smooth surface at the level of the ocean underlying the surface of the land; then the area of a tract of land is equal to the contents of a figure formed by projecting the boundaries of the tract on the horizontal surface below. 3. Were the real surface considered, it would be impossible in many instances either to compute its area or represent its figure on paper. ( 4 ). The extremities of lines in surveying are called corners, and each corner marks the vertex of an angle formed by the meet- ing of two lines. The corner is a mathematical point, and may or may not be marked with a monument. ( 5 ). Lines are surveyed either with a solar or a magnetic in- strument. With the former, where more precision than expedi- tion is required ; and with the latter, where expedition is of more importance than precision. ( 6 ). The principal magnetic instruments in use are the transit and compass. 1. The transit is provided with a telescope, and at the present time is so constructed as to be adapted to the measurement of both horizontal and vertical angles. It serves many important purposes independent of the assistance of the magnetic needle, and is not strictly a magnetic instrument. 2, The compass is either supplied with sights or a telescope, and is strictly a magnetic instrument. It is not usually adapted to measuring vertical angles. The lightness, simplicity, and con- venience of the compass have brought it into almost general use in common surveying, and the following chapter is devoted to a description of it. The transit may be found described in almost any comprehensive work on Surveying. ( 7 ). Measurements of lines in surveying are made with an iron or steel chain* usually 33 feet or 2 rods long and divided into 50 links. It has a handle at each end by which it is carried during the survey, and the successive chain-lengths are maked *For convenience in counting the links, this chain is divided into five parts of ten links each by four brass or copper tags. The real chain is 100 links or 66 feet in length, and called a Gunter's chain, from its inventor. Ihe half-chain of 50 links is used for convenience. A link is 7.92 inches in length. In Government surveys, a chain 66.06 feet in length is used. The M foot being added to make up for " slack," etc. MANUAL OF PLANE SURVEYING. 19 with pins of iron or steel wire, generally 10 or 12 inches in length, sharpened at one end and bent into the form of a ring at the'other. In the ring is sometimes tied a bright ribbon or piece of cloth to render the pin more conspicuous, in order that it may be easily found by the person who carries the rear end of the chain, and such colors should always be chosen as will contrast most with the surface to be surveyed. ( 8 ). In chaining up and down hill, the chain must be kept taut and horizontal, as on a level surface. In order to do this, it is sometimes necessary to drop the pin from the front end of the chain, or elevate the rear end of the chain to a point exactly over the pin that sticks in the ground. (9). A straight staff, about 1 inches in diameter and 8 or 10 feet high, surmounted by a small flag of brilliant color, is used in alignment; and a good set of drawing instruments, drawing board, t-square, triangle, protractor, ruler, etc., are necessary in drawing and plotting. (10). In field-work the surveyor generally needs four assist- ants ; two chain-men, one flag-man, and a marker. The first two measure the line with the chain, the third carries the flag, and the fourth assists the chain-men by marking the line with stakes at the proper distances. To this force it is sometimes necessary to add one or more ax-men, as bushes may have to be cut out of the way and trees marked. QUESTIONS ON CHAPTER I. 1. Define Surveying. 2. Into what three branches is it divided? 3. What does each branch embrace ? 4. How are measurements made in surveying? 5. Why is the actual area of a tract of land nearly always greater than its computed area? 6. What is a corner? 7. Name the two kinds of instruments used in surveying. 8. Name the principal magnetic instruments. 9. Describe the transit. The compass. 10. How are lines measured in surveying? 11. How should the chain be held in chaining over hills? 12. How many assistants does the surveyor usually need? CHAPTER II. DESCRIPTION OF THE COMPAS& (11). The compass consists essentially of the compass circle, the magnetic needle, and the sights. The circle has its circum- ference raised and divided into 360 equal parts or degrees, and these are usually subdivided to half or quarter degrees. ( 12 ). At the center of the compass circle" is placed a perpen- dicular pin, called the center pin. Upon this pin the magnetic needle is balanced in such a way as to mark opposite points of tht divided circumference of the circle. ( 13 ). The degrees on the circumference are marked from twa opposite points, called points, up to 90 to the right and left. The 90 points are, therefore, opposite one another also. One of the points is called the north point, and the other the south point, and one of the 90 points is called the east point, and the other the west point. ( 14). The compass circle is enclosed in what is known as the compass box, and this rests upon the compass plate. The box is usually between 5 and 7 inches in diameter, and the plate about 14 or 16 inches long. At the ends of the plate perpendicular sights are placed. These sights have slits in them, and are so placed that the line of sight from one of them to the other will strike opposite points of the graduated circumference of the com- pass circle. Between the compass box and the sights are usually placed two spirit levels, one at right angles to the other. These are used in leveling the compass. The compass rests upon a tri- pod or Jacob's staff', at the head of which is a ball and socket joint, enabling a person to move the compass as he may wish. (15). The needle is a magnetized steel bar, very delicately (20) MANUAL OF PLANE SURVEYING. 21 balanced upon the point of the center pin, and it is a little shorter than the diameter of the compass circle. The delicacy of the needle is determined by the number of horizontal vibrations it will make before coming to rest after being disturbed. Needles 5 or 5 2 inches in length are generally preferred by surveyors to longer or shorter ones. ( 16 ). Horizontal angles are measured with the compass by turning the sights from one line of the angle to the other and noting the number of degrees passed over by the end of the needle. The sights are sometimes arranged for the measurement of verti- cal angles also. ( 17 ). The letters "E" and " W" are reversed on the compass face, but it will be plainly seen that the arrangement enables the surveyer to take the direction (bearing) of a line more readily from the compass face, and reduces the liability to err in the reading. ( 18 ). This may be illustrated by Fig. 1. FIG. 1. Suppose the sights set in the direction of the points of the circle, and that the compass be turned until the north end of the needle marks the point midway between the north point (gener- 22 MANUAL OF PLANE SURVEYING. ally marked with afleur de Its, instead of the letter N) and E, or to 45. The sights have moved in the direction of the second hand of a watch, and the bearing of the line marked by them is N 45 E, which is read directly from the north end of the needle. ( 19 ). The following is the general rule for all readings: Note the letters between which the end of the needle comes, and to what number. Then name the letter N or S (as the case may be) that is the nearer to the end of the needle from which you are reading, next the number of degrees to which the needle points, and lastly the letter E or W that is the nearer to the same end of the needle. ( 20 ). If the preceding reading had been taken from the south end of the needle, the bearing would have been S 45 W, the re- verse of X 45 E, but equivalent to it, and indicating the bearing taken from the opposite end of the line. S 45 W is called a southerly bearing, and N 45 E is called a northerly bearing. (21). No matter how the lines run, the same end of the com- pass (the north end is preferred) should be kept in front. (22). In running lines east, the E point of the compass circle will be turned toward the north, and in running west, the W point will be turned north. All bearings should be read from the north end of the needle. (23 ). In measuring angles observe the following rules: 1. When both readings are in the same quadrant, as between N and E, N and W, S and E, or S and W, the angle is equal to the difference between the two readings. Thus, the angle between N 56 E and N. 43 E is equal to 13. 2. When one reading is in each of either the two north quad- rants or the two south quadrants, the included angle is equal to the sum of the two readings. Thus, the included angle of N 35 E and N 23 W is equal to 35 + 23 = 58. 3. When one reading is in each of either the two east quadrants or the two west quadrants, the included angle is equal to the sum of the readings subtracted from 180. Thus, the angle included between N 50 E and S 37 E is equal to 180 (50 + 37) = 93. 4. When one reading is in each of two opposite quadrants, the angle is equal to the difference of the readings subtracted from MANUAL OF PLANE SURVEYING. 180. Thus, the included angle of N 16 E and S 12 W is equal to 180 (16 12) = 176. ( 24:). The reasons for these rules will be seen in Fig. 2. FIG. 2. The angle included between the courses A B and A C is equal to 59 28 = 31, as both readings are in the same quadrant. The angle included between the courses A B and A D is equal to 28 + 36 = 64, because one reading is in each of the two north quadrants. The angle included between the courses A C and A F is equal to 180 (59 + 33) = 88, because one is in each of the two east quadrants. The angle included between the courses A D and A F is equal to 180 (36 33) = 177, because the courses are in opposite quadrants. ( 25 ). The compass box is protected by a glass covering over which fits a brass lid. Care should be taken while the brass lid is off that no electricity be excited in the glass by the friction of the hand or a cloth upon its surface, as it interferes with the work- 24 MANUAL OF PLANE SURVEYING. ing of the needle and may cause a serious error. However, when the fluid does exist, it may be removed by breathing on the glass or touching it in various places with the moistened finger. ( 26 ). The action of the needle is also affected by pieces of iron or steel brought or kept near it. This materially interferes with its use at sea, particularly on iron ships. While surveying, noth- ing having a tendency to affect the action of the needle should be carried upon the person or allowed near the compass. ( 27 ). Two kinds of compasses are in use the sight compass and the telescope compass. Each of these may be either a plain compass or a vernier compass. The plain compass is not very ex- tensively used, as all readings are made from its face alone, and can not be depended on for precision. In the plain compass, the line of sight lies in the direction of the points of the compass circle. The vernier compass differs from the plain compass in having its compass circle, to which a " vernier" is attached, mov- able, generally through a short arc, about its center, thus enabling the surveyor to set the zeros or points of the circle at an angle with the line of sight. This angle is read from the vernier. The movement of the circle is effected by means of a thumb-screw that gives it a slow motion. When the required angle is set off, the vernier is clamped to the plate of the compass and the readings taken. The vernier enables a surveyor to take a certain class of readings "closer "or with greater precision than is possible without it, as it gives him the advantage of a double index ; yet readings down to a very small angle, say V or 30", are hardly ever trustworthy, owing to the difficulty a surveyor meets in setting the needle to exactity. The fault, however, is not in the vernier. ( 28 ). The VERNIER* consists of an arc divided into a certain number of equal parts, and moving within another arc whose di- visions are somewhat larger or smaller than its own. The first arc (vernier), as stated before, is attached to the compass circle and moves with it around a common center ; the second arc is called the " limb," and is generally on the brass plate of the com- pass upon which the circle moves, so that the outer edge of the vernier coincides with the inner edge of the limb. *The vernier is so called from its inventor, and on this account the word is sometimes written with an initial capital. It was first applied to the compass by David Rittenhouse of Philadelphia. MANUAL OF PLANE SURVEYING. 2D ( 29). Let us now suppose the divisions on the limb to equal half-degrees, and that the vernier-arc, corresponding to twenty- nine divisions of the limb, is divided into thirty equal parts. It is plain, since 30 divisions of the vernier equal 29 divisions of the limb, that one division of the limb equals ? divisions of the vernier, and that one division of the vernier equals f di- vision of the limb. But each division of the limb equals 30' or one-half of one de- OA/\xOQ gree ; therefore, one division of the vernier will equal HQ = ^9' > which is one minute less than a division of the limb. Now, suppose the zero of the vernier to correspond with the zero of the limb ; then the points of the compass circle lie in the line of sight. If now we turn the vernier until its first division from zero coincides with the first division from zero on the limb, and on the same side of zero as the division of the vernier, the points of the compass will make an angle of V with the line of sight; if the second division of each coincide, the angle will be 2'; if the third, it will be 3', and so on by the same increase, so that if we make the twenty-ninth division of each correspond, the an- gle will be 29' ; and if we turn still further until the first division of the limb coincides with zero of the vernier, the angle will be SO'. In the same manner, 30' acting as a base, the angle may be increased to 1, and so on. ( 30 ). Sometimes verniers read lower than V, but they are not of much practical use on magnetic instruments. They may also differ in construction from the kind described above, but they all work on the same principle. The plate on page 26 represents a vernier compass; the vernier may be observed on the compass plate in front of the box. ( 31 ). As a general thing, when a compass needs repairing, either from wear or on account of some mishap, it is best to for- ward it to some maker of mathematical instruments. This, how- ever, may not always be convenient or practicable, and it may be well enough to give some directions, which may be of service in case of an emergency : 1. To Re-magnetize the Xeedk. When the needle works lazily, on account of losing a portion of its magnetism, it may be re- magnetized with a common bar or horse-shoe magnet by passing the south pole of the magnet along the north end of the needle 26 MANUAL OF PLANE SURVEYING. from the center to the extremity and bringing the magnet back to the starting point in a circle of five or six inches radius. The south end of the needle should be treated in the same manner, except that the north pole of the magnet should be used on this end. From twenty to thirty passes will give it an ample charge. 2. To Sharpen the Center-pin. Sometimes the needle moves slug- ishly when the cenler-pin upon which it turns becomes dull. When this is the case, take out the plate in which the center-pin is set and then unscrew the pin. It may then be sharpened on a very fine stone and finished on a piece of smooth leather. Care must be taken to grind equally from every side of it. MANUAL OF PLANE SURVEYING. 27 3. To Replace a Spirit-level. Eemove the brass tube from the plate and take off the caps at the ends of it. Then with some pointed instrument, as an awl or a penknife, scrape out the plas- , ter or other substance that holds the vial in place, and next force out the old vial by pressing on one end of it. Xow slide the new vial into place, keeping the proper side up, and if it is too small for the tube, wedge it up with pieces of wood or paper. Notice carefully its position with regard to the opening in the tube, and when it is set in its proper place press some beeswax, boiled plas- ter, or putty of the proper consistency, around the ends of it, so as to fasten it firmly to the sides of the tube; then put on the brass caps and replace the tube on the compass plate. To re-adjust the level, press on the compass plate until the bubble stands in the center of the opening in the tube; then turn the compass one-half round, and if it remains there, the level is properly placed, but if it runs toward the end of the vial, and it probably will, the end toward which it settles is too high, and should be lowered or the other end raised, whichever is necessary in order to keep the tube parallel with the compass plate. After this, give the compass another half-turn and repeat the process given above until the bubble will remain in the middle of the opening in the tube in every horizontal position of the plate. 4. To Adjust a Ifeiv Sight. Fit it to its place on the plate, and notice how the slit lines with that of the old one on the opposite end. If it inclines to one side, remove it and file off its base on the opposite side where it rests on the plate. Then try it again, and keep up the operation until the two slits coincide throughout their whole length. If both sights need adjusting, hang a plumb, using a fine thread or hair, and regulate both sights by it. The com- pass should be perfectly level whenever an observation of the thread is taken, and the sights will be properly adjusted when- ever they correspond with the plumb-line. 5. To Straighten the Center-pin. Remove it with its base from the rest of the compass and bend it with a pair of pincers or wrench made for the purpose, always grasping it about an eighth of an inch below its point. 6. To Straighten the Needle. It sometimes happens that the nee- dle of the compass does not " cut " opposite degrees on the circle, as, for instance, when its north point is placed at its south point inclines either to the right or left of the opposite 0; when this is 28 MANUAL OF PLANE SURVEYING. the case the error may be corrected by bending the needle with the fingers. 7. To put in a new Glass. First take off the brass ring that con- tains it (bezzle ring) and remove the putty. Then take out the old glass and put in the new by reversing the process. If the new glass is so large that it will not go in readily, hold the edge on a grindstone and grind it down. The manner in which it should be ground may generally be seen by noticing the glass just taken out. 8. The motion of the ball at the head of the Jacob's staff may be regulated by a screw-cap that fits down upon it. It should be kept reasonably tight in order that the compass may not be too easily jarred out of level. If it works loosely, screw the cap down tighter. After long usage the ball may not fit the cavity well, and in this case it may be taken out and a small piece of sheet brass placed under it, or even a piece of paper will answer for a short time. ( 32 ). In carrying the compass it need only be lifted from the staff and put under the left arm so that one of the sights may pro- ject up behind the shoulder, and the staff makes a good walking stick; but in transportation over the country the sights should be taken off and all packed snugly in a box or something else that will answer the purpose. A suitable box is usually furnished with the compass by the manufacturer. ( 33 ). All compasses are provided with a lever or spring with which to raise the needle from the center-pin when the compass is not in use, and this should not be neglected. ( 34 ). The compass when not in use should be placed in a hor- izontal position and the needle allowed to assume its natural di- rection. If this precaution is taken, the needle will better retain its polarity. ( 35 ). The telescope compass is gradually growing into favor with surveyors and seems to be taking the place of the sight com- pass in many localities. The telescope enables the surveyor to- set a flag at longer ranges, at greater elevations and depressions, and discern it more easily among trees and bushes. It is not quite so convenient to handle as the sight compass, however, but this is no great disadvantage. Either may be used on a tripod, instead of a Jacob's staff. For plate of a telescope compass see frontis- piece. This engraving represents the very fine instrument manu- factured by T. F. Randolph, Cincinnati. MANUAL OF PLANE SURVEYING. 