yx $kc~~~~~~C 0 A"3antred ~ 0N~ ___~~~~~~~~~~~~~~-........._ St. ~ ~ I;.i H ~~~~~~~~~~/~~~~E'V -~~~~~~~~~~~~7~~~~<~~~~~N~"Ij Y-A-3at, IS~~~~~~~~~~~~~~~~~ s UITED STAE~ WE AMTER. JOUR. SCIENCE. Oct. 1840. TOPOGR.&PHICAL CONVENTPIONAL S1GN-S (To accompanyHELLER & BRIGHTLYS1"'Remarls on Mninoer's Survoyinig hishrments."1) COMP1LED BY LEWVIS M. HAUPT, PROFESSOR OF CIVIL ENGINMEERING, VNIVERSITY OF PENNVSYLVANIA. ss~~~~ — 9;: 1;.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~W l~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,Af Lattd.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ WI.r e~ttin fl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~"Z. 4'_ \oii e ad geaton. e.-ad SA Veast Fraton.l Marsh.t Grss n.a~e Cors~~~~~~urovv }lzora I I I I I t | *e c; (cJ } |~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Q. Q i~~~~~~~~~~ie II/ _R; |_.}i. Fl l || 06 tgWAI pas S mi |1 arksDimskShlls,|A LakePverondS|ea.t 1 Ozirer Bes Mangrove | C1C^vre;Atztoare l I l7rze| fLai-id.srl I fiLa Waer Vgttion.oo o oo I. I |t Lantl 8:lfater Communicatiorls J I II't])1OVOIien[SW I L, FcncesBoundaries &c. Cok e oe d:~~~~~~~~~~~~~~~~~~~~~~~: ~~~fijr ~~~~~~~~~~~f ~ ~ Y Ss n~~~~~~~~~~~~~~~0~/ ~n r\~~~~ I~~~h~CIe: a:10 CI -— QA oh d t~~~~~~O ~~~~I~~~~~, a~~~~~~~~rz ~~~~~~~~~~~s~~~~~~I~~~~~-1~~~~~~~-1-7A6C6 11-61::::~~~;:" ~O;A, J:7- 1-, ~~~~~~~~~~ 1 x Brusl-lwoocl. Orcl-tard. Palmetto. Rice. ff op S. Tobacco.~1~2~;1 TREATISE ON LAND-SURVEYING: COMPRISING THE THEORY DEVELOPED FROM FIVE ELEMENTARY PRINCIPLES; AND THE PRACTICE WITH THE CHAIN ALONE, THE COMPASS, THE TRANSIT, THE THEODOLITE, THE PLANE TABLE, &c. ILLUSTRATED BY FOUR HUNDRED ENGRAVINGS, AND A MAGNETIC CHART. BY W. M. GILLESPIE, A. M., Civ. ENG., PROFESSOR OF CIVIL ENGINEERING IN UNION COLLEGE. AUTHOR OF " A MANUAL OF ROAD-MAKING," ETC. NEW-YORK: D. APPLETON & CO. 346 & 348 BROADWAY, AND 16 LITTLE BRITAIN, LONDON. M DCCC LV. Entered, according to the Act of Congress, in the year 1855, by WILLIAM MITCHELL GILLESPIE, In the Clerk's Office of the District Court of the United States for the Southern District of New-York. PREFACE. LAND-SURVEYING is perhaps the oldestof the mathematical arts. Indeed, Geometry itself, as its name —" Land-measuring "-implies, is said to have arisen from the efforts of the Egyptian sages to recover and to fix the land-marks annually swept away by the inundations of the Nile. The art is also one of the most important at the present day, as determining the title to land, the foundation of the whole wealth of the world. It is besides one of the most useful as a study, from its striking exemplifications of the practical bearings of abstract mathematics. But, strangely enough, Surveying has never yet been reduced to a systematic and symmetric whole. To effect this, by basing the art on a few simple principles, and tracing them out into their complicated ramifications and varied applications (which extend from the measurement of " a mowng lot " to that of the Heavens), has been the earnest endeavor of thepresent writer. The work, in its inception, grew out of the author's own needs. Teaching Surveying, as preliminary to a course of Civil Engineering, he found none of the books in use (though very excellent in many respects) suited to his purpose. He was therefore compelled to teach the subject by a combination of familiar lectures on its principles and exemplifications of its practice. His notes continually swelling in bulk, gradually became systematized in nearly their present form, and in 1851 he printed a synopsis of them for the use of his classes. His system has thus been fully tested, and the present volume is the result. IV LAND-SURVEYING. A double object has been kept in view in its preparation; viz. to produce a very plain introduction to the subject, easy to be mastered by the young scholar or the practical man of little previous acquirement, the only pre-requisites being arithmetic and a little geometry; and at the same time to make the instruction of such a character as. to lay a foundation broad enough and deep enough for the most complete superstructure which the professional student may subsequently wish to raise upon it. For the convenience of those wishing to make a hasty examination of the book, a summary of some of its leading points and most peculiar features will here be given I. All the operations of Surveying are deduced from only five simple principles. These principles are enunciated and illustrated in Chapter 1, of Part I. They will be at once recognized by the Geometer as familiar systems of " Co-ordinates;" but they were not here arbitrarily assumed in advance. They were arrived at most practically by analyzing all the numerous and incongruous methods and contrivances employed in Surveying, and rejecting, one after another, all extraneous and non-essential portions, thus reducing down the operations, one by one and step by step, to more and more general and comprehensive laws, till at last, by continual elimination, they were unexpectedly resolved into these few and simple principles; upon which it is here attempted to build up a symmetrical system. II. The three operations common to all kinds of Land-surveying, viz. Making the Measurements, Drawing the Maps, and Calculating the Contents, are fully examined in advance, in Part I, Chapters 2, 3, 4, so that when the various methods of Surveying are subsequently taken up, only the few new points which are peculiar to each, require to be explained. Each kind of Surveying, founded on one of the five fundamental principles, is then explained in its turn, in the successive Parts, and each carefully kept distinct from the rest. Preface. v III. A complete system of Surveying with only a chain, a rope, or any substitute, (invaluable to farmers having no other instruments,) is very fully developed in Part II. IV. The various Problems in Chapter 5, of Part II, will be found to constitute a course of practical Geometry on the ground. As some of their demonstrations involve the "Theory of Transversals, etc," (a beautiful supplement to the ordinary Geometry), a carefully digested summary of its principal Theorems is here given, for the first time in English. It will be found in Appendix B. V. In Compass Surveying, Part III, the Field work, in Chapter 3, is adapted to our American practice; some new modes of platting bearings are given in Chapter 4, and in Chapter 6, the rectangular method of calculating contents is much simplified. VI. The effects of the continual change in the Variation of the magnetic needle upon the surveys of old lines, the difficulties caused by it, and the means of remedying them, are treated of with great minuteness of practical detail. A new table has been calculated for the time of "greatest Azimuth," those in common use being the same as the one prepared by Gummere in 1814, and consequently greatly in error now from the change of place of the North Star since that date. VII. In Part IV, in Chapter 1, the Transit and Theodolite are,explained in every point'; in Chapter 2, all forms of Verniers are shewn by numerous engravings; and in Chapter 3, the Adjustments are elucidated by some novel modes of illustration. VIII. In Part VII, will be found all the best methods of overcoming obstacles to sight and to measurement in angular Surveying. IX. Part XI contains a very complete and systematic collection of the principal problems in the Division of Land. X. The Methods of Surveying the Public Lands of the United States, of marking lines and corners, &c., are given in Part XII, from official documents, with great minuteness; since the subject interests so many land-owners residing in the Eastern as well as in the Western States. VI LAND-SURVEYING. XI. The Tables comprise a Traverse Table, computed for this volume, and giving increased accuracy in one-fifteenth of the usual space; a Table of Chords, appearing for the first time in English, and supplying the most accurate method of platting angles; and a Table of natural Sines and Tangents. It was thought best not to increase the size of the volume with Logarithmic Tables, not absolutely necessary for its purposes, and to be found in all Trigonometries. The tables are printed on tinted paper, on the eye-saving principle of Babbage. XII. The great number of engraved illustrations, most of them original, is a peculiar feature of this volume, suggested by the experience of the author that one diagram is worth a page of print in giving clearness and definiteness to the otherwise vague conceptions of a student. XIII. The practical details, and hints to the young Surveyor, have been made exceedingly full by a thorough examination of more than fifty works on the subject, by English, French and German writers, so as to make it certain that nothing which could be useful had been overlooked. It would be impossible to credit each item (though this has been most scrupulously done in the few cases in which an American writer has been referred to), but the principal names are these: Adams, Ainslie, Baker, Begat, Belcher, Bourgeois, Bourns, Brees, Bruff, Burr, Castle, Francoeur, Frome, Galbraith, Gibson, Guy, Hogard, Jackson, Lamotte, Lefevre, Mascheroni, Narrien, Nesbitt, Pearson, Puille, Puissant, Regnault, Richard, Serret, Simms, Stevenson, Weisbach, Williams. Should any important error, either of printer or author, be discovered (as is very possible in a work of so much detail, despite the great care used) the writer would be much obliged by its prompt communication The present volume will be followed by another on LEVELLING AND HIGHER SURVEYING: embracing Levelling (with Spirit-Level, Theodolite, Barometer, etc.); its applications in Topography or Hill-drawing, in Mining Surveys, etc.; the Sextant, and other reflecting instruments; Maritime Surveying' and Geodesy, with its practical Astronomy. GENERAL DIVISION OF THE SUBJECT. [A full Analytical Table of Contents is given at the end of the volume.] Page. PART I. GENERAL PRINCIPLES AND OPERATIONS Page. CHAPT. 1. DEFINITIONS AND METHODS........................... 9 CHAPT. 2. MAKING THE MEASUREMENTS.......................... 16 CHAPT. 3. DRAWING THE MAP................................ 25 CHAPT. 4. CALCULATING THE CONTENT.......................... 38 PART II. IIHAIN SURVEYING. CHAPT. 1. SURVEYING BY DIAGONALS........................... 57 CHAPT. 2. SURVEYING BY TIE-LINES...................... 66 CHAPT. 3. SURVEYING BY PERPENDICULARS..................... 68 CHAPT. 4. SURVEYING BY THESE METHODS COMBINED............. 8'2 CHAPT. 5. OBSTACLES TO MEASUREMIENT.......................... 96 PART III, COMPASS SURVEYING, CHAPT. 1. ANGULAR SURVEYING IN GENERAL................... 122 CHAPT. 2. THE COMIPASS...................................... 127 CHAPT. 3. THE FIELD-WORK.................................. 138 CHAPT. 4. PLATTING THE SURVEY....157............ 157 CHArT. 5. LATITUDES AND DEPARTURES........................ 169 CHAPT. 6. CALCULATING THE CONTENT.......................... 180 CHAPT. 7. MAGNETIC VARIATION............................... 189 CHAPT. 8. CHANGES IN THE VARIATION........................ 203. PART IV. TRANSIT AND TIIEODOLITE SURVEYING, CIIAPT. 1. THE INSTRUMENTS................................... 211 CHAPT. 2. VERNIERS.......................................... 228 CHAPT. 3. ADJUSTMENTS...................................... 240 CHAPT. 4. THIE FIELD-WORK................................... 2.0 PART V. TRIANGULAR SURVEYING.................. 260o PART VI. TRILINEAR SURVEYING.................... 275 PART VII. OBSTACLES IN ANGULAR SURVEYING. CIIAPT. 1. PERPENDICULARS AND PARALLELS.................... i279 CHAPT. 2. OBSTACLES TO ALINEMENT........................... 282 CIIAPT. 3. OBSTACLES TO MIEASUREMENT......................... 287 CHAPT. 4. SUPPLYING OIISSIONS................................ 297 PART VIII, PLANE-TABLE SURVEYING............... 30o PART IX. SURVEYING WITHIOUT INSTRUMENTS....... 311 PART. X. IAPPING, CITAPT. 1. COPYING PLATS................................... 316; CHAPT. 2. CONVENTIONAL SIGNS.............................. 322 CHAPT. 3. FINISHING THE MAP............................. 328 PART Xi. LAYINt OUT ATD DIVIDIXG U[? LA1D), CHAPT. 1. LAYING OUT LAND.................................. 330 CHAPT. 2. PARTING OFF LAND.................................. 334 CHAPT. 3. DIVIDING ur LAND................................. 347 PART XII. UNITED STATES' PUBLIC LANDS........... 36 APPENDIX A-SYNOPSIS OF PLANE TRIGONOMETRY................... 379 APPENDIX B-DEMONSTRATIONS OF PROBLEMIS; &C.................'.3 87 APPENDIX C-INTRODUCTION TO LEVELLING........................ 409 ANALYTICAL TABLE OF CONTENTS........................... 415 TO TEA CHERS AND STUDENTS. As it is desirable to obtain, at the earliest possible period, a sufficient knowledge of the general principles of Surveying to commence its practice, the Student at his first reading may omit the portions indicated below, and take them up subsequently in connection with his review of his studies. T2ze same omissions may be made by Teachers whose classes have only a short time for this study. In PART I, omit onlyg Articles (46), (47), (48), (51), (72), (84), (85). In PART II, omit, in Chapter IV, (127), (128), (129), (130); and ils Chapter V, learn at first tnder each Problem, only one or two of the simpler methods. In PART III, omit only (225), (226), 232), (233), (244), (251), (280), (297), (322). Then pass over PART IV; and in PART V, take only (379), (380); and (391) to (395). Then pass over PART VI; and go to PART VII, (if the student has studied Trigonometry), and omit (423); (434) to (438); and all of Chapter IV, ezxcept (439) and (440), PART VIII may be passed over; and PARTS IX and X may be taken in fill. In PART XI, take all of Chapter I; and in Chapters II and Il, take only the sitmpt! ra(:t structions, not omitting, however, (517), (518) and (538). In PART XII, take (560), (561), (565), (566). Appendiz C, on LEVELLING, may conclude this abridged course. LAND-SURVEYING. PART I. GENERAL PRINCIPLES AND FUNDAMENTAL OPERATIONS. CHAPTER I. DEFINITIONS AND METHODS. (1) SURVEYING is the art of making such measurements as will determine the relative positions of any points on the surface of the earth; so that a Map of any portion of that surface may be drawn, and its Content calculated. (2) The position of a point is said to be determined, when it is known how far that point is from one or more given points, and in what direction there-from; or how far it is in front of them or behind them, and how far to their right or to their left, &c; so that the place of the first point, if lost, could be again found by repeating these measurements in the contrary direction. The " points" which are to be determined in Surveying, are not the mathematical points treated of in Geometry; but the corners of fences, boundary stones, trees, and the like, which are mere points in. comparison with the extensive surfaces and areas which they are the means of determining. In strictness, their centres should be regarded as the points alluded to. 10 GENERAL PRINCIPLES. [PART I. (3) A straight Line is "determined," that is, has its length and its position known and fixed, when the points at its extremities are determined; and a plane Surface has its form and dimensions determined, when the lines which bound it are determined. Consequently, the determination of the relative positions of points is all that is necessary for the principal objects of Surveying; which are to make a map of any surface, such as a field, farm, state, &c., and to calculate its content in square feet, acres, or square miles. The former is an application of Drafting, the latter of Mensuration. (4) The position of a point may be determined by a variety of methods. Those most frequently employed in Surveying, are the following: (5) First Method. By measuring the distances from the required point to two given points. Thus, in Fig. 1, the point S is " determined," if it is known to be one inch from A, and half an inch from B: for,... its place, if lost, could be found by de- A' - scribing two arcs of circles, from A and B as centres, and with the given distances as radii. The required point would be at the intersection of these arcs. In applying this principle in surveying, S may represent any station, such as a corner of a field, an angle of a fence, a tree, a house, &c. If then one corner of a field be 100 feet from a second corner, and 50 feet from a third, the place of the first corner is known and determined with reference to the other two. There will be two points fulfilling this condition, one on each side of the given line, but it will always be known which of them is the one desired. In Geography, this principle is employed to indicate the position of a town; as when we say that Buffalo is distant (in a straight line) 295 miles from New-York, and 390 from Cincinnati, and thus convey to a stranger acquainted with only the last two places a correct idea of the position of the first. CHAP I.] Definitions and Methods. 11 In Analytical Geometry, the lines AS and BS are known as "Focal Co-ordinates;" the general name "co-ordinates" being applied to the lines or angles which determine the position of a point. (6) Second Method. By measuring the perpendicular distance front the required point to a given line, and the distance thence along the line to a given point. Thus, in Fig. 2, if the perpendicular dis- Fig. 2. tance SC be half an inch, and CA be one inch, the point S is " determined": for, its place could be again found by measuring one inch from A to C, and half an inch from C, A. at right angles to AC, which would fix the point S. The Public Lands of the United States are laid out by this method, as will be explained in Part XII. In Gleography, this principle is employed under the name of Latitude and Longitude. Thus, Philadelphia is one degree and fifty-two minutes of longitude east of Washington, and one degree and three minutes of latitude north of it. In Analytical Geometry, the lines AC and CS are known as " Rectangular Co-ordinates." The point is there regarded as determined by the intersection of two lines, drawn parallel to two fixed lines, or'" Axes," and at a given distance from them. These Axes, in the present figure, would be the line AC, and another line,perpendicular to it and passing through A, as the origin. (7) Third Method. By measuring the angle between a given line and a line drawn from any given point of it to the required point; and also the length of this latter line. Thus, in Fig. 3, if we know the angle Fig. 3. BAS to be half a light angle, and the line s? AS to be one inch, the point S is determined; for, its place could be found by drawing from A, a line making the given angle with A. AB, and measuring on it the given distance. 12 GENERAL PRINCIPLES. [PART I. In applying this principle in surveying, S, as before, may represent any station, and the line AB may be a fence, or any other real or imaginary line. In " Compass Surveying," it is a north and south line, the direction of which is given by the magnetic needle of the compass. In Geography, this principle is employed to determine the relative positions of'places, by " Bearings and distances"; as when we say that San Francisco is 1750 miles nearly due west from St. Louis; the word " west" indicating the direction, or angle which the line joining the two places makes with a north and south line, and the number of miles giving the length of that line. In Analytical Geometry, the line AS, and the angle BAS, are called " Polar Co-ordinates." (8) Fourth Mlethod. By measutring the angles made with a given line by two other lines starting from given points upon it, and passing through the required point. Thus, in Fig. 4, the point S is deter- Fig. 4. mined by being in the intersection of the two lines AS and BS, which make respectively angles of a half and of a third of a right angle with the line AB, which Aa:-s is one inch long; for, the place of the point could be found, if lost, by drawing from A and B lines making with AB the known angles. In Geography, we might thus fix the position of St. Louis, by saying it lay nearly due north from New-Orleans, and due west from Washington. In Analytical Geometry, these two angles would be called "Angular Co-ordinates." (9) In Fig. 5, are shown together all Fig. 5. the measurements necessary for determin- S mg the same point S, by each of the four preceding methods. In the First Method, we measure the distances AS and - BS; in the Second Method, the distances AC and CS, the latter at right angles to the former; in the Third Method, the distance CHAP. 1.1 Definitions and Methods. 13 AS, and the angle SAB; and in the Fourth Method, the angles SAB and SBA. In all these methods the point is really determined by the intersection of two lines, either straight lines or arcs of circles. Thus, in the First Method, it is determined by the intersection of two circles; in the Second, by the intersection of two straight lines; in the Third, by the intersection of a straight line and a circle; and in the Fourth, by the intersection of two straight lines. (10) Fifth Method. By measuring the angles made with each other by three lines of sight passing from the required point to three points whose positions are known. Thus, in Fig. 6, the point S is deter- Fig. 6. mined by the angles ASB, and BSC H-i C. made by the three lines SA, SB and = SC. —! Geographically, the position of Chicago would be determined by three —,.' straight lines passing from it to Washington, Cincinnati, and Mobile, and mak- 8 ing known angles with each other; that of the first and second lines being about one-third, and that of the second and third lines, about one-half of a right angle. From the three lines employed, this may be named the Method of Trilinear co-ordinates. (11) The position of a point is sometimes determined by the intersection of two lines, which are themselves determined by their extremities being given. Thus, in Fig. 7, Fig. 7. the point S is determined by its being sit-,'r uated in the intersection of AB and CD This method is sometimes employed to fix the position of a Station on a Rail-Road c. line, &c., when it occurs in a place where a stake cannot be driven, such as in a pond; and in a few other cases; but is not used frequently enough to require that it should be called a sixth principle of Surveying. It is said to be employed 14 GENERAL PRINCIPLES. [PART I. by smugglers to fix the spot where they have sunk contraband goods. (12) These five methods of determining the positions of points, produce five corresponding systems of Surveying, which may be named as follows: I. DIAGONAL SURVEYING. II. PERPENDICULAR SURVEYING. III. POLAR SURVEYING IV. TRIANGULAR SURVEYING. V. TRILINEAR SURVEYING. (13) The above division of Surveying has been made in harmony with the principles involved and the methods employed. The subject is, however, sometimes divided with reference to the instruments employed; as the chain, either alone or with crossstaff; the compass; the transit or theodolite; the sextant; the plane table, &c. (14) Surveying may also be divided according to its objects. In Land Surveying, the content, in acres, &c., of the tract surveyed, is usually the principal object of the survey. A map, showing the shape of the property, may also be required. Certain signs on it may indicate the different kinds of culture, &c. This land may also be required to be divided up in certain proportions; and the lines of division may also be required to be set out on the ground. One or all of these objects may be demanded in Land Surveying. In Topographical Surveying, the measurement and graphical representation of the inequalities of the ground, or its " relief," i. e. its hills and hollows, as determined by the art of " Levelling," is the leading object. In ]Maritime or Hydrographical Surveying, the positions of rocks, shoals and channels are the chief subjects of examination. In Mining Surveying, the directions and dimensions of the subterranean passages of mines are to be determined. CHAP. I.] Definitions and Methods. 15 (15) Surveying may also be divided according to the extent of the district surveyed, into Plane and Geodesic. Geodesy takes into account the curvature of the earth, and employs Spherical Trigonometry. Plane Surveying disregards this curvature, as a needless refinement except in very extensive surveys, such as those of a State, and considers the surface of the earth as plane, which may safely be done in surveys of moderate extent. (16) Land Surveying is the principal subject of this volume; the surface surveyed being regarded as plane; and each of the five Methods being in turn employed. For the purposes of instruction, the subject will be best divided, partly with reference to the Methods employed, and partly to the Instruments used. Accordingly, the First and Second Methods (Diagonal and Perpendicular Surveying) will be treated of under the title "' Chain Surveying," in Part II. The Third Method (Polar Surveying) will be explained under the titles " Compass Surveying," Part III, and " Transit and Theodolite Surveying," Part IV. The _Fourth and Fifth Methods will be found under their own names of " Triangular Surveying," and " Trilinear Surveying," in Parts V and VI. (17) In all the methods of Land Surveying, there are three stages of operation: 10 Measuring certain lines and angles, and recording them; 20 Drawing them on paper to some suitable scale; 3~0 Calculating the content of the surface surveyed. The three following chapters will treat of each of these topics in their turn. 16 FUNDAMENTAL OPERATIONS. [PART I. CHAPTER II. MAKING THE MEASUREMENTS. (18) THE Measurements which are required in Surveying, may be of lines or of angles, or of both; according to the Method employed. Each will be successively considered. MEASURING STRAIGHT LINES. (19) The lines, or distances, which are to be measured, may be either actual or visual. Actual lines are such as really exist on the surface of the land to be surveyed, either bounding it, or crossing it; such as fences, ditches, roads, streams, &c. Visual lines are imaginary lines of sight, either temporarily measured on the ground, such as those joining opposite corners of a field; or simply indicated by stakes at their extremities or otherwise. If long, they are "ranged out" by methods hereafter to be described. Lines are usually measured with chains, tapes or rods, divided into yards, feet, links, or some other unit of measurement. (20) Gunter's Chain. This is the measure most commonly used in Land surveying. It is 66 feet, or 4 rods long.