29 QUESTIONS ON CHAPTER II. " 1. Describe the compass. 2. How are the degrees numbered on the circumference of the compass circle ? 3. How are the sights arranged ? 4. Describe the magnetic needle. How is the delicacy of a needle determined? 5. Explain the method of measuring horizontal angles. 6. How are vertical angles sometimes approximately meas- ured? 7. Why are the letters E and W reversed on the compass face? 8. State the rule for taking the readings from the compass circle. 9. The north end of the needle points 20 to the right of N. What is the bearing of the line of sight? What is its reverse bearing? 10. What is a northerly bearing ? A southerly bearing? 11. In running lines east what letter on the compass face should be turned north? In running west, what one? Why?' 12. Give the four rules for measuring angles. 13. What is the included angle in each of the following cases : N 40 E and N 62 E? S 15 W and S 39 E? N 29 W and S 43 W ? 14. How is electricity excited in the glass of the compass ? How may it be removed ? 15. How do iron and steel affect the action of the needle ? 16. What two kinds of compasses are in use ? 17. How many kinds of sight compasses are there ? 18. What is the difference between a plain compass and a ver- nier compass? 19. Describe the " vernier." Of what advantage is it ? 20. How is the needle re-magnetized ? 21. Explain the manner in which the center-pin is sharpened. 22. How is a spirit level replaced ? 23. In what way is a new sight adjusted ? 24. How do you determine, when the two extremities of the needle do not " cut " opposite degrees, whether it is the needle or the center-pin that is bent? Answer Turn the compass and notice the amount of the error in several 30 MANUAL OF PLANE SURVEYING. positions. If it decreases in certain places and increases in others, it is the center-pin. If it remains about the same in every position, it is probable that the needle alone is bent. 25. Why should the needle be raised against the glass when the compass is not in use ? 26. What is the proper position for the compass during the in- terval between surveys? 27. What are the advantages of the telescope compass over the sight compass? CHAPTER III. THE VARIATION OF THE MAGNETIC NEEDLE. ( 36 ). The meridian of any point on the earth's surface is a due north and south line connecting the point with the poles of the earth. ( 37 ). This is called the true meridian of the place, in order to distinguish it from the magnetic meridian, which will be considered further on. ( 38 ). Various methods are employed in determining the true meridian, but only two of the most simple and satisfactory will be described : 1. By a Shadow Cast by a Perpendicular Otgect. Erect a perpen- dicular staff on a level surface, so that its shadow will remain on the surface from about 8 o'clock, A. M., till 4 o'clock, p. M., as in- dicated in the horizontal projection, Fig. 3, in which S N repre- sents the staff. (31) 32 MANUAL OF PLANE SURVEYING. Three or four hours before noon, with a radius, P S, shorter than the length of the shadow, and from the point S as a center, describe an arc through the point P and produce it beyond, opposite P. Then mark the point P, where the shadow last touches the arc, and in the afternoon the other point P where it first touches it again, and connect these two points with a line PP. Bisect this line at O, and the line SO, produced in either or both directions, will represent the true meridian of the place. It is not exactly correct, except at certain times during the year (at the solstices), but it is always sufficiently accurate for ordinary purposes. 2. By the Polar Star. The Polar star (Polaris) is situated about 1J degrees from the north pole of the heavens, and appears to re- volve around the pole once in 23 hr. 56 min. If, now, we suppose a vertical plane to pass through the north pole and the eye of the observer, then twice during the time of revolution Polaris will be in this plane, and consequently in the meridian of the observer (once when above the pole and again when below it). These are called its upper and lower culminations, respectively. ( 1 ). On the opposite side of the pole from Polaris is a star known as Alioth, or more commonly as Epxilcm, of the constellation of the Great Bear or " Dipper" This is the first star in the handle of the Dipper, and is situated next to the four that form the quad- rilateral, the outside two of which, Dubhe and Merak, are called " the pointers," because they indicate the position of Polaris. ( 2 ). Since these stars, Polaris and Alioth, are almost exactly on opposite sides of the pole, it is evident that they will both be on the meridian of the observer when one of them is above the other, and it is more convenient to make the observation during the upper culmination of Polaris. (3). Suspend a plumb-line from some elevated projection, as the limb of a tree or a strip nailed to the side of a building or high post, and at a point south, not so distant that Polaris will rise above the point of suspension of the plumb, arrange a short board horizontally east and west, a little below the level of the eye. On this board place some kind of a contrivance containing an opening across which a thread may be stretched so that it will be parallel to the plumb-line when the instrument is in use, and slide the instrument along the board until the thread ranges with MANUAL OF PLANE SURVEYING. the plumb-line and Polaris. Continue to move the instrument west as Polaris moves to its point of superior culmination, and watch also the approach of Alioth to the meridian. As soon as the plumb-line falls on both stars, fasten the instrument to the POLARIS. NORT.fr (- ALOTH. FIG. 4. OUBHE. WERAK board, and you have two points in the true meridian. The line through these may be produced at pleasure and permanently marked. Fig. 4 represents the plumb-line covering both stars. (4). Still greater accuracy may be reached by following Polaris for twenty-two minutes after the plumb-line falls on it, and then marking the line. (5)- When the upper culmination occurs during the day, the lower culmination must be used, but Alioth is then very high. ( 39 ). The following table shows the time of the upper culmin- ation of Polaris for each tenth day. The time for intermediate days may be approximated by interpolation. The time is given to the nearest minute: 3 34 MANUAL OF PLANE SURVEYING. MONTH. 1st Day. llth Day. 21st Day. January . . . h. m. 6 : 30 P. M. h. m. 5: 50P.M. h. m. 4: 50P.M. February . . 4 : 27 " 3 : 47 " 2 : 48 " March .... 2 : 32 " 1:53 " 12:50 " April May 12 : 26 " 10 : 29 A.M. 11: 47A.M. 9 : 49 " 10: 49 A.M. 8: 50 " June 8:27 " 7 : 48 " 6 : 49 " July 6:29 " 5: 50 " 4 : 51 " August . . . 4 :28 " 3:48 " 2:49 " September . 2: 26 " 1 : 47 " 12:48 " October . . . 12 : 29 " 11: 49 P.M. 10: 50P.M. November. . 10 : 27 P. M. 9:47 " 8:48 " December . . 8:28 " 7 : 49 " 6: 50 " (40). If a magnetic compass be placed on the true meridian thus established, the needle of the compass, pointing toward thenorth magnetic pole, instead of toward the north poleofthe earth, marks a line called the magnetic meridian, which coincides with the true merid- ian in comparatively few places on the earth. The angle formed by the difference in direction of these two lines is called the varia- tion or declination of the needle. (41). If the north point (south pole) of the needle points to the east of the true meridian, the variation is said to be east, and if it points to the west, the variation is said to be west. The amount ot variation is determined by the size of the angle. ( 42 ). In the United States there is a line extending southeast through the eastern part of Michigan, western part of Lake Erie, eastern Ohio, central West Virginia, Virginia, and North Carolina, reaching the Atlantic ocean near Wilmington, that is called the agmic line or line of no variation, because at all points thereon the magnetic meridian and true meridian coincide. Places east of this line have west variation, and those west of it have east varia- tion. MANUAL OF PLANE SURVEYING. 35 ( 43 ). Isogonic lines or lines of equal variation run through places having the same variation. For instance, the line of three degrees west variation passes through Chesapeake Bay, Maryland, central Pennsylvania, and western New York, and the line of three de- grees east variation passes through western South Carolina, eastern Georgia, Tennessee, Kentucky and Indiana, and western North Carolina, Ohio, and Michigan.- The variation increases in both directions from the agonic line, reaching about 18 degrees west variation in eastern Maine and 22 degrees east variation in the northern part of Washington Territory. ( 44 ). The isogonic lines converge toward the north magnetic pole, situated at present in longitude about 96 west from Green- wich, and latitude about 70 north. This pole has been gradually moving westward for several years, and will perhaps continue to do so for several years to come. This causes the agonic line and all the isogonic lines to move in the same direction, so that west variation is constantly increasing and east variation constantly decreasing throughout the eastern and central portions of the United States. ( 45 ). The change that takes place in the variation of the needle at any place from this cause is called its secular change. This varies in different localities, and is generally greater in the northern part of the United States than in the southern part. It is determined by comparing the variation at the time of any ob- servation with that of a preceding or succeeding one. ( 46 ). But this is not the only change to which the needle is subject. Its action is modified by other influences. The north end of the needle moves westward from about 6 o'clock A. M., until about 2 o'clock p. M., and then gradually returns to the starting point. This is called its diurnal change, and it sometimes amounts to 10 or 12 minutes of a degree. This change is about twice as great in summer as in winter, hence an annual change must be taken into consideration. ( 47 ). The following table, taken from the Keport of the United States Coast Survey, illustrates these changes. The mean magnetic meridian is the average position of the needle for the day : MANUAL OF PLANE SURVEYING. ^ d HOUK. A 1 a a S i 3 'S .S OJ "*! ^ / 1 i f 6 A. M. 3 4 2 i 7 " " 4 ' 5 3 i 3 4 70 71 5 3 5 4 36 s 15 20 2 3 31 32 33 34 35 36 6 3 5? 35 iS FIG. 17. 66 MANUAL OF PLANE SURVEYING. The manner in which the sections are surveyed will now be ex- plained. 4. The surveyor goes to the south-west corner of section thirty- six to begin the survey. At this point he sets his compass at the bearing of the east line of the township, which should be, but sel- dom is, the true meridian, and runs north forty chains. Here he establishes a quarter-section corner between sections thirty-five and thirty -six, and then proceeds forty chains or one- half mile further to the corner of the section, or rather to the corner of sec- tions twenty-five, twenty-six, thirty-five and thirty-six, since these four sections have a common corner here. Distances from the starting point at which brooks, creeks and other objects of im- portance are met on the line, are carefully noted. From this cor- ner he runs a random line to the east line of the township. If this line intersects the township line at the first mile corner, it is marked back as the true north line of section thirty-six, but if it does not, the distance which it misses the corner, either right or left, is noted, and the line changed accordingly. 5. Having returned to the north-west corner of section thirty- six, he next proceeds to survey section twenty-five in the same manner, and he follows the route indicated by the figures until he completes the survey of the eastern tier of sections. It will be ob- served that when he completes the survey of section twelve, he then finishes up the survey of section one by running north on a random line and correcting back to the south-west quarter. 6. He next surveys the second tier of sections by beginning at the south-west corner of section thirty-five, and proceeding north in the same manner as in the survey of the first tier, and thus he continues until he reaches the fifth tier. Here, after surveying section thirty-two, he runs west from its north-west corner to the range line or meridian and completes the survey of section thirty- one, and continues to work north, surveying the fifth and sixth tiers together, until he reaches the north line of the township. (115). The township is now divided into sections, each of which, except those in the north and west tiers, called fractional sections, is sold as containing six hundred and forty acres of land, more or less, and each quarter-section as containing one hundred and sixty acres, more or less. The fractional sections generally contain a greater or less quantity of land than the other sections, because all excesses and deficiencies fall to them. In surveying MANUAL OF PLANE SURVEYING. 67 them, -however, the quarter-section corners between them are so placed that the excesses and deficiencies fall to the exterior quar- ters, and the interior quarters, or those touching the other sections of the township, are sold as containing the proper amount of land one hundred and sixty acres each. The exterior quarters are sold as containing whatever the measurements of the survey indicate that they contain. Section six is sometimes called the "double fractional," and usually contains only one exact quarter. (116). Whenever, in the course of a survey, an impassable bar- rier, such as a lake or navigable river, is met, the surveyor estab- lishes what is called a meander corner on its margin, and then runs a meander line from this corner along the edge of the obsta- cle. The rivers and lakes thus meandered are reserved in the sale of the public lands. (117). The surveyor, from the time the survey of the principal meridian is begun until the township is divided into sections, marks every half-mile of true line that he surveys with a corner, and keeps an account in his field-notes of every important object he meets in the survey, as well as a topographical description of the country. He also makes two sets of corners on the correction parallels, one for townships north, and the other for townships south of the line. Aside from this it is not unusual to find two sets of corners on interior parallels and meridians, owing to dis- crepancies between contiguous surveys. (118). The monuments used in marking corners are always adapted to the country in which the survey is made, and their position is generally witnessed by one or more bearing trees, or mounds of earth thrown up around a stake or a stone, whose courses and distances from the corner are carefully noted in the field-books of the survey. These witnesses, if trees, are always marked facing the corner, which enables them to be more easily found at any subsequent time. (119). This completes the work of the Government Surveyor, or deputy, as he is usually called. He returns his notes to the Surveyor General of his district, and these notes become the basis of all subsequent surveys. The work of dividing and sub-dividing the sections, which belongs to the county and private surveyors, we shall consider in the next chapter. (120). This beautiful system of land surveying, not unlike the old Roman system, was devised about the year 1785, for the pur- NOTE ON ARTICLE 117. In a few instances three sets of corners have been established on range lines, but at the present time only one set is established except on the base-line and correction parallels. MANUAL OF PLANE SURVEYING. pose of preparing the North West Territory for settlement, and has answered the purpose in an admirable manner. It is the re- sult of mature deliberation, and exhibits no mean knowledge of engineering skill, and, like many other great inventions, its beauty and utility consist in its extreme simplicity. It has long since outgrown the limits for which it was intended, and soon nearly the whole territory between the western boundary of Pennsylvania and the Pacific ocean will be united in one complete net-work of sections. The readiness with which it enables a surveyor to re-trace old lines and determine the location of lost corners prevents an end- less amount of litigation common to States not surveyed accord- ing to this system. QUESTIONS ON CHAPTER VI. 1. What are the fundamental lines of a survey? 2. What is their point of intersection called? 3. Upon what principal meridian is the survey of Indiana based? 4. What is a range? A township? 5. Draw a diagram representing town 4 south, range 3 east. 6. How is the error caused by the convergence of the merid- ians arrested? 7. How are townships north of the base-line and east of the principal meridian surveyed ? South of the base-line and west of the principal meridian? 8. What is the difference between a congressional township and a civil township? 9. How may a surveyor ascertain the bearing of the lines that bound a township? 10. How many sections in a township? How are they num- bered? 11. Where does a surveyor begin work when he divides a town- ship into sections ? 12. Describe the method of surveying the eastern tier of sec- tions. The western. 13. Name the fractional sections. The double-fractional section. 14. What causes fractional sections? 15. How many acres in a section? MANUAL OF PLANE SURVEYING. 69 16. What quarters of fractional sections are generally full? 17. What is a meander line? 18. Why are two sets of corners needed on correction lines? 19. How far apart are the corners on lines surveyed by the gov- ernment surveyor? 20. For what purpose was the rectangular system devised? When? NOTE ON CHAPTER VI. The system of surveying described in this Chapter was not set forth in its present completeness at the beginning. The first act of Congress on the subject was that of May 20, 1786, and under this act the first public surveys were made. The territory covered by these surveys forms a part of the State of Ohio, and is generally called the " Seven Ranges " In it the sections are numbered north and south from the south- east corner of the township. The Geographer of the United States directed the surveys. Although the principles of the system have virtually re- mained unchanged, the system itself has been made definite and concise since iis inception. CHAPTER VII. THE DIVISION AND SUB-DIVISION OF THE SECTION. ( 121). The county or other surveyor makes all his surveys in accordance with the field-notes of the original survey, a copy of which forms a part of the public records of each county. Subsequent surveys may prove that great irregularities exist in the original survey, but none of the lines or corners can be changed. (122). The ideal section of the young surveyor frequently 3 If FIG. 18. differs greatly from the real section he meets in practice. The ideal section is very nearly a perfect square varying a little on account of the convergence of the meridians ; it is bounded by four straight lines, and contains almost exactly 640 acres of land. (70) MANUAL OF PLANE SURVEYING. 71 The real section may sometimes be far from square; its boundary lines may deflect every half-mile on its perimeter, and its area may exceed by several acres the area of another section adjoining. The surveyor, however, must adhere as closely as possible to the original survey, and let the section and divisions and sub-divisions of the section contain whatever the government deputy saw proper to put into them. (123). Let us now see what the principal divisions and sub- divisions of the section are, and then consider the method of sur- veying them, locating the corners, etc. (124). Fig. 18 represents a section, with some of the main di- visions and sub divisions laid off on its face. They are described .as follows : 1. S. W. qr. 2. N. W. qr. 3. S. } N. E. qr. 4. E. \ S. E. qr. 5. W. } S. E. qr. 6. N. E.JN. Kqr. 7. S. J N. W. J N. E. qr. 8. N. E. i N. W. i N. E. qi. 9. N. W. \ N. W. \ N. E. qr. Of these tracts, Nos. 1 and 2 each contain 160 acres; 3, 4 and 5 each 80 acres ; 6, 40 acres ; 7, 20 acres ; and 8 and 9 each 10 acre? ( 125 ). Whenever an instrument of writing, such as a deed or mortgage, implying responsibility to the amount of the descrip- tion, bears on a tract of laud, the area is usually qualified by the compound term " more or less." For instance, No. 1, above, would be described as containing 160 acres, more or less ; No. 3, as con- taining 80 acres, more or less, and so on. ( 126 ). The corners of the section, or of the different parts of it, are named from their position. The principal ones are the sec- lion corners, J corners, \ corners, and J- J corners. 72 MANUAL OF PLANE SURVEYING. The numbers on the diagram correspond with the names after similar numbers below. to !l a / S Z 3 / 3 1 3 z S 2 B IS 7 If FIG. 19. 1. Section Corners : (1). N. E. Corner. (2). S. E. " (3). S. W. " (4). N.W. " 2. Quarter-Section Corners: (5). N. \ Corner. (6). E. \ (7). S. \ " (9). Center MANUAL OF PLANE SVRVEYING. 73 3. Half-Quarter Corners: (10). N. H W. Corner. (11). N. HE. (12). E. i i K " (13). E. \ S. " (14). S. J i E. (15). S. J i W. (16). W.H& ' (17). W.H>'. " (18). N. H (19). E. H (20). S. H (21). W.H 4. Fourth-Quarter Corners: (22). N. W. Center Corner. (23). N. E. " " (24). S. E. (25). S. W. " ( 127 ). The exterior lines of the section are called section lines, and the two lines that cross at the center of the section are called center lines. ( 128 ). We are now ready to examine the method by which the position of each of the various classes of corners is determined. 1. The section corners, and the exterior quarter-section corners, except in an occasional case on a town or range line, are set by the Government deputy, and the surveyor who follows him sets the re- maining quarter-section corner (center) and all the minor corners. 2. The section is divided and corners set according to instruc- tions from the proper authority, and the following is the author- ized method of setting the center corner : (1). A line is run connecting the N. J corner with the S } cor- 74 MANUAL OF PLANE SURVEYING. ner, and another connecting the E. | corner with the W. corner. The point of intersection of these two lines is taken for the center of the section. The preceding is the method in general use, and is perhaps the most equitable one that could be devised, but the following has been used in some places : (2). A line is surveyed connecting the E. corner with the \V. } corner, and the middle point of this line is taken for the center of the section. If the section lines were straight from one corner of the section to the other, and the quarter-section corner midway between the section corners, the corner determined by this method would coin- cide in position with the one determined by the other; but as this is not always the case, they may differ considerably in position. 3. To Set a Half-Quarter Corner. Kun a line along the quarter- section, on the side upon which the corner is to be set, and from one corner to the other. Bisect this line for the corner. For example : To set the E. J $ corner, we connect the center corner and E. J corner with a line, and the middle point of this line is the required corner. 4. To Set a Fourth- Quarter Corner. -First set a i } corner on each of two opposite sides of the quarter-section. Then connect these two corners with a line, and bisect this line for the } corner. To illustrate : In order to set the N. E. center corner, it is nec- essary first to set the N. i E. corner and the E. i J- corner, or the E. J J N. corner and the N. } corner. The middle point of a line connecting one corner with the other, in either set, for it is immaterial which is taken, will be the required corner. 5. The same methods are employed for setting corners to the divisions of the fourth of quarter-sections. For instance, to set the north-east corner of the S. $ N. W. J N. E. qr., it is necessary only to bisect the line between the N. E. center corner and N. E. corner. EXAMPI/2S. ( 129 ). (1). How would you set the S. H corner? (3). How is the S. J J W. corner set? (4). Describe the method of setting the S. W. center corner. ( 130). We are now prepared to understand how the divisions and sub-divisions of the section are surveyed. 1. A quarter-section is surveyed by running the two exterior MANUAL OF PLANE SURVEYING. 75 Lalf-mile lines, and the two center lines of the section, in order to locate its interior corner and fix its interior boundaries. 2. A half-quarter is surveyed by first surveying as many of the lines of the quarter as enter wholly or in part into its boundaries, then whatever other lines are necessary to determine its remain- ing corners, and finally the lines connecting these corners with one another, or with others. For instance, to survey the S. S. E. qr., it would be necessary to run the south line of the quarter, the east line, the two center lines of the section, and the E. and W. center line of the south- east quarter. 3. A fourth-quarter is surveyed on the same principle as the half-quarter. For example, in surveying the S. E. J S. E. qr., it is necessary to survey the south and east lines of the quarter, the E. and W. and N. and S. center lines of the section, either the E. and W. or the N. and S. center line of the S. E. qr., and finally the north line of the fourth-quarter, if the N. and S. center line has been sur- veyed, or the west line, if the E. and W. center line has been sur- veyed. ( 131 ). As a general rule for the survey of a tract of land bear- ing a relation to the section, the following is submitted : Run lines to connect the known corners of the tract when no unknown corner intervenes between them, then such lines as are necessary to determine the location of unknown corners, and finally lines to connect these newly located corners with one another, or with others. ( 132 ). In many cases some, if not quite all, the boundary lines of a tract of land are known. There is seldom any need of re- surveying these, except where it must be done in order to establish other lines or corners. EXAMPLES. (133). What lines would a surveyor have to run in order to establish all the boundaries of each of the following described tracts? (1). S. W. qr. (2). N..JN. E. qr. (3). S. } N. W. qr. (4). N. E. J S. W. qr. (5). N.W.JN.W.qr. /6 MANUAL OF PLANE SURVEYING. ( 134 ). So far we have dealt exclusively with corne-s, lines, and tracts which may be called independent : The corners, because their position is determined by the division of certain lines, and is not definitely fixed, so far as distance is concerned, by any other point in the section. The lines, because they connect the corners, and may, therefore, vary in a limited degree either in course or distance, or both. The tracts, because they are limited by the lines and are not definitely fixed as to area or figure. ( 135 ). To illustrate the preceding still further, suppose the north line of a quarter-section to be 36 chains long, instead of 40 chains long, and the contents of the quarter-section to be 148 acres, instead of 160 acres. The J J corner on the north line will then be 18 chains from the section corner, and the same distance from the \ corner, and each of the lines will be but 18 chains in length, and a fourth-quarter out of the quarter-section may fall short three or four acres. If, now, the north line of the quarter- section had been longer, the N. 5 \ corner would have been further from each of its north corners, and the area of the fourth-quarter would have been greater. (136). There is a class of corners, lines, and tracts, however, that may be called dependent : The corners, because their distance is fixed from some given point, and is not obtained by bisecting a line. The lines, because they connect the corners, and are conse- quently of a definite course and distance. The tracts, because they are bounded by the lines, and their area, therefore, is not affected by the excess or deficiency of land in the section. Dependent corners, then, are those whose position is definitely fixed ; dependent lines connect dependent corners, and dependent tracts are bounded by dependent lines. (137). Dependent lines and tracts are surveyed without any reference whatever to the division or sub-division of the section, although they may depend on some corner in the section as a base. A tract may be partly dependent and partly independent. ( 138). The following are examples of descriptions of depend- ent tracts: (1). N. 45 E. 16.00; thence N. 45 W. 10.00; thence S. 45 W. 16.00; thence S._45 E. 10.00, to the place of beginning. MANUAL OF PLANE SURVEYING. 77 (2). Commencing at the S. E. corner of section 22, town 3 N., range 4 E. and running thence N. 15 E. 12.00; thence S. 45 E. 12.00; thence S. 75 W. 12.00, to the place of beginning. ( 139). In a dependent tract the course and distance of each of its boundary lines are usually given in the description of it, and it is then said to be described by " metes and bounds." ( 140). The rules given for setting corners, running lines, and surveying tracts, in full sections, also apply to fractional sections, except where their fractional sides do not contain half, or much more than half, the usual amount of land found in these parts of sections, or where the amount they contain considerably ex- ceeds the usual amount. In the first case, the outside tier of fourths of-quarter is omitted, and in the second the excess is usually thrown to them, and the inside tier of fourths-of -quarter in the outside quarters of the sections, are left of their usual size. If the deficiency is very great, perhaps the entire outside tier of quarters is wanting. By act of Congress approved April 24, 1820, fractional sections containing less than 160 acres are not to be di- vided in the original survey. In the division of fractional sec- tions, generally, such rules must be adopted as the exigencies of the case seem to require. ( 141 ). When a tract of land lies partly in one quarter of a section and partly in another, each part should generally be de- scribed and surveyed separately ; and the same may be said of tracts extending into two or more sections, and sometimes also of those extending into different fourths of the same quarter. The following are examples : (1). N. } N. E. qr., and N. E. J N. W. qr. (2). S. W. i N. E. qr., and N. J N. W. J S. E. qr (3). N. E. qr. Sec. 4, and N. W. qr. Sec. 3. (4). S. E. i S. E. qr. Sec. 35, and W. \ S. W. qr. Sec. 36. (5). S. E. J N. W. qr., and E. } N. E. \ N. W. qr. In some of the cases an occasional line may be a boundary to each part of the tract, as the east line of No. 5, described above. QUESTIONS ON CHAPTER VII. 1. Does the section always contain exactly 640 acres? 2. Draw a diagram of a section and represent the following tracts on it : N. E. qr.; N. W. } N. W. qr.; S \ S. W. qr.; N. } N. E. } N. W. qr. 3. Why is the phrase " more or less " used in descriptions of land in deeds, etc.? 78 MANUAL OF PLANE SURVEYING. 4. Name the quarter-section corners. The J \ corners. The J } corners. 5. What are the " center lines "? 6. What eight corners does the Government deputy establish to nearly every section ? 7. Does he ever set an interior corner of a section? 8. Give the first method of finding the center of a section? 9. Prove that the first and second methods would agree, if the section lines were straight from one section corner to another. 10. How do you set a \ \ corner? A \ \ corner? 11. How would you survey a quarter section? A half-quarter? A fourth-quarter? 12. Give a general rule for the survey of a tract of land bearing a relation to the section. 13. What is the difference between independent and dependent corners? Lines? Tracts? 14. Describe an independent tract. A dependent tract. 15. When do the general methods for dividing and sub-dividing- sections not hold good in fractional sections? 16. When should different parts of a tract be surveyed sepa- rately? CHAPTER VIII. FIELD-NOTES. ( 142 ). The field-notes of sectional surveys by the Government deputy show : 1 . The witnesses taken at section and quarter-section corners. 2. The length of the fractional lines in fractional sections. 3. The number of acres in each of the fractional quarters of fractional sections. 4. The offsets between section corners in one township and the corresponding ones in the adjoining township. These offsets are sometimes found on town and range lines as well as on correction parallels. 5. The distances from the starting point of a line at which brooks and creeks are crossed, and trees and other objects met with on the line. 6. A description of the timber, surface, soil, etc. 7. The courses and distances of meander lines surveyed along rivers, lakes, etc. (143). Each full section is supposed to contain 640 acres of land, and it is always taken for granted that the distance between a section corner and a quarter-section corner is 40 chains. The lines are also supposed to run due north and south and east and west. These suppositions, however, are strictly correct only in comparatively few cases. Quarter-section corners, like section corners, on town and range lines answer for sections on each side of the line, but where an offset occurs aud the closing section corner is set either at one side or the other of the corner already on the line, the quarter-section corner for the closing section is omitted. In this case the closing section has only seven corners instead of eight. NOTE ON ARTICLE (142). The field-notes of several .-tates have been turned over to the State authorities, and may be procured from them when needed. (79) 80 MANUAL OF PLANE SURVEYING. ( 144 ). For convenience of reference a plot of the township is drawn after the survey is completed, and whatever is essential to subsequent surveys is represented on it. ( 145 ). Fig. 20 will give an idea of the manner in which a plot of this kind is drawn, although space will not permit its being made as complete as it should be. A J I ^ / K L l 6 < 3 5 < > 4 * > 3 c > 2 < 7 > > 8 > n 9 i r 10 i i 11 i > 12 18 ' 17 * (6 - * 15 i- 14- * 13 19 , j 20 < o 21 < i 22 t , 23 * > 24 30 c a 29 , j 28 r j 27 , 2G < j 25 3! - - 32 - - 33 - - 34 - - 35 - - 36 w; u T FIG. 20. The capital letters on the margin designate the section corners on the town and range lines, and the small letters the quarter- section corners. The interior section corners are designated by the numbers of sections. Thus, the southwest corner of section one is numbered MANUAL OF PLANE SURVEYING. 81 1, 2, 11, 12, because it serves as a corner to each of these sections. The interior quarter-section corners are designated as corners 1 to 6, respectively, on the line B E, I W, or whatever line it may be. ( 146 ). Attached to the plot of the township is a list of the wit- nesses at each of the corners, generally on the principle of the fol- lowing : 1. Exterior corners. Sections. At A. Be 6 N 14 W 32, Ash 16 S 12 W 17. At B. Oak 10 S 19 E 11, Hickory 18 N 5 W 6. At C. Maple 14 N 12 E 41, Poplar 28 S 72 W 19. Qu arter-sections. At a. Oak 14 N 61 W 14, Sugar 15 S 16 W 13. At b. Elm 26 S 12 E 25, Ash 9 S 63 W 14. 2. Interior corners. Sections. (1). Cor.ofl,2,ll,12,Maplel5N10E19,Elml6S71E12. (2). Cor. of 2, 3, 10, 11, Oak 36 S 15 W 12, Ash 20 N 82 W41. On the line B R (Quarter-section corners). (1). At 1. Be. 12 N 16 E 42, Gum 14 S 15 W 22. (2). At 2. Pop. 28 S. 46 E 27, Ash 20 N 29 W 31. In a similar manner the quarter-section corners on the lines H X, I W, J V, etc., are also numbered and the witnesses given to each. Each of these lists is extended so as to include all the corners of that particular class. ( 147 ). The following particulars are usually shown on the face of the plot : Length of fractional lines : B to 6, 39.07. C to 6, 38.49. D to 6, 38.03, and so on with all the other fractional lines. 82 MANUAL OF PLANE SURVEYING. 1. Area of fractional quarters. N. E. qr. sec. 1, 159.17 acres. N. W. qr. sec. 1, 157.51 acres. N. E. qr. sec. 2, 155.70 acres. 2. Creeks, etc. N from R 31.42, creek running S. W., 43 links wide. E from 25, 26, 35, 36; 22.16, creek running S.W., 41 links wide. 3. Offsets. B 41 links E. of corner in town north. L 59 links S. of corner in range west. Only a few instances are cited in each case to show the general plan. SUBSEQUENT NOTES. ( 148 ). Every surveyor should be provided with a copy of the original field-notes of his county, together with notes of all the surveys made by his predecessors. These notes, with the addi- tions he himself makes from time to time (provided he and his predecessors are authorized surveyors), constitute the surveyor's records of the county. ( 149 ). These records should contain a plot of each piece of land surveyed, showing its area and the course and distance of each of its boundary lines. The manner of drawing these plots is explained in the chapter on " PLOTTING." ( 150 ). From the regular county record is usually drawn a pocket record for field use (1) of the original survey, and (2) of the subsequent surveys. The notes of the original survey generally fill but a small book, and may be arranged according to the method given above ; but, unless the county is unusually small, it is more convenient to have a separate book for each range in which to enter the notes of the subsequent surveys. ( 151 ). Each of these books should contain at least twice as many pages as there are sections in the range. The left-hand page in each one sh6uld contain a plot of a section about 4 inches square divided into quarters, and the right-hand or opposite page should be left blank, so that notes of the successive surveys in the MANUAL OF PLANE SURVEYING. section may be entered upon it. These refer to the plot by num- bers or letters in the manner shown in the figure. 1. (Left-hand page). SECTION , TOWN , RANGE t C I k tO. 20 ^ * A ^ $ 75 . Oa tO . 18 *> ti ^ h 3 " A to . /f > 40. S/ g FIG. 21. (This plot is only one-fourth the size suggested above, but will serve to illustrate.) 2. (Eight hand page). a. Be. 16 N 12 E 19, Ash 14 S 6 E 12. B. Pop. 28 S 60 W 19, Walnut 30 N 16 E 41. c. Be. 13 S 16 E 17, Elm 18 N 22 E 31. d. Pop. 16 N 19 E 27, Ash 18 N 23 W 21. A. Be. 23 S 17 W 40, Elm 16 N 21 W 14. ' Bearings of principal lines : a f. N 89 35 7 E. c e. N 1 16 X W. ( 152 ). In these cases the bearings are all given on the basis of the true meridian. When the magnetic bearings are given, they should be accompanied with the date at which they were taken. 84 MANUAL OF PLANE SURVEYING. ( 153 ). All interior surveys in the quarter-sections should be represented by proper lines on the plot. The dotted line through the north-east quarter in the figure indicates that this quarter has been divided into north and south halves. The courses of roads and creeks may also be shown on the plot. (154). When the bearing of any independent line is not known, it may generally be approximated by comparing it with other lines in the section or adjoining sections. For instance, there is but little difference between the bearing of the line a b in the figure and that of the line c d, since the distance between their northern extremities differs but 4 links from the distance be- tween their southern extremities, and both lines are about of the same length. (155). The bearing of the line ab may be determined by the following method, which has been explained in a preceding chap- ter : 70 : 4 : : 60 / ^z=3'-f- = amount of correction. 