* Eighty such chains make one mile. Fig. 8. It is composed of one hundred pieces of iron wire, or links, each bent at the end into a ring, and connected with the ring at the end of the next piece by another rifg. Sometimes two or three rings are placed between the links. The chain is then less liable to ~ This length was chosel (by Mr. Edward Gunter) because 10 square chains of 66 feet make one acre, (as will be shown in Chapter IV,) and the computation of areas is thus greatly facilitated. For other Surveying purposes, particularly for Rail-road work, a chain of 100 feet is preferable. On the Uliiterd States Coast Survey, the unit of measurement (which at some future time will be the universal one) is the French Metre, equal to 3.281 feet, nearly. CHAP. II.] Making the Measurements. 17 twist and get entangled, or " kinked." Two or more swivels are also inserted in the chain, so that it may turn around without twisting. Every tenth link is marked by a piece of brass, having one, two, three, or four points, corresponding to the number of tens which it marks, counting from the nearest end of the chain.' The middle or fiftieth link is marked by a round piece of brass. The hundredth part of a chain is called a link.t The great advantage of this is, that since links are decimal parts of a chain, they may be so written down, 5 chains and 43 links being 5.43 chains, and all the calculations respecting chains and links can then be performed by the common rules of decimal Arithmetic. Each link is 7.92 inches long, being = 66 x 12 ~- 100. The following Table will be found convenient: CHAINS INTO FEET. FEET INTO LINKS. CHAINS. FEET. CHAINS. FEET. FEET. LINKS. FEET. LINKS. 0.01 0.66 1.00 66. 0.10 0.15 10. 15.2 0.02 1.32 2. 132. 0.20 0.30 15. 22.7 0.03 1.98 3. 198. 0.25 0.38 20. 30.3 0.04 2.64 4. 264. 0.30 0.45 25. 37.9 0.05 3.30 5. 330. 0.40 0.60 30. 45.4 0.06 3.96 6. 396. 0.50 0.76 33. 50.0 0.07 4.62 7. 462. 0.60 0.91 35. 53.0 0.08 5.28 8. 528. 0.70 1.06 40. 60.6 0.09 5.94 9. 594. 0.75 1.13 45. 68.2 0.10 6.60 10. 660. 0.80 1.21 50. 75.8 0.90 1.36 55. 83.3 0.20 13.20 20. 1320. 1.00 1.52 60. 90.9 0.30 19.80 30. 1980. 2. 3.0 65. 98.5 0.40 26.40 40. 2640. 3. 4.5 70. 106.1 0.50 33.00 50. 3300. 4. 6.1 75. 113.6 0.60 39.60 60. 3960. 5. 7.6 80. 121.2 0.70 46.20 70. 4620. 6. 9.1 85. 128.8 0.80 52.80 80. 5280. 7. 10.6 90. 136.4 0.90 59.40 90. 5940. 8. 12.1 95. 143.9 1.00 66.00 100. 6600. 9. 13.6 100. 151.5 " To prevent the very common mistake, of calling forty, sixty; or thirty, seventy; it has been suggested to make the 11th, 21st, 31st and 41st links of brass; which would at once show on which side of the middle of the chain was the doubtful mark. This would be particularly useful in Mining Surveying. t This must not be confounded with the pieces of wire which have the same name, since one of them is shorter than the " link" used in calculation, by half a ring, or more, according to the way in which the chain is made. 2 18 FUNDAMENTAL OPERATIONS. [PART I. To reduce links to feet, subtract from the number of links as many units as it contains hundreds; multiply the remainder by 2 and divide by 3. To reduce feet to links, add to the given number half of itself, and add one for each hundred (more exactly, for each ninety-nine) in the sum. The chain is liable to be lengthened by its rings being pulled open, and to be shortened by its links being bent. It should therefore be frequently tested by a carefully-measured length of 66 feet, set out by a standard measure, on a flat surface, such as the top of a wall, or on smooth level ground, between two stakes, their centres being marked by small nails. It may be left a little longer than the true length, since it can seldom be stretched so as to be perfectly horizontal and not hang in a curve, or be drawn out in a perfectly straight line.* Distances measured with a perfectly accurate chain will always and unavoidably be recorded as longer than they really are. To ensure the chain being always strained with the same force, a spring, like that of a spring-balance, is sometimes placed between one handle and the rest of the chain. If a line has been measured with an incorrect chain, the true length of the line will be obtained by multiplying the number of chains and links in the measured distance by 100, and dividing by the length of the standard distance, as given by measurement of it with the incorrect chain. The proportion here employed is this: As the length of the standard given by the incorrect chain Is to the true length of the standard, So is the length of the line given by the measurement To the true length. Thus, suppose that a line has been measured with a certain chain, and found by it to be ten chains long, and that the chain is afterwards found to have been so stretched that the standard distance, measured by it, appears to be only 99 links long. The measured line is therefore longer than it had been thought to be, and its true length is obtained by multiplying ten by 100, and dividing by 99.' The chain used by the Government Surveyors of France, which is 10 Metres, or about half a Gnnter's chain in length, is made from one-fifth to two-fifths of an inch longer than the standard. An inaccuracy of one five hundredth of its length (= -1 inches on a Gunter's chain) is the utmost allowed not to vitiate the survey. CHAP. II.] Making the Measurements, 1 (21) Pins. Ten iron pins or " arrows," usually accompany the chain.' They are about a foot long, and are made of stout iron wire, sharpened at one end, and bent into a ring at the other. Pieces of red and white cloth should be tied to their heads, so that they can be easily found in grass, dead leaves, &c. They should be strung on a ring, which has a spring catch to retain them. Their usual form is shown in in Fig. 9. Fig. 9. Fig. 10. Fig. 10 shows another form, made very large, and therefore very heavy near the point, so that when held by the top and dropped, it may fall vertically. The uses of this will be seen presently. (22) On irregular ground, two stout stakes about six feet long are needed to put the forward chainman in line, and to enable whichever of the two is lowest, to raise his end of the chain in a truly vertical line, and to strain the chain straight. A number of long and slender rods are also necessary for ~" ranging out " lines between distant points, in the manner to be explained hereafter; in Part II, Chapter V. (23) How to Chain. Two men are required; a forward chainmain, and a hind chain-man; or leader and follower. The latter takes the handles of the chain in his left hand, and the chain itself in his right hand, and throws it out in the direction in which' it is to be drawn. The former takes a handle of the chain and one pin in his right hand, and the other pins (and the staff, if used,) in his left hand, and draws out the chain. The follower then walks beside it, examining carefully that it is not twisted or bent. He then returns to its hinder end, which he holds at the beginning of the line to be measured, puts his eye exactly over it, and, by the words "' Right," " Left," directs the leader how to put his staff, or the pin which he holds up, " in line," so that it may seem to cover and hide the flag-staff, or other object at the end of the line. * Eleven pins are sometimes used, one being of brass. Nine of iron, with four or eight of brass, may also be employed. Their uses will be explained in the following article. 20 FUNDAMENTAL OPERATIONS. [PART I. The leader all the while keeps the chain tightly stretched, and his end of it touching his staff. Every time he moves the chain, he should straighten it by an undulating shake. When the staff (or pin) is at last put "in line," the follower says "Down." The leader then puts in the single pin precisely at the end of the chain, and replies "Down." The follower then (and never before hearing this signal that the point is fixed) loosens his end of the chain, retaining it in his hand. The leader draws on the chain, making a step to one side of the pin just set, to avoid dragging it out. He should keep his eye steadily on the object ahead, or, in a hollow, should line himself approximately by looking back. The follower should count his steps, so as to know where to look for the pin in high grass, &c. As he approaches the pin, he calls " Halt." On reaching it, he holds the handle of the chain against it, pressing his knee against both to keep the pin firm. He then, with his eye over the pin, " lines" the leader as before. When the " Down" has been again called by the follower, and answered by the leader, the former pulls out the pin with the chain-hand, and carries it in his other hand, and they go on as before.* The operation is repeated till the leader has arrived at the end of the line, or has put down all his pins. When the leader has put down his tenth pin, he draws on the chain its length farther, and after being lined, puts his foot on the handle to keep it firm, and calls "Tally." The follower then drops his end of the chain, goes up to the leader and gives him back all the pins, both counting them to make sure that none have been lost. One pin is then put down at the forward end of the chain, and they go on as before. Some Surveyors cause the leader to call " tally" at the tenth pin, and then exchange pins; but then the follower has only the hole made by the pin, or some other indefinite mark, to measure from. Eleven pins are sometimes preferred, the eleventh being of brass, or otherwise different from the rest, and being used to mark the * When a chain's length would end in a ditch, pool of water, &c. and the chainmen are afraid of wetting their feet, they can measure part of a chain, to the edge of the water, then stretch the chain across it, and then measure another portion of a chain, so that with the former portion, it may make up a full chain. CHAP. II.] Making the Measurements. 21 end of the eleventh c!:;,~ another being substituted for it before the leader goes on. The two chain-men may change duties at each change of pins, if they are of equal skill, but the more careful and intelligent of two laborers should generally be made " follower." When the leader reaches the end of the line, he stops, and holds his end of the chain against it. The follower drops his end and counts the links beyond the last pin, noting carefully on which side of the " fifty" mark it comes. Each pin now held by the follower, including the one in the ground, represents 1 chain; each time "tally" has been called, and the pins exchanged, represents 10 chains, and the links just counted make up the total distance. (24) Tallies. In chaining very long distances, there is danger of miscounting the number of "tallies," or tens. To avoid mistakes, pebbles, &c., may be changed from one pocket into another at each change of pins; or bits of leather on a cord may be slipped from one side to the other; or knots tied on a string; but the best plan is the following. Instead of ten iron pins, use nine iron pins, and four, or eight, or ten pins of brass, or very much longer than the rest. At the end of the tenth chain, the iron pins being exhausted, a brass pin is put down by the leader. The follower then comes up, and returns the nine iron pins, but retains the brass one, with the additional advantage of having this pin to measure from. At the end of the twentieth chain, the same operation is repeated; and so on. When the measurement of the line is completed, each brass pin held by the follower counts ten chains, and each iron pin one, as before. (25) Chaining on Slopes. All the distances employed in Land-surveying must be measured horizontally, or on a level; for reasons to be given in chapter IV. When the ground slopes, it is therefore necessary to make certain allowances or corrections. If the slope be gentle, hold the up-hill end of the chain on the ground, and raise the down-hill end till the chain is level. To ensure the elevated end being exactly over the desired spot, raise it along a staff kept vertical, or drop a pin held by the point with the ring 22 FUNDAMENTAL OPERATIONS. [PART I. downwards, (if you have not the heavy pointed ones shown in Fig. 10), or, which is better, use a plumb-line. A person standing beside the chain, and at a little distance from it, can best tell if it be nearly level. If the hill be so steep that a whole chain cannot be held up level, use only half or quarter of it at a time. Great care is necessary in this operation. To measure down a steep hill, stretch the whole chain in line. Hold the Fig. 11. upper end fast on the ground. Raise up the 20 or 30 link-mark, so that that portion of the chain is level. Drop a plumb-line or pin. Then let the follower come forward and hold down that link on this spot, and the leader hold up another short portion, as before. Chaining down a slope is more accurate than chaining up it, since in the latter case the follower cannot easily place his end of the chain exactly over the pin. (26) A more accurate, though more troublesome, method, is to measure the angle of the slope; and make the proper allowance by calculation, or by a table, previously prepared. The correction being found, the chain may be drawn forward the proper number of links, and the correct distance of the various points to be noted will thus be obtained at once, without any subsequent calculation or reduction. If the survey is made with the Theodolite, the slope of the ground can be measured directly. A " Tangent Scale," for the same purpose, may be formed on the sides of the sights of a Compass. It will be described when that instrument is explained. In the following table, the first column contains the angle which the surface of the ground makes with the horizon; the second column contains its slope, named by the ratio of the perpendicular to the base; and the third, the correction in links for each chain measured on the slope, i. e. the difference between the hypothenuse, which is the distance measured, and the horizontal base, which is the distance desired. CHAP. II.] Making the Measurements. 23 TABLE FOR CHAINING ON SLOPES. ANGLE. SOPE. CORRECTION CORRECTION ANGLE. SLOPE. ILNK.ANGLE. SLOPE. IN LINKS. IN LINKS. 30 1 in 19 0.14 130 1 in 41 2.56 40 1 in 14 0.24 14~ 1 in 4 2.97 50 1 in 11 0.38 15~ 1 in 4 3.41 6~ 1 in 91 0.55 16~ 1 in 33[.87 70 1 in 8 0.75 170 1 in 3, 4.37 80 1 in 7 0.97 180 1 in 3 4.89 90 1in 61 1.23 193 1 in 3 5.45 100 1 in 6 1.53 20~ 1 in 21 6.03 110 1 in 5} 1.84 250 1 in 2 9.37 12~ 1 in 43 2.19 30 1 tin 1 I 13.40 (27) Chaining is the fundamental operation in all kinds of Surveying. It has for this reason been very minutely detailed. The " follower" is the most responsible person, and the Surveyor will best ensure his accuracy by taking that place himself. If he has to employ inexperienced laborers, he will do well to cause them to measure the distance between any two points, and then remeasure it in the opposite direction. The difference of their two results will impress on them the necessity of great carefulness. To " do up'.' the chain, take the middle of it in the left hand, and with the right hand take hold of the doubled chain just beyond the second link; double up the two links between your hands, and continue to fold up two double links at a time, laying each pair obliquely across the others, so that when it is all folded up, the handles will be on the outside, and the chain will have an hour-glass shape, easy to strap up and to carry. (28) Tape. Though the chain is most usually employed for the principal measurements of Surveying, a tape-line, divided on one side into links, and on the other into feet and inches, is more convenient for some purposes. It should be tested very frequently, particularly after getting wet, and the correct length marked on it at every ten feet. A " Metallic Tape," less liable to stretch, has 24 FUNDAMENTAL OPERATIONS. [PART I. been recently manufactured, in which fine wires form its warp. When the tape is being wound up, it should be passed between two fingers to prevent its twisting in the box, which would make it necessary to unscrew its nut to take it out and untwist it. While in use, it should be made portable by being folded up by arm's lengths, instead of being wound up. (29) Substitutes for a chain or a tape, may be found in leather driving lines, marked off with a carpenter's rule, or in a cord knotted at the length of every link. A well made rope, (such as a "'patent wove line," woven circularly with the strands always straight in the line of the strain), when once well stretched, wetted and allowed to dry with a moderate strain, will not vary from a chain more than one foot in two thousand, if carefully used. (30) Rods. When unusually accurate measurements are required, rods are employed. They may be of well seasoned wood, of glass, of iron, &c. They must be placed in line very carefully end to end; or made to coincide in other ways; as will be explained in Part V, under the title of "Triangular Surveying," in which the peculiarly accurate measurement of one line is required, as all the others are founded upon it. (31) Pacing, Sound, and other approximate means, may be used for measuring the length of a line. They will be discussed, in Part IX, under the title "Surveying without Instruments." (32) A Perambulator, or "Measuring Wheel," is sometimes used for measuring distances, particularly Roads. It consists of a wheel which is made to roll over the ground to be measured, and whose motion is communicated to a series of toothed wheels within the machine. These wheels are so proportioned, that the index wheel registers their revolutions, and records the whole distance passed over. If the diameter of the wheel be 31A inches, the circumference, and therefore each revolution, will be 81 feet, or half a rod. The roughnesses of the road and the slopes necessarily cause the registered distances to exceed the true measure. OalA. III.] Making the Measurements. 25 MEASURING ANGLES. (33) The angle made by any two lines, that is, the difference of their directions, is measured by vaxious instruments, consisting essentially of a circle divided into equal parts, with plain sights, or telescopes, to indicate the directions of the two lines. As the measurement of angles is not required for " Chain Surveying," which is the first Method to be discussed, the considera-r tion of this kind of measurement will be postponed to Part III. NOTING THE MEASUREMENTS. (34) The measurements which have been made, whether of lines, or of angles, require to be very carefully noted and recorded. Clearness and brevity are the points desired. Different methods of notation are required for each of the systems of surveying which are to be explained, and will therefore be given in their appropriate places. CHAPTER III. DRAWING THIE MtP (35) A MAP of a survey represents the lines which bound the surface surveyed, and the objects upon it, such as fences, roads, rivers, houses, woods, hills, &c., in their true relative dimensions and positions. It is a miniature copy of the field, farm, &c., as it would be seen by an eye moving over it; or as it would appear, if from every point of its irregular surface, plumb lines were dropped to a level surface under it, forming what is called in geometrical language, its horizontal projection. (36) Platting. A plat of a survey is a skeleton, or outline map. It is a figure " similar'" to the original, having all its angles equal, and its sides proportional. Every inch on it represents a foot, a yard, a rod, a mile, or some other length, on the ground; 26 FUNDAMENTAL OPERATIONS. [PART I. all the measured distances being diminished in exactly the same ratio. PLATTING is repeating on paper, to a smaller scale, the measurements which have been made on the ground. Its various operations may therefore be reduced, in accordance with the principles established in the Fig. 12. first chapter, to two, viz: drawing a straight line in a given direction and of a given length; and describ- - ing an arc of a circle with a radius whose length is also given. The only instruments absolutely necessary for this, are a straight ruler, and a pair of " dividers," or " compasses." Others, however, are often convenient, and will be now briefly noticed. (37) Straight Lines. These are usually drawn by the aid of a straight-edged ruler. But to obtain a very long straight line upon paper, stretch a fine silk thread between any two distant points, and mark in its line various points, near enough together to be afterwards connected by a common ruler. The thread may also be blackened with burnt cork, and snapped on the paper, as a carpenter snaps his chalk line; but this is liable to inaccuracies, from not raising the line vertically. (38) rcs,. The arcs of circles used in fixing the position of a point on paper, are usually described with compasses, one leg of which carries a pencil point. A convenient substitute is a strip of pasteboard, through one end of which a fine needle is thrust into the given centre, and through a hole in which, at the desired distance, a pencil point is passed, and can thus describe a circle about the centre, the pasteboard keeping it always at the proper distance. A string is a still readier, but less accurate, instrument. (39) Parallels, The readiest mode of drawing parallel lines is by the aid of a triangular piece of wood and a ruler. Let AB CHAP. II.] Drawing the Map. 27 be the line to which a parallel is to Fig. 13. be drawn, and C the point through A which it must pass. Place one side of the triangle against the line, and place the ruler against another side of the triangle. Hold the ruler firm and immovable, and slide the triangle along it till the side of the triangle which had coincided with the given line, passes through the given point. This side will then be parallel to that given line, and a line drawn by it will be the line required. Another easy method of drawing parallels, is by means of a T square, an instrument very valuable for many other purposes. It is nothing but a ruler let into a thicker piece of wood, very truly at right angles to it. For this use of it, one side of the cross-piece must be even, or " flush," with the ruler. To use it, lay it on the paper so that one edge of the Fig. 14. ruler coincides with the given line AB. Place another ruler against,x the cross-piece, hold it firm, and' - - slide the T square along, till its edge passes through the given point C, as shown by the lower part of the figure. Then draw by this edge the desired line parallel to the given line. (40) Perpendiculars. These may be drawn by the various problems given in Geometry, but more readily by a triangle which has one right angle. Place the longest Fig. 15. side of the triangle on the given line, and place a ruler against a second side of the triangle. Hold the ruler fast, and turn the triangle so as to bring its third side against the ruler. Then will the long side be perpendicular to the 28 FUNDAMENTAL OPERATIONS. [PART I. given line. By sliding the triangle along the ruler, it may be used to draw a perpendicular from any point of the line, or from any point to the line. (41) Angles. These are most easily set out with an instrument called a Protractor, usually a semi-circle of brass. But the description of its use, and of the other and more accurate modes of laying off angles, will be postponed till they are needed in Part III, Chapter IV. (42) Drawing to Scale. The operation of drawing on paper lines whose length shall be a half, a quarter, a tenth, or any other fraction, of the lines measured on the ground, is called " Drawing to Scale." To set off on a line any given distance to any required scale, determine the number of chains or links which each division of the scale of equal parts shall represent. Divide the given distance by this number. The quotient will be the number of equal parts to be taken in the dividers and to be set off. For example, suppose the scale of equal parts to be a common carpenter's rule, divided into inches and eighths. Let the given distance be twelve chains, which is to be drawn to a scale of two chains to an inch. Then six inches will be the distance to be set off. If the given distance had been twelve chains and seventy five links, the distance to be set off would have been six inches and three-eighths, since each eighth of an inch represents 25 links. If the desired scale were three chains to an inch, each eighth of an inch would represent 371 links; and the distance of 1275 links would be represented by thirty-four eighths of an inch, or 4; inches. A similar process will give the correct length to be set off for any distance to any scale. If the scale used had been divided into inches and tenths, as is much the most convenient, the above distances would have become on the former scale 636T%7 inches, or nearly 6 40 inches; and on the latter scale 4 2%- inches, coming midway between the 2d and 3d tenth of an inch. CHAP. III.] Drawing the Map, 29 (43) Conversely, to find the real length of a line drawn on paper to any known scale, reverse the preceding operation. Take the length of the line in the dividers, apply it to the scale, and count how many equal parts it includes. Multiply their number by the number of chains or links which each represents, and the product will be the desired length of the line on the ground. This operation and the preceding one are greatly facilitated by the use of the scales to be described in Art. (48) (44) Scales. The choice of the scale to which a plat should be drawn, that is, how many times smaller its lines shall be than those which have been measured on the ground, is determined by several considerations. The chief one is, that it shall be just large enough to express clearly all the details which it is desirable to know. A Farm Survey would require its plat to show every field and building. A State Survey would show only the towns, rivers, and leading roads. The size of the paper at hand will also limit the scale to be adopted. If the content is to be calculated from the plat, that will forbid it to be less than 3 chains to 1 inch. Scales are named in various ways. They should always be expressedfractionally; i. e. they should be so named as to indicate what fractional part of the real line measured on the ground, the representative line drawn on the paper, actually is. When custom requires a different way of naming the scale, both should be given. It would be still better, if the denominator could always be some power of 10, or at least some multiple of 2 and 5, such as 5 0, 1-ooo, 2O606, 2W, &c. For convenience in printing, these may be written thus: 1: 500, 1:1000, 1:2000, 1: 2500, &c. Plats of _Farm Surveys are usually named as being so many chains to an inch. Maps of Surveys of States are generally named as being made to a scale of so many miles to an inch. Maps of Rail-road Surveys are said to be so many feet to an inch, or so many inches to a mile. (45) Farm Surveys, If these are of small extent, two chains to one inch (which is = 1:1584) is convenient. 