1 16' 3 / =l 13'= bearing of line a b. (156). This method of determining the bearing of one line from that of another depends on the following principles: (1) Two parallel lines have the same bearing, and (2) the difference in bearing of two lines not parallel is in proportion to their incli- nation to one another. The error caused by convergence in north and south lines may be disregarded when the lines are short. By reversing this rule we may determine the distance between two lines at successive points when their distance apart at one point and their difference of bearing are known. ( 157 ). The surveyor's field-book is the memorandum he keeps of his field-work. It contains only the rough entries, which are changed in form and transmitted to the records. ( 158 ). In surveys of independent tracts, perhaps the following method of keeping it is as good as any : Let us suppose a survey of the south-east quarter of section 9, town 8, range 10, commencing at the south-east corner of the sec- tion, and made March 29, 1881. If all the corners to the quarter- section have been established previously, and the surveyor runs the east line first, the entries may be somewhat as follows : Mar. 9, 1881. 9 8 10. Commenced S E cor. and ran N 2 15' W, 40.20 to E } cor., Be MANUAL OF PLANE SURVEYING. 85 20 N 15 E 36, Ash 10 N 41 W 12, M E 16 links. At 16.30 from S E cor. crossed brook 8 links wide flowing S E. Com. E J ran S 88 40' W, 40.12 to center of section. 19.00 crossed brook 6 links flowing S E. Com. cen. ran S 2 30' E, 40.20 to S J. Com. S J ran N 88 40' E, 40.15, M L 5 links. The first line terminated 16 links east of the corner, showing that the assumed bearing was about } degree too small. The second and third lines struck the corners, but the fourth ran 5 links to the north, perhaps on account of some slight error in set- ting the compass, as the assumed bearing was correct, if we com- pare with the second line run. (159 ). Wherever the witnesses are found in bad condition new ones are taken, as was done at the E corner in this survey. (160). In dependent surveys it is generally best to have the page of the field-book ruled in five vertical columns: The first giving the relative name or number of the station or corner at which the line begins; the second, its course; the third, its dis- tance; the fourth, the number of links missed to the right; the fifth, the number of links missed to the left, as follows : Sec. 5, Town 6, Kange 4. Mar. 18, 1881. Sta. Course. Dis. R. L. A N10W 16.00 4 B N 4W 4.00 C N 16 3/4 40.21 FIG. 24. is a considerable distance from A. the supposed approximate loca- tion of the old corner. The dotted lines represent the old lines, and the numbers below the new line show the length of each sec- tion of it. (4). If the corner of a section be lost, a new one is set by sur- veying the exterior lines of the adjacent quarter-sections as if they did not bend at the section corner. The new corner will be at the point of intersection of the two lines thus surveyed. For instance, to set a new S E corner to section 2, connect the S J corner of section 1 with the S ^ corner of section 2, and the E J corner of section 2 with the E } corner of section 11. The point at which the lines cross will be the new corner. (5). As in the former case, this corner may not be identical with the old corner. Fig. 25 represents a possible case in which they are some distance apart. In actual work, however, such ex- treme cases as we have noticed will seldom, if ever, come up. ( 6 ). In any case, in setting a new section corner, if any J cor- ner can not be found, the line must be produced to the next corner that can be found. This may cause one of the lines to be li or even 2 miles long, but the corner is set at the point of intersection, same as before. (166 ). In re-locating the original corners to the variable quar- MANUAL OF PLANE SURVEYING. ters of fractional sections any of the first four methods given above may be employed, but when a new corner must be established the oth or last does not always hold good, except for the \ corner on the town or range line, which is set midway between the section corners, as in full sections. ( 167 ). To set the i corner between two fractional sections, run Set. Z. / '# cor, Sec, l t Sec, II. ^ COT. FIG. 25. a line from the interior section corner westward or northward, as the case maybe, to the exterior section corner on the town or range line, and locate the corner 40 chains from the starting point. ( 168 ). To set an exterior corner to a fractional section, or to any exterior section. 1. If there be an offset between the corner of one section and that of the corresponding section in the other town or range, the 92 MANUAL OF PLANE SURVEYING. corner may be re-located by measuring this offset along the town or range line, or correction parallel, in the proper direction. 2. When there is no offset, the corner must be set by crossing the lines, according to the method used in interior sections. EXAMPLES. ( 169 ). 1. How may a new E } corner be set to section 11, pro- viding the section corners on that side can both be found? 2. What must be done, if one or both of the section corners are lost, before the quarter-section corner between them can be set? 3. How do you establish a new interior section corner? 4. If one or more of the exterior corners to the adjacent quar- ter-sections be lost, what must be done in order to establish the section corner? 5. How do you establish the ^ corner section between two frac- tional sections ? 6. How do you re-locate an exterior section corner when there is an offset on the town or range line? (170). When possible, all corners, whether independent or de- pendent, are re-located according to the rules by which they Vrere located at first. QUESTIONS ON CHAPTER IX. 1. When it is found impossible to re-locate a corner, what must be done? [corner? 2. Why is not the new corner always identical with the original 3. Explain each of the different methods of re-locating original corners. 4. In re-locating an original J corner by "projection," the ran- dom line was found to be 16 links to the left of the $ ^ corner. In what direction and how far will the re-located corner be from the extremity of the random line? 5. How many corners must be found on the side of a quarter- section before the remaining one can be re-located ? Why ? 6. What section in the township north corresponds with section 2? With section 5? What one in the township west cor- responds with 7? With 30? 7. What townships touch T 2 N, R 3 E? T4S, B5W? Tft N, E 1 W? 8. How are subsequent corners re-located? CHAPTER X. DESCRIPTIONS OF LAND. (171 ). No piece of land can be sold or surveyed, unless its de- scription is known, and this description should be just as concise and simple as possible. (172). It would be better, if the length of lines and area of tracts were always given in surveyor's measure, instead of in or- dinary linear and square measure; yet when this is not done, they may be reduced to their equivalents in surveyor's measure by the following tables : (173). LINEAR MEASURE. 100 links = 1 chain = 4 rods. 1,000 " =10 " =1 furlong. 8,000 " =80 " =1 mile. SQUARE MEASURE. 1 sq. chain = 16 sq. rods. 10 " " =1 acre. 6,400 " " = Isq. mile. It is plain that rods maybe reduced to chains by dividing by 4; furlongs to chains by multiplying by 10 ; and miles to chains by multiplying by 80. EXAMPLES. (174). 1. Reduce 15 rods to chains. 2. Reduce 1 fur. 3 rods to chains. 3. Reduce 1 mi. 3 fur. 24 rods to chains. (175). Fractional parts of a chain should be expressed in links: Thus, lOf chains should be written 10 chains and seventy- five links, or simply 10. 75. (93) 94 MANUAL OF PLANE SURVEYING. (176). As a general :mle for reducing from ordinary long or linear measure to surveyor's measure, perhaps it would be well to use the following: Eeduce the denominations expressing the length of the line to rods, and multiply by .25. The product will be the length of the line expressed in chains and links. To illustrate, suppose the length of a line to be 7 fur. 16 rods = 296J rods. = 296.5 rods. This multiplied by .25 equals 74.125, or 74 chains 12J links. (177). The area of tracts in surveyor's measure is always given in acres and hundredths, instead of in acres, roods, rods, etc., as ordinarily. This will be explained in the chapter on Computation of Area. (178). Whenever an independent tract is to be described, nothing whatever should be said of the course and distance of any of its boundary lines, and it should be described simply as such a division or sub-division of the section ; as, for instance, the south- west quarter, or the north half of the north-east quarter, or the north-west fourth of the south-east quarter, or the north half of the south-east fourth of the north-west quarter, etc., etc. ( 179). Errors like the following are frequently made in de- scriptions: Forty acres in the form of a square in the south-east corner of the section ; eighty acres off the south side of the north- east quarter; one hundred and sixty acres in the north-east corner of the section ; a strip twenty chains wide off the north side of the south-west quarter; and so on. Each of these descriptions is faulty, because the independent division intended to be described may overrun or fall short in the amount of land named in the description. If the tracts were not independent, the descriptions would be good. Correct the following descriptions : ( 180 ). 1. 160 acres off west side of section. 2. Forty acres in the south-west corner of the north-east quarter. 3. Commencing at the X E cor. of the section ; thence running south 20 chains; thence west 40 chains; thence north 20 chains to the N } cor.; thence east to the place of beginning. Containing 80 acres. 4. 60 acres off the south side of the south-east quarter. MANUAL OF PLANK SURVEYING. 95- (181 ). Sometimes descriptions contain errors that render them worthless. The following are a few examples; tell where the error lies in each one : 1. NEJNWqr. 2. S W } N E qr., containing 80 acres. 3. SJNJXEqr. 4. 60 acres in N E qr. 5. N W qr sec. 28, containing 80 acres. 6. Running north ; thence east 50 chains. 7. Eunning S 43 E, 11.21 ; thence N 32 W, 5.26 ; thence S 8 3V E, 16.32, to the place of beginning. Mistakes like the preceding are frequently made by persons who are careless, or do not understand how lands should be described> and sometimes give rise to vexatious litigation. ( 182 ). It is best in nearly all cases to qualify the area of the tract described by the phrase " more or less," as, perhaps, no two surveys of the same tract, particularly if it be large, will ex- actly coincide throughout, and of course the area will vary with the length of the lines. ( 183 ). In describing dependent tracts the course and distance of each of their boundaries should be given, except, perhaps, in occasional cases where they have a natural or artificial boundary, as for instance, a creek or road, whose course and distance may be determined at any time; but it is always best to be definite in re- gard to boundaries when possible. ( 184 ). The description should also state whether the bearings are based on the true meridian or on the magnetic meridian. If based on the magnetic meridian, the date at which they were taken should be given. ( 185 ). Where a line is described as running north, a due north and south line is meant, and the same is true of south. Similarly, an east line means one running due east, and a west line one running due west. ( 186). The survey of a tract of land is always made in ac- cordance with the description, except where an obvious mistake occurs, in which case the surveyor will have to exercise his judg- ment in regard to the course to be pursued, as no rule can be given that will apply to all cases. However, the decisions in the "Ap- pendix " may assist him somewhat in arriving at a conclusion. 96 MANUAL OF PLANE SURVEYING. Sometimes the mistake is made in writing the original descrip- tion of the tract, and at others in copying from preceding titles and deeds. In the latter case, a comparison of the deeds will show in what it consists. As soon as a mistake is discovered in a deed or mortgage, or in any other instrument in which a great deal may depend upon the description, steps should be taken by those interested to have it corrected. None but competent persons should be chosen to write descrip- tions of land. QUESTIONS ON CHAPTER X. 1. Why can not a tract of land be sold or surveyed without a description ? 2. How many links in a rod? Chains in a mile? 3. Write 17 chains and 46^ links decimally. 4. Give the general rule for reducing from ordinary long or linear measure to surveyor's measure. 5. How is the area of a tract of land expressed in surveyor's measure? 6. Why should not the metes and bounds of an independent tract be given in a description ? 7. Why should the phrase " more or less " be inserted in a de- scription ? 8. Why is it necessary to state whether the bearings are based on the true meridian or on the magnetic meridian? 9. If on the magnetic meridian, why should the date at which they were taken be given ? CHAPTER XI. OBSTACLES TO ALIGNMENT AND MEASUREMENT. ( 187 ). It frequently happens in the course of a survey that the line strikes an obstacle of some kind as, for instance, a building, or a large pond, or creek that obstructs the measurement, if not both line of sight and measurement. (188). These obstacles may be divided into two classes: (1) Ob- stacles that may be spanned by measurements along their sides or margins, as a building, a pond, etc. ; (2) obstacles that can not be spanned in this way, as rivers and lakes. ( 189). Various methods are employed for spanning obstacles, but only a few will be given, in order to prevent confusion. FIRST CLASS OF OBSTACLES. 1. By Perpendiculars. Fig. 26 represents an obstacle on the line FIG. 26. A B which runs nearly to the side of it. At the extremity B, a perpendicular, B C, is measured long enough to permit the line C D to pass the obstacle. In this case the perpendicular is fifty link* long. The line C D is then run at the bearing of the line A B, and is, consequently, parallel to it. From the extremity, D, of this line another perpendicular, D E, of the same length as the first, 7 (97) 98 MANUAL OF PLANE SURVEYING. is measured, which, of course, terminates on the original line pro- duced through the obstacle. The survey of the line may then be continued from E in the direction E F at pleasure, and the length of C D added to the regular sections, A B and E F. 2. By an Equilateral Triangle. The line A B terminates some- what further from the side of the obstacle than before, and the FIG. 27. line B C is then laid off at an angle of 60 with the line A B pro- duced and measured to a suitable distance. In the case before us it is 1 chain and 25 links in length. From the extremity, C, of this line, the line C D, of equal length with it, is surveyed at an angle of 60 with C B. We then have an equilateral triangle, and the side B D is also 1 chain and 25 links in length. The line may then be continued, and the distance through the obstacle, 1 chain and 25 links, added to the other sections, as before. 3. (a) By a Right-angled Triangle. This method is similar to the preceding one, and differs from it only in having a right-angle FIG. 28. at C, and angles of 45 at B and D in the triangle used. The side B C is first surveyed, and then C D at right-angles to it and of MANUAL OF PLANE SURVEYING. equal length. The distance from B to D is found by extracting the square root of the sum of the squares of B C and C D, as the side B D is the hypothenuse of the triangle. In this case the dis- tance from B to D is equal to y(100)+ (100)* = 1.414. (6) When the obstacle is a pond, or something that does not obstruct the line of sight, the following method will be found most convenient : C FIG. 29. The line is measured to B, near the margin of the pond, and the flag set at D on its continuation on the opposite side. C B is then measured perpendicular to A B, and lastly the line C D is meas- ured. We have now a right-angled triangle whose base is required and may be found by extracting the square root of the difference between the squares of C D and B C. In this example the base B D equals /(125) 2 (75) 2 = 1.00. In every case the line is to be continued from D at the bearing of the first section, A B, and the distance through the obstacle must be added. 4. By Symmetrical Triangles. When, as in the last case, the line of sight is not obstructed, the following method may sometimes be used : F A FIG. 30. 100 MANUAL OP PLANE SURVEYING. From the extremity, B, of the line A B measure a line to C and produce it to F, an equal distance beyond, and then from D meas- ure the line D E so that C will be in the center. The line E F will then be equal to the line B D. 5. When a fence is built on a line to be surveyed, it is best to take an offset either to one side or the other, and allow for it when the stakes are set on the true line, or the stakes may be moved back a distance equal to the offset as they are set. They will thus be on the random line, and may be corrected the same as if no offset had been taken. It is customary, after an offset has been taken, to measure back to the random line as soon as the obstruction is cleared, but if the corner be reached before this is done, the offset must not be forgot- ten in measuring the distance the line runs to the right or left of it. In doing this observe the following rules : (1). When the offset is taken either to the right or left and the offset line terminates on the opposite side of the corner, the dis- tance missed by the random line will be equal to the distance missed by the offset line, plus the offset, and it will terminate on the same side of the corner as the offset line. A. IQ\ FIG. 31. In the figure an offset, A B, of 10 links was taken to the right, and the offset line, B C, ran 10 links to the left of the corner A. The random line, A B, will therefore terminate 20 links to the left of the corner. (2). When the offset line terminates on the same side as that on which the offset is taken. This involves two cases: (a) When the distance missed is greater than the offset, and (6) when the offset is greater than the distance missed. (a). Subtract the offset from the distance that the offset line misses the corner; the remainder will be the distance missed by the random line. The termination of the random line will lie on the same side of the corner as the termination of the offset line. (6). Subtract the distance missed by the offset line from the MANUAL OF PLANE SURVEYING. 101 offset ; the difference will equal the distance missed by the random line. The termination of the random line will be on the opposite side of the corner from the termination of the offset line. The offset line is always parallel to the random line. In correcting the stakes on the offset line, it is best to correct as if they were on the random line, and then move them a distance equal to the size of the offset, and in a direction opposite to that in which the offset is taken. This will put them on the true line. 6. Sometimes, when surveys are made over hills, it is impossi- ble for the chain-men to see the compass or flag to which they are running. In this case a stake should be put up at some promi- nent point on the line by the surveyor, to which they may meas- FIG. 32. ure until they come in sight of the compass or flag. For instance, if the chain-men are down in the valley A, of the figure, a stake or flag should be set on the ridge, as it is impossible for them to see the compass at B. In chaining up and down hill it is frequently necessary to double the chain or divide it into two sections, so that it may be held in a horizontal position. A light steel chain is always pref- erable to a clumsy iron one, as it will not sag so much. SECOND CLASS OF OBSTACLES. (190). 1. Suppose the obstacle to be a large creek. The line is surveyed up somewhere near the edge and the flag set on the line on the opposite shore. In Fig. 33, let A represent the point to which the line is measured, and B the flag set on the opposite shore. From the point A, a line of indefinite length is sighted at right angles to A B. The compass is then set at any point not too near A, as C, on this line, and turned so the sights will strike B. The size of the angle A C B is then noted, and the point D on the 102 MANUAL OF PLANE SURVEYING. line A E sighted at an equal angle on the other side of A C. The distance from A to B will then equal the distance from A to D. FIG. 33. 2. From the point A, a perpendicular, A C, may be sighted and another, C D, set off from its extremity. The point E, on the line FIG. 34. A C, is then found, and each of the sections, E C and E A, meas- MANUAL OF PLANE SURVEYING. 103 red. The distance from A to B may then be found by the fol- lowing proportion : CE : EA :: CD : (x = A B) ; E AXCD whence A B = CE 3. A perpendicular is set off from the line B F at F, and another at A, extended to the line B D. The distances, A F, A C, and D FIG. 35. whence A B = D F, are measured and the distance A B, found as follows : (D F A C) : AC : : A F : (x = A B) ; ACXAF ' AC ( 191 ). A great many other methods might easily be given, but these will suffice. These methods will, of course, answer equally well where the point B is inaccessible and at the termination of a line. In field-work the method that seems best adapted to the peculi- arities of the case should be adopted. QUESTIONS ON CHAPTER XI. 1. What is meant by an obstacle to measurement? To align- ment? 2. How many classes of obstacles are there? Name one of each class. 104 MANUAL OF PLANE SURVEYING. 3. Describe the method of spanning an obstacle by perpen- diculars. By an equilateral triangle. 4. In the survey of a certain line an obstacle is met. A line is then surveyed 1 chain and 40 links, bearing from the termination of the main line so as to pass the obstacle. From the end of this line a perpendicular 90 links long is measured back to the original line produced through the obstacle. What is the distance through the obstacle? 5. Describe the method by symmetrical triangles. 6. What is an offset? 7. What is the difference between the offset line and the ran- dom line? 8. An offset of 12 links is taken to the right, and the offset line misses 13 links to the left. How far will the random line miss the corner? 9. What two cases arise when the offset and termination of the offset line both lie on the same side of the corner? 10. Give the rule for correcting the stakes on an offset line. 11. How do you set an intermediate stake for the flag-men to run to when an elevation of land prevents them from see- ing the compass or flagstaff? 12. When is it necessary to double the chain or divide it into two sections ? 13. Explain each of the methods used in the second class of obstacles. 14. In Fig. 34, A E = 1.40, C E = 90, and C D = 1.10. What is the length of A B ? 15. In Fig. 35, AC = 1.22, A F = 98, and DF= 1.54. What is the length of A B? CHAPTER XII. COMPUTATION OF AREA. ( 192 ). In computing areas the length of all lines should be expressed in chains, chains and links, or links, as the case may be, and the areas may then be reduced to acres and decimals of an acre by the rules for multiplication and division of decimals. Thus: (1). 11.25 X 2.50 = 28.1250 sqch.; 28.1250 -=-10 = 2.81250 acres. (2). 21.32 X 8= 170.56 sq.ch.; 170.56 -i- 10 = 17.056 acres. (3). 22X15 = 330 sq.ch.; 330 -=-10 = 33.0 acres. ( 193). The following special rules are deduced from the pre- ceding processes : (1). When each dimension contains hundredths of a chain (links), the product may be reduced to acres by pointing off five decimal places. (2). When one contains hundredths and the other is expressed in full chains, the product is reduced to acres by pointing off three decimal places. (3). When each dimension is in full chains, reduce to acres by pointing off one decimal place. If either dimension contain a fraction of a link, point off as many additional places in the prodnct as are necessary to express the frac- tion. Thus : (1 ). 5.125 X 3.20 = 1 .640000 acres. (2). 2.3725X5 = 1.18625 acres. (105) 106 MANUAL OF PLANE SURVEYING. EXAMPLES. (194). (1). 21.52 X 12.40 = how many acres? (2). 40.24 X 20.31 = " (3). 16.42 X18 = " (4). 12 X13 = " " " (5). 12.165 X 15-30 = " " " (6). 15.2425X12.125= " ( 195). Every tract of land is in figure a polygon, and its area is computed according to the rule for finding the area of the par- ticular polygon representing its contour. ( 196 ). EECT ANGLES. 1. Multiply the length by the breadth and the product will be the area. 2. If the rectangle be a square, the area may be found by squar- ing one side. ( 197 ). PARALLELOGRAMS. Multiply the length of one side by the perpendicular distance between this side and the opposite. The product will equal the area. In Fig. 36 the line A B represents the perpendicular, and the AC /5.60 FIG. 36. area of the parallelogram is therefore equal to the product of 15.60 by 7.55= 11.778 acres. If either of the sides B C or D E be given instead of B E or C D, the perpendicular must be measured in the direction of the length of the figure. MANUAL OF PLANE SURVEYING. 107 ( 198 ). TRAPEZOIDS. Multiply the sum of the parallel sides by the perpendicular distance between them, and half the product will be the area. 7*. 30 FIG. 37. In Fig. 37 the line A B represents the distance between the par- allel sides, and the area Is equal to ( (16.24 -\- 14.90) X 6.40) H- 2 = 9.4648 acres. When the trapezoid contains two right-angles, the line between them represents the perpendicular distance between the parallel sides. ( 199 ). TRIANGLES. In computing the area of triangles, two general classes will be considered : 1. Triangles whose base and altitude are given. The area of triangles of this class is found by multiplying the base by the altitude and taking half the product. FIG. 38. Fig. 38 represents a triangle whose base is 18.05 and altitude 5.90. Its area, therefore, is (18.05 X 5.90) -=- 2 = 5.32475 acres. Eight-angle triangles belong to this class. 108 MANUAL OF PLANE SURVEYING. When the three sides are given, isosceles triangles may be brought under this class by the following rule: Square one of the equal sides and subtract the square of half of the odd side. The square root of the difference will equal the altitude. The altitude of an equilateral triangle is found by extracting the square root of the square of one side minus the square of half of one side. EXAMPLES. (1 ). The base is 14.75 and the altitude 2.90. What is the area of the triangle? (2). The base and perpendicular of a right-angle triangle are 5.60 and 7.42, respectively. What is its area? (3). In an isosceles triangle the even sides are each 9.16 in length and the odd side 7.45. What is the area? (4). What is the area of an equilateral triangle each of whose sides is 7.25 in length? 2. Triangles whose altitude is not given. The general rule for triangles of this class is the following : (a). Take half the sum of the three sides. (6). Subtract from the half sum each side severally. (c). Multiply the half sum and three remainders together. (d). Extract the square root of the product for the area. FIG. 39. Let Fig. 39 represents a triangle whose area is to be computed. Then, 10.54 -f 14.60 + 8.72 = 33.86 -4- 2 = 16.93 16.93 10.54 = 6.93 16.93 14.60 = 2.33 16.93 8.72 = 8.21 ^716.93 X 6.39 X 2.33 X 8.21 = area. MANUAL OF PLANE SURVEYING. 109 EXAMPLES. 1. What is the area of a triangle, the sides of which are 4.20, 2.65, 3.71, respectively ? 2. The sides of a triangle are 2.91, 6.90, and 5.42, respectively. What is its area? It is sometimes more convenient to measure the altitude, and 'lius place the triangle under the first class. ( 200 ). TRAPEZIUMS. Divide the trapezium into two triangles. The sum of their areas will be the area of the trapezium. To do this, measure a diagonal of the trapezium. to. 7? FIG. 40. Fig. 40 represents a trapezium, one of whose diagonals has been A 110 MANUAL OF PLANE SURVEYING. measured. It will be seen that its area will equal the sum of the* areas of the triangles A B D and B C D. A serious mistake is sometimes made by incompetent persons by multiplying together the half-sums of the opposite sides for the area. When one angle of the trapezium is re-entrant, as in Fig. 41, the area may be found by subtracting the area of the triangle BCD from that of the triangle A B D ; or it may be computed the same as when the angles are all salient by omitting the triangle BCD and measuring a diagonal from A to C. ( 201 ). ANY FIGURE. Divide the figure into triangles and compute their areas sepa- rately. The sum of the areas of the triangles will be the area of the figure. The area of the tract of land represented in Fig. 42 is equal to- FIG. 42. the sum of the areas of the triangles A B F, B E F, B C E, and CDE. Sometimes, when a tract of land is narrow and has one irregn- lar boundary, its area may be approximated by dividing it into trapezoids. Fig. 43 represents a tract of this kind bounded on one side by a creek. In cases of this kind the area of the tract m equal to the sum of the areas of the trapezoids that compose it MANUAL OF PLANE SURVEYING. Ill FIG. 43. (202). COMPUTATION OF AREA BY LATITUDES AND DEPARTURES. The method of latitudes and departures now to be developed is simple, precise, expeditious, and universal in application, if the course and distance of each of the boundary lines of the tract whose area is to be computed are given. (203). In plane surveying, meridians, like parallels of lati- tude, are supposed to be parallel to one another, and the latitude of a course is the distance between two parallels running through its extremities, while the departure of a course is the distance be- tween two meridians drawn through its extremities. In Fig. 44 the latitude of the course A B is represented by B C, and its de- rl_ m FIG. 44. 112 MANUAL OF PLANE SURVEYING. parture by A C. It is evident that the latitude of a course is equal to the difference of latitude of its extremities, and that its depar- ture is equal to the difference .of longitude of its extremities. ( 204). The latitudes of courses bearing north are called north latitudes or northings, and of those bearing south, south latitudes or southings. Likewise the departures of courses bearing east are called east departures or eastings, aud of those bearing west, west de- f Tortures or westings. In Fig. 45 the latitudes of A B and A F are northings, and of FIG. 45. A C and A D southings ; while the departures of A B and A C are eastings, and of A D and A F westings. (205). North latitudes are additive and are marked with the sign -)-, plus, while south latitudes are subtractive and marked with the sign , minus. In the same manner, east departures are additive and marked +, and west departures are subtractive and marked . ( 206 ). If we now refer to Fig. 12, we shall see that the radius A F, may represent the course A C, Fig. 46, whose latitude and departure we wish to find. Then will C E, the departure, equal MANUAL OF PLANE SURVEYING. 113 the sine of the angle BAG, and E A, the latitude, equal the cosine of the angle B A C. ( 207 ). A Table of Natural* Sines and Cosines is given in the FIG. 46. APPENDIX, by which the latitude and departure of any course may be easily found. In this table the length of the sine and cosine is given for a radius equal to unity, for each degree and minute of arc between and 90; and, hence, to find the latitude of any course, it is necessary only to multiply the cosine of its bearing by the length of the course, and to find the departure of any course, to mul- tiply the sine of its bearing by the length of the course.!' For instance, suppose it is required to find the latitude and de- parture of a course bearing N 42 33' E, and 20.22 in length. By referring to the Table, we find the cosine of the bearing to to be .73669, and the sine of the bearing to be .67623. Therefore, the latitude of the course will equal .73669 X 20.22 = 14.8958, and the departure of the course will equal .67623 X 20.22 = 13.6733. In using the Table, when the bearing is 45 or less, take the de- grees from the top of the page and the minutes from the left-hand * Called natural sines and cosines to distinguish them from logarithmic sines and cosines. fin this rule observe that the angles are measured to the right and left of the vertical radii. If, as in Fig. 12, they were measured from horizontal radii, the word "sine" would be used for "cosine," and "cosine" for "sine" in the rule. 8 114 MANUAL OF PLANE SURVEYING. column, and when the bearing is greater than 45, use the degrees at the bottom of the page and the minutes in the right-hand column. ( 208 ). EXAMPLES. The course and distance are given in each of the following cases. Find the latitudes and departures : (1). N 52 16' W, 10.12. This bearing is greater than 45; so the degrees must be taken from the bottom of the page. Having found the double column marked 52, ascend it to the line marked 16' on the right. We now find the cosine to be .61199, and the sine to be .79087; there- fore the Latitude =.61199 X 10.12, and the Departure = .79087 X 10.12. (2). S 15 40 / E, 11.41. (3). N 21 32' W, 19.71. (4). S 88 56' E, 73.98. (5). N 66 25' E, 46.12. ( 209 ). It will be seen that the columns in the table marked "sine" at the top are marked "cosine" at the bottom, and that those marked "cosine" at the head are marked "sine" below. Care must be taken to use the heading for bearings read from the top, i. e., for bearings not greater than 45; and the bottom mark- ings for bearings read from below, i. e., for bearings greater than 45. ( 210 ). TRAVERSE TABLES are sometimes used instead of the Table of Natural Sines and Cosines in determining the latitudes and departures of courses, and somewhat facilitate calculations in many cases; but they are usually computed only to quarter- degrees, and it has been thought best to use in the present work only the more accurate method of natural sines and cosines. (211). In the survey of every tract of land, the sum of the north latitudes should equal the sum of the south latitudes, and the sum of the east departures should equal the sum of the west departures; and, hence, in plotting a survey, or making prepara- tions to compute the area of the tract, we have an almost infalli- ble means of testing the accuracy of the survey by which the course and distance of each of its boundaries were determined. (212). Let us now make an application to the survey of the MANUAL OF PLANE SUBVEYISG. 115 following described tract of land : Running N 10 E, 5.60 ; thence S 35 3V E, 4.00; thence S 55 30' W, 4.00, to the place of begin- ning- Taking each course separately, we find the respective latitudes and departures. (1). Latitude of first course equals .98481 X 5.60 = 5.51. Departure equals .17366 X 5.60 = .97. (2). Latitude of second course equals .81412 X 4.00 = 3.25. Departure equals .58070 X 4.00 = 2.32. (3). Latitude of third course equals .56641 X 4.00 = 2.26. Departure equals .82413 X 4.00 = 3.29. The latitude of the first course is a north latitude, and must be marked +, and the latitudes of the second and third courses are south latitudes, and take the sign . Likewise, the departures of the first and second courses are east departures and should be marked +, while the departure of the third course is a west de- parture and should be marked . The separate courses, with the latitude and departure for each one, may be entered in a diagram similar to the one used in keep- ing field-notes (Art. 160), and a space left at the bottom for the footings, as follows : Sta. Dis Lat. Dep. + + . A N 10 OP E. 5.60 5.51 .97 B 835 3^ E. 4.00 3.25 2.32 c C 55 3ff W 4.00 2.26 329 5.51 5.51 3.29 3.29 FIG. 47. ( 213 ). The reason why the east departures should balance the west departures, and the north balance the south latitudes will be seen by noticing Fig. 48, which represents the above tract of land. The north and south lines represent the latitudes, and the east and west lines the departures. ( 214). When the + latitudes balance the latitudes, and the -f- departures balance the departures, as in the case just con- 116 MANUAL OF PLAXE SURVEYING. sidered, the survey is said to " close." Usually, however, owing to slight inaccuracies in sighting the flag, reading the bearing of the line, measuring the line, or, perhaps, all combined, neither the latitudes nor departures balance. If the disagreement is consider- able, a re-survey should be made, as there is probably an error FIG. 48. somewhere in the work; but if it is only slight, as, for instance, 1 or 2 links in 7 or 8 chains, it is probably due to some unavoidable inaccuracy in the survey, and may be corrected by the following rule: Find the amount of the error for each chain, and distribute it among the latitudes or departures, as the case may be, in proportion to their re- spective lengths. Adding to those that are too small, and subtracting from those that are too large. MANUAL OF PLANE SURVEYING. 117 This will cause them to balance and answer all ordinary pur- ( 215 ). The longitude or meridian distance of a line is its mean distance from an initial line or meridian. Preparatory to finding the area of a tract of land, this meridian is conceived to be drawn through its extreme western or eastern corner usually the west- ern and the longitude of each of the courses of the tract is com- puted from this meridian as a base. In Fig. 49 this meridian is drawn through the western corner FIG. 49. of the tract, and the lines, A B, C D, E F, G H, and M O, repre- sent the longitudes of the various courses. (216). It will be observed that there is a difference between longitudes and departures: The former show the mean distance of the line from the meridian, while the latter indicate the differ- ence in longitude of the two ends of the line. ( 217 )- By referring to the figure, it will be seen that the lon- gitude. A .6, of the first course is equal to half of its departure, 118 MANUAL OF PLANE SURVEYING. a b; and also that the longitude, C D, of the second course is equal to c d, which equals the longitude of the first course, plus half the departure of the first course, plus half the departure of the second course, and it may easily be shown that the longitude of any course is equal to the longitude of the preceding course, plus half the departure of the preceding course, plus half the departure of the course itself. ( 218 ). It must be borne in mind that the algebraic sum is meant, and that west departures, having the minus sign, are really subtractive. (219). In order to simplify the rule, and at the same time avoid fractions, it will be preferable to double each of the pre- ceding expressions and use double longitudes. The following will then be the general rule for finding the double longitudes of courses. The double longitude of the first course is equal to its departure. The double longitude of the second course is equal to the double longi- tude of the first course -\- the departure of the first course + the departure of the second course. The double longitude of any course is equal to the double longitude of the preceding course + the departure of the preceding course + the de- parture of the course itself. COMPUTATION OF AREA. (220). We are now prepared to compute areas by means of longitudes. Take for example the tract of land described in Art. 212. The area of the triangle ABC, Fig. 50, is equal to the area of the trapezoid E A B D,plus the area of the triangle BCD, minus the area of the triangle ACE. Finding the area of each of these figures, respectively, we have : Area of trapezoid EABD = DEXab = the product of the latitude of the course A B by its longitude = (3.25 X 2.13) = .692 acre. (See Fig. 48.) Area of triangle BCD = CDXef = the product of the lati- tude of the course B C by its longitude = (2.26 X 1-645) = .371 acre. Area of triangle ACE = CEXcd = the product of the lati- tude of A C by its longitude = (5.51 X -485) = .267 acre. MANUAL OF PLANE SURVEYING. 