30 FUNDAMENTAL OPERATIONS. [PART I. A scale of one chain to one inch (1: 792) is useful for plans of build ings. Three chains to one inch (1:2376) is suitable for larger farms. It is the scale prescribed by the English Tithe Commissioners for their first class maps. In France, the Cadastre Surveys are lithographed on a scale about equivalent to this, being 1: 2500. The original plans are drawn to a scale of 1: 5000. Plans for the division of property are made on the former scale. When the district exceeds 3000 acres, the scale is 1:10,000. When it exceeds 7,500 acres, the scale is 1: 20,000. A common scale in France for small surveys is 1:1000; about 1' chains to 1 inch. Fig. 16. ONVE AC]RE ON SCALE OF I CHAIN TO ] INCH[. The choice of the most suitable scale for the plat of a farm survey, may be facilitated by the Figure given above, which shows the actual space occupied by one acre, (the customary unit of land measure), laid out in the form of a square, on maps drawn to the various scales named in the figure. CHAP. III.] Drawing the Map. 31 (46) State Surveys. On these surveys, smaller scales are necessarily employed. On the admirable United States Coast Survey, all the scales are expressed fractionally and decimally. "The surveys are generally platted originally on a scale of one to ten or twenty thousand, but in some instances the scale is larger or smaller. These original surveys are reduced for engraving and publication, and when issued, are embraced in three general classes. 1~, small Harbor charts; 20, charts of Bays, Sounds, and 30, of the Coast General Charts. The scales of the first class vary from 1:10,000 to 1: 60,000, according to the nature of the Harbor and the different objects to be represented. Where there are many shoals, rocks, or other objects, as in Nantucket Harbor and Hell-Gate, or where the importance of the harbor makes it necessary, a larger scale of 1: 5,000, 1:10,000, and 1:20,000 is used. But where, from the size of the harbor, or its ease of access, a smaller one will point out every danger with sufficient exactness, the scales of 1:40,000 and 1:60,000 are used, as in the case of New-Bedford Harbor, Cat and Ship Island Harbor, New-Haven, &c. The scale of the second class, in consequence of the large areas to be represented, is usually fixed at 1: 80,000, as in the case of New-York Bay, Delaware Bay and River. Preliminary charts, however, are issued, of various scales from 1: 80,000 to 1: 200,000. Of the third class, the scale is fixed at 1:400,000, for the General Chart of the Coast from Gay Head to Cape Henlopen, although considerations of the proximity and importance of points on the coast, may change the scales of charts of other portions of our extended coast."` The National Survey of Great Britain is called, from the corps employed on it, the " Ordnance Survey." The " Ordnance Survey" of the southern counties of England was platted on a scale of 2 inches to 1 mile, (1: 31,680), and reduced for publication to that of one inch to a mile, (1: 63,360). The scale of 6 inches to a mile (1:10,560) was adopted for the U Communicated from the U. S. Coast Survey office. 32 FUNDAMENTAL OPERATIONS. [PART I. northern counties of England and for the southern counties of Scotland. The same scale was employed for platting and engraving in outline the "Ordnance Survey" of Ireland. But a map on a scale of 1 inch to 1 mile (1: 63,360) is about to be published, the former scale rendering the maps too unwieldy and cumbrous for consultation. The Ordnance Survey of Scotland was at first platted on a scale of six inches to one mile, (1':10,560). That scale has since been abandoned, and it is now platted on a scale of two inches to 1 mile, (1: 31,680), and the general maps are made to only half that scale. The Ordnance Survey scale for the maps of London and other large towns, is 5 feet to 1 mile, (1:1056), or 1~ chains to one inch. In the " Surveys under the Public Health act" of England, the scale for the general plan is two feet to one mile, (1: 2,640); and for the detailed plan, ten feet per mile, (1: 528), or two-thirds of a chain per inch. The Government Survey of France is platted to a scale of 1:20,000. Copies are made to 1:40,000; and the maps are engraved to a scale of 1: 80,000, or about 3 inch to 1 mile. Cassini's famous map of France was on a scale of 1: 86,400. The French War Department employ the. scales of 1:10,000; 1:20,000; 1:40,000; and 1:80,000; for the topography of France. (47) Rail-road Surveys. For these the New-York General Rail-road Law of 1850 directs the scale of maps which are to be filed in the State Engineer's Office, to be five hundred feet to onetenth of a foot, (- 1: 5000.) For the New-York Canal Maps a scale of 2 chains to 1 inch (1: 1584) is employed. The Parliamentary "standing orders" prescribe the plans of Rail-roads, prepared for Parliamentary purposes, to be made on a scale of not less than 4 inches to the mile, (1: 15840): and the enlarged portions (as of gardens, court-yards, &c.) to be on a scale not smaller than 400 feet to the inch, (1: 4800.) Accordingly the practice of English Railway Engineers is to draw the whole plan to a scale of 6 chains, or 396 feet to the inch, (1: 4752) as being just within the Parliamentary limits. craP. III.] Drawing the Map. 83 In France, the Engineers of " Bridges and Roads"' (Corps des Ponts et Chaussees) employ for the general plan of a road a scale of 1: 5000, and for appropriations 1: 500. (48) In the United States Engineer service, the following scales are prescribed: General plans of buildings, 1 inch to 10 feet, (1; 120). Maps of ground, with horizontal curves one foot apart, 1 inch to 50 feet, (1:600). Topographical maps, one mile and a half square, 2 feet to one mile, (1: 2,640). Do. comprising three.miles square, 1 foot to one mile, (1:5,280). Do. between four and eight miles square, 6 inches to one mile, (1: 10,560). Do. comprising nine miles square, 4 inches to one mile, (1: 15,840). Maps not exceeding 24 miles square, 2 inches to one mile, (1: 31,680). Maps comprising 50 miles square, 1 inch to one mile, (1: 63,360). Maps comprising 100 miles square, J inch to one mile, (1: 126,720.) Surveys of Roads, Canals, &c., 1 inch to 50 feet, (1: 600). (49) The most convenient scales of equal parts are those of boxwood, or ivory, which have a fiducial or feather edge, along which they are divided, so that distances can be at once marked off from this edge, without requiring to be taken off with the dividers; or the length of a given line can be at once read off. Box-wood is preferable to ivory as much less liable to warp, or to very in length with changes in the moisture in the air. The student can, however, make for himself platting scales of drawing paper, or Bristol board. Cut a straight strip of this material, about an inch wide. Draw a line through its middle, and set Fig. 17. off on it a number of equal parts, each representing a chain to the desired scale. Sub-divide the left hand division into ten equal parts, each of which will therefore represent ten links to this scale. Through each point of division on the central line, draw (with the T square) perpendiculars extending to the edges, and the scale is made. It explains itself. The above figure is a scale of 2 chains to I inch. On it the distance 220 links would extend 3 34 FUNDAMENTAL OPERATIONS. [PART I. between the arrow-heads above the line in the figure; 560 links extends between the lower arrow-heads, &c. A paper scale has the great advantage of varying less from a plat which has been made by it, in consequence of changes in the weather, than any other. The mean of many trials showed the difference between such a scale and drawing paper, when exposed alternately to the damp open atmosphere, and to the air of a warm dry room, to be equal to.005, while that between box-wood scales and the paper was.012, or nearly 2- times as much. The difference with ivory would have been even greater. Some of the more usual plotting scales are here given in their actual dimensions. In these five figures, different methods of drawing the scales have been given, but each method may be applied to any scale. The first and second, being the most simple, are generally the best. In the third the subdivisions are made by a diagonal line: the distances between the various pairs of arrow heads, beginning with the uppermost, are, respectively, 310, 540, and 270 links. Fig. 18. Scale of 1 chain to 1 inch. 0 1 Fig. 19. Scale of 2 chains to 1 inch. -:T re':d('l' 1J 2.1- 34 - ~:'"' 41' l 7 ——: 1_ Fig. 20. Scale of 3 chains to 1 inch. 0 1 2 3 4 5 6 7 8 9 In the fourth figure the distances between the arrow heads are respectively 310, 270, and 540 links. Fig. 21. Scale of 4 chains to I inch. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 CHiP. III.] Drawing the Map, 35 In the fifth figure the scale of 5 chains to 1 inch is subdivided diagonally to only every quarter chain, or 25 links. The distance between the upper pair of arrow-heads on it is 121 chains, or 12.25; between the next pair of arrow-heads, it is 6.50; and between the lower pair, 14.75. Fig. 22. Scale of 5 chains to 1 inch. I I o 05 _ 10 A diagonal scale for dividing an inch, or a half-inch, into 100 equal parts, is found on the "Plain scale" in every case of instruments. i:) Vernier Scale. This is an ingenious substitute for the diagonal scale. The one given in the following figure divides an inch into 100 equal parts, and if each inch be supposed to represent a chain, it gives single links. Fig. 23. 33 6554 22 Make a scale of an inch divided into tenths, as in the upper scale of the above figure. Take in the dividers eleven of these divisions, and set off this distance from the 0 of the scale to the left of it. Divide the distance thus set off into 10 equal parts. Each of them will be one tenth of eleven tenths of one inch; i. e. eleven hundredths, or a tenth and a hundredth, and the first division on the short, or vernier scale, will overlap, or be longer than the first division on the long scale, by just one hundredth of an inch; the second division will overlap two hundredths, and so on. The principle will be more fully developed in treating of " Verniers," Part IV, Chapter'I. Now suppose we wish to take off from this scale 275 hundredths of an inch. To get the last figure, we must take five divisions on the lower scale, which will be 55 hundredths, for the reason just given. 220 will remain which are to be taken from the upper 36 FUNDAMENTAL OPERATIONS. [PART I. scale, and the entire number will be obtained at once by extending the dividers between the arrow-heads in the figure from 220 on the upper scale (measuring along its lower side) to 55 on the lower scale, 254 would extend from 210 on the upper scale to 44 on the lower. 318 would extend from 230 on the upper scale to 88 on the lower. Always begin then with subtracting 11 times the last figure from the given number; find the remainders on the upper scale, and the number subtracted on the lower scale. (51) A plat is sometimes made by a nominally reduced scale in the following manner. Suppose that the scale of the plat is to be ten chains to one inch, and that a diagonal scale of inches, divided into tenths and hundredths, is the only one at hand. By dividing all the distances by ten, this scale can then be used without any further reduction. But if the content is measured from the plat to the same scale, in the manner explained in the next chapter, the result must be multiplied by 10 times 10. This is called by old Surveyors " Raising the scale," or'' Restoring true measure." (52) Sectoral Scales. The Sector, (called by the French " Compass of Proportion"), is an instrument sometimes convenient for obtaining a scale of equal parts. It is in two portions, turning on a hinge, like a carpenter's pocket rule. It contains a great number of scales, but the one intended for this use is lettered at its ends L in English instruments, and consists of two lines running from the centre to the ends of the scale, and each divided into ten equal parts, each of which is again subdivided into 10, so that each leg of the scale contains 100 Fig. 24. equal parts. To illustrate its use, suppose that a scale of 7 chains to 1 inch is re- i quired. Take 1 inch in the dividers, and open the sec- I tor till this distance will just reach from the 7 on one leg to the 7 on the other. The sector is then " set" for this CHAP. III.] Drawing the Map. 37 scale, and the angle of its opening must not be again changed. Now let a distance of 580 links be required. Open the dividers till they reach from 58 to 58 on the two legs, as in the dotted line in the figure, and it is the required distance. Again, suppose that a scale of 21 chains to one inch is desired. Open the sector so that 1 inch shall extend from 25 to 25. Any other scale may be obtained in the same manner. Conversely, the length of any known line to any desired scale can thus be readily determined. (53) Whatever scale may be adopted for platting the survey, it should be drawn on the map, both for convenience of reference, and in order that the contraction and expansion, caused by changes in the quantity of moisture in the atmosphere, may affect the scale and the map alike. When the drawing paper has been wet and glued to. a board, and cut off when the map is completed, its contractions have been found by many observations to average from one-fourth to one-half per cent. on a scale of 3 chains to an inch, (1: 2376), which would therefore require an allowance of from one-half perch to one perch per acre. A scale made as directed in Art. (49), if used to make a plat on unstretched paper, and then kept with the plat, will answer nearly the same purpose. Such a scale may be attached to a map, by slipping it through two or three cuts in the lower part of the sheet, and will be a very convenient substitute for a pair of dividers in measuring any distance upon it. (54) Scale omitted. It may be required to find the unknown scale to which a given map has been drawn, its superficial content being known. Assume any convenient scale, measure the lines of the map by it, and find the content by the methods to be given in the next chapter, proceeding as if the assumed scale were the true one. Then make this proportion, founded on the geometrical principle that the areas of similar figures are as the squares of their corresponding sides: As the content found Is to the given content So is the square of the assumed scale To the square of the true scale. 38 FUNDAMENTAL OPERATIONS. [PART I. CHAPTER IV. CALCULATING THE CONTENT. (55) The CONTENT of a piece of ground is its superficial area, or the number of square feet, yards, acres, or miles which it contains. (56) Horizontal Measurement. All ground, however inclined or uneven its surface may be, should be measured horizontally, or as if brought down to a horizontal plane, so that the surface of a hill, thus measured, would give the same content as the level base on which it may be supposed to stand, or as the figure which would be formed on a level surface beneath it by dropping plumb lines from every point of it. This method of procedure is required for both Geometrical and Social reasons. Greometrically, it is plain that this horizontal measurement is absolutely necessary for the purpose of obtaining a correct plat. In Fig. 25, let ABCD, and BCEF, Fig. 25. zA n be two square lots of ground, platted horizontally. Suppose the ground to slope in all directions from the point C, which is the summit of a hill. "' Then the lines BC, DC, measured on D the slope, are longer than if measur- Fi ed on a level, and the field ABCD, of Fig. 25, platted with these long lines, would take the shape ABGD in Fig. 26; and the field BCEF,...of Fig. 25, would become BHEF of Fig. 26. The two adjoining fields would thus overlap each other; and the same difficulty would occur in every case of platting any two adjoining fields by the measurements made on the slope. CHAP. IV.] Calculating the Content. 39 Let us suppose another case, Fig. 27. Fig. 28. more simple than would ever oc- / cur in practice, that of a threesided field, of equal sides and composed of three portions each sloping down uniformly, (at the rate of one to one) from one point in the centre, as in Fig. 27. Each slope being accurately platted, the three could not come together, but would be separated as in Fig. 28. We have here taken the most simple cases, those of uniform slopes. But with the common irregularities of uneven ground, to measure its actual surface would not only be improper, but impossible. In the Social aspect of this-question, the horizontal measurement is justified by the fact that.no more houses can be built on a hill than could be built on its flat base; and that no more trees, corn, or other plants, which shoot up vertically, can grow on it; as is represented by the vertical lines in the Fit. 29. Figure.* Even if a side hill should produce more of certain creeping plants, the {L l LL increased difficulty in their cultivation;,might perhaps balance this. For this reason the surface of the soil thus measured is sometimes called the productive base of the ground. Again, a piece of land containing a hill and a hollow, if measured on the surface would give a larger content than it would after the hollow had been filled up by the hill, while it would yet really be of greater value than before. Horizontal measurement is called the " Method of Cultellation," and Superficial measurement, the " Method of Developement."t An act of the State of New-York prescribes that " The acre, for land measure, shall be measured horizontally." * This question is more than two thousand years old, for Polybius writes, Some even of those who are employed in the administration of states, or placed at the head of armies, imagine that unequal and hilly ground will contain more houses than a surface which is flat and level. This, however, is not the truth. For the houses being raised in a vertical line, form right angles, not with the declivity of the ground, but with the flat surface which lies below, and upon which the hills themselves also stand." t The former from Cultellum, a knife, as if the hills were sliced off; the latter so named because it strips off or unfolds, as it were, the surface. 40 FUNDAMENTAL OPERATIONS. [PART I. (57) Unit of Content. The Acre is the unit of land-measurement. It contains 4 Roods. A Rood contains 40 Perches. A Perch is a square Rod; otherwise called a Perch, or Pole. A Rod is 51 yards, or 161 feet. Hence, 1 acre - 4 Roods - 160 Perches =4,840 square yards = 43,560 square feet. One square mile = 5280 x 5280 feet = 640 acres. Since a chain is 66 feet long, a square chain contains 4356 square' feet; and consequently ten square chains make one acre.* In different parts of England, the acre varies greatly. The statute acre, as in the United States, contains 160 square perches of 16- feet, or 43,560 square feet. The acre of Devonshire and Somersetshire, contains 160 perches of 15 feet, or 36,000 square feet. The acre of Cornwall is 160 perches of 18 feet, or 51,840 square feet. The acre of Lancashire is 160 perches of 21 feet, or 70,560 square feet. The acre of Cheshire and Staffordshire, is 160 perches of 24 feet, or 92,160 square feet. The acre of Wiltshire is 120 perches of 161 feet, or 32,670 square feet. The acre in Scotland consists of 10 square chains, each of 74 feet, and therefore contains 54,760 square feet. The acre in Ireland is the same as the Lancashire. The chain is 84 feet long. The French units of land-measure are the Are = 100 square Metres, = 0.0247 acre, = one fortieth of an acre, nearly; and the Hectare=100 Ares= 2.47 acres, or nearly two and a half. Their old land-measures were the " Arpent of Paris," containing 36,800 square feet; and the "Arpent of Waters and Woods," containing 55,000 square feet. (58) When the content of a piece of land (obtained by any of the methods to be explained presently) is given in square links, as is customary, cut off four figures on the right, (i. e. divide by 10,000), to get it into square chains and decimal parts of a chain; cut off the right hand figure of the square chains, and the remaining figures will be Acres. Multiply the remainder by 4, and the figure, if any, outside of the new decimal point will be Roods. * Let the young student beware of confounding 10 square chains with 10 chains square. The former make one acre; the latter space contains tell acrc's. CHAP. IV.] Calculating the Content. 41 Multiply the remainder by 40, and the outside figures will be Perches. The nearest round number is usually taken for the Perches; fractions less than a half perch being disregarded.* Thus, 86.22 square chains = 8 Acres 2 Roods 20 Perches. Also, 64.1818 do. = 6 A. 1 R. 27 P. " 43.7564 do. -- 4 A. 1 R. 20 P. 71.1055 do. -- 7 A. 0 R. 18 P. " 82.50 do. 8 A. 1 R. 0 P. " 8.250 do. -- A. 3 R. 12 P. C" 0.8250 do. — 0 A. 0 R. 13 P. (59) The following Table gives by mere inspection the Roods and Perches corresponding to the Decimal parts of an Acre. It explains itself. ROODS. ROODS. _ 0 | 1 |2 3 Perches. 0 2 3.000.250.500.750 +- 0.131.381[.631.881 +21.006.256.506.756 + 1.137.387.637.887 +22.012.262.512.762 + 2.1441.394/.644.894 +23.019.269.519.769 + 3.150.400.650.900 +24.025.275.525.775 + 4.156.406.656.906 +25.031.281.531.781 + 5.1621.412/.662.912 +26 s.037.287.537.787 + 6 X.169.4191.669.919 +27.044.294.544.794 + 7 <.175.425.675.925 +28 e.050.300.550.800 + 8.181.431.681.931 +29.056.306 556.806 + 9.187}.437[.687.937 +30 cd.062.312.562.812 +10.194.444.694.944 +31 1~ 9.069.3 819 +1 11 8.2001.450.700.950 +32.075.325.575.825 +12 E.2061.456/.706.956 +33..081.331.581.831 +13.212.462.712.962 +34 a.087.337.587.837 +14 ].219.469.719.969 +35.094.344.594.844 +15.225.475.725.975 +36 l.100.350.600.850 1 +16.231.481.731.981 +37 I.1 06.356.606.856 +17.237.4871.737.987 +38 6.112 362.612.862 +18.244.4941.744.994 +39.119.369.619.8691 +19.250.500.750 1.000 +4011.1251.375.625.875 +20 __l (60) Chain Correction. When a survey has been made, and the plat has been drawn, and the content calculated; and after* To reduce square yards to acres, instead of dividing by 4840, it is easier, and very nearly correct, to multiply by 2, cut off four figures, and add to this product one-third of one-tenth of itself. 