119 Therefore, the area of the triangle A B C = .692+ .371 .267 = .796 acre. FIG. 50. ( 221 ). In computations of this kind the product of a longitude by a north latitude is called a north product, and by a south latitude, is called a south product, and the difference between the north products and south products is the area of the tract. ( 222 ). Hereafter double longitudes will be used, and the differ- ence between the north products and the south products will then be double the area of the tract. ( 223 ). The differeht steps in the process of computation may be shown very nicely, and the work kept in compact form, by rul- ing a sheet of paper in fourteen columns, adding seven to the right of the seven shown in Fig. 47. In the first four of the added seven write the corrected latitudes and departures, in the fifth the double longitudes, and in the sixth and seventh the north and south products or areas marked -j- and , same as latitudes. 120 MANUAL OF PLANE SURVEYING. The following will serve as an illustration, and at the same time indicate the process used in computation. i + I' S 8 s s S IS e g |S o * t>: FIG. 51. In this example the error in latitudes and departures amounts to very little in each case, and might have been disregarded in the calculation. MANUAL OF PLANE SURVEYING. 121 ( 224 ). The following is the general rule for computing areas by double longitudes : Multiply the double longitude of each course by its latitude. If the latitude is north or plus, write the product in the column of plus areas. If south or minus, u'rite the product in the column of minus areas. Half the difference between the sums of the areas of these two columns will be the area of the tract. This rule holds good for any tract of land bounded by straight lines. ( 225 ). When the most westerly corner of the tract can not be determined readily it is best to draw a plot of the tract according to the directions given in the chapter on PLOTTING. ( 226 ). Compute the areas of the following tracts : (1). N3415'E, 2.73. N 85 00' E, 1.28. S 5645 / E, 2.20. S 34 15' W, 3.53. N 56 3(K W, 3.20. (2). S 7315'E, 19.08. S 19 30' W, 13.68. N 69 15' W, 10.34. S 20 15' W, 11.36. N6800 / W, 9.06. N 20 15' E, 23.56. (3). South, 3.75; S 35 OO' E, 1.04; S 86 30' E, 5.02; N 82 00' E, 1.72 ; S 34 30' E 2.46 ; S 77 30' E, 4. 25 ; N 45 30' E, 9.78; N 2 40 X W, 233; West, 2.18; N 4 00' E, 1.30 ; N 83 45' W, 5.35 ; S 76 (KK W, 1.94 ; S 60 15' W, 2.27 ; S 76 00' W, 3.47 ; N 73 30' W, 2.90 ; N 57 30' W, 1.65 ; S 21 00' W. 2.67. ( 227 ). It is, of course, plain that a due north or south course has no departure, and that its latitude is equal to its length. Likewise, that a due east or west course has no latitude, and its departure is equal to its length. ( 228 ). It is not absolutely necessary that the meridian should be drawn through the most westerly station in calculating the con- tents, but it is generally more convenient to compute from a me- ridian so drawn. Sometimes the surveyor imagines the meridian to pass through the most easterly station. 122 MANUAL OF PLANE SURVEYING. If necessary, the areas may be expressed in the ordinary de- nominations of land measure (acres, roods, and rods), instead of in acres and decimals of an acre, by reducing the decimals to in- tegers. Thus, .82 of an acre = (.82 X 4) = 3.28 R. .28 X 40 = 11.20 sq. rods. Hence, .82 acre = 3 K. 11.2 rods. QUESTIONS ON CHAPTER XII. 1. Why should the length of lines be written in chains and links? 2. When links are multiplied by links, how many decimals are pointed off in reducing the product to acres? Links by chains ? Chains by chains ? 3. Give the rule for finding the area of a rectangle. A paral- lelogram. A trapezoid. 4. How is the area of a triangle found when the base and al- titude are given? When the three sides are given? 5. How may the altitude of an isosceles triangle be found ? Of an equilateral triangle? 6. State the rule for finding the area of a trapezium. 7. When may the area of a figure be computed by dividing it into trapezoids? 8. What is meant by the latitude of a course? Departure? 9. W T hat are north latitudes? South latitudes? East de- partures ? West departures ? 10. Describe the Table of Natural Sines and Cosines. 11. How do you find the latitude of a course from the Table? The departure? 12. Give the rule for correcting latitudes and departures. 13. What is meant by the longitude of a line? What is the difference between the longitude of a line and its depar- ture? 14. State the rule for finding the double longitudes of courses. 15. W T hat are north products or areas? South products or areas ? 16. Give the general rule for computing areas by double longi- tudes. CHAPTER XIII. LAYING OUT AND DIVIDING UP LAND. ( 229 ). No general rule can be given either for laying out or dividing up land, and in the present chapter only the most com- mon cases that arise in practice will be considered. This is nec- essary in order to keep our work within its intended limits as well as to avoid the confusion that a multiplication of details would cause. As a general thing, a little ingenuity on the part of the surveyor will enable him to devise a method to meet the exigencies of the case he may have on hand when it can not be reached by any of the rules given in this chapter. ( 230 ). In the problems now to be taken up, the area and one or more of the boundaries are in nearly all cases supposed to be known, and it is required to find from these the length of certain other boundary lines necessary to a survey of the tract. The pro- cesses used in work of this kind are generally the reverse of those employed in the last chapter, so that a careful study of operations in computing areas will materially assist in the work now before us. LAYING OUT LAND. ( 231 ). To Lay Out a Square. The square root of the area ex- pressed in square chains and decimals will represent the length of one of its sides. Thus, each side of a square tract of land containing 5 acres equals v/50 = 7.07. EXAMPLES. 1. What is the length of each side of a square tract containing 1 acre? 2. A piece of land in the form of a square contains 11 A. 3 R. 26 P. What is the length of its sides? (123) 124 MANUAL OF PLANE SURVEYING. ( 232 ). To Lay Out a Rectangle. Divide the area by the length of the given side. The quotient will be the length of the required side. Thus, if a rectangle contain 4 acres, and the given length be 5 chains, the length of the required side will equal (40 -*- 5) = 8 chains. EXAMPLES. 1. The area of a rectangle is 14 acres, and the length 15.00, what is the breadth ? 2. The area of a rectangular tract of land equals 10 A. 2 R 20 P., and it is 63 rods long. What is the breadth ? Process 10 A. 2 E. 20 P. = 10.625 acres. 63 rods = 15.75. (10.625 X 10) -s- 15.75 = 6.746. ( 233 \. To Lay Owl a Parallelogram. (a). Divide the area by the given length. The quotient will be the perpendicular distance between the given sides. (Art. 197). (6). Find one of the angles of the parallelogram according to the methods explained in Articles (22) and (23). ( c ). Divide the length of the perpendicular by the sine of the angle thus found, and the quotient will equal the required side. If the angle of the parallelogram be greater than 90, its sup- plement* must be used in its stead. Let Fig. 52 represent the parallelogram to be laid out ; its area BACKS A C 12 . o o FIG. 52. being 6 acres, and the length of the side A B, 12.00. Dividing (6 X 10) = 60 sq. chains by 12, we find the perpendicular C D, to be 5.00 long. * The sine of an angle is always equal to the sine of the supplement of the angle. MANUAL OF PLANE SURVEYING. 125 Suppose the angle A D E to equal 108; then its supplement will be (180 108) = 72, the sine of which is .95106. Dividing 5.00 by .95106, we find the length of the required side, A D, to be 5.257. Had the angle BAD been used, instead of A D E, the supple- ment need not have been taken, as it is less than 90. EXAMPLES. 1. In a parallelogram, the area is 12 acres, the length of one side 14.00, and the measured angle equal to 61. What is the length of the required side? 2. A field in the form of a parallelogram is 22.00 long and con- tains 25 acres. The size of one of the angles is 96 45' ; what is the length of the required side? ( 234 ). To Lay Out a Eight-Angled Triangle. Let it be required S FIG. 53. to lay out a right-angled triangle containing .3 acre by a line per- pendicular to A B, Fig. 53. (a). Measure the angle BAG and find its sine. (6). Multiply the sine by any length of base, as A D, less than the required base. The product will represent the altitude D E of the triangle A D E. (c). Compute the area of this triangle, and the length of the side A B may be found by the following proportion : Area A D E : Area A B C : : ( A D) 2 : (A B) J . If A D = 1.20, and D E .80, the area of the triangle A D E will be .048 acre. Then, .048 : .3 : : (1.20) 2 : (A B) 2 ; whence (AB) 2 = 9, and A B = 3 chains. 126 MANUAL OF PLANE SURVEYING. The length of the side A C may be found by a similar propor- tion: Area A V E : Area A B C : : (A E) 2 : (A C) 2 . ( 235 ). To Lay Out a Trnpezoid. Approximate the distance be- tween the parallel sides by treating it as a parallelogram. The distance thus found will be too short if the sides not parallel con- verge, and too long if they diverge. Let Fig. 54 represent a tract to be laid out or parted off. B FIG. 54. By dividing the area by the length of the line A B, the perpen- dicular is found to equal C D. The guess line E D is then meas- ured, and the area of the trapezoid A B D E computed. The de- ficiency of area is then added outside of D E, and the trapezoid A B F G will then contain the required amount of land. If it still vary a little from the exact amount, the line F G may be moved further out or in, as the case may be. In case of divergence of the sides, the overplus of area must be- subtracted from the computed area, and the guess line moved back instead of further out. FIG. 55. MANUAL OF PLANE SURVEYING. 127 When the difference in length of the parallel sides can be deter- mined without a measurement, no guess line need be surveyed, providing the distance between the parallel sides be known. Let A B C D, Fig. 55, represent a tract of land to be laid out or parted off the main tract by a line perpendicular to its parallel sides. Divide its area by the perpendicular distance between its par- allel sides. The quotient will be the mean length of the trapezoid. From this subtract half the distance D E for the shorter side, and add for the longer side. (236). To Lay Out any Figure. 'When the underlying princi- ples of the particular problem differ from those obtaining in any of the cases considered, it will probably be best to depend on cor- rections made from guess lines, as in Art. 235, and thus reach the result by approximations. Yet, in many instances, easy and beau- tiful solutions may be reached by close observation and study. DIVIDING UP LAND. (237 ). Problems in dividing up land are such as grow out of division of estates, generally among heirs. This division is made with reference to the value of the respective shares (considering location, improvements, quality of soil, etc.), and not with regard to the quantity of land each share contains. If, however, taking all these things into consideration, the value of the land is uni- form throughout the tract to be divided, and the shares of the per- sons among whom it is to be partitioned are equal to each other, each should receive the same quantity. In making a partition of land no share should be taken out in such a way that it will injure any other share, when it possibly can be avoided. (238). The problems in dividing land introduced into this chapter are of the nature of those that usually come up in prac- tice where the land has been surveyed according to the Rectangu- lar System. Only simple ones have been chosen. ( 239 ). To Divide a Rectangle into Equal Parts by Lines Parallel to a Side. Divide each of the lines upon which all of these parts are to rest into as many equal sections as there are shares. Connect the extremities of these sections by perpendiculars, and these per- pendiculars will be the division lines of the shares. 128 MANUAL OF PLANK SURVEYING. Let Fig. 56 represent the south half of a quarter-section con- ace 10.00 10.00 10.00 10.00 20 20 A. i I 20 A. 20 A. 10.00 ! 10.00 | 10.00 10.00 i d / FIG. 56. taining exactly 80 acres. The perpendiculars, a b, e d, and e f, divide it into four parts, each containing 20 acres. ( 240 ). To divide a rectangle into any number of unequal parts bear- ing a given relation to one another, by lines running parallel to a side. Suppose that, on account of the varying value of the land in the tract to be divided, the shares are to be to one another as the num- bers 1, 2, and 5. In this case, divide the base lines into parts bearing the same relation to one another as the shares, and con- 5.00 10.00 25.00 10 20 A. ' 50 A, 5.00 10.00 25.00 FIG. 57. nect the points of division by perpendiculars, as shown in the division of the 80 acre tract in the figure. Sometimes it is possible to divide land of varying quality so that each share shall contain its portion both of the best and worst. If we consider the unequal division in Fig. 57 to have been caused by the difference in quality of the land in various parts of the tract, it might have been possible to make the shares all equal by dividing the tract in the direction of its length. MANUAL OF PLANE SURVEYING. 129 The figure of the shares may be almost as variable as the quan- tity of land they contain. The rectangular form is preferred, but of course can not always be preserved, even in the territory sur- veyed according to the Rectangular System. ( 241 ). Problems. 1. Divide a quarter-section of land into five shares in the series 1, 2, 3, 4, 5, by lines running parallel to a side. 2. The commissioners, in a certain partition of a quarter-sec- tion, set off the widow's dower of 30 acres in the form of a square in the south-east corner, and divided the remainder, by lines running north and south, equally among five children. What was the width of each share ? 3. In a sale of a quarter-section for taxes, the lowest bid was for fifteen acres, and this amount was set off in the form of a square in the north-west corner. A few years afterward the remainder of the quarter was offered again, and the lowest bid was for twelve acres, which area was set off next to that first sold. What were the dimensions of each piece ? 4. Divide the following described tract into four equal shares by north and south lines: South half north-east quarter, and east half north-east fourth north-east quarter, and south half north-west fourth north-east quarter. 5. Divide a quarter-section into 7 equal shares by lines running north and south, and write a description of each share, giving metes and bounds. 6. Divide the following tract into 5 equal shares by north and south lines, and write a description of each share : North half south- west quarter, and south-west fourth north-west quarter, and west half north-west fourth south-east quarter, giving metes and bounds. 7., Divide the north-west quarter, and north half north-east quarter, and north half north-east fourth south-west quarter, by east and west lines, into 3 shares that will be to each other as 1, 2, and 3, and write a description of each share, giving metes and bounds. In the above examples each tract is supposed to contain exactly the prescribed amount of land, and the boundaries to run due east and west, or north and south, as the case may be. 130 MANUAL OF PLANE SURVEYING. QUESTIONS ON CHAPTER XIII. 1. Why will a study of methods used in computation of area assist in laying ou.t and dividing up land? 2. How do you determine a side of a square from the area? Of a rectangle ? 3. Give the rule for finding the required side of a parallelogram. 4. Explain the method given for laying out a right-angle tri- angle. 5. How do you lay out a trapezoid from the area and 'length of one of the parallel sides? 6. In partitioning land, which is considered, quantity of land or value? 7. Explain the method of dividing a rectangle into equal parts by lines running parallel to a side. 1 8. In dividing lands, what figure for the shares is preferred ? NOTE ON CHAPTER XIII. The cases in dividing upland given in this chapter do not apply except where the land has been surveyed according to the Rectangular System. In States east of the Mississippi river, excepting Michigan, Wisconsin, Illinois, Indiana, Ohio, Mississippi, Alabama and Flor- ida, and even in many instances in these States, as well as in the ones west of the river, the surveyor must modify the method as the case may require. CHAPTER XIV. SURVEYING TOWN LOTS. (242 ). The dimensions of town lots are usually given in feet, instead of in chains and links, and, as a general thing, the lots are all of the same size and numbered in regular order from 1 up, as shown in Fig. 58. The larger figures indicate the numbers of the FIG. 58. blocks, and sometimes the lots in each block are numbered sepa- rately ; as lot 5, block 2 ; lot 2, block 9, and so on. As the town grows from the original plot, the lots in each addition are fre- (131) 132 MANUAL OF PLANE SURVEYING. quently numbered and referred to the particular addition to which they belong ; as lot 8, Brown's addition ; lot 7, Johnson's addi- tion, etc. ( 243). The survey of the town is generally based on some in- dependent corner of the section in which it is situated, and im- portant corners in various parts of the town should be marked with durable monuments. Fig. 59 shows a few lots in a town lo- cated on a section line, and the distance is given from the section corner to the south-east corner of lot number 1. It will be seen s e * 3 i 2 8 J 3 .ZS c o FIG. 59. that stones are placed at the north-east corner of lot number 4 and the south-west corner of lot number 8. These will enable the sur- veyor at any subsequent time to make a survey in the town with- out going to the section corner to find a starting point, thus saving him time and trouble. ( 244 ). Lots are usually rectangular in shape and about twice as long as they are wide, but this is not always the case. They may be any reasonable shape or size that adapts them to the plan of the town. Likewise, streets and alleys generally cross one another at right angles, though by no means always. ( 245 ). The plot of a town should always be accompanied by full explanations showing, ( 1 ). The size of each of the lots. ( 2 ). The width of each of the streets and alleys. ( 3 ). The name of each of the additions. ( 4 ). Any other explanations necessary to determine the bear- ings of any of the lines which would have to be run in a survey of the town. MANUAL OF PLANE SURVEYING. 133 ( 246 ). Suppose that in Fig. 58 the lots are each 100 feet long and 50 feet wide, the streets 50 feet wide, and the alleys 16 feet wide. Since the lots are rectangular, the north and south lines are at right angles to the east and west lines. It is evident that after finding a starting point, the surveyor need experience no dif- ficulty in the survey of any of the lots. If, for instance, he wishes to survey lot number 13, and can find no corner except the one marked with a stone at the south-east cor- ner of the town, he may start at the center of this stone, run west 266 feet to the west side of Main street, and thence north 166 feet to the south-east corner of the lot to be surveyed. He can then survey the lot without any trouble. Instead of running first west and then north, he may run first north to the south-east corner of lot number 29, and thence west to the corner of the lot to be sur- veyed; or he may take other routes. ( 247 ) Fig. 60 represents a portion of a town in which the lots are of different sizes and shapes. All the lots west of Main street are 50 ft. wide, except number 12, which is 60 ft wide, and num- ber 14, which is 75 feet wide. Main street bears N 40 W. 134 MANTTAL OF PLANE SURVEYING. Lots number 15, 16, 17, 18 and 19 do not belong to the regular plot of the town, and are called out-lots. The two alleys running north into Main street, and the one be- tween lots 4 and 5 are each 20 ft. wide, and Main street and the short street between lots 2 and 3 are each 50 ft. wide. A stone monument marks the south-east corner of lot number 13. The width of each tier of lots is marked in feet at the foot of the tier. ( 248 ). It is now a very easy matter to survey any of the lots in the regular plot of the town. Take, for example, number 11. To survey this lot, measure first west 280 ft., and thence north 200 ft. to its south-west corner. From this point set off a perpendicu- lar and extend it to the street ; then measure north 50 ft. further and set off another perpendicular as before. ( 249 ). EXAMPLES. 1. How would lot number 1 be surveyed? 2. Explain a method of surveying lot number 14. 3. If the out-lots east of Main street were separated by lines perpendicular to the street, what would be the bearing of the lines? ( 250 ). Town lots are measured either with a chain or tape. The chain used for this purpose is usually 50 or 100 feet long and divided into links, each 1 foot in length. It is made light, and as greater accuracy is generally required in surveying town lots than in ordinary surveying, its length should be frequently tested by comparison with a standard measure. The length of the chain is affected by wear, temperature, and accidents. All measures used in surveying should be subjected to frequent tests. Even in surveys where tolerable accuracy is sufficient, there is no excuse for neglecting anything that would be conducive to greater accuracy. Tapes used in measuring town lots usually consist of a jointed steel ribbon, but sometimes a linen tape through which a fine brass wire is interwoven with the thread, is used. Common linen tapes contract when wet and are not trustworthy. The steel tape is the best. MANUAL OF PLANE SURVEYING. 135 QUESTIONS ON CHAPTER XIV. 1. How are town lots numbered ? 2. Upon what is the survey of a town usually based ? 3. What advantage is there in having important corners marked by monuments? 4. What is the usual shape of town lots ? 5. State the explanations that should accompany the plot of a town. 6. A street bears N 29 32' E. What is the bearing (obverse and reverse) of a line perpendicular to it? 7. What kind of measures are employed in the survey of town lots? 8. How is the length of the chain affected ? CHAPTER XV. PLOTTING. ( 251 ). Plotting is the operation of drawing to a scale upon paper the lines of a survey, so that the plot will be a correct rep- resentation of the actual lines surveyed. ( 252 ). The instruments used in plotting are a drawing board, t-square, ruler, drawing pen, dividers, protractor, and a diagonal scale. 1. The drawing board should be made of pine, and its surface should be perfectly smooth and level. The paper is fastened to the drawing board while the plot is drawing. Perhaps the most suitable size for a drawing board is about 30 inches square, but 24 by 28 inches makes a very nice board. The paper should be stretched evenly, and the edges pulled down over the edges of the board and glued or tacked. 2. The t-square, so called on account of its resemblance to the letter T, consists of a thin blade with parallel edges, to which is attached a cross-head somewhat thicker than the blade, so as to form a shoulder. The blade is usually about 24 inches in length, and the cross-head about 10 inches. By laying the blade on the paper and pressing the shoulder against the edge of the drawing board, as shown in Fig. 61, perpendiculars may be drawn to any edge of the paper. 3. A good box-wood ruler, about 12 inches long, divided to 16ths of an inch, will answer every purpose in plotting. This ruler should have one beveled edge, upon which the divisions are marked, and one projecting edge, along which the pen should be pressed in drawing lines. 4. The drawing pen consists of two steel blades, whose distance apart is regulated by a thumb-screw. A little practice will en- (136) MANUAL OF PLANE SURVEYING. 137 able any person to draw nice smooth lines of any desirable width with the drawing pen. India ink should be used, as it flows more smoothly from the pen than common ink. I FIG. 61. 5. The dividers or compasses is an instrument used in drawing arcs, sub-tending angles, etc., and consists of two arms which open and shut by a hinge joint at the end. Each of these arms termi- nates in a sharp point, and one, if not both, is usually jointed so as to permit the point to be taken out and a drawing pen put in its stead. In drawing large arcs a lengthening bar is inserted in FIG. 62. the jointed arm. Fig. 62 represents a pair of plain dividers. It will be seen that the arms are not jointed in this pair. 6. The protractor is an instrument used in laying out angles. It is usually nothing more than a semicircle divided to degrees, half-degrees, or quarter-degrees. The degrees are numbered from to 180 in one or both directions from opposite extremities of the arc. The best protractors are made of silver or German silver, but the more common ones are made of brass or horn, and some- times of paper. Fig. 63 represents a small protractor. 138 MANUAL OF PLANE SURVEYING. Where great accuracy is required, protractors are supplied with an arm to which a vernier, like the compass vernier, is attached. Sometimes rectangular protractors are used instead of semi- circular. 7. The diagonal scale of equal parts is a flat scale a given num- FIG. 63. ber of units, say inches, in length, and has the space devoted to one unit at the end divided by diagonals as shown in Fig. 64. These diagonals with the assistance of the lines running parallel to the edges of the scale, enable a person to take the length of a 3,8 J. 8.5.^. 3.2.7 FIG. 64. line to T ^y of the unit of the scale. If the unit of the scale be 1 inch, then the length of a line may be taken to T J^ of an inch. ( 253 ). In drawing plots and maps a unit of the scale repre- sents a certain number of units of the line to be represented on the plot. Suppose the real line is 20 chains long, and it is to be plot- ted to a scale of 5 chains to the inch. The line on the plot will therefore be (20 -4- 5) = 4 inches long. Fig. 65 represents a line 1 chain in length plotted to different scales. MANUAL OF PLANE SURVEYING. 139 1 in. = 5.00. 1 in. = 2.00. 1 in. = 1.00. FIG. 65. ( 254 ). Let us now employ the drawing instruments in plot- ting lines. Suppose an east and west line 2.27 in length is to be drawn to a scale of 1 chain to an inch. Arrange the drawing paper with its edges parallel to the edges of the board, and then place the t-square, as shown in Fig. 61, with its shoulder fitting squarely to the left-hand edge, and the edge of the blade just moved up to the point from which the line is to be drawn. Then spread the dividers so that when one arm is placed two units from the inner edge of the divided unit and on the line marked .07, the other will just reach the point where this line crosses the line marked .2 at the top of the scale. The arms then embrace the proper length of line. Next place one arm of the dividers against the t-square with its point on the point from which the line is to be drawn and swinging the free arm round in the proper direction until it too touches the same edge of the blade. Connect these two points by a line with the drawing pen, and it will be the required line. In like manner, a line 1.25 long to the same scale may be em- braced by the dividers by placing one point at a on the diagonal scale, and the other at e; and a line 1.40 long by placing one point at E and the other at the point marked .4 at the top of the scale. If the diagonal scale is not long enough to permit the required line to be taken off, it may be extended by means of a ruler. (255). EXAMPLES. Draw lines representing the following distances : (1). 2.50; scale 1 in. = 1.00. (2). 3.79 ; scale 1 in. = 1.00. (3). 4.75; scale 1 in. = 2.00. (4). 6.42; scale 1 in. == 5.00. (5). 10.00; scale 1 in. = 5.00. (6). 12.31 ; scale 1 in. = 10.00. 140 MANUAL OF PLANE SURVEYING. ( 256). Rectangular tracts of land may be plotted with the in- struments used in drawing lines already described. Take, for instance, a rectangular tract 7.15 long, 4.35 wide. First draw a line representing the length of the tract, then another perpendicular to this at one end representing the breadth, then from the end of this another parallel to the first and of equal length, and close by connecting the extremities of the first and third with one another. If no diagonal scale is at hand, a common ruler will answer for rough work. ( 257 ). The protractor is used in nearly all cases where the courses and distances of the boundaries of the tract to be plotted are given, and its use will now be explained. The bearing of a line represents the angle the line makes either with the magnetic meridian or the true meridian drawn through the point from which the bearing is taken ; and to deter- mine this angle and represent the line on the plot, meridians should be drawn through each station of tne survey as soon as the station is located. Begin at any important station to draw the plot by laying out a meridian on the proper part of the paper and locating the station on this meridian; then draw the first course at the proper angle with this 1 meridian, producing it the required distance, as explained in Art. 254; then draw another meridian through the other ex- tremity of the course, and lay out the second course in the same way ; proceed in this way until the lines are all drawn, and the last line should terminate at the station taken as the starting point in the plot. ( 258 ). Let us now draw a plot of the following tract of land: N 62 45' E, 9.25 ; thence S 36 E, 7.60 ; thence S 45 30' W, 10.40 ; thence N 31 30' W, 10.00. In this case we may commence at the first station in the de- scription. Draw a meridian as N S, Fig. 66 and locate the sta- tion at some point, as A, on the meridian. Then place the pro- tractor so its center will fall on the station and its edge coincide with the meridian, and with the point of a pin mark the termina- tion of an arc of 66 45' from the north end of the protractor. Then draw the line A B from the station through this point and determine its length by the method explained in Art. 254. Draw MANUAL OF PLANE SURVEYING. 141 B C, C D, and D A in the same manner, and the plot will be com- plete. The plot, Fig. 66, is constructed to a scale of 5 chains to the inch. In northerly courses the angle or bearing should be read from the north end of the protractor, and in southerly courses from the south end. N B D fo FIG. 66. If the last course lack but a little of terminating at the first station, the discrepancy may be the result of the imperfection of the instruments employed; but if the extremities of the lines are a considerable distance apart, it is probable that a mistake has been made somewhere. If it be tested by latitudes and departures, and they balance (Art. 212), the mistake is in the plot, but if they do not balance, the error is in some of the previous work. 142 MANUAL OP PLANE SURVEYING. The descriptions given in Art. 226 may be used as examples in plotting. Various other methods are also used in drawing plots, but the one given is, perhaps, the most speedy and simple, and will answer every purpose. (259). The pantograph is an instrument used for copying plots, etc., either in a reduced or enlarged form. It consists of four rulers arranged somewhat in the form of a parallelogram. By fastening the instrument on the drawing-board and moving a point on one arm along the plot to be copied, another arm to which a pencil is attached sketches a precise copy on the sheet placed under it. (260 ). Buildings, springs, etc., may be located on the plot if their courses from certain points on the boundaries of the tract are known. For example, in surveying the west half of a quarter-section, a line from the north-east corner to the north-west corner of a house was found to bear S 45 W, and one from a point 12.00 west of the north-east corner was found to bear S 13 E. While constructing the plot these lines may be laid out from the proper places with the protractor, and the place where they meet will be the north- west corner of the house, as shown in Fig. 67. FIG. 67 MANUAL OF PLANE SURVEYING. 143 ( 261 ). Plots and maps may be colored with crayon pencils or with water-colors; but when water-colors are used care must be taken to keep them from running into one another and injuring the shades. The paper should be dampened preparatory to ap- plying them. For inexperienced persons, crayons will prove the most satisfactory. QUESTIONS ON CHAPTER XV. 1. What is plotting? 2. Name the instruments used in plotting. 3. Descr-ibe the diagonal scale. 4. Describe the method of using the diagonal scale. 5. Give the length of each of the following lines plotted ta scales of 1 chain to an inch, 2 chains to an inch, and 10 chains to an inch: 12.50; 15.00; 18.375; 11.25. 6. Describe the method of using the protractor. 7. Why should the last course in a plot terminate at the first station ? 8. For what is the pantograph used ? 9. How may buildings and other objects be located on a plot? 10. How are plots and maps colored? 11. Draw plots of the tracts described in Article (226). CHAPTER XVI. SURVEYING WITHOUT A COMPASS. (262). A great many surveys can be made without the com- pass, and a few pages will now be devoted to the consideration of the most common cases in which it may be dispensed with. It must be borne in mind, however, that the compass could be ad- vantageously used in nearly all the cases here cited, and that the methods given are intended for use only in emergencies. (263). Setting Oorners. Where two witness trees taken to a corner can be found, the corner may be located from them by their distances measured from the sides upon which the blazes are made. Suppose one tree is 15 links from the corner, and the other 19 links. Measure off 15 links on one end of a cord and 19 links on the other end, and tie a knot where the two measurements ter- minate. Then have the long end of the cord held against the WITNESS blaze on the more distant tree, and the short end against the blaze on the other. Stretch both ends of the cord tightly, and the knot will mark the corner, as shown in Fig. 68. The corner is always in front of the blazes on the trees. The distances of the witness trees from the corner may be found from the field-notes of the tract. (144) OF PLANE SURVEYING. 145 Where only one witness can be found, the corner can not be located with certainty without a compass. This same method may be employed, slightly modified, in lo- cating a corner by the two lines meeting there, when the length of each line and the location of each of the corners at the other end of each, are known. ( 264 ). Establishing Lines. When one corner is visible from the coiner at the other end of the line, a stake may be put up, and intermediate points on the line may be marked at pleasure. When one corner is not visible from the other, but its direction is approximately known, the line may be " ranged '' from one to the other. To do this, put up a stake or flag at the corner from which the line is ranged, and at a certain distance, say 50 or 100 steps, in the direction of the other corner, set up another stake or flag. Then walk ahead an equal distance and set another stake in line with the first and second. Proceed in the same manner, always setting the stakes at equal distances from one another, and ranging the last one with the two previously put up, until the other corner is reached. The distance that the line misses, either to the right or left, can then be noted, and the stakes corrected in a manner entirely similar to that already explained. For instance, if there are 12 stakes on the line, and it termi- nates 30 links to the right of the corner, each stake must be moved to the left. The distance it is to be moved is found by dividing the distance missed by the number of stakes and multiplying the quotient by the number of the stake from the starting point. In ( 1 1 'v' ^0 1 this case, the llth stake must be moved to the left I ^ ! =27 links, the 10th [ 10 * 3 ] = 25 links, and so on. The stake put down at the corner at starting is not counted, and the next is called the first. Of course, the location of the corners must be known before the line can be established. ( 265 ). Setting Out Perpendiculars. Almost any kind of a con- trivance with two lines of sight at right angles to one another, will answer for this purpose. It may be a sort of cross-staff with four upright sights provided with slits or threads, two marking each line of sight. The sights need not be more than 18 inches 10 146 MANUAL OF PLANE SURVEYING. apart, and the apparatus should be made to rest on a staff about 4 feet high. It may be rude in construction, but the lines of sight should be exactly at right angles to one another. ( 266 ). Rectangular tracts of land may be readily surveyed in many instances with this instrument, but it should be used with care in independent divisions of the section, as they are not often exactly rectangular in form. One of the lines of sight may also be used in sighting lines, and is a good substitute for the method of " ranging " described in the preceding article. (267). Measurements. Lines may be measured with a cord, tape-line, or pole, and distances may be given in feet or links, as best suit the case at hand. APPENDIX. ABSTRACT OF DECISIONS, (147) ^ /-.-^~~ (rp^^^ju tf r-t si^^- &^t_ ^ ABSTRACT OF DECISIONS OF THE UNITED STATES AND VARIOUS STATE COURTS RELATING TO CONTRACTS, SURVEYS, ETC. (Nearly all of the following Decisions have been taken by per- mission from DUNN'S LAND DECISIONS, a valuable book for sur- veyors, published by George H. Frost, New York.) BOUNDARIES. 1. Course and distance must yield to natural and artificial ob- jects of description. Gaveny m. Hinton, 2 Greene (Iowa) 344. 2. Boundaries marked on the land are to govern courses and distances. Blaisdell vs. Bissell, 6 Barr (Pa.) 478. 3. The lines marked on the ground constitute the actual sur- vey and control the return of the surveyor, even where a natural or other fixed boundary is called for by the survey, though the space between the two is but twelve perches in breadth. Walker TO. Smith, 2 Barr (Pa.) 43; Hall m Tanner, 4 Barr (Pa.) 244. 4. A grant called for a certain number of poles " to a stake, crossing the river." Held, that the line must cross the river, though the distance terminated before entering it. Whiteside i. Singleton, 1 Meigs (Tenn.) 207j 5. A survey must be closed in some way or other. If this can be done only by following the course the proper distance, then it would seem that distance should prevail ; but when distance falls (149) 150 APPENDIX. short of closing, and the course will do it, the reason for observing distance fails. Doe vs. King, 3 How. (Miss) 125. 6. Where a deed describes lands by its admeasurements, and, at the same time, by known and visible monuments, these latter shall govern. Mayhew vs. Norton, 17 Peck (Mass.) 357; Massen- gille vs. Boyles, 4 Humph. (Tenn.) 205; Woods vs. Kennedy, 5 Monr. (Ky.) 174; Nelson vs. Hall, 1 McLean (U. S.) 518; Camp- bell vs. Clark, 8 Mis. 553. 7. The rule that monuments control in boundaries is, however, not inflexible ; and in case where no mistake could reasonably be supposed in the courses and distances, the reasons of the rule were held to fail, and the rule itself was not applied. Davis vs. Rains- ford, 17 Mass. 207. 8. A line is to be extended to reach a boundary in the direc- tion called for, disregarding the distance. Witherspoon vs. Blanks, 1 Taylor (N. C.) 110. 9. If a vendor hold two tracts adjoining, and sell a certain quantity by metes and bounds, though the deeds call for one tract, yet if the metes and bounds run into the other, the purchaser shall hold according to the metes and bounds. Wallace vs. Max- well, 1 J. J. Marsh (Ky.) 447; Mundell vs. Perry, 2 Gill & Johns (Md.) 206. 10. Posts set up at corners, between adjoining owners of land, control the calls for course and distance and establish the bound- ary where they are mentioned and recognized in the deeds. Al- shire vs. Hulse, 5 Ham. (Ohio) 534. 11. Where land is described as running a certain distance by admeasurement, to an ascertained line, though without a visible boundary, such line will control the admeasurement and de- termine the extent of the grant. Flagg rs. Thurston, 13 Pick. (N. Y.) 145; Carroll t. Norwood, 5 Har. & J. (Md.) 163. 12. Where the line or course of an adjoining tract, being suffi- ciently established, are called up in a patent or deed, the lines shall be extended to them without regard to distance. Cherry vs. Slade, 3 Murph. (N. C.) 82. 13. Where the boundaries of land are fixed, known, and un- questionable monuments, although neither courses, nor distances, nor the computed contents correspond, the monuments must gov- ern. Pernam rs. Wead, 6 Mass. 131 ; Calhoun vs. Wall, 2 Har. & McHen. (Mo.) 416. APPENDIX. 151 14. If a deed from the government of the U. S., or an individ- ual, describes land as partly bounded by a river, the river bound- ary will be adhered to, though it does not correspond with estab- lished corners and monuments. Shelton a iis. Mauphin, 16 Mo. 124. 15. If nothing exists to control the call for courses and dis- tances, the land must be bounded by the course and distances of the grant, according to the magnetic meridian ; but courses and distances must yield to natural objects. 16 Ga. 141. 17. The corners established by the original surveyors of public lands under the authority of the United States, are conclusive as to the boundaries of sections and divisions thereof, and no error in placing them can be corrected by any survey made by individ- uals or by a state surveyor. Arnier vs. Wallace, 28 Miss. 556. 18. Whenever natural or permanent objects are embraced in the calls of either a survey or a patent, these have absolute con- trol, and both course and distance must yield to them. Brown vs. Huger, 21 Howard (U. S.) 305. 19. In determining boundaries under a grant, natural objects, .as landmarks, are to be considered before courses and distances. Daggett vs. Wiley, 6 Fa. 482. 20. Where adjoining proprietors abut on opposite banks of a stream, their boundary line will follow the natural and imper- ceptible alterations in its course, but not changes caused by arti- ficial means. Halsey vs. McCormick, 3 Kernan (N. Y.) 296. 21. When lands are described in a deed or grant as bounded by river not navigable, the center of the stream is to be considered the boundary. Claremont vs. Carlton, 2 N. Hamp. 369 ; Palmer DS. Mulligan, 3 Caines (N. Y.) 407, 319; Hayes vs. Bowman, 1 -Hand (Va.) 417; Ingraham vs. Wilkinson, 4 Pick. (Mass.) 268; Gavil vs. Chambers, 3 Ham. (Ohio) 496 ; Brown vs. Kennedy, 5 Har. & J. (Md.) 195; Arnold vs. Mundy, 1 Halst. (N. J.) 1. QUANTITY OF LAND. 22. A conveyance by metes and bounds will carry all the land contained in them. Belden vs. Seymour, 8 Conn. 19; Jackson vs. Ives, 9 Cow. (N. Y.) 661. Although it be more or less than is stated in the deed. Butler vs. Widger, 7 Cow. (N. Y.) 723. 23. Where a specified tract of land is sold for a gross sum, the boundaries of the tract control the description of the quantity it contains, and neither party can have a remedy against the other for an excess or deficiency in the quantity, unless such excess or deficiency is so great as to furnish evidence of fraud or misrepre- sentation. Voorhees vs. De Meyer, 2 Bar. Sup. Ct. Rep. (N. Y.) 87. 24. Where a person purchases land by metes and bounds said to contain a certain number of acres, more or less, he is entitled to all the land within the limits, whatever the number of acres may be. Bratton ts. Clawson, 3 Strobh. (S. C.) 127. 24. Quantity, although the least reliable and last to be resorted to of all descriptions in a deed, in determining the boundaries of the premises conveyed, may sometimes be considered in corrobora- tion of other proof. McClintock vs. Rogers, 11 Ills. 279. FIGURE OP TRACTS OF LAND. 25. If the order for a survey of land do not certainly determine the form in which it should be made, the survey ought to be in a square. Kennedy vs. Paine, Hardin, 10. 26. "Seventy acres, being and lying in the south-west corner" of a section, is a good description, and the land will be in a square. 2 Ham. (Ohio) 327; Cockrell vs. McQuinn, 4 Monr. (Ky.) 63. The rectangular figure will be preserved in preference to any other in fixing locations. Massie is. Watts, 6 Cranch, 148 ; Holmes vs. Trout, 7 Pet. 171. ACQUIESCENCE IN BOUNDARIES. 28. Acquiescence for a long time (e. g. for eighteen years), in an erroneous location, is conclusive on the party making or acqui- escing in such location. Rockwell vs. Adams, 6 Wend. (N. Y.) 469. 29. Where a boundary is disputed between parties who own ad- joining tracts, and the parties employ a surveyor, who runs out the line, and marks it on a plat in their presence, as a boundary, after twenty years corresponding possession, they are concluded by it. Boyd vs. Graves, 4 Wheat. 513. 30. An acquiescence for twenty years is, as a general rule, nec- essary to support an implied agreement in respect to a boundary different from that clearly expressed in the title deeds. Ball vs. Cox, 7 Ind. 453. 31. Where two persons own equal parts of a lot of land in sev- erally, but not divided by visible monuments, if both are in pos- session of their respective parts for fifteen years, acquiescing in an APPENDIX. 153 imaginary line of division during that time, that line is thereby established as a divisional line. Beecher vs. Parmele, 9 Ver. 352 ; also 18 Ver. 395. 32. Maintaining a fence for many years is strong but not con- clusive evidence of limitation of claim to the boundary. Potts tw. Everhart, 26 Pa. 493. 33. A party is precluded upon principles of public policy, from setting up or insisting upon a boundary line, in opposition to one which has been steadily adhered to upon both sides for more than forty years. Baldwin vs. Brown, 16 N. Y. 359. 34. A division fence of more than twenty-one years' standing, although crooked, constitutes the line between adjacent land own- ers, even though the deeds of both parties call for a straight line between acknowledged landmarks. McCoy vs. Hance, 28 Tenn. 149. 35. Where adjoining proprietors, being unable to ascertain the division line, agree verbally upon a certain line, the agreement is binding, and improvements by one up to the line is notice thereof to a purchaser from the other. Houston vs. Sneed, 15 Texas, 307. 36. An ancient line of division marked on the ground by ad- joining owners, and afterwards acted upon by them, will become the boundary between the lots, although different from the line described in the original deeds. Hathaway vs. Evans, 108 Mass. 267. 37. A possession for twenty years of a part of the land in dis- pute, in reference to a line conflicting with another tract, of which another party may be also in actual possession, but outside of the disputed territory, may be enough to presume the execution of a deed conveying the land in dispute to the party in possession. Amick vs. Holman, 13 Shobh. (S. C.) 132. 38. Parties are not bound by a consent to boundaries, which have been fixed under an evident error, unless perhaps by the pre- scription of thirty years. Gray vs. Couvillon, 12 La. Ann. 730. TABLE OF NATURAL SINES AND COSINES. (See Articles (88), (207), (208), (209), (210), etc.) (155) 156 TABLE OF NATURAL SINKS AND COSINES. TABLE OF NATURAL SIXES AND COSINES. 157 > M. Sine. .!.89-,> MfttJ .l>f-*.n 14C85 .98914 14723 !<8<<1 14752:. 989(16 14781 .98902 14810 .98897 14838 98893 14867 .98889 14896 .98884 14925:. 98880 14954!. 14982- .98871 15011 lf,040l 15069 '.98858 l.%97 .98854 15126 .98849 151551.98845 16878 16906 1(899 15327 1556. 98814 15414 15-142 .9S-00 15471 .98796 l."rf)0 9K791 98787 15557 ^ p-' . Co. In . I M. 15701 ir,7.M 15758 ].->: 15816 1684C 15873 15902 16968 lr('17 10041 16074 16103 16132 Kil(X) 16181 168ft 1688 l,: j ,; n.y,( 16411 1644' 16476 16505 .16533 loia 16691 ];M- it;-7 16763 M7M 17136 171M 1719:^ 17222 17250 98501 1727!" .98496 80 98671 ( 37 9857 36 98652 35 98648: 34 98643! 33 98619 28 !614 27 98609 26 98604 25 18600 24 98595 28 OBM TABLE OF NATURAL SINES AND COSINES. TABLE OF NATURAL SIXES AND COSINES. Oin. 95iu7; 159 M. .94552 60 '.4542 ..",1040 95061 3X694 .31068 B6069 89799 .31095 96048 32749 .81198 96088 89771 .31151 95094 32804 9:(nr, 8988S ! 8190(1 89869 .81988 114997 89867 .81961 919-.- 89914 .312-!! 9-1979 :-,2942 .31316 94970 32%9 .81344 .94961 .32997 .31 :,-;, 94969 94943 [88061 '.3i':i"7 9J933 48079 .31454 .'.M924 88106 .31482 .'.'1915 48184 .94908 48161 181685 .94897 .81666 .94888 48911 .81693 .94878 .:> 244 .31020 .94860 48971 .31C48 .94861 .a 2! s .31075 .31703 !94841 ; J ! J . V f .' J ; .31730 .94832 .33381 .31758 .94823 .33408 .31781 .94814 .:::!."( .31.-1: .94806 i :MM' : ; .31841 .'. 1795 .88491 .31808 .94786 .88618 .818% ! 947 77 48641 .31923 4476E 18867I .31951 .9475,- .3r6(( .91719 ;-j; fj^l ! 82001 .94740 .8E66E .890 .947^0 .88685 .8' 061 .14721 *vi7 1 ( .89C8! .94712 .aawi 321 i( .94709 .88764 ! .32144 .94693 .33792 .32171 .94684 .33819 32199 .94fi74 .88844 32227 94666 .88874 89964 94- M .88901 32282 MM, 88928 89801 88961 89881 94627 88981 89864 84011 88883 .33419 94599 '.'. \( l >it- .32447 94690 840M .32474 915-n .34120 .89609 94571 34147 .32529 94661 31175 .89667 94669 34202 1 Co>ln. | Sine. 71" 0in. | TO TABLE OF NATURAL SINES AND COSINES. 91355 tt) ) mm . 93:253 30133 .93243 3616 2 36 190 i. 93222 36-217 .93211 30244 .93^)1 TABLE OF NATURAL SINES AND COSINES. 161 .42473 .42400 .90530 .44072 ..,2525 '.90507 .14-10-1 .42-552 .9049.-) .44124 .42578'. 9048V '.44151 .43604 .90470 .44177 .42631 .90458 ,.44203 .42037 .901 IK .44220 28 29 i Sine. Cwin TBST c',-r,:\ 4>i047 88295 48481 vt-tm 4',! 173 88281 1 s -:' t; b744s J'i'f'.*'.* 47n24 S "*',!()" 8B8M 16689 .48557 >7-io I 87420 47060 .8BMI 48688 87406 47076 .88994 48606 87891 .47101 .88213 (6684 87:-:77 47127 .88191 486M S7: f,3 47168 .8818C .4W184 .S7340 4717,s >M7- .48710 >I888 .47204 .88168 487% xmm 47999 >.-H4 .48761 .87806 47966 >.-!:-;< .4b78H .87999 .47281 >Hr .48811 .>7i7x .47306 .88101 .48887 .^72C4 .47332 .880b9 .48862 .87250 .47358 >07f> .46888 .87235 .47:,-:; 8806! .48918 .87991 [47408 .earn .48888 .b7i07 .47434 >N ?A .48064 >71!tH 47460 .86091 .48988 .KI178 .47488 .88001 .41.014 .87164 .47511 .87901 .49C40 .87150 .47681 .87971 .4; i r,:, .H7136 .4-.B6J .8796E .49090 .87121 .47588 .Kid .49116 .87107 .47014 .87937 .49141 .87093 .47639 .87925 .49166 .87079 .47660 .87901 .49192 .87064 .47690 .87891 .49217 .87060 .47716 JOOK .49242 .87036 .47741 .87868 .49268 .87021 .47767 .87864 .49898 .8T007 .47793 .878* .49316 >f.'.,!.3 47818 .67891 .49344 .6978 47844 jtnstii .49869 .86964 47860 .87798 49804 .86949 I.47.-95 .87784 .49419 o:;o 47880 .8777) 4944B .8K!)21 .47946 .87761 49470 mn (i .47971 .87741 4949C 86899 .47997 .87729 49521 86878 4-122 .87715 49646 SBft 3 48048 .K7(H .1107! M:WO 48078 48080 >7fi^7 .87672 4!l5!di 4%22 M S:M M>20 .48124 .87659 .49647 66805 481CO .87645 49672 f-1.791 4, -17.-. .87631 49697 K,777 48901 .S7f,17 49799 Mi762 48996 .87C03 49748 80748 4-2.V2 87689 407 7:; 88788 4 ,-277 87878 49798 .719 48308 KT.V,1 49894 M1704 4<:;-S 87646 'l!tV 1'* B6690 48354 87',: 2 49874 NiC.75 48878 K.Ms 40699 vrt-fil 48406 87504 49994 .-;t;46 48480 87490 49960 B6689 48466 S747H 49975 8rr,i7 .48461 87462 50000 80608 Colin. Sine. Cosin. | Sine. J ) ) 50 58 57 56 55 54 I 10 9 ? 8 1 / 6 6 ? i : - i ' TABLE OF NATVRAL SINES AND COSINES. Sine. 50000 50025 50126 .86530 501511.86515 86501 30 50503 50588 50553 50578 501103 501128 50654 5')'179 50S-29 5IK5! 50879 51104 51129 5122!) 51854 51304 51329 .85821 31 C 85926 85! ID 51504 51690 51551 5i 5; '.) 51(104 51028 51653 51678 517(1:! 51728 518,03 51X28 5185-2 51877 51902 51! I. '7 51(15-' 51977 52002 53088 5-2-2 10 52225 .52250 5-2275 52291) 52!>t 5284U 52 I IS 59* i 528 I I Cosi .a5717 .85702 .85687 .85672 .85657 .85ti J2 85597 S558-2 , ,85506 ,851; 11 s.Mlti ,85431 85370 .S5325 ,85310 .-(KM 53189 58314 53-2H3 53288 53730 53754 53779 510-21 5404'.) 5 I .' 1 1 5l'2(i!t 542: 13 54317 ,5434-2 .513IH1 ,54391 5I115 .8I5SS .811:;:; .84417 .' I -2! 1-2 .M-21-, 33 34 Sine Cosin Sine. fV^TI. .54464 83807 55919 82904 5 1-188 83851 559 13 82887 51513 83835 55968 82871 54537 83S19 50992 .82855 54561 83804 56016 .82839 5458i 1 83788 5601(1 54610 .83772 .8-J8I 6 54635 .8:1756 '56088 .82790 5-165!) .837-10 5611-2 .82773 51683 .83; 21 .56136 .82757 .54708 ..-3708 .56160 .82741 517::-2 .83(15)2 .56184 .82721 51756 >:;o:r .8-2 ;i>- .54781 '.56232 .82692 5-18(15 '.8SMI ..-(1256 .82675 .54829 .83629 .56280 .82659 .54854 .83613 .56305 .82643 .5-1878 >3:97 ..-(1329 .821 26 .54902 .88681 .5635:; .82010 .64112! .56377 .51951 !&354 .56101 >25 7 7 .51975 .56125 .82661 .54999 .83517 .561-19 .8251 1 .55(124 .83501 .5617:, .8V52X .55048 .56497 .82511 .55072 !8. : i-l(i! .56521 .82495 .55(197 .83-15: .565 15 .82-178 .55121 .83437 .5656!) .82462 .5ii.- .8:121 .5, 15! 13 .82416 .83-10: .56617 .82-129 !55UM .83369 .56641 .82113 .55218 .83373 .566(15 .82396 .66842 .88361 .56689 .82380 .66801 .8334C .56713 .82363 .552!)! .6786 .82347 .553ir ! 83308 ..-,671.0 .8233d . 55339 .83292 .56781 .82314 .55363 .8:127 ( .82297 .65888 .8326C :-,6S,2 .82281 .55112 .83241 .56856 .8226 1 8 "Ms .'rci'iu . S.'i^S .83212 .5(i880 .50901 1 8223*1 .55-184 .6560) .83195 .8317!) ! 569-28 .56952 .82214 .5553: .83163 .56976 '.82181 .55557 .83147 .57000 .82165 .55581 .83131 .57024 .82148 .55605 .83115 .57047 .82132 556:! i >3i 63 .57071 .57(195 .82115 .82098 55678 .57119 .82182 55702 ieaoeo .57143 .82065 55726 .83034 .57167 .8201-- 55750 .83017 .57191 .82032 .55775 .83001 .57215 .82'115 5579!) .82985 ,57238 .819!)!) .55823 .82969 TM "Tl'' .81982 5-8 17 82963 .57286 .819(15 .55871 >->!<:i6 .5731(1 .81919 .55*95 .82920 .57334 .81932 .55919 .82901 .57358 .81915 Cosin. Sine. Cosin. Mne. 66 55 19 ( 18 > 17 ) 16 b 15 ) 14 \ 13 ( 12 ( 11 } 10 5 TABLE OF NATURAL SINKS AND COSINES. 773^9 .77310 38 87 .77273: 36 .77255! 35 .77236 34 .77218 33 .77199 32 .771811 31 .77102: 30 164 TABLE OF NATURAL SINKS AND COSINSS. BY 5. U. (SOOMBS. PRICE, .25. CONTENTS OF THE NORMAL READER. INTRODUCTION. PART I. How to teach a child to read. PART II. Dictionary Work. 1. Pronunciation. 1. Key to Pronunciation. 3. Elementary Sounds. 4. Principles of Pronunciation. 5. Articulation. 6. Words Often Mispronounced. 1. How to Teach Reading. 2. Examples for Practice. PART IV. Elocution. Art of Delivery. Outline of Elocution. Plan of Studies. Elements. Respiration. Breathing. Formulas. Articulation. Orthoepy. Vocal Culture. Exercises for Drill. Quality Vocal Volume. Rate. Gesture. Su 1. 2. To Ministers. To Lawyers. PART V. One Hundred and Twelve of the best Selections of Prose and Poetry from the Best English and Amer- can Authors. Liberal terms for first introduction by the quantity. Sample copy sent for $1.00 to those who desire to examine with a view to an adop- tion. No free copies. Please don't ask for them. Address, J. E. SHERR1LL, Pub., DANVILLE, IND. - ): DALE'S :( - OUTLINE OF ELOCUTION. BY G. WALTER The purpose of this book is to afford a complete and philosoph- ical treatment of all the principles underlying the art of human expression. It is designed to. analyze the art into its elements, and discuss these elements in such a manner as that students who do not have the advantage of aid from the living teacher may suc- cessfully acquaint themselves in a practical manner with this fine art. Its scope is bounded only by the possibilities of the subject, which makes it a work of extraordinary value, in that it contains the entire subject, logically treated. The definitions, discussions and exercises are concise, explicit and pertinent. They are adapted to students in any grade, from the primary to the high school, seminary or college. The twelve appended Essays cover ground discussed in no other similar work, and involve, among others, " Care of the Voice," " Primary Teaching," and a multi- tude of ;< Hints and Suggestions." It is an elocutionary library in itself, as it represents the science and art of expression by voice and action. As a book of reference it is standard literature, and a classic of its kind. A student of ordinary intelligence can take this book and study elocution without the aid of an instructor, which he can not do by the aid of any other book. This is be- cause the subject is placed before the student with such clearness, in such minuteness of detail, that there is no room for misunder- standing. It is the most popular book published on the subject of Elocution. IT CONTAINS ESSAYS ON : 1. Emphasis. 2. Projection of Sound. 3. Timbre. 4. Care of the Toice. o. A Course of Reading. 6. Dramatic Reading and Recitations. 7. Impersonation of Old Age. 8. Primary Teaching. 9. Hints and Suggestions, etc., etc. The last 200 pages are filled with the best collection of selections for reading ever embodied in a book of this kind. The matter is new, the selections abundant and fresh, and the tone of the book immeasurably above that of the ordinary reading book. It is the work of one of the most successful teachers of reading, who is himself a living example of the high class of in- struction he gives in his book. Liberal terms for introduction. Agents wanted all over the world. The most favorable induce- ments offered. Write for terms. Address, J. E. SHERRILL, Pub., IPUD. THE Iijdiarja This is an entirely new and original departure in books of this class, and can not fail to be a happy hit and meet with warm approval and a hearty welcome from the public in general. As its name indicates, it is made up of selections from the very best productions of the very best authors in Indiana. We mean no offense to authors in limiting the work to this State ; neither do we think it necessary to offer any apologies, for her bright galaxy of distinguished writers has al- ready placed the " Hoosier State " in the front rank of the literary world. The book contains writings of much merit from almost every class of our educated people, comprising new, original, attractive and patri- otic recitations' and declamations for every grade of pupils. This book will displace the old and hack- neyed pieces in our school rooms by supplying orig- inal and the brightest and best thoughts and speeches of our own people. Send for a copy at once. Address, J.E.S^e Qa^ville, Jrjdiarja. THF XORJI.VI. DIALOGUE BOOK. PRICE, FIFTY CENTS. Arranged with a Tien to giving something highly entertaining, and at the same time something suit able and practicable for the school exhibition. CONTENTS OF THE NORMAL DIALOGUE BOOK. Miscellaneous. Another Arrangement. Aunt Betsy's Beau. Which Will You Choose? Maud's Command ; or, Yielding to Temptation. The Smoke Fiend. Charade. Phan-Tom. Fourth of July Oration. A Woman's Business Meeting. Boarding School Accomplishments. Leap Year in the Village with One Gentleman. Too Greedy by Half. The Way to Wyndham. Little Folks' Department. Vacation Fun. The Old Maid. The Secret. Little Mischief. An Interrupted Recitation. A Stitch in Time Saves Nine. How He Had Him. The Stolen Pets. Tableaux. Tableau 1. Little Jack Homer. " 2. The Old Woman who Lived in a Shoe. 3 ) 4 f 5. When I was a Bachelor. " 6. Cinderella. ?. Mischief in School. " 8. The Four Seasons. " 9. The Witches in Macbeth. Shadow Acts and Pantomimes. Box and Cox. Courting Under Difficulties. Hospital Practice. Chapter of Instructions. Great Inducements to Agents. Liberal Terms to the Trade. Address, J. G, SffEJWfcfc, Pub,, DANVILJUE, IND. UNIVERSITY OF CALIFORNIA AT LOS ANGELES THE UNIVERSITY LIBRARY This book is DUE on the last date stamped below AUG 1 Form L-9-15m-3,'34 UNIVERSITY of CALIFORNIA AT LOS ANGELES 545 Bagot - U4m A manual of plane ing. i mi ii 3 1158 00849 991