42 FUNDAMENTAL OPERATIONS. [PART I. wards the chain is found to have been incorrect, too short or too long, the true content of the land, may be found by this proportion: As the square of the length of the standard given by the incorrect chain Is to the square of the true length of the standard So is the calculated content To the true content. Thus, suppose that the chain used had been so stretched that the standard distance measured by it appears to be only 99 links long; and that a square field had been measured by it, each side containing 10 of these long chains, and that it had been so platted. This plat, and therefore the content calculated from it, will be smaller than it should be, and the correct content will be found by the proportion 992: 1002:: 100 sq. chains:: 102.03 square chains. If the chain had been stretched so as to be 101 true links long, as found by comparing it with a correct chain, the content would be given by this proportion: 1002: 1012:: 100 square chains: 102.01 square chains. In the former case, the elongation of the chain was 1-j- true links; and 1002: (101-)2:: 100 square chains: 102.03 square chains.' (61) Boundary Lines. The lines which are to be considered as bounding the land to be surveyed, are often very uncertain, unless specified by the title deeds. If the boundary be a brook, the middle of it is usually the boundary line. On tide-waters, the land is usually considered to extend to low water mark. Where hedges and ditches are the boundaries of fields, as is almost universally the case in England, the dividing line is generally the top edge of the ditch farthest from the hedge, both hedge and ditch belonging to the field on the hedge side. This varies, however, with the customs of the locality. From three to six feet from the roots of the quickwood of the hedges are allowed for the ditches. CHAP. IV.] Calculating the Content. 43 METHODS OF CALCULATION. (62) The various methods employed in calculating the content of a piece of ground, may be reduced to four, which may be called Arithmetical, Geometrical, Instrumental, and Trigonometrical. (63) FIRST METHOD. —ARITHMETICALLY. PFrom direct measurements of the necessary lines on the ground. The figures to be calculated by this method may be either the shapes of the fields which are measured, or those into which the fields can be divided by measuring various lines across them. The familiar rules of mensuration for the principal figures which occur in practice, will be now briefly enunciated. (64) Rectangles. If the piece of ground be rectangular in shape, its content is found by multiplying its length by its breadth. (65) Triangles. When the given quantities are one side of a triangle and the perpendicular distance to it from the opposite angle; the content of the triangle is equal to half the product of the side and the perpendicular. When the given quantities are the three sides of the triangle; add together the three sides and divide the sum by 2; from this half sum subtract each of the three sides in turn; multiply together the half sum and the three remainders; take the square root of the product; it is the content required. If the sides of the triangle be designated by a, b, c, and their sum by s, this rule will give its area- [1s (Is-a) (As -b) (as c)].* k When two sides of a triangle, and the included Fig. 30. angle are given, its content equals half the product s3 of its sides into the sine of the included angle. Designating the angles of the triangle by the capital letters A,B,C, and the sides opposite them by the corresponding small letters a,b,c, the area =- be sin. A. AC When one side of a triangle and the adjacent angles are given, its content equals the square of the given side multiplied by the sines of each of the given angles, and divided by twice the sine of the sum of sin. B. sin. C these angles. Using the same symbols as before, the area =a2 2 sin. (B + C) When the three angles of a triangle and its altitude are given, its area, referring sin. B to the above figure, = ~ BD2 sin. A. in. C 44 FUNDAMENTAL OPERATIONS. [PART I. (66) Parallelograms; or four-sided figures whose opposite sides are parallel. The content of a Parallelogram equals the product of one of its sides by the perpendicular distance between it and the side parallel to it. (67) Trapezoids; or four-sided figures, two opposite sides of which are parallel. The content of a Trapezoid equals half the product of the sum of the parallel sides by the perpendicular distance between them. If the given quantities are the four sides a, b, c, d, of which b and d are parallel; then, making q = ~ (a + b + c- d), the area b6+d of the trapezoid will b + [q (q — a) (q —c) (q-b + d).] (68) Quadrilaterals, or Trapeziums; four-sided figures, none of whose sides are parallel. A very gross error, often committed as to this figure, is to take the average, or half sum of its opposite sides, and multiply them together for the area: thus, assuming the trapezium to be equiva, lent to a rectangle with these averages for sides. In practical surveying, it is usual to measure a line across it from corner to corner, thus dividing it into two triangles, whose sides are known, and which can therefore be calculated by Art. (65).t * When two parallel sides, b and d, and a third side, a, are given, and also the angle, C, which this third side makes with one of the parallel sides, then the content of the trapezoid= - + d a. sin. C. t When two opposzte sides, and all the angles are given, take one side and its adjacent angles, (or their supplements, when their sum exceeds 180~), consider them as belonging to a triangle, and find its area by the second formula in the note on page 43. Do the same with the other side and its adjacent angles. The difference of the two areas will be the area of the quadrilateral. When three sides and their two included angles are given, multiply together the sine of one given angle and its adjacent sides. Do the same with the sine of the other given angle and its adjacent sides. Multiply together the two opposite sides and the sine of the supplement of the sum of the given angles. Add together the first two products, and add also the last product, if the sum of the given angles is more than 180~, or subtract it if this sum be less, and take half the result. Calling the given sides, p, q, r; and the angle between p and q = A; and the angle between q and r = B; the area of the quadrilateral =- [p. q. sin. A + q. r. sin. B A — p. r. sin. (180~ - A - B)]. When the four sides and the sum of any tawo opposite angles are given, proceed thus: Take half the sum of the four given sides, and from it subtract each side in turn Multiply together the four remainders, and reserve the product. Multiply together the four sides. Take half their product, and multiply it by the cosine of tae given sum of the angles increased by unity. Regard the sign of CHAP. IV.] Calculating the Content. 45 (69) Surfaces bounded by irregularly curved lines. The rules for these will be more appropriately given in connection with the surveys which measure the necessary lines; as explained in Part TI, Chap. III. (70) SECOND METHOD. —GEOMETRICALLY. From m. easurements of the necessary lines upon the plat. (71) Division into Triaigles. The plat of a piece of ground having been drawn from the measurements made by any of the methods which will be hereafter explained, lines may be drawn upon the plat so as to divide it into a number of triangles. Four Fig. 31. Fig. 32. Fig. 33. Fig. 34. ways of doing this are shown in the figures: viz. by drawing lines from one corner to the other corners; from a point in one of the sides to the corners; from a point inside of the figure to the corners; and from various corners to other corners. The last method is usually the best. The lines ought to be drawn so as to make the triangles as nearly equilateral as possible, for the reasons given in Part V. One side of each of these triangles, and the length of the perpendicular let fall upon it, being then measured, as directed in Art. (43,) the content of these triangles can be at once obtained by multiplying their base by their altitude, and dividing by two. The easiest method of getting the length of the perpendicular, without actually drawing it, is, to set one point of the dividers at the angle from which a perpendicular is to be let fall, and to the cosine. M UtlplJy this producttby the reserved product, and take the square root of the resulting:product. It will be the area of the quadrilateral. When the four sides, and the angle of intersection of the diagonals of the quadrilateral are given; square each side; add together the squares of the opposite sides; take the difference of the two sums; multiply it by the tangent of the angle of intersection, and divide by four. The quotient will be the area. When the diagonals of the quadrilateral, and their included angle are given, multiply together the two diagonals and the sine of their included angle, and divide by two. The quotient will be the area. 46 FUNDAMENTAL OPERATIONS. [PART I. open and shut their legs till an arc described by the other point will just touch the opposite side. Otherwise; a platting scale, (described in Art. (49) may be placed so that the zero point of its edge coincides with the angle, and one of its cross lines coincides with the side to which a perpendicular is to be drawn. The length of the perpendicular can then at once be read off. The method of dividing the plat into triangles is the one most commonly employed by surveyors for obtaining the content of a survey, because of the simplicity of the calculations required. Its correctness, however, is dependant on the accuracy of the plat, and on its scale, which should be as large as possible. Three chains to an inch is the smallest scale allowed by the English Tithe Commissioners for plats from which the content is to be determined. In calculating in this way the content of a farm, and also of its separate fields, the sum of the latter ought to equal the former. A difference of one three-hundredth (30) is considered allowable. Some surveyors measure the perpendiculars of the triangles by a scale half of that to which the plat is made. Thus, if the scale of the plat be 2 chains to the inch, the perpendiculars are measured with a scale of one chain to the inch. The product of the base by the perpendicular thus measured, gives the area of the triangle at once, without its requiring to be divided by two. Another way of attaining the same end, with less danger of mistakes, is, to construct a new scale of equal parts, longer than those by which the plat was made in the ratio V2: 1; or 1.414:1. When the base and perpendicular of a triangle are measured by this new scale and then multiplied together, the product will be the content of the triangle, without any division by two. In this method there is the additional advantage of the greater size.and consequent greater distinctness of the scale. When the measurement of a plat is made some time after it has been drawn, the paper will very probably have contracted or expanded so that the scale used will not exactly apply. In that case a correction is necessary. Measure very precisely the present length of some line on the plat of known length originally. Then CHAP. IV.] Calculating the Content. 47 make this proportion: As the square of the present length of this line Is to the square of its original length, So is the content obtained by the present measurement To the true content. (72) Graphical Multiplication. Prepare a strip of drawing paper, of a width exactly equal to two chains on the scale of the plat; i. e. one inch wide, as in the figure, for a scale of two chains to 1 inch; two-thirds of an inch wide for a scale of 3 chains; half an inch for 4 chains; and so on. Draw perpendicular lines across the paper at distances representing one-tenth of a chain on the scale of the triangle to be measured, thus making a platting scale. Apply it to the triangle so that one edge of the scale shall pass through one corner, A, of the triangle, and the other edge through another Fig. 35. Ad C jcorner, ill an nt vr pr cisely 1w d o l!I corner, B; and note very precisely what divisions of the scale are at these points. Then slide the scale in such a way that the points of the scale which had coincided with A and B, shall always remain on the line BA produced, till the edge arrives at the point C. Then will A'C, that is, the distance, or number of divisions on the scale, from the point to which the division A on the scale has arrived, to the third corner of the triangle, express the area of the triangle ABC in square chains.' "For, from C draw a parallel to AB, meeting the edge of the scale in C', and draw C'B. Then the given triangle ABC = ABC'. But the area of this last triangle = AC' multiplied by half the width of the scale, i. e. = AC' X 1 = AC'. But, because of the parallels, A'C = AC'. Therefore the area of the given triangle ABC = A'C; i. e. it is equal in square chains to the number of linear chains read off from the scale. This ingenious operation is due to il. Cousinery. 48 FUNDAMENTAL OPERATIONS. [PART I. (73) Division into Trapezoids. A line may be drawn across the field, as in Fig. 36, and perpen- Fig. 36. diculars drawn to it. The field will thus be divided into trapezoids, (excepting a triangle at each end),. and their content can be calculated by Art. (67). Otherwise; a line may be drawn outside of the figure, and per- Fig.37. pendiculars to it be drawn from each angle. In that case the difference between the trapezoids formed by lines drawn to the outer angles of the figure, and those drawn to the inner angles,: will be the content.: l a This method is very advantageously applied to surveys by the compass; as will be explained in Part III, Chap. VI. (74) Division into Squares. Two sets of parallel lines, at right angles to each other, Fig. 38. one chain apart (to the scale - -/ of the plat) may be drawn - over the plat, so as to divide - it into squares, as in the figure. The number of squares which fall within the plat represent so many square chains; and the triangles and trapezoids which fall outside of these, may then be calculated and added to the entire square chains which have been counted. Instead of drawing the parallel lines on the plat, they may better be drawn on a piece of transparent " tracing paper," which is simply laid upon the plat, and the squares counted as before. The CHAP. IV.] Calculating the Contents 49 same paper will answer for any number of plats drawn to the same scale. This method is a valuable and easy check on the results of other calculations. To calculate the fractional parts, prepare a piece of tracing paper, or horn, by drawing on it one square of the same size as a square of the plat, and subdividing it, by two sets of ten parallels at right angles to each other, into hundredths. This will measure the fractions remaining from the former measurement, as nearly as can be desired. (75) Division into Parallelograms, Draw a series of parallel lines across the plat at equal distances depending on the scale. Thus, for a plat made to a scale of 2 chains to 1 inch, the distance between the parallels should be 21 inches; for a scale of 3 chains to i inch, 1I inch; for a scale of 4 chains to 1 inch,. inch; for a scale of 5 chains to 1 inch, -A inch; and for any scale, make the distance between the parallels that fraction of an inch which would be expressed by 10 divided by the square of the number of chains to the inch. Then apply a common inch scale, divided on the edge into tenths, to these parallels; and every inch in length of the spaces included between each pair of them will be an acre, and every tenth of an inch will be a square chain.' To measure the triangles at the ends of the strips between the parallels, prepare a piece of transparent horn, or stout tracing paper, of a width equal to the width between the parallels, and draw a line through its middle longitudinally. Apply it to the oblique line at the end of the space between Fig. 39. two parallels, and it will bisect the line, and thus reduce the triangl6 to an equivalent nA rectangle, as at A in the figure. When an angle occurs between two parallels, as at B [ ~ A in the figure, the fractional part may be measured by any of the preceding methods. * For, calling the number of chains to the inch, = n, and making the width be10 10 10 tween the parallels inch, this width wili represent -X. = - chains; and as the inch length represents n chains, their product, 10 X n = 10 square chains -1 acre. 4 50 FUNDAMENTAL OPERATIONS. [PART I. A somewhat similar method is much used by some surveyors, particularly in Ireland: the plat being made on a scale of 5 chains to 1 inch, parallel lines being drawn on it, half an inch apart, and the distances along the parallels being measured by a scale, each large division of which is s inch in length. Each division of this scale indicates an acre; for it represents 4 chains, and the distance between the parallels is 2- chains. This scale is called the " Scale of Acres." (76) Addition of Widths. When the lines of the plat are very irregularly curved, as in the Fig. 40. figure, draw across it a num- I ber of equi-distant lines as near together as the case may seem to require. Take a straightedged piece of paper, and apply one edge of it to the middle of the first space, and mark its length from one end; apply the same edge to the middle of the next space, bringing the mark just made to one end, and making another mark at the end of the additional length; so go on, adding the length of each space to the previous ones. When all have been thus measured, the total length, multiplied by the uniform width, will give the content. (77) THIRD.METHOD.-INSTRUMENTALLY. By performing certain instrumental operations on the polat. (78) Reduction of a many sided figure to a single equivalent triangle. Any plane figure bounded by straight lines may be reduced to a single triangle, which shall have the same content. This can be done by any instrument for drawing parallel lines, such as those described in Art. Fig. 41. (39). Let the trapezium, or four sided figure, shown in Fig. 3 41, be required to be reduced to a single equivalent triangle. Produce one side of the figure, as 4 -. Draw a line from / the first to the third angle of 4 1 cHAP. Iv.] Calculating the Content. 51 the figure. From the second angle draw a parallel to the line just drawn, cutting the produced side in a point 1'. From the point 1' draw a line to the third angle. A triangle (1'- 3 -4 in the figure) will thus be formed, which will be equivalent to the original tra,pezium.' The content of this final triangle can then be found by measuring its perpendicular, and taking half the product of this perpendicular by the base, as in the first paragraph of Art. (65). (79) Let the given figure have five sides, as in Fig. 42. For brevity, the angles Fig. 42. of the figure will be / named as numbered in the engraving. / Produce 5-1. 1 Join 1 — 3. From 2 draw a parallel to 7 1-3, cutting the 2 --- 1 produced base in 1'. Join 1' —4. From 3 draw a parallel to it, cutting the base in 2'. Join 2'-4. Then will the triangle 2' —4 —5 be equivalent to the five sided figure 1-2- 3-4-5, for similar reasons to those of the preceding case. (80) Let the given figure be 1 —2-3-4 —5 — 6 - 7-8, as shown in Fig. 43, given at the top of the following page. All the operations are shown by dotted lines, and the finally resulting triangle 5'- 7- 8, is equivalent to the original figure of eight sides. It is best, in choosing the side to be produced, to take one which has a long side adjoining it on the end not produced; so that this long side may form one side of the final triangle, the base of which will therefore be shorter, and will not be cut so acutely by the final line drawn, as to make the point of intersection too indefinite. " For, the triangle 1-2-3 taken away from the original figure is equivalent to the triangle 1'-1-3 added to it; because both these triangles have the same base and,-also the same altitude, since the vertices of both lie in the same line parallel to the base. 52 FUNDAMENTAL OPERATIONS. [PART I. Fig. 43.,o'o 8 I' 2' I 4 3' (81) General Rule. When the given figure has many sides, with angles sometimes salient and sometimes re-entering, the operations of reduction are very liable to errors, if the draftsman attempts to reason out each step. All difficulties, however, will be removed by the following General Rule: 1. Produce one side of the figure, and call it a base. Call one of the angles at the base the first angle, and number the rest in regular succession around the figure. 2. Draw a line from the 1st angle to the 3d angle. Draw a parallel to it from the 2d angle. Call the intersections of this parallel with the base the 1st mark. 3. Draw a line from the 1st mark to the 4th angle. Draw a parallel to it from the 3d angle. Its intersection with the base is the 2d mark. 4. Draw a line from the 2d mark to the 5th angle. Draw a parallel to it from the 4th angle. Its intersection with th.e base is the 3d mark. 5. In general terms, which apply to every step after the first, draw a line from the last mark obtained to the angle whose number is greater by three than the number of the mark. Draw a parallel to it through the angle whose number is greater by two than that of the mark. Its intersection with the base will be a mark whose number is greater by one than that of the preceding mark.* In the concise language of Algebra, draw a line from the nth mark to the u+3 angle. Draw a parallel to it through the n+2 angle, and the intersection with the base will be the n+1 mark. maP. rv.] Calculating the Content. 53 6. Repeat this process for each angle, till you get a mark whose number is such that the angle having a number greater by three is the last angle of the figure, i. e. the angle at the other end of the base. Then join the last mark to the angle which precedes the last angle in the figure, and the triangle thus formed will be the equivalent triangle required. In practice it is unnecessary to actually draw the lines joining the successive angles and marks, but the parallel ruler is merely laid on so as to pass through them, and the points where the parallels cut the base are alone marked. (82) It is generally more convenient, for the reasons given at the end of Art. (80), to reduce Fig. 44. half of the figure on one side and half on the other, as is shown in Fig. 44, which represents the same field as Fig. 42. The equivalent 2, triangle is here 1'-3-2'. When the figure has many angles, they should not be numbered con- a I5, secutively all the way around, but, after the numbers have gone around as far as the angle where it is intended to have the vertex 10 I. Fig. 45. of the final triangle, the numbers should be continued from the 54 FUNDAMENTAL OPERATIONS. [PART I. other angle of the base, as is shown in Fig. 45. In it only the intersections are marked.' (83) It is sometimes more convenient, not to produce one of the sides of the figure, but to draw at one end of it, as at the point 1 in Fig. 46, an indefinite line, usually a perpendicular to a line Fig. 46. joining wo disan angles of he figure, and make his line he base joining two distant angles of the figure, and make this line the base of the equivalent triangle desired. The operation is shown by the dotted lines in the figure. The same General Rule applies to it, as to the previous figures. (84) Special Instruments; A variety of instruments have been invented for the purpose of determining areas rapidly and correctly. One of the simplest is the "' Computing Scale,"' which is on the same principles as the Method of Art. (75). It is represented in Fig. 47, given on the following page. It consists of a scale divided for its whole length from the zero point into divisions, each representing 21 chains to the scale of the plat. The scale carries a slider, which moves along it, and has a wire drawn across its centre at right angles to the edges of the scale. On each side of this wire, a portion of the slider equal in length to one of the primary, or 21 chain, divisions of the scale, is laid off and divided into 40 equal parts. This instrument is used in connection with a sheet of transparent paper, ruled with parallel lines at distances apart each equal to one chain on the scale of the plat. It is plain, that when the * A figure with curved boundaries may be reduced to a triangle in a similar manner. Straight lines must be drawn about the figure, so as to be partly in it and partly out, giving and taking about equal quantities, so that the figure which these lines form, shall be about equivalent to the curved figure. This having been done, as will be further developed in Art. (124), the equivalent straight lined figure is reduced by the above method. CHAP. IV.] Calculating the Content. 55 instrument is laid on this paper, with its edgeon one of the Fig 47. parallel lines, and the slider is moved over one of the divisions of 21 chains, that one rood, or a quarter of an acre, has been measured between two of the parallel lines on the ~ o paper (since 10 square chains make one acre); and that one of the smaller divisions measures one perch between the same parallels. Four of the larger divisions give one acre. The scale is generally made long enough to ] measure at once five acres. To apply this to the plat of a field, or farm, lay the 1 _ transparent paper over it in such a position that two of the ruled lines shall touch two of the exterior points of the boundaries, as Fig. 48. * v T ~ atAandB. Lay.A the scale, with the,' i slide set to zero, on the paper, in a direction parallel to the ruled lines, and so that the wire of the slide cuts the left hand oblique line so as -- to make the spaces c and d about equal. Hold the scale firm, and move the slider till the wire cuts the right hand oblique line in such a way as to equalize the spaces e and f. Without changing the slide, move the scale down the width of a space, and to the left hand end of the next space; begin there again, and proceed as before. So go on, till the whole length of the scale is run out, (five acres having been measured), and then begin at the right hand side and work backwards to the left, reading the lower divisions, which run up to 10 acres. By continuing this process, the content of plats of any size can be obtained. A still simpler substitute for this is a scale similarly divided, but without an attached slide. In place of it there is used a piece of 56 FUNDAMENTAL OPERATIONS. [PART I. horn having a line drawn across it and rivetted to the end of a short scale of box-wood, divided like the former slide. It is used like the former, except that at starting, the zero of the short scale and not the line on the horn is made to coincide with the zero of the long scale. The slide is to be held fast to the instrument when this is moved. The Pediometer is another less simple instrument used for the same object. It measures any quadrilateral directly. (85) Some very complicated instruments for the same object have been devised. One of them, Sang's Planometer, determines the area of any figure, by merely moving a point around the outline of the surface. This causes motion in a train of wheel work, which registers the algebraic sum of the product of ordinates to every point in that, perimeter, by the increment of their abscissas, and therefore measures the included space. Instruments of this kind have been invented in Germany by Ernst, Hansen, and Wetli. A purely mechanical means of determining the area of any surface by means of its weight, may be placed here. The plat is cut out of paper and weighed by a delicate balance. The weight of a rectangular piece of the same paper containing just one acre is also found; and the " Rule of Three" gives the content. A modification of this is to paste a tracing of the plat on thin sheet lead, cut out the lead to the proper lines and weigh it. (87) FOURTH METHOD.-TRIGONOMETRICALLY. By calculating, from the observed angles of the boundaries of the piece of ground, the lengths of the lines needed for calculating the content. This method is employed for surveys made with angular instruments, as the compass, &c., in order to obtain the content of the land surveyed, without the necessity of previously making a plat, thus avoiding both that trouble and the inaccuracy of any calculations founded upon it. It is therefore the most accurate method; but will be more appropriately explained in Part HI, Chapter VI, under the head of " Compass Surveying." PART II. CHAIN-SURVEYING; By the First and Second Methods: OR DIAGONAL AND PERPENDICULAR SURVEYING. (88) The chain alone is abundantly sufficient, without the aid of any other instrument, for making an accurate survey of any surface, whatever its shape or size, particularly in a district tolerably level and clear. Moreover, since a chalin, or some substitute for it, formed of a rope, of leather driving reins, &c., can be obtained by any one in the most secluded place, this method of Surveying deserves more attention than has usually been given to it in this country. It will, therefore, be fully developed in the following chapters. CHAPTER I. SURVEYING BY DIAGONILS: OR By the First Method. (89) Surveying by Diagonals is an application of the First Method of determining the position of a point, given in Art. (5,) to which the student should again refer. Each corner of the field or farm which is to be surveyed is " determined" by measuring its distances from two other points. The field is then "platted" by repeating this process on paper, for each corner, in a contrary order, and the " content" is obtained by some of the methods explained in Chapter IV of Part I. 58 CHIAIN SURVEYING. [PARr II. The lines which are measured in order to determine the corners of the field are usually diagonals of the irregular polygon which is to be surveyed. They therefore divide up this polygon into triangles; whence this mode of surveying is sometimes called " Chain Triangulation." A few examples will make the principle and practice perfectly clear. Each will be seen to require the three operations of mreasur. ing, platting, and calculating. (90) A three-sided field; as Fig. 49. Fig. 49. Field-work. Measure the three sides, AB, BC, and CA. Measure also, as a proof line, the distance from one of the cor- A D B ners, as C, to some point in the opposite side, as D, at which a mark should have been left, when measuring from A to B, at a known distance from A. A stick or twig, with a slit in its top, to receive a piece of paper with the distance from A marked on it, is the most convenient mark. Platting. Choose a suitable scale as directed in Art. (44). Then, by Arts. (42) and (49), draw a line equal in length, on the chosen scale, to one of the sides; AB for example. Take in the compasses the length of another side as AC, to the same scale, and with one leg in A as a centre, describe an arc of a circle. Take the length of the third side BC, and with B as a centre, describe another arc, intersecting the first arc in a point which will be the third corner C. Draw the lines AC and BC; and ABC will be the plat, or miniature copy — as explained in Art. (35)of the field surveyed. Instead of describing two arcs to get the point C, two pairs of compasses may be conveniently used. Open them to the lengths, respectively, of the last two sides. Put one foot of each at the ends of the first side, and bring their other feet together, and their point of meeting will mark the desired third point of the triangle. To " prove " the accuracy of the work, fix the point D, by setting off from A the proper distance, and measure the length of the line CHAP. I.] Surveying by Diagonals, 59 DC, by Art. (43). If its length on the plat corresponds to its measurement on the ground, the work is correct.* Calculation. The content of the field may now be found as directed in Art. (65), either from the three sides, or more easily though not so accurately, by measuring on the plat, by Art. (43), the length of the perpendicular CE, let fall from any angle to the opposite side, and taking half the product of these two lines. Example 1. Figure 49, is the plat, on a scale of two chains to one inch, of a field, of which the side AB is 200 links, BC is 100 links, and AC is 150 links.. Its content by the rule of Art. (65), is 0.726 of a square chain, or OA. OR. 12P. If the perpendicular AD be accurately measured, it will be found to be 721 links. Half the product of this perpendicular by the base will be found to give the same content. Ex. 2. The three sides of a triangular field are respectively 89.39, 54.08, and 45.98. Required its content. Ans. 100A. OR. 10P. (91) A four-sided field; Fig. 50. as Fig. 50. D Field-work. Measure the four sides. Measure also a diagonal, as AC, thus dividing the four-sided field into two triangles. Mea- B sure also the other diagonal, or BD, for a "Proof line." Platting. Draw a line, as AC, equal in length to the diagonal, to any scale, by Arts. (42) and (49). On each side of it, construct a triangle with the sides of the field, as directed in the preceding article. To prove the accuracy of the work, measure on the plat the length of the " proof line," BD, by Art. (43), and if it agrees with the length of the same line measured on the ground, the field work and platting are both proved to be correct. ~ It is a universal principle in all surveying operations, that the work must be tested by some means independent of the original process, and that the same resuit must be arrived at by two different methods. The necessary length of this proof line can also easily be calculated by the principles of Trigonometry. 60 CHAIN SURVEYING. [PART 1. Calculation. Find the content of each triangle separately, as in the preceding case, and add them together; or, more briefly, multiply either diagonal (the longer one is preferable) by the sum of the two perpendiculars, and divide the product by two. Otherwise: reduce the four-sided figure to one triangle as in Art. (78); or, use any of the methods of the preceding chapter. Example 3. In the field drawn in Fig.-50, on a scale of 3 chains to the inch, AB = 588 links, BC = 210, C(D = 430, DA = 274, the diagonal AC = 626, and the proof diagonal BD = 500. The total content will be 1A. OR. 17P. -Ex. 4. The sides of a four-sided field are AB 12.41, BC 5.86, CD = 8.25, DA = 4.24; the diagonal BD = 11.55, and the proof line AC = 11.04. Required the content. Ans. 4A. 2R. 38P. -Ex. 5. The sides of a four-sided field are as follows: AB — 8.95, BC - 5.33, CD -10.10, DA = 6.54; the diagonal from A to C is 11.52; the proof diagonal from B to D is 10.92. Required the content. Ans. Ex. 6. In a four-sided field, AB = 7.68, BC = 4.09, CD - 10.64, DA = 7.24, AC = 10.32, BD = 10.74. Required the content. Ans. (92) A many-sided field, as Fig. 51. Fig. 51. B TC r 0 _..t;- I CHAP. I.] Surveying by Diagonals. 61 Field-Work. Measure all the sides of the field. Measure also diagonals enough to divide the field into triangles; of which there will always be two less than the number of sides. Choose such diagonals as will divide the field into triangles as nearly equilateral as possible. Measure also one or more diagonals for " Proof lines." It is well for the surveyor himself to place stakes in advance at all the corners of the field, as he can then select the best mode of division. Platting. Begin with any diagonal and plat one triangle, as in Art. (90). Plat a second triangle adjoining the first one, as in Art. (91). Plat another adjacent triangle, and so proceed, till all have been laid down in their proper places. Measure the proof lines as in the last article. Calculation. Proceed to calculate the content *of the figure, precisely as directed for the four-sided field, measuring the perpendiculars and calculating the content of each triangle in turn; or taking in pairs those on opposite sides of the same diagonal; or using some of the other methods which have been explained. Example 7. The six-sided field, shown in Fig. 51, has the lengths of its lines, in chains and links, written upon them, and is divided into four triangles, by three diagonals. The diagonal BE is a " proof-line." The Figure is drawn to a scale of 4 chains to the inch. The content of the field is 5A. 3R. 22P. -Ex. 8. In a five-sided field, the length of the sides are as follows: AB = 2.69, BC = 1.22, CD = 2.32, DE 3.55, EA 3.23. The diagonals are AD = 4.81, BD -3.33. Required its content. Ans. (93) A field may be divided up into triangles, not only by measuring diagonals as in the last figure, but by any of the methods shown in the four figures of Art. (71). The one which we have been employing, corresponds to the last of those figures. Still another mode may be used when the angles cannot be seen from one another, or from any one point within. Take three or more convenient points within the field, and measure from them to the corners, and thus form different sets of triangles. 62 CHAIN SURVEYING. [PART II. KEEPING THE FIELD NOTES. (91) By Sketch. The most simple method is to make a sketch of the field, as nearly correct as the unassisted hand and eye can produce, and note down on it the lengths of all the lines, as in Fig. 51. But when many other points require to be noted, such as where fences, or roads, or streams are crossed in the measurement, or any other additional particulars, the sketch would become confused, and be likely to lead to mistakes in the subsequent platting from it. The following is therefore the usual method of keeping the Field-notes. A long narrow book is most convenient for it. (95) In Columns. Draw two parallel lines about an inch apart from the bottom to the top of the page of the field-book, as in the margin. This column, or pair of lines, may be conceived to represent the measured line, split in two, its two halves being then separated, an inch apart, merely for convenience, so that the distances measured along the line, may be written between these halves. Hold the book in the direction of the measurement. At the bottom of the page write down the name, or number, or letter, which represents the station at which the survey is to begin. A " station" is marked with a triangle or circle, as in the margin. The latter is more easily made. In the complicated cases, which will be hereafter explained, and in which one long base line is measured, and also many other subordinate lines, it will be well, as a help to the memory, to mark the stations on the Base line with a triangle, and the stations on the other lines with the ordinary circle. The station from which the measure- o) to B ments are made is usually put on the left 562 of the column; and the station which is measured to, is put on the right. CHAP. I.] Surveying by Diagonals. 63 But it is more compact, and avoids interfering with the notes of "offsets" (to be explained here- B after) to write the name or number of the station A in the column, as in the margin. B The measurements to different points of a line are 400 written above one another. The numbers all refer 250 to the beginning of the line, and are counted from it. 100 A The end of a measured line is marked by a line drawn across the page above the numbers which- _ indicate the measurements which have been made. If the chaining does not continue along the adjoining line, but the chain-men go to some other part of the field to begin another measurement, two lines are drawn across the page. When a line has been measured, the marks r r or 1 are made to show whether the followlowing line turns to the right or to the left. A line is named, either by the names of the stations between which it is measured, as the line AB; or by its length, a line 562 links long, being called the line 562; or it is recorded as Line No. 1, Line No. 2, &c; or as Line on page 1, 2, &c., of the Field-book. When a mark is left at any point of a line, 562 as at D, in Fig. 49, with the intention of com- 200 S ing back to it again, in order to measure to O some other point, the place marked is called a 562 False Station, and is marked in the Field-book F. S.; or has a line drawn around it, to distinguish it; or has a station mark A placed outside 0 of the column, to the right or left, according to 562 the direction in which the measurement from it is 200 to be made. Examples of these three modes are given in the margin. 64 CIHAIN SURVEYING. [PART II. A False Station is named by its position on the line where it belongs; as thus-" 200 on 562." When a gate occurs in a measured line, the distance from the beginning of the line to the side of the gate first reached, is the one noted. When the measured line crosses a fence, brook, road, &c., they are drawn on the field-notes in their true direction, as nearly as possible, but not in a continuous line across the column, as in the first figure in the margin, but as in the second figure, so that the two parts would form a - continuous straight line, if the halves of the " split line" were brought together. It is convenient to name the lines, in the margin, as being Sides, Diagonals, Proof lines, &c. (96) The Field-notes of the triangular field platted in Fig. 49, are given below, according to both the methods mentioned in the preceding Article, pages 62 and 63. In the Field-notes in the column on the right hand, it is not absolutely necessary to repeat the B and C. z z C J 89 to C 89 ~arom D F.. from on 200 _ _ _ _,.. 150 to A A 0From C 1 100 V'Arom B ~ a3 f200 to B l 200 80 F.S 6l) From A 0 ( A A COrP. I.] Surveying by Diagonals. 65 (97) The Field-notes of the survey platted in Fig. 51, are given below. They begin at the bottom of the left hand column. zE -j fi 770 0 i 532 |_ 300 Crate. A E r z 0. 1142 E 0 400 Brook. 40 i775/ X 270 RIoad. 1 I0 80 Road D a< 420 300 7 27~80 E 7 737 2<0 13 B ~ ( __ Road. - < 210 a 703 _150 Gate._A B z 270 Bi r r p A Road. 562 F r A CHAPTER II. SURVEYING BY, TIE-LINES. (98) Surveying by Tie-lines is a modification of the method explained in the last chapter. It frequently happens that it is impossible to measure the diagonals of a field of many sides, in consequence of obstacles to measurements, such as woods, water, houses, &c. In such cases,'"Tie-lines," (so called because they tie the sides together), are employed as substitutes for diagonals. Thus, in the four-sided field shown in the Figure, the diagonals cannot be measured because of woods inter- Fig. 52. vening. As a substitute, measure off from D any convenient corner of the field, as B, any /s A, C distances, BE, BF, along the sides of the - field. Measure also the "tie-line" EF. / Measure all the sides of the field as usual. A To plat this field, construct the triangle BEF, as in Art. (90). Produce the sides BE, and BF, till they become respectively equal to BA and BC, as measured on the ground. Then with A and C as centres, and with radii respectively equal to AD and CD, describe arcs, whose intersection will be D, the remaining corner of the field. (99) It thus appears that one tie-line is sufficient to determine a four-sided field; two, a five-sided field, and so on. But, as a check on errors, it is better to measure a tie-line for each angle, and the agreement, in the plat, of all the measurements will prove the accuracy of the whole work. Since any inaccuracy in the length of a tie-line is increased in proportion to the greater length of the sides which it fixes, the tielines should be measured as far from the point of meeting of these sides as possible, that is, they should be as long as possible. The radical defect of the system is that it is " working from less to greater," (which is the exact converse of the true principle), thus magnifying inaccuracies at every step. CHAP. II.] Surveying by Tie-lines. 67 A tie-line may also be employed as a " proof line," in the place of a diagonal, and tested in the same manner. Fig. 53. If any angle of the field is re-entering, as at B in the figure, measure a tie-line across the salient angle ABC. (100) Chain Angles, It is convenient, though not necessary, to measure equal distances along the sides; BE, BF, in Fig. 5", and BA, BC, in Fig. 53. " Chain Angles" are thus formed.* (101) Inaccessible Areas. The method of tie-lines can be applied to measuring fields which cannot be entered. Thus, in the Figure, ABCD is an inac- Fig. 54. cessible wooded field, of four sides. To E C survey it, measure all the sides, and at /,, - any corner, as D, measure any distance --- DE, in the line of AD produced. Mea- / /'? B sure also another distance DF in the line G' -- of CD produced. Measure the tie-line EF, and the figure can be platted as in the case of the field of Fig. 52, the sides of the triangle being produced in the contrary direction. The same end would be attained by prolonging only one side, as shown at the angle A of the same figure, and measuring AG, AH, and GH. It is better in both cases to tie all the angles in a similar manner. This method may be applied to a figure of any number of sides by prolonging as many of them as are necessary; all of them, if nossible. " Chain angles may be reduced to angles measured in degrees, by observing that the tie-line is the chord of the angle to a radius equal to one of the equal distances measured on the sides. Therefore, divide the length of the tie-line by the length of this distance. The quotient will be the chord of the angle to a radius of one. In the TABLE OF CHORIs, at the end of this volume, find this quotient, and the number of degrees and minutes corresponding to it gives the angle required. Otherwise; since the chord of any angle equals twice the sine of half the angle, we have this rule: Divide half the tie-line by the measured distance; find in a table of natural sines the angle corresponding to the quotient, and multiply this angle by two, to get the angle desired. 68 CHAIN SURVEYING. [PART II. (102) If the sides CD and AD were prolonged by their full length, the content of the figure could be calculated without any plat; for the new triangle DEF would equal the triangle DAC; and the sides of the triangle ACB would then be known. Fig. 55. This principle may be extended still farther. I For a five-sided field, as in Fig. 55, produce \\ two pairs of sides, a distance equal to their length, forming two new triangles, as shown by; Br the dotted lines, and measure the sides B'D',.\and A'D". The three sides of each of these A D triangles will thus be known, and also the three sides of the triangle BAD, since AD = A'D", and BD — B'D'.,, / D o...........:' Fig. 56. The method of this article may be employed:: for a figure of six sides as shown in Fig. 56,,a (in which the dotted lines within the wooded'hi\ field have their lengths determined by the tri- >/. angles formed outside of it,) but not for figures of a greater number of sides. CHAPTER III. SURVEYING BY PERPENDICULARS: OR By the Second Method. THE method of Surveying by Perpendiculars is founded on the Second Method of determining the position of a point, explained in Art. (6). It is applied in two ways, either to making a complete Survey by " Diagonals and Perpendiculars," or to measuring a crooked boundary by " Off-sets." Each will be considered in turn. CIAP. III.] Surveying by Perpendiculars. 69 The best methods of getting perpendiculars on the ground must, however, be first explained. TO SET OUT PERPENDICULARS. (104) Surveyor's Cross. The simplest instrument Fig. 57. for this purpose is the Surveyor's Cross, or Cross-Staff, shown in the figure. It consists of a block of wood, of any shape, having in it two saw-cuts, made very precisely at right angles to each other, about half an inch deep, and with centre-bit holes made at the bottom of the cuts to assist in finding the objects. This block is fixed on a qr pointed staff, on which it can turn freely, and which 6 should be precisely 8 links (63k inches) long, for the convenience of short measurements. To use the Cross-staff to erect a perpendicular, set it 4 at the point of the line at which a perpendicular is want- 3 ed. Turn its head till, on looking through one saw-cut, l you see the ends of the line. Then will the other sawcut point out the direction of the perpendicular, and thus guide the measurement desired. To find where a perpendicular to the line, from some object, as a corner of a field, a tree, &c., would meet the line, set up the cross-staff at a point of the line which seems to the eye to be about the spot. Note about how far from the object the perpendicular at this point strikes, and move the cross-staff that distance; and repeat the operation till the correct spot is found. (105) To test the accuracy of the in- Fig. 58. B strument, sight through one slit to some / point A, and place a stake B in the line of sight of the other slit. Then turn its A ---—............ —- N head a quarter of the way around, so that the second slit looked through, points to A. Then see if the other slit covers B again, as it will if correct. If it does not do so, but sights to some other point, as B', the apparent error is double the real one, for it now points as far to the right of the true point, C as it did before to its left. 70 CHAIN SURVEYING. LPART II. This test is the first example we have had of the invaluable principle of Reversion, which is used in almost every test of the accuracy of Surveying and Astronomical instruments, its peculiar merit being that it doubles the real error, and thus makes it twice as easy to perceive and correct it. (106) The instrument, in its most finished form, is made of a hollow brass cylinder, which has two pairs of slits exactly opposite to each other, one of each pair being narrow and the other wide, with a horse-hair stretched from the top to the bottom of the latter. It is also, sometimes, made with eight faces, and two more pairs of slits added, so as to set off half a right angle. Fig. 59. Another form is a hollow brass sphere, as in the figure. This enables the surveyor to set off perpenpendiculars on very steep slopes. Another form of the surveyor's cross consists of two pairs Fig. 60. of plain " Sights," each shaped as in the figure, placed at.the ends of two bars at right angles to each other. The slit, and the opening with a hair stretched from its top to its bottom, are respectively at the top of one sight and at the bottom of the opposite sight.* This is used in the same manner as the preceding form, but is less portable and more liable to get out of order. A temporary substitute for these instruments may be Fig. 61. made by sticking four pins into the corners of a square piece of board; and sighting across them, in the direction of the line and at right angles to it. (107) Optical Square. The most convenient and accurate ilnstrument is, however, the Optical Square. The figures give a perspective view of it, and also a plan with the lid removed. It is a small circular box, containing a strip of looking-glass, from the upper half of which the silvering is removed. This glass is placed'' t French call the nlrrow opening ailleton, and the wide one croisie, CHAP. III.] To set out Perpendiculars 71 so as to make precisely half a right Fi. 62. angle with the line of sight, which passes through a slit on one side of the box, and a vertical hair stretched across the opening on the other side, or a mark on the glass. The box is held in the hand over the spot where the perpendicular is desired, (a plumb line in the hand will give perfect accuracy) and the observer applies his eye to the. - slit A, looking through the upper or unsilvered part of the glass, and turns the box till he sees the other D end of the line B, through the opening C. The assistant, with a rod, moves along in the direction where the perpendicular is desired, being seen in the silvered parts of the glass, by reflection through the opening D, till his rod, at E, is seen to coincide with, or to be exactly under, the object B. Then is the line DE at right angles to the line AB, by the optical principle of the equality of the angles of incidence and reflection. To find where a perpendicular from a distant object would strike the line, walk along the line, with the instrument to the eye, till the image of the object is seen, in the silvered part of the glass, to coincide with the direction of the line seen through the unsilvered part. The instrument may be tested by sighting along the perpendicular, and fixing a point in the original line; on the principle of " Reversion." The surveyor can make it for himself, fastening the glass in the box by four angular pieces of cork, and adjusting it by cutting away the cork on one side, and introducing wedges on the other side. The box should be blackened inside. 72 CHAIN SURVEYING. [PART II. (108) Chain Perpendiculars. Perpendiculars may be set out with the chain alone, by a variety of methods. These methods generally consist in performing on the ground, the operations executed on paper in practical geometry, the chain being used, in the place of the compasses, to describe the necessary arcs. As these operations, however, are less often used for the method of surveying now to be explained, than for overcoming obstacles to measurement, it will be more convenient to consider them in that connection, in Chapter V. DIAGONALS AND PERPENDICULARS. (109) In Chapter I, of this Part, we have seen that plats of surveys made with the chain alone, have their contents most easily determined by measuring, on the plat, the perpendiculars of each of the triangles, into which the diagonals measured on the ground have divided the field. In the Method of Surveying by -Diagonals and Perpendiculars, now to be explained, the perpendiculars are measured on the ground. The content of the field can, therefore, be found at once, (by adding together the half products of each perpendicular by the diagonal on which it is let fall,) without the necessity of previously making a plat, or of measuring the sides of the field. This is, therefore, the most rapid and easy method of surveying when the content alone is required, and is particularly applicable to the measurement of the ground occupied by crops, for the purpose of determining the number of bushels grown to the acre, the amount to be paid for mowing by the acre, &c. (110) A three-sided field. Measure the Fig. 63. longest side, as AB, and the perpendicular, A CD, let fall on it from the opposite angle C. \ Then the content is equal to half the product...... of the side by the perpendicular. If obsta, E D < cles prevent this, find the point, where a perpendicular let fall from an angle, as A, to the opposite side produced, as BC, would meet it, as at E in the figure. Then half the product of AE by CB is the content of the triangle. CHAP. III.] Diagonals and Perpendiculars. 73 (111) A four-sided field. Fig. 64. Measure the diagonal AC. Leave marks at the points on this diagonal at which perpendiculars from B A 2:. 4 —— 4 c and from D would meet it; finding these points by trial, as previously directed in Arts. (101) and (107). The best marks at these D "False Stations," have been described in Art. (90). Return to these false stations and measure the perpendiculars. When these perpendiculars are measured before finishing the measurement of the diagonal, great care is necessary to avoid making mistakes in the length of the diagonal, when the chainmen return to continue its measurement. One check is to leave at the mark as many pins as have been taken up by the hind-chainman in coming to that point from the beginning of the line. Example 9. Required the content of the field of Fig. 64. Ans. OA. 2R. 29P. The field may be platted from these measurements, if desired, but with more liability to inaccuracy than in the first method, in which the sides are measured. The plat of the figure is 3 chains to 1 inch. The field-notes may be taken by writing the measurements on a sketch, as in the figure; or in more complicated cases, by the column method, as below. A new symbol may be employed, this mark, 1 —, or -, to show the False Station, from which a perpendicular is to be measured. ]Example 10. Calculation. 110 to B sq. lks. Prom 200 on 480 F. S. q ABC = 1 x 480 x 110 = 26400 ADC- x 480 x 175 — 42000 F Prom 280 on 480 F. S. |- sq. chains 6.8400 _ 480 to C Acres 0.684 280 I- It is still easier to take the two -8 200 triangles together; multiplying From A ( the diagonal by the sum of the perpendiculars and dividing by two. 74 CHAIN SURVEYING. [PART II (112) A many-sided field. Fig. 65, and the accompanying field-notes represent the field which was surveyed by the First Method and platted in Fig. 51. Fig. 65. B ZOI -\!, O. - I —... E' -Example 11. Calculation. 1.54 to t The content of the triangles may Fronz 5.07 on 7.37 F. S., be expressed thus: 2.53 So D sq. iks. From 1.60 on 7.75 F. S. ABC = I x 1142 x 267=152457 4.93 to E AEC = -2 x 1142 X493=281503 From 5.45 on 11.42 F. S. CDE = X 775X253= 98037 -.67 to B AEF = IX 737x154= 56749 From 4.95 on 11.42 1F S. s. chains 58.8746 7.37 to A A]cres 5.88746 H 5.07 or, 5A. 3R. 22P. From E 0 r The first two triangles might H 1.60 have been taken together, as in From C ( r the previous field. 11.42 to C Content calculated from the 5.45 I- perpendiculars will generally var - 4.95 F~rom A cO ry slightly from that obtained by measuring on the plat. CHAP. III.] Offsets, 75 (113) A small field which has many sides, may sometimes be conveniently surveyed by taking one diagonal and measuring the perpendiculars let fall on it from each angle of the field, and thus dividing the whole area into triangles and trapezoids; as in Fig. 36, page 48. The line on which the perpendiculars are to be let fall, may also be outside of the field, as in Fig. 37, page 48. Such a survey can be platted very readily, but the length of the perpendiculars renders it less accurate. This procedure supplies a transition to the method of " Offsets," which is to be explained in the next article. OFFSETS. (114) Offsets are short perpendiculars, measured from a straight line, to the angles of a crooked or zigzag line, near which the straight line runs. Thus, in the figure, Fi,. 66. let ABCD be a crooked fence, D bounding one side of a field. Chain A _ 1.0 o:5.0 along the straight line AB, which runs from one end of the fence to the other, and, when opposite each corner, note the distance from the beginning, or the point A, and also measure and note the perpendicular distance of each corner C and D from the line. These corners will then be " determined" by the Second Method, Art. (6). The Field-notes, corresponding to [Fig. 66, are as in the margin. The measurements along the line are written in the 0 3001to B. column, as before, counting from the beginning of the line, and the offsets are DI 25 250 written beside it, on the right or left, oppo- CI 30 100 site the distance at which they are taken. A sketch of the crooked line is also usually iFrionm A 0 (0 made in the Field-notes, though not absolately necessary in so simple a case as this. The letters C and D would not be used in practice, but are here inserted to show the connection between the Field-notes and the plat. 76 CHAIN SURVEYING. [PART II. In taking the Field-Notes, the widths of the offsets should not be drawn proportionally to the distances between them, but the breadths should be greatly exaggerated in proportion to the lengths. (115) A more extended example, with a little different notation, is given below. In the figure, which is on a scale of 8 chains to one inch for the distances along the line, the breadths of the offiets are exaggerated to four times their true proportional dimensions. Fig. 67. B 1500 0 1250 20 O 1000 0 30 750 50 500 40 250 0 -A (116) The plat and Field-notes of the position of two houses, determined by offsets, are given below, on a scale of 2 chains to 1 inch. Fig. 68. 30 185 3 20' 150 90 o10 50 10 F7rom A. 30 (117) Double offsets are sometimes convenient; and sometimes triple and quadruple ones. Below are given the notes and the plat, 1 chain to 1 inch, of a road of varying width, both sides of which are determined by double offsets. It will be seen that the line AB crosses one side of the road at 160 links from A, and the other side of it at 220. CHAP. III.] Offsets. 77 Two methods of keeping the Field-notes are given. In the first form, the offsets to each side of the road are given separately and connected by the sign +. In the second form, the total distance of the second offset is given, and the two measurements connected by the word " to." This is easier both for measuring and platting. Fig. 69 B B 260 30+60 260 30 to 90 240 10+70 240 10 to 80 0 220 50 0 220 50 20 200 30 20 200 30 40 180 10 40 180 10 45 160 0 45 160 0 50 0 140 50 to 0 140 55+ 5 120 60 to 5 120 50~-20 100 70 to 20 100 45 +15 80 60 to 15 80 50 +10 60 60 to 10 60 50 +20 40 70 to 20 40 55 +20 20 75 to 20 20 60+ 01 A 60 to 0 A (118) These offsets may generally be taken with sufficient accuracy by measuring them as nearly at right angles to the base line as the.eye can estimate. The surveyor should stand by the chain, facing the fence, at the place which he thinks opposite to the corner to which he wishes to take an offset, and measure " square" to it by the eye, which a little practice will enable him to do with much correctness. 78 CHAIN SURVEYING. [PART II. The offsets may be measured, if short, with an Offset-staff, a light stick, 10 or 15 links in length, and divided accordingly; or if they are long, with a tape. They are generally but a few links in length. A chain's length should be the extreme limit, as laid down by the English " Tithe Commissioners," and that should be employed only in exceptional cases. When the " Cross-staff" is in use, its divided length of 8 links, renders the offset-staff needless. When offsets are to be taken, the method of chaining to the end of a line, described in Art. (23), page 21, is somewhat modified. After the leader arrives at the end of the line, he should draw on the chain till the follower, with the back end of the chain, reaches the last pin set. This facilitates the counting of the links to the places at which the offsets are taken. The offsets are to be taken to every angle of the fence or other crooked line; that is, to every point where it changes its direction. These angles or prominent bends can be best found by one of the party walking along the crooked fence and directing another at the chain what points to measure opposite to. If the line which is to be thus determined is curved, the offsets should be taken to points so near each other, that the portions of the curved line lying between them may, without much error, be regarded as straight. It will be most convenient, for the subsequent calculations, to take the offsets at equal distances apart along the straight line from which they are measured. In the case of a crooked brook, such as is shown in the figure given below, offsets should be taken to the most prominent angles, such as are marked a a a in the figure, and the intermediate bends may be merely sketched by eye. Fig. 70. a(gi When offsets from lines measured around a field are taken inside of these bounding lines, they are sometimes distinguished as Insets. CHAP. III.] Offsets. 79 (119) Platting. The most rapid method of platting the offsets, is by the use of a Platting Scale (described in Art. 49) and an Offset Scale, which is a short scale divided on its edges like a platting scale, but having its zero in the middle, as in the figure. Fig. 71 q.. i The platting scale is placed parallel to the line, with its zero point opposite to the beginning of the line. The offset scale is slid along the platting scait, till its edge comes to a distance on the latter at which an offset had been taken, the length of which is marked off with a needle point from the offset scale. This is then slid on to the next distance, and the operation is repeated. If one person reads off the field-notes, and another plats, the operation will be greatly facilitated. The points thus obtained are joined by straight lines, and a miniature copy of the curved line is thus obtained; all the operations of the platting being merely repetitions of the measurements made on the ground. If no offset scale is at hand, make one of a strip of thick drawing paper, or pasteboard; or use the platting scale itself, turned crossways, having previously marked off from it the points from which the offsets had been taken. In plats made on a small scale, the shorter offsets are best estimated by eye. On the Ordnance Survey of Ireland, the platting of offsets is facilitated by the use of a combination of the offset scale and the platting scale, the former being made to slide in a groove in the latter, at right angles to it. (120) Calculating Content, When the crooked line determined by offsets is the boundary of a field, the content, enclosed 80 CHAIN SURVEYING. [PART II. between it and the straight line surveyed, must be determined, that it may be added to, or subtracted from, the content of the field bounded by the straight lines. There are various methods of effecting this. The area enclosed between the straight and the crooked lines is divided up by the offsets into triangles and trapezoids, the content of which may be calculated separately by Arts. (65) and (67), and then added together. The content of the plat on page 75, will, therefore, be 1500 + 4125 + 625 = 6250 square links = 0.625 square chain. The content of the plat on page 76, will in like manner be found to be, on the left of the straight line 30,000 square links, and on its right 5,000 square links. (121) When the offsets have been taken at equal distances, the content may be more easily obtained by adding together half of the first and of the last offset, and all the intermediate ones, and multiplying the sum by one of the equal distances between the offsets. This rule is merely an abbreviation of the preceding one. Thus, in the plat of page 76, the distances being equal, the content of the offsets on the left of the straight line will be 120 x 250 = 30,000 square links, and on the right 20 x 250 = 5,000 square links; the same results as before. When the line determined by the offsets is a curved line, " Simp son's rule" gives the content more accurately. To employ it, an even number of equal distances must have been measured in the part to be calculated. Then add together the first and last offset, four times the sum of the even offsets, (i. e. the 2d, 4th, 6th, &c.,) and twice the sum of the odd offsets, (i. e. the 3d, 5th, 7th, &c.,) not including the first and the last. Multiply the sum by one of the equal distances between the offsets, and divide by 3. The quotient will be the area. -Example 12. The offsets from a straight line to a curved fence, were 8, 9, 11, 15, 16, 14, 9, links, at equal distances of 5 links. What was the content included between the curved fence and the straight line? Ans. 371.666 CHAP. II. Offsets. 81 (122) Many erroneous rules have been given on this part of the subject. One rule directs the surveyor to divide the sum of all the offsets by one less than their number, and multiply the quotient by the whole length of the straight line; or, what is the same thing, to multiply the sum of all the offsets by the common distance between them. This will be correct only when the offsets at each end of the line are nothing, i. e. when the curved line starts from the straight line and returns to it at the beginning and end of one of the equal distances. In all other cases it will give too much. A second rule directs the surveyor to divide the sum of all the offsets by their number, and then to multiply the quotient by the whole straight line. This may give too much, or too little, according to circumstances. Suppose offsets of 10, 30, 20, 80, 50, 30, links, to have been taken at equal distances of a chain. The correct content of the enclosed space is 200 x 100 = 2 square chains. The first of the above rules would give 2.2 square chains, and the second would give 1.8333 chains. (123) Reducing to one triangle the many-sided figure which is formed by the offsets, is the method of calculation sometimes adopted. This has been fully explained in Part I, Art. (78), &c. The method of Art. (83) is best adapted for this purpose. (124) Equalizing, or giving and taking, is an approximate mode of calculation much used by practical surveyors. A crooked line, determined by offsets, having been platted, a straight line is drawn on the plat, across the crooked line, leaving as much space outside of the straight line as inside of it, as nearly as can be estimated by the eye, "Equalizing" it, or "Giving and taking" equal Fi,. 72. portions. The straight line is best determined by laying across the irregular outline the straight edge of a piece of transparent horn, or tracing paper, or glass, or a fine thread or horse-hair 82 CHIAIN SURVEYING. [PART II. stretched straight by a light bow of whalebone. In practical hands, this method is sufficiently accurate in most cases. The student will do well to try it on figures, the content of which he has previously ascertained by perfectly accurate methods. Sometimes this method may be advantageously combined with the preceding; short lengths of the croooked boundary being "Equalized," and the fewer resulting zigzags reduced to one line by the method of Art. (78), &c. CHAPTER IV. SURVEYING BY THE PRECEDING- MIETHODS COMBINED. 125) All the methods which have been explained in the three preceding chapters — Surveying by iDiagonals, by Tie-lines, and by Perpendiculars, particularly in the form of offsets —are frequently required in the same survey. The method by Diagonals should be the leading one; in some parts of the survey, obstacles to the measurement of diagonals may require the use of Tie-lines; and if the fences are crooked, straight lines are to be measured near them, and their crooks determined by Offsets. (126) Offsets are necessary additions to almost every other method of surveying. In the smallest field, surveyed by diagonals, unless all the fences are perfectly straight lines, their bends must be determined by offsets. The plat (scale of 1 chain to 1 inch), and field-notes, of such a case are given below. A sufficient num oEAP. IV.] Diagonals, Tie-lines and Offsets, 83 her of the sides, diagonals, and proof-lines, to prove the work, should be platted before platting the offsets. Fig. 73. B 3. 60. - I 10: 2iS: < [ 310 | 0 6 8 0 340 B 010 270 __c,,0 5' i2 2 152, 0 150 0 3 s15 to A r 80 5 A 65tent of8 the0 248 B OF r 11 180 0 0 105 0 B 65 5 125 he difficulty o f keep 11 90 23 62 0 135 12 22 15 110 a 13 90 co 0 50 0 Exaamnpl 13. Recquired the con- 30 9 tent of the above field. Ans. C 0 F (127) Field-books. The difficulty and the importance of keeping the Field-notes clearly and distinctly, increase with each new combination of methods. For this reason, three different methods of keeping the Field-notes of the same survey will now be given, (from Bourns' Surveying), and a careful comparison by the student of the corresponding portions of each will be very profitable to him. 84 CHAIN SURVEYING. [PAT II. FIELD-BOOK No, 1. 110 ~ 100,; " c ~oT 0 % ~',,'?,, )o: in At'-. I9' - - ( 2600,,,,, (o= / Ficl-dBook- No. 1 (Fig. 74) shews the Sketch method, explained in Art. (91). caPr. Iv.] Diagonals, Tie-lines and Offsets. 85 FIELD-BOOK No. 2. 45oo A 9........ f...1300 10 4oo,, 6 1200 3480 1020 200 \, 3060 X 140 680 190 / 240 From C to A 24-50k & 2300 C 160 620 20 640 60 >20 580 fComA o X roo A ed in Art. (95). 86 CHAIN SURVEYINW. [PaRT IX. FIELD-BOOK No. 3../'o' 62\ 05[ 1 /.230 - -, 0,a1120 099 j o 2,oo, = - E'/ 00-.. 640 1/01 20 o 0 C01 2I0"30 COcl /.oeo g,,Book No. 3 (Fig. 76 ) is a convenien~ 0ombinaton of the 1220 50 this figure, at A. 10'150 t7 Field Book No. 3 (Fig. 76) is a convenient combination of the two preceding methods. The bottom of the Book is at the side of this figure, at A. CHAP. Iv.] Diagonals, Tie-lines and Offsets. 87 (128) It will easily appear from the sketch of Field-book No. 1, how much time and labor may be saved, or lost, by the manner of doing the work. Thus, beginning at A, and measuring 750 links, a pole should be left there, and the line to the right measured to 17 chains, or C, leaving a pole at 12.30 as a new starting point by and by. Then from C measure 19 chains to A again; then measure from A to B, and from B back to the pole left at 7.50 on the main line. (129) The example which will now be given shows part of the Field-notes, the plat, (on a scale of 6 inches to 1 mile [1: 10,560]), and'a partial calculation of the "Filling up" of a large triangle, the angular points of which are supposed to have been determined by the methods of Geodesic Surveying. They should be well studied.' Fig. 77. D * Capt. FROME, in his " Trigonometrical Survey," from which this example has been condensed, remarks, " It may, perhaps, be thought that too much stress is laid on forms; but method is a most essential part of an undertaking of magnitude: and without excellent preliminary arrangements to ensure uniformity in all the most trifling details, the work never could go on creditably." 88 CHAIN SURVEYING. [PART II. 2564 80 D F2452 C3 100 4050 Q 1700 0 62 3890 N 2324 1530 84 42 3730 1420 40 0 3540 0 0 1340 0 3420 30 1112640 FromA A to D) V v. A 72 2484 S 1240 52 A 1140 86 A 40 2332 950 100 60 2206 77260 0 2056 0 0 604 0,~ 1805 40 34 502 M 3296 01550 50 50 450 - 70 342 3275 54 X 1442 0 82 220 3120 62 From C A to A 2940 85 ]FromD A to C. I C 2572 60 D In the above specimen of a field-book, (which resembles that on page 85), all offsets, except those having relation to the boundary lines, are purposely omitted, to prevent confusion, the example being given solely to illustrate the method of calculating these larger divisions. Rough diagrams are drawn in the field-book not to any scale, but merely bearing some sort of resemblance to the lines measured on the ground, for the purpose of showing, at any period of the work, their directions and how they are to be connected; and also of eventually assisting in laying down the diagram and content plat. On these rough diagrams are written the distinctive letters by which each line is marked in the field-book, and also its length, and the distances between points marked upon it, from which other measurements branch off to connect the interior portions of the district surveyed. (130) Calculations. The calculations of one of the figures, 1i, is given below in detail. It is composed of the triangle DPQ, with offsets along the sides PQ; and of the triangle DWX, with offsets CHAP. IV.] Diagonals, Tie-lines and Offsets. 89 along the sides IW:anWitl WX. From the content thus obtained must be subtracted the offsets on PQ, belonging to the figure A, and those on WX belonging to the figure Z. When the offsets are triangles, (right angled, of course), the base and perpendicular are put down as two sides; when they are trapezoids, the two parallel sides and the distance between them occupy the columns of" sides." TRIANGLE IST 2D 8 CONTENT DIVISION. OR SIDE. SIDE. SIDE. IN TRAPEZOID. CHAINS. DPQ 11680 1698 1078 86.2650 52 250.6500 PQ 2 30 80.3280 30 216.3240'.~~~~! ~~1.3020 Additives. DWX 1370 14421 770 51.8339 PW 30 I 310.4650 ( 56 114.3192) WX t 56 36 104.4784 36 90.1620.9596 Total Additives,| 140.8255 50 1741.4350 PQ 50 30 292 1.1680 30 66.0990 1.7020 E2 I _1 — 52 142.3692 Subtractives. WX 52 64 232 1.3456 ( 64- 88.2816 1.9964 Total Subtractives, 3,6984 Total Additives, 140.8255 L.ifference, 137.1271 90 CHIIN SURVEYING. [PAIT II. The other figures, comprised within the large triangle, are recorded and calculated in a similar manner. An abridged register of the results is given below. DIFFERENCE IN DIVISION. ADDITIVES. SUBTRACTIVE. S QUARE CHAINS DNS DWX and offsets. NUV 140.4893 _________ _and offsets. IDNO DPQ 100.1882 and offsets. and offsets. IT ANO}N WM 103.9778 and offsets. and offsets. 13.9m8 HTN D {| NRM Offsets. 81.6307 NRM and offsets. CNS HTN 109.5064 and offsets. and offsets. DPQ DWX Offsets. 137.1271 and offsets. Total, - - - - - - 672.9195 The accuracy of the preceding calculations of the separate figures must now be tested by comparing the sum of their areas with that of the large triangle ACD, which comprises them all. Their area must previously be increased by the offsets on the lines CS and CH, which had been deducted from II, and which amount respectively to 3.5270 and 2.8690. The total areas will then equal 679.3155 square chains. That of the triangle ACD is 679.5032; a difference of less than a fifth of a square chain, or a fiftieth of an acre; or about one-fortieth of one per cent. on the total area. (131) The six lines. In most cases, great or small, six fundamental lines will need to be measured; viz. four approximate boundary lines, forming a quadrilateral, and its two diagonals. Small triangles, to determine prominent points, can be formed within and without these main lines by the FIRST METHOD, Art. (5), and the lesser irregularities can be determined by offsets. CHAP. rV.] Diagonals, Tie-lines and Offsets. 91 Fig. 78. A. X; /0 T D 3 Thus, in the above figure, two straight lines AB and CD are measured through the entire length and breadth of the farm, or township, which is to be surveyed. The connecting lines AC, CB, BD and DA are also measured, uniting the extremities of the first two lines. The last four lines thus form a quadrilateral, which is divided into two triangles by one of the first measured lines, while the second serves as a proof-line. The distance from the intersection of the two diagonals to the extremities of each, being measured on the ground and on the plat, affords an additional test. Other points of the district surveyed (as E, G, K., &c., in the figure,) are determined by measuring the distances from them to known points (as M, N, P, R, &c., in the figure) situated on some of the six fundamental lines, thus forming the triangles T, T. The intersection 0 of the main diagonals, and also the intersections of the various minor lines with the main lines and with each other, should all be carefully noted, as additional checks when the work comes to be platted. 92 CHAIIN SURVEYING, [PART II. The larger figures are determined first, and the smaller ones based upon them, in accordance with this important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. The unavoidable inaccuracies are thus subdivided and diminished. The opposite course would accumulate and magnify them. These additional lines, which form secondary triangles, should be so chosen and ranged as to pass through and near as many objects as possible, in order to require as few and as short offsets as the position of the lines will permit; the smaller irregularities being determined by offsets as usual. It is better to measure too many lines than too few, and to establish unnecessary " false stations," rather than not to have enough. (132) Exceptional cases. The preceding arrangement of lines, though in most cases the best, may sometimes be varied with advantage. Unless the farm surveyed be of a shape nearly as broad as long, the two diagonals will cross each other obliquely, instead of nearly at right angles, as is desirable. Fig. 79. When the farm is much:B longer than it is wide, two G" - systems, of six lines each, — r may be used with much advantage, as in Fig. 79.. Several such may be com- Z. bined when necessary. 1ig. 80. In a case like that in Fig. ~ 80, five lines will be better than six, and will tie one another together, their points of intersection being carefully noted. o, CHAP. IV.] Diagonals, Tie-lines and Offsets. 93 Fig. 81. In the farm represented in ((Fig. 81, the system of lines ", -~ there shown is the best, and they will also tie one another..?..B (133) Much difficulty will often be found in ranging and measuring the long lines required by this method in extensive surveys. Various contrivances for overcoming the obstacles which may be met with, will be explained in the following chapter. It will often be convenient to measure the minor lines along roads, lanes, paths, &c., although they may not lie in the most desirable directions. Steeples, chimneys, remarkable trees, and other objects of that character, may often be sighted to, and the line measured towards them, with much saving of time and labor. The point where the measured lines cross one another should always be noted, and they will thus form a very complete series of tie-lines.* A view of the district to be surveyed, taken from some elevated position, will be of much assistance in planning the general direction of the lines to be measured. (134) Inaccessible Areas. Fig. 82. A combination of offsets and,. — 7 tie-lines supplies an easy me- /p.... thod of surveying an inaccessible area, such as a pond, swamp, forest, block of houses, &c., as appears from the fi-,i gure; in which external bounding lines are taken at will and ____ _1 ___ ____'_____ * To find the exact point of intersection of these lines, which are only" visual lines," (explained in Art. (19),) three persons are necessary: one stands at some point of one of the lines and sights to some other point on it; a second does the same on the second line; by signs they direct, to right or left, the movements of a third person, who holds a rod, till he is placed in both of the lines and thus at their intersection, on the principle of Art. (11). 94 CIlAIN SURVEYING. [PART II. measured, and tied by "tie-lines"' measured between these lines, prolonged when necessary, as in Art. (101), while offsets from them determine the irregularities of the actual boundaries of the pond, &c. These offsets are insets, and their content is, of course, to be subtracted from the content of the principal figure. Even a circular field might thus be approximately measured from the outside. If the shape of the field admits of Fig. 83. it, it will be preferable to measure four lines about the field in such directions as to enclose it in a rect-..:- angle, and to measure offsets from the - sides of this to the angles of the field. (135) When one of the lines with which Fig. 84 an inaccessible field is surrounded, as in i, the last two figures, cuts a corner of the field, as in Fig. 84, the triangle ABC is c to be deducted from the content of the enclosing figure, and the triangle CDE M added to it. The triangle DEF is also D4-.- ___ to be added, and the triangle FGH deducted. To do this directly, it would be necessary to find the points of intersection' C and F. But this may be difficult, and can be dispensed with by obtaining the difference of each pair of triangles. The p difference of ABC and CDE will be obtained at once by multiplying the difference of the offsets AB and DE by half of BE; and the difference of DEF and FGH by multiplying the difference of DE and GH by half of EG.* * For, making the triangle Dmn = ABC, then mnEC - En X j (min + CE) -(DE - AB) X J EB; and so with the other pair of triangles. CHAP. IV.] Diagonals, Tie-lines and Perpendiculars. 95 (136) Roads, A winding Road may also be surveyed thus, as is shown in Fig. 85; straight lines being measured in the road, Fig. 85. Z rI ~do their changes in direction determined by tie-lines, tying one line to the preceding one prolonged, as explained in Chapter II, of this Part, and points in the road-fences, on each side of these straight lines, being determined by offsets. A River may also be supposed to be represented by the above winding lines; and the lower set of lines, tied to one another as before, and with offsets from them to the water's edge, will be sufficient for making an accurate survey of one side of the river. (137) Towns. ~A town could be surveyed and mapped in the same manner, by measuring straight lines through all the streets, determining their angles by tie-lines, and taking offsets from them to the blocks of houses. 96 CHAIN SURVEYING. [PART II. CHAPTER V. OBSTACLES TO MEASUREMENT IN CHAIN SURVEYING. (138) In the practice of the various methods of surveying which have been explained, the hills and vallies which are to be crossed, the sheets of water which are to be passed over, the woods and houses which are to be gone through —all these form obstacles to the measurement of the necessary lines which are to join certain points, or to be prolonged in the same ditection. Many special precautions and contrivances are, therefore, rendered necessary; and the best methods to be employed, when the chain alone is to be used, will be given in the present chapter. (139) The methods now to be given for overcoming the various obstacles met with in practice, constitute a LAND-GEOMETRY. Its problems are performed on the ground instead of on paper: its compasses are a chain fixed at one end and free to swing around with the other; its scale is the chain itself; and its ruler is the same chain stretched tight. Its advantages are that its single instrument, (or a substitute for it, such as a tape, a rope, &c.) can be found anywhere; and its only auxiliaries are equally easy to obtain, being a few straight and slender rods, and a plumb-line, for which a pebble suspended by a thread is a sufficient substitute. Many of these problems require the employment of perpendicular and parallel lines. For this reason we will commence with this class of Problems. The Demonstrations of these problems will be placed in an Appendix to this volume, which will be the most convenient arrangement for the two great classes of students of surveying; those who wish merely the practice without the principles, and those who wish to secure both. The elegant " Theory of Transversals" will be an important elerment in some of these demonstrations. All of them will constitute excellent exercises for students. CHAP. v.] Obstacles to Measurement. 97 PROBLEMS ON PERPENDICULARSo" Problem 1. To erect a perpendicular at any point of a line. (140) -First Method. Let A be the Fig. 86. point at which a perpendicular to the line is to be set out. Measure off equal distances AB, AC, on each side of the point. Take _I c a portion of the chain not quite 1I times as long as AB or AC, fix one end of this at B, ( and describe an arc with the other end. Do the same from C. The intersection of these arcs will fix a point D. AD will be the perpendicular required. Repeat the operation on the other side of the line. If that is impossible, repeat it on the side with a different length of chain (141) Second Method. Mleasure off as be- Fi. 87 fore, equal distances AB, AC, but each about only one-third of the chain. Fasten the ends of the chain with two pins at B and C. Stretch B it out one side of the line and put a pin at the middle of it, D. Do the same on the other side of the line, and set a pin at E. Then is DE a perpendicular to BC. If it is impossible to perform the operation on both sides of the line, repeat it on the same side with a different length of chain, as shown by the lines BF and CF in the figure, so as to get a second point. (142) Other Methods. All the methods to be given for the next problem may be applied to this. i Many of these methods would seldom be required in practice, but cases sometimes occur, as every surveyor of much experience in Field-work has found to his serious inconvenience, in which some peculiarity of the local circumstances forbids any of the usual methods being applied. In such cases the collection here given will be found of great value. In all the figures, the given and measured lines are drawn with fine full lines, the visual lines, or lilies of sight, with broken lines, and the lines of the result with heavy full lines. The points which are centres around which the chain is swung, are enclosed in circles. The alphabetical order of the letters attached to the points, shows in what order they are taken. 7 98 CHAIN SURVEYING. [PART II. Problem 2. To erect a perpendicular to a line at a given point, when the point is at or near the end of the line. (143) First Method. Measure Fig. 88. 5U 40 links along the line. Let one assistant hold one end of the chain at that point; let a second hold the 20 link mark which is nearest the other end, at the given point A, and let a B40 third take the 50. link mark, and ~ t0 tighten the chain, drawing equally on both portions of it. Then will the 50 link mark be in the perpendicular desired. Repeat the operation on the other side of the line so as to test the work. The above numbers are the most easily remembered, but the longer the lines measured the better; and nearly the whole chain may be used, thus: Fix down the 36th link from one end at A, and the 4th link from the same end on the line at B. Fix the other end of the chain also at B. Take the 40th link mark from this last end, and draw the chain tight, and this mark will be in the perpendicular desired. The sides of the triangle formed by the chain will be 24, 32 and 40. (144) Otherwise: using a 50 feet Fig. 89. 8 tape, hold the 16 feet mark at A; hold the 48 feet mark and the ringend of the tape together on the line; take the 28 feet mark of the tape, and aD. draw it tight; then will the 28 feet'8 a A. mark be in the perpendicular desired. o 16 (145) Second Method. Hold one end Fig. 90. of the chain at A and fix the other end at a D point B, taken at will. Swing the chain around B as a centre, till it again meets the line at C. Then carry the same end around (the other end remaining at B) till it comes in the line of CB at D. AD is the perpendicular required. CHAP. V.] Obstacles to Measurement. 99 (146) Third Method. Let A be the given Fig. 91. point. Choose any point B. Measure BA. Set off, on the given line, AC = AB. On CB produced set off from C, a distance = (CB This will fix the point D, and AD will be the perpendicular required. c (147) -Fourth Method. From the Fig. 92. given point A set off on the given line r P any distance AB. From B, in any convenient direction, set off BC = AB. D A Then on the given line, set off AD = AC. On CB prolonged, set off CE = C AD. Join DE; and on DE, from D, set off DF = 2 AB. Then will the line AF be perpendicular to the line AD at the point A. Problem 3. To erect a perpendicular to an inaccessible line, at a given point of it. (148) First lfethod. Get points in the direction of the inaccessible line prolonged, and from them set out a parallel to the line, by methods which are given in Art. (165), &c. Find by trial the point in which a perpendicular to this second line (and therefore to the first line) will pass through the required point. (149) Second Method. If the line is not only inaccessible, but cannot have its direction prolonged, the desired perpendicular can be obtained only by a complicated trigonometrical operation. Problem 4. To let fall a perpendicular from a given point to a given line. (150) First Method. Let P be Fig. 93. the given point, and AB the given line. Measure some distance, a chain or less, from C to P, and then fix one end of the chain at P, and swing it A. C around till the same distance meets 100 CHAIN SURVEYING. [PART II. the line at some point D. The middle point E of the distance CD will be the required point, at which the perpendicular from P would meet the line. (151) Second Method. Stretch a chain, or a portion of it, from the given point P, to some point, as A, of the Fig. 94. given line. Hold the end of the distance at A, and swing round the other end of the chain from P, so as to set off the same distance along the given line from A to some point B. Mea- ]. sure BP. Then will the distance BC from B to the foot of the BP2 desired perpendicular = 2AB' (152) Other Methods. All the methods given in the next problem can be applied to this one. Problem 5. To let fall a perpendicular to a line, from a point nearly opposite to the end of the line. (153) First Method. Stretch a chain from the given point P, to some point, as A, of the given line. Fix to ): the ground the middle point B of the chain P AP, and swing around the end which was at B P, or at A, till it meets the given line in a point C, which will be the foot of the required perpendicular. A:,/ (154) Second Method. Takeanypoint, Fig. 96 p as A, on the given line. Measure a dis- d tance AB. Let the end of this distance en the chain be held at B, and swing around / the end of the chain, till it comes in the A B D line of AP at some point C, thus making BC - AB. Measure AC and AP. Then the distance AD, from A to the foot of the APxAC perpendicular required = 2 AB CHAP. V.] Obstacles to Measurement. 101 (155) Third 2Method. At any convenient Fig. 97. P point, as A, of the given line, erect a perpendicular, of any convenient length, as AB, and mark a point C on the given line, in the line of P and B. Measure CA, CB and OP. A D Then the distance from C to the foot of the CAxOP perpendicular, i. e. CD = CB Problem 6. To let fall a perpendicular to a line, from an inaccessible point. (156) First Method. Let P be the given Fig. 98. point. At any point A, on the given line, set out P a perpendicular AB of any convenient length. Prolong it on the other side of the line the / same distance. Mark on the given line a - point D in the line of PB; and a point E in — 7 7A / the line of PC. Mark the point F at the intersection of DC and BE prolonged. The line C""!" FP is the line required, being perpendicular to the given line at the point G. (157) Second lliethod. Let A and B Fig. 99. be two points of the given line. From A,_ let fall a perpendicular, AC, to the visual line BP; and from B let fall a perpendicular, BD, to the visual line AP. Find the point at which these perpendiculars intersect, as at E (seeArt.(133)), and the, 3!.B line PE, prolonged to F, will give the perpendicular required. Problem 7. To let fall a perpendicular from a given point to an inaccessible line. 102 CHAIN SURVEYING. [PART II. (158) First liethod. Let P be Fig. 100. the given point and AB the given A i —— _ _, line. By the preceding problem, let " <: fall perpendiculars from A to BP, at / C; and from B to AP, at D; the line PE, passing from the given point P to the intersection of these perpendiculars, is the desired perpendicular to the inaccessible line AB. This method will apply when only two points of the line are visible. (159) Second 3lethod. Through the given point, set out, by the methods of Art. (16.5), &c., a line parallel to the inaccessible line. At the given point erect a perpendicular to the parallel line, and it will be the required perpendicular to the inaccessible line. PROBLEMS ON PARALLELS. Problem 1. To run a line, from a given point, parallel to a given line. (160) First Method. Let fall a perpendicular from the point to the line. At another point of the line, as far off as possible, erect a perpendicular, equal in length to the one just let fall. The line joining the end of this line to the given point will be the parallel required. (161) Second Method. Let AB be Fig. 101. the given line, and P the given point. A D C Take any point, as C, on the given line, and from it set off equal distances, as long as possible, CD on the given line, and CE, on the line CPo. Measure P G DE. From P set off PF = CE; and from F, with a distance = DE, and from P, with a distance = CD, describe arcs intersecting in G. PG will be the parallel required. If it is more convenient, PC may be prolonged, and the equal triangle, CDE, be formed on the other side of the line AB. CoAP. V.] Obstacles to Measurement. 103 (162) Third Method. Measure from Fig. 102. P to any point, as C, of the given line, and A-. B put a mark at the middle point, D, of that line. From any point, as E, of the given line, measure a line to the point D, and continue it till DF DE. Then will the line PF be parallel to AB. (163) Fourth Method. Measure from Fig. 103. P to any point C,' of the given line, and D continue the measurement till CD = CP. From D measure to any point E of the given line, and continue the measurement A- till EF = ED. Then will the line PF be parallel to AB. If more convenient, P AF CD may be made one-half, or any other fraction, of CP,. and EF be then made twice, &c., DE. (164) Fifth Method. From any Fig 104. point, as C, of the line, set off equal A -- -; /'B distances along the line, to D and E. Take a point F, in the line of PD. Stake out the lines FC and FE, and, also the line EP, crossing the line CF in the point G. Lastly, prolong the line DG, till it meets the line EF in the point H. PH is the parallel required. Problem 2. To run a line from a given point parallel to an inaccessible line. (165) First Method. Let AB Fig. 105. be the given line, and P the given A B point. Set a stake at C, in the line: of PA, and another at any conven-'.-., ient point, D. Through P, set out, D\ by the preceding problem, a parallel to DA, and set a stake at the point, P as E, where this parallel intersects DC prolonged. Through E 104 CHAIN SURVEYING. [PART II. set out a parallel to BD, and set a stake at the point F, where this / parallel intersects BC prolonged. PF is the parallel required. (166) Second Method. Set a stake Fig. 106. at any point, C, in the line of AP, and A: another at any convenient place, as at D. _ Through P set out a parallel to AD, (not P,shewn in the figure,) intersecting CD in /\ -- E. Through E set out a parallel to DB, intersecting CB in F. The line PF will be the parallel required. (167) Alinement and Measurement. We are now prepared, having secured a variety of methods for setting out Perpendiculars and Parallels in every probable case, to take up the general subject of overcoming Obstacles to Measurement. Before a line can be measured, its direction must be determined. This operation is called Ranging the line; or Alining it; or Boning it.* The word Alinementt will be found very convenient for expressing the direction of a line on the ground, whether between two points, or in their direction prolonged. This branch of our subject naturally divides itself into two parts, the first of which is preliminary to the second; viz: I. Of Obstacles to Alinement; or how to establish the direction of a line in any situation. II. Of Obstacles to Measurement; or how to find the length of a line which cannot be actually measured. I. OBSTACLES TO ALINEMENT. (168) All the cases which can occur under this head, may be reduced to two; viz: A. To find points in a line beyond the given points, i. e. to prolong the line. B. To find points in a line between two given points of it, i. e. to interpolate points in the line. * This word, like many others used in Engineering, is derived from a French word, Borner, to mark out, or limit; indicating that the Normans introduced the art of Surveying into England. t Slightly modified from the French Alignement. HABP. V.q Obstacles to Measurement. 105 A. TO PROLONG A LINE (169) By Ranging with rods. When two points in a line are given, and it is desired to Fig. 107. prolong the line by ranging D" it out with rods, three per- A sons are required, each furnished with a straight slender rod, and with a plumb-line, or other means of keeping their rods vertical. One holds his rod at one of the given points, A, in the figure, and another at B. A third, C, goes forward as far as he can without losing sight of the first two rods, and then, looking back, puts himself " in line" with A and B, i. e. so that when his eye is placed at C, the rod at B hides or covers the rod at A. This he can do most accurately by holding a plumb-line before his eye, so that it shall cover the first two rods. The lower end of the plumb-bob will then indicate the point where the third rod should be placed; and.so with the rest. The first man, at A, is then signalled, and comes forward, passes both the others, and puts himself at D, " in line" with C and B. The man at B, then goes on to E, and "lines" himself with D and C: and so they proceed, in this " hand over hand" operation, as far as is desired. Stakes are driven at each point in the line, as soon as it is determined. (170) The rods should be perfectly straight, either cylindrical or polygonal, and as slender as they can be without bending. They should be painted in alternate bands of red and white, each a foot, or link in length. Their lower ends should be pointed with iron, and a projecting bolt of iron will enable them to be pressed down by the foot into the earth, so that they can stand alone. When this is done, one man can range out a line. A rod can be set perfectly vertical, by holding a plumb-line before the eye at some distance from the rod, and adjusting the rod so that the plumb-line covers it from top to bottom; and then repeating the operation in a direction at right angles to the former. A stone dropped from top to bottom of the rods will approximately attain the same end. When the lines to be ranged are long, and great accuracy is required, the rods may have attached to them plates of tin with open 106. -CHAIN SURVEYING. [PART II. ings cut out of them, and black horse-hairs stretched from Fig. 108. top to bottom of the openings. A small telescope must then be used for ranging these hairs in line. In a hasty survey, straight twigs, with their tops split to receive a paper folded as in the figure, may be used. (171) By Perpendiculars. Fig. 109. The straight line, AB in the y —- -.L figure, is supposed to be stop- C D F' E ped by a tree, a house, or other obstacle, and it is desired to prolong the line beyond this obstacle. From any two points, as A and B, of the line, set off (by some of the methods which have been given) equal perpendiculars, AC and BD, long enough to pass the obstacle. Prolong this line beyond the obstacle, and from any two points in it, as E and F, measure the perpendiculars EH and FG, equal to the first two, but in a contrary direction. Then will G and H be two points in the line AB prolonged, which can be continued by the method of the last article. The points A and B should be taken as far apart as possible, as should also the points E and F. Three or more perpendiculars, on each side of the obstacle, may be set off, in order to increase the accuracy of the operation. The same thing may also be done on the other side of the line, as another confirmation, or test, of the accuracy of the prolonged line. (172) By equilateral triangles. Fig. 110 The obstacles, noticed in the last arti- A BG cle, may also be overcome by means of three equilateral triangles, formed by the chain. Fix one end of the chain, C and also the end of the first link from / its other end, at B; fix the end of the 33d link at A; take hold of the 66th D link, and draw the chain tight, pulling equally on each part, and put a pin at the point thus found, C, in the figure. An equilateral triangle will thus be formed, each side being 33 links. Prolong the line AC, past the obstacle, to some point, as D. Make another CHAP. v.] Obstacles to Measurement. 107 equilateral triangle, DEF, as before, and thus fix the point F. Prolong DF, to a length equal to that of AD, and thus fix a point G. At G form a third equilateral triangle GHK, and thus fix a point K. Then will KG give the direction of AB prolonged. (173) By symmetrical triangles. Let AB be the line to be prolonged. Take any conve- Fig. 111. nient point, as C. Range B _. out the line AC, to a point'4 A', such that CA'=CA. Range out CB, so that CB', = CB. Range backwards A'B', to some point D, such D that DC prolonged will pass' the obstacle. Find, by ranging, the intersection, at E, of DB and AC. From C, measure, on CA', the distance CE' CE. Then range out DC and B'E' to their intersection in P, which will be a required point in the direction of AB prolonged. The symmetrical points are marked by corresponding letters. Several other points should be obtained in the same manner. In this, as in all similar operations, very acute intersections should be avoided as far as possible. (174) By transversals. Let AB be Fig. 112. the given line. Take any two points C P and D, such that the line CD will pass the obstacle. Take another point, E, in the intersection of CA and DB. E Measure AE, AC, CD, BD and BE. Then the distance from D to P, a point in the required prolongation, will be.DP CDxBDxAE A BE BxAC-BD x AE Other points in the prolongation may be obtained in the same manner, by merely moving the single point C, in the line of EA; in which case the new distances CA and CD will alone require to be measured. 108 CHAIN SURVEYING. [PART II. CDxBD If AE be made equal to AC, then is DP = BE-BD CDxAE If BE be made equal to BD, then is DP = AC —AE The minus sign in the denominators must be understood as only meaning that the difference of the two terms is to be taken, without regard to which is the greater. (175) By harmonic conjugates. Fig. 113. Let AB be the given line. Set a B%_ _ _ _ _ _'. P stake at any point C. Set stakes at - /,f —i-, points, D, on the line CA, and at,, E, on the line CB; these points, "...: " D and E, being so chosen that the',. L:' " line DE will pass beyond the obsta-. - cle. Set a fourth stake, F, at the;,' intersection of the lines AE and DB. Set a fifth stake, G, any- c where in the line CF; a sixth stake, I, at the intersection of CB and DG prolonged; and a seventh, K, at the intersection of CA and EG prolonged. Finally, range out the lines DE and KH, and their intersection at P, will be in the line AB prolonged. (176) By the complete Quadrilateral.' Let AB be the given line. Take any conven- Fig. 114 ient point C; measure -I; -------------------- C -------------- from it to B, and onward, in the same line prolonged, an equal distance to D. Take any other convenient D point, E, such that CE and H DE produced will clear the obstacle. Measure from E to A, and onward, an equal distance, to F. Range out the lines FC and DE to their intersection in G. Range out FD and CE to intersect in H. Measure GH. Its middle point, P, is the required point in the line of AB prolonged. The unavoidable acute intersections in this construction are objectionable. CHAP. v.] Obstacles to Measurement. 109 B,. TO INTERPOLATE POINTS IN I LINE. (177) The most distant given point of the line must be made as conspicuous as possible, by any efficient means, such as placing there a staff, bearing a flag; red and white, if seen against woods, or other dark back-ground; and red and green, if seen against the sky. A convenient and portable signal is shown in the figure. Fig. 115. Front View. Side View. Back View. The figure represents a disc of tin, about six inches in diameter, painted white and hinged in the middle, to make it more portable. It is kept open by the bar, B, being turned into the catch, C. A screw, S, holds the disc in a slit in the top of the pole. Another contrivance is a strip of tin, which has its ends bent horizontally in contrary directions. As the wind will take strongest hold of the side which is concave towards it, the bent strip will continually revolve, and thus be very conspicuous. Its upper half should be painted red and its lower half white. A bright tin cone set on the staff, can be seen at a great distance when the sun is shining. 178) Ranging to a point, thus made conspicuous, is very simple when the ground is level. The surveyor places his eye at the nearest end of the line, or stands a little behind a rod placed on it, and by signs moves an assistant, holding a rod at some point as nearly in the desired line as he can guess, to the right or left, till his rod appears to cover the distant point. 110 CHAIN SURVEYING. [PART II. (179) Across a Valley. When a valley, or low spot, intervenes between the two ends Fig. 116. of the line, A and Z in the figure, a rod held in the low place, as at B, would seldom be high enough to Z be seen, from A, to cover _ = the distant rod at Z. In such a case, the surveyor at A should hold up a plumb-line over the point, at arm's length, and place his eye so that the plumb-line covers the rod at Z. He should then direct the rod held at B to be moved till it too is covered by the plumb-line. The point B is then said to be " in line" between A and Z. In geometrical language, B has now been placed in the vertical plane determined by the vertical plumb-line and the point Z. Any number of intermediate points can thus be " interpolated," or placed in line between A and Z. (180) Over a Hill, When a hill rises between two points and prevents one being seen from the other, as in the figure, (the upper Fig. 117. A Z C -E 0=1==v of which shows the hill in "Elevation," and the lower part in "Plan"), two observers, B and C, each holding a rod, may place themselves on the ridge, in the line between the two points, as nearly as they can guess, and so that each can at once see the other and the point beyond him. B looks to Z, and by signals puts C CHAP. V.] Obstacles to Measurement. 111 -in line." C then looks to A, and puts B in line at B'. B repeats his operation from B', putting C at C', and is then himself moved to B", and so they alternately " line" each other, continually approximating to the straight line between A and Z, till they at last find themselves both exactly in it, at B"' and C"'. (181) A single person may put himself in line between two points, on the same principle, by laying a straight stick on some support, going to each end of it in turn, and making it point successively to each end of the line. The " Surveyor's Cross," Art. (104), is convenient for this purpose, when set up between the two given points, and moved again and again, until, by repeated trials, one of its slits sights to the given points when looked through in either direction. (182) On Water. A simple instru- Fig. 118. ment for the same object, is represented a in the figure. AB and CD are two tubes, about 11 inches in diameter, con- Mi N nected by a smaller tube EF. A piece I of looking-glass, GH, is placed in the lower part of the tube AB, and another, - KL, in the tube CD. The planes of the two mirrors are at right angles to I each other. The eye is placed at A, and the tube AB is directed to any distant object, as X, and any other object behind the observer, as Z, will be seen, apparently under the first object in the mirror GH, by reflection from the mirror KL, when the observer has succeeded in getting in line between the two objects. M, N, are screws by which the mirror KL may be adjusted. The distance between the two tubes will cause a small parallax, which will, however, be insensible except when the two objects are near together. 112 CHAIN SURVEYING. [PART II. (183) Through a Wood. When a wood intervenes between any two given Fig. 119. points, preventing one B - I seen from the -C' - other, as in the figure, in which A and Z are the given points, proceed thus: Hold a rod at some point B' as nearly in the desired line from A as can be guessed at, and as far from A as possible. To approximate to the proper direction, an assistant may be sent to the other end of the line: and his shouts will indicate the direction; or a gun may be fired there; or, if very distant, a rocket may be sent up after dark. Then range out the " random line" AB', by the method given in Art. (169), noting also the distance from A to each point found, till you arrive at a point Z', opposite to the point Z, i. e. at that point of the line from which a perpendicular there erected would strike the point Z. Mleasure Z'z. Then move each of the stakes, perpendicularly from the line AZ', a distance proportional to their distances from A. Thus, if AZ' be 1000 links, and Z'Z be 10 links, then a stake B', 200 links from A, should be moved 2 links to a point B, which will be in the desired straight line AZ; if C' be 400 links from A, it should be moved 4 links to C, and so with the rest. The line should then be cleared, and the accuracy of the position, of these stakes tested by ranging from A to Z. (184) To an invisible intersection. Let AB and CD be two lines, which, if prolong- Fig. 120. ed, would meet in a B. —%~-.z point Z, invisible from' > X~J either of them; and let - -,E,, P be a point, from which i >d a line is required to be' set out, tending to this, invisible intersection. C Set stakes at the five given points, A, B, C, D, P. Set a sixth stake at E, in the alinements of AD and CP; and a seventh stake CHar. V.] Obstacles to Measurement. 113 at F, in the alinements of BC and AP. Then set an eighth stake at G, in the alinements of BE and DF. PG will be the required line. Otherwise; Through P range out a parallel to the line BD. Note the points where this parallel meets AB and CD, and call these points Q and R. Then the distance from B, on the line BD, to a point which shall be in the required line running from P to the BDxQP invisible point, will be = QR II. OBSTACLES TO MEASUREMENT. (185) The cases, in which the direct measurement of a line is prevented by various obstacles, may be reduced to three. A. When both ends of the line are accessible. B. Whien one end of it is inaccessible. C. When both ends of it are inaccessible. A. WHEN BOTH ENDS OF THE LINE ARE ACCESSIBLE. (186) By Perpendiculars. On Fig. 121. reaching the obstacle, as at A in A D the figure, set off a perpendicular, AB; turn a second right angle at B, B C and measure past the obstacle; turn a third right angle at C; and measure to the original line at D. Then will the measured distance, BC, be equal to the desired distance, AD. If the direction of the line is also unknown, it will be most easily obtained by the additional perpendiculars shown in Fig. 109, of Art. (171). Fig. 121'. (187) By equilateral triangles. A B G The method given in Art. (172), for B determining the direction of a line through an obstacle, will also give its C length; for in Fig. 121' (Fig. 110 re- F peated) the desired distance AGis equal to the measured distances AD, or DG. D 114 CHAIN SURVEYING. [PART II. (188) By symmetrical triangles. Fig. 122. Let AB be the distance required. Measure from A obliquely to some - B point C, past the obstacle. Mea, sure onward, in the same line, till CD is as long as AC. Place stakes at C and D. From B measure to C, and from C measure onward, in E D the same line, till CE is equal to CB. Measure ED, and it will be equal to AB, the distance required. If more convenient, make CD and CE equal, respectively, to half of AC and CB; then will AB be equal to twice DE. (189) By transversals, Let Fig. 123. AB be the required distance. Set A a stake, C, in the line prolonged; X. set another stake, D, so that C and F B can be seen from it; and a third stake, E, in the line of BD prolonged, and at a distance from D equal to the distance from D to B. E Set a fourth stake, F, at the intersection of EA and CD. Measure AC, AF and FE. Then is AB A (FE-AF). Fig. 124. (190) In a Town. Casesmayoccur, \ in the streets of a compactly built town, in which it is impossible to measure along any other lines than those of the streets. The figure represents such a case, in which is required the distance, AB, between points situated on two streets which meet at the point C, and between which runs a cross-street, DE. In this case measure AC, CE, CD, DE and CB. Then is the required distance cnrP. v.] Obstacles to Measurement. 11.5 AB / (AC _ BC)2 + [DE2_ (CE - cD)2 AC xBC} CDxCE As this expression is somewhat complicated, an example will be given: Let AC = 100, CE = 40, CD 30, DE 21, and CB = 80; then will AB= 51.7. B. WHEN ONE END OF THE LINE IS INACCESSIBLE. (191) By Perpendiculars. This principle Fig. 125. may be applied in a variety of ways. In Fig. B 125, let AB be the required distance. At the point A, set off AC, perpendicular to AB, and of any convenient length. At C, set off a perpendicular to CB, and continue it to a point, D, in the line of A and B. Measure DA. Then is AC2 D AB = (192) Otherwise: At the point A, in Fig. Fig. 126. 126, set off a perpendicular, AC. At C set off another perpendicular, CD. Find a point, - E, in the line of AC, and BD. Measure AE, and EC. Then is AB AE x CD If EC be made equal to AE, and D be set in the line of BE, and also in the perpendicular D from C, then will CD be equal to AB. If EC = AE, then CD = AB. Fig. 127. (193) Otherwise: At A, in Fig. 127, measure a perpendicular, AC, to the line AB; and at any point, as D, in this line, set off a perpendicular to DB, and continue it to a point E, in the line of CB. Measure DE and also DA. A Then is AB = AC x AD DE - AC' 116 CHAIN SURVEYING. [PART II. Fig. 128. (194) By Parallels. From A measure AC, in any convenient direction. From a point D, in the line of BC, measure a line parallel to CA, to a point E, in the line of AB. Measure also AE. C/ Then isAB= x AE DE — AC A (195) By a Parallelogram. Set a stake, C, Fig. 129 in the line of A and B, and set another stake, ID, wherever convenient. With a distance equal to CD, describe from A, an arc on the ground; and,, with a distance equal to AC, describe another I arc from D, intersecting the first arc in E. Or, Et[ - - A take AC and CD, so that together tley make,. one chain; fix the ends of the chain at A and D;0 - c take hold of the chain at such a link, that one part of it equals AC, and the other CD, and draw it tight to fix the point E. Set a stake at F, in the intersection of AE and DB. Measure AF and EF. Then is AB -AC AF or, CB xCD EF orCB EF (196) By Symmetrical Triangles. Fig. 130. Let AB be the required distance. From, I A measure a line, in any convenient direction, as AC, and measure onward, in To the same direction, till CD = AC. Take / A any point E in the line of A and B. d' Measure from E to C, and onward in the D same line, till CF = CE. Then find by trial a point G, which shall be at the same time in the line of C and B, and in G the line of D and F. Measure the distance from G to D, and it will be equal to the required distance from A to B. If more convenient, make CD = AC, and CF= ~ CE, as shown by the finely dotted lines in the figure. Then will DG = AB. CHAP. V.] Obstacles to Measurement. 117 (197) Otherwise: Prolong BA to Fig. 131. some point C. Range out any convenient line CA', and measure CA' - _ CA. The triangle CA'B, is now to be reproduced in a symmetrical triangle, / situated on the accessible ground. B' —D For this object, take, on AC, some point D, and measure CD' = CD. Find the c point E, at the intersection of AD' and A'D. Find the point F, at the intersection of A'B and CE. Lastly, find the point B', at the intersection of AF and CA'. Then will A'B' — AB. The symmetrical points have corresponding letters affixed to them. (198) By transversals. Set a stake, C, Fig. 132. in the alinement of BA; a second, D, at any convenient point; a third, E, in the line CD; and a fourth, F, at the intersection of the; alinements of DA and EB. Measure AC, CE, ED, DF and FA. Then is \ AO X AF x DE D / AB = CE x DF - AP x DE' If the point E be taken in the middle of CD, (as it is in the figure) then AB =A x AF DP - AF' AC x DE If the point F be taken in the middle of AD, then AB = CE DE The minus signs must be interpreted as in Art. (174). (199) By harmonic division. Set Fig.133. stakes, C and D, on each side of A, and. so that the three are in the same straight line. Set a third stake at any point, E, of the line AB. Set a fourth, F, at the -F +. G intersection of CB and DE; and a fifth,,/.G, at the intersection of DB and CE. Set a sixth stake, H, at the intersection C / — Dof AB and FG. Measure AE and EH. Then is AB AE x Al AE - ER 118 CHAIN SURVEYING. [PART II. (200) To an inaccessible line. The Fig. 134. shortest distance, CD, from a given point,,&,; C, to an inaccessible straight line AB, is required. From C let fall a perpendicular - to AB, by the method of Art. (158). Then set a stake at any point, E, on the i! / line AC; set a second, F, at the intersection of EB and CD; a third, G, at c the intersection of AF and CB; and a fourth, IH, at the intersection of EG and CD. Measure CH and HF. Then is CCx CF CH+HF CHxCF CD CHIxCF; or CD=CH.;or CD = 2CHCF Otherwise; When the inaccessible line is determined by the method of Art. (205) or (206), the distance from any point to it, can be at once measured to its symmetrical representative. (201) To an inaccessible intersection. When two lines (as AB, CD, in the figure) meet in a Fig. 135. river, a building, or any other inaccessible point, the distance _ _ from any point of either to their A - <+ intersection, DE, for example, may be found thus. From any point B, on one line, measure cAc - BD, and continue it, till DF -IDB. From any other point, G. of the former line, measure GD, and continue the line till DI — GD. Continue HF to meet DC in some point K. Measure KID. KD will be equal to the desired distance DE. BE can be found by measuring FK, which is equal to it. If DF and DH, be made respectively equal to one-half, or onethird, &c., of DB and DG, then will KD and KF be respectively equal to one-half or one-third, &c., of DE and BE. CHAP. V.] Obstacles to Measurement, 119 C. WHEN BOTH ENDS OF THIE LINE ARE INACCESSIBLE. (202) By similar triangles. Let AB Fig. 136. be the inaccessible distance. Set a stake at any convenient point C, and find the distan- D ces CA and CB, by any of the methods just given. Set a second stake at any point, D, on the line-CA. Measure a distance, equal to B x CAD from C, on the line CB, to some point E. Measure AC x DE DE. Then is AB AD-. Fig. 137. If more convenient, measure CD in the A- -.-, 3 contrary direction from the river, as in Fig. - 137, instead of towards it, and in other respects proceed as before. D (203) By parallels. Let AB be the in- Fig. 138. accessible distance. From any point, as C, /. range out a parallel to AB, as in Art. (165), &c. Find the distance CA, by Art. (191), &c. Set a stake at the point E, the intersection of CA and DB, and measure CE. / D C Then is AB = 0D (AC - CE) CE (201) By a parallelogram. Set.Fig. 139. a stake at any convenient point C. A Set stakes D and E, anywhere in the alinements CA and CB. With D as a centre, and a length of the |<' E chain equal to CE, describe an arc; and with E as a centre, and a length c of the chain equal to CD, describe another arc, intersecting the former one at F. A parallelogram, CDEF, will thus be formed. Set stakes at G and H, where the alinements DB and EA intersect the sides of this parallelogram. Measure CD, DF, GF, FH, 120 CHAIN SURVEYING. [PART II. CD x DF x Gil and HG. The inaccessible distance AB - FG FH CDCE x FFI (205) By symmetrical triangles. Take any convenient point, as C. Set stakes at two other Fig. 140. points, D and D', in the same _ _ -L line, and at equal distances = from C. Take a point E, in the line of A.D; measure fr.om / it to C, and onward till CE', > CE. Take a point F in J As the line of BD; measure from, it to C, and onward till CF' - /' a\ CF. Range out the lines AC / and E'D', and set a stake at, /' their intersection, A'. Range -- - out the lines BC and F'D', and set a stake at their intersection, B'. Measure A'B'. It will be equal to the desired distance AB. (206) Otherwise: Take Fig. 141. any convenient point, as C, A\ and set off equal distances 7 on each side of it, in the line of CA, to D and D'. Set \\ I off the same distances from' C, in the line of CB, to E and E'. Through C, set out a parallel to DE, or D'E', and' set stakes at the points F /. and F' where this parallel t, intersects AE' and BD'. n"/ A' Range out the lines AD' and EF', and set a stake at their intersection A'. Range out the lines BE' and DF, and set a stake at their intersection B'. Measure A'B', and it will be equal to the desired distance AB. CHAP. v.] Obstacles to Measurement. 121 The easiest method of setting out the parallel in the above case, is to fix the middle of the chain at the point C, and its ends on the lines CD, CE'; then carry the middle of the chain from C towards F, and mark the point to which it reaches; and repeat this on the other side of C, as shown by the finely dotted lines in the figure. INACCESSIBLE AREAS. (267) Triangles. In the case of a triangular field, in which one side cannot be measured, or determined by any of the methods just given, the two accessible sides may be prolonged to their full length, and an equal symmetrical triangle formed, all of whose sides can be measured. Thus in Fig. 102, page 103, if CDE be the original triangle, of which the side EC is inaccessible, DFP will be equal to it. But if this also be impossible, por- Fig.142. tions of the sides may be measured, asAD, AE, B C in the figure in the margin, and also DE, and the area of this triangle found by any of the D E methods which have been given. Then is the desired area of the triangle ABC = area of AB x A. AD x AE' (208) Quadrilaterals. In the case Fig.143. of a four-sided field, whose sides cannot be measured, or prolonged, but whose B — - diagonals can be' measured, the area, may be obtained thus. Measure the diagonals AC and BD; and also the portions AE, EC, into which one of 1, them is divided by the other. Calculate the area of the triangle BCE, by the preceding method, or any of those heretofore given. Then the area of the quadrilateral AC x BD ABCD = area of BCE x AC x BD BE x CE (209) Polygons. Methods for obtaining the areas of inaccessible fields of more than four sides, have been given in Arts. (101,) &c. PART IIT. COMPASS SURVEYING; OR By the Third Method CHAPTER I. ANGULAR SURVEYING IN GENERAL. (210) Angular Surveying determines the relative positions of points, and therefore of lines, on the THIRD PRINCIPLE, as explained in Art. (7), which should now be referred to. (211) When the two lines which form an angle lie in the same horizontal or level plane, the angle is called a horizontal angle.* When these lines lie in a plane perpendicular to the former, the angle is called a vertical angle. When one of the lines is horizontal and the other line from the eye of the observer passes above the former, and in the same vertical plane, the angle is called an angle of elevation. When the latter line passes below the horizontal line, and in the same vertical plane, the angle is called an angle of depression. When the two lines which form an angle, lie in other planes which make oblique angles with each of the former planes, the angle is called an oblique angle. Horizontal angles are the only angles employed in common land surveying. * A plane is said to be horizontal, or level, when it is parallel to the surface of standing water, or perpendicular to a plumb-line. A line is horizontal when it lies in a horizontal plane. [CHAP. I. Angular Surveying in general. 123 (212) The angles between the directions of two lines, which it is necessary to measure, may be obtained by a great variety of instruments. All of them are in substance mere modifications of the very simple one which will now be described, and which any one can make for himself. (213) Provide a circular piece of Fig. 144. wood, and divide its circumference s (by any of the methods of Geometri-' cal Drafting) into three hundred and sixty equal parts, or "' Degrees," and number them as in the figure. The divisions will be like those of a watch face, but six times as many. These divisions are termed graduations. The figure shows only every fifteenth one. In the centre of the circle, fix a needle, or sharp-pointed wire, and upon this fix a straight stick, or thin ruler placed edge-wise, (called an alidade), so that it may turn freely on this point and nearly touch the graduations of the circle. Fasten the circle on a staff, pointed at the other end, and long enough to bring the alidade to the height of the eyes. The instrument is now complete. It may be called a aoniometer, or Angle-measurer. (214) Now let it be required to measure Fig. 145. the angle between the lines AB and AC. Fix X~ the staff in the ground, so that its centre shall / be exactly over the intersection of the two lines. Turn the alidade, so that it points, (as --. — cdetermined by sighting along it) to a rod, or other mark at B, a point on one of the lines, and note what degree it covers, i. e. " The Reading." Then, without disturbing the circle, turn the alidade till it points to C, a point on the other line. Note the new reading. The difference of these readings, (in the figure, 45 degrees), is the difference in the directions of the two lines, or is the angle which one makes with the other. If the dis 124 COMPASS SURVEYING. [PART III. tance from A to C be now measured, the point C is " determined," with respect to the points A and B, on the Third Principle. Any number of points may be thus determined. (215) Instead of the very simple and rude alidade, which has been supposed to be used, needles may be fixed on each end of the alidade; or sights may be added, such as those described in Art. (103); or a small straight tube may be used, one end being covered with a piece of pasteboard in which a very small eye hole is pierced, and threads, called " cross-hairs," being stretch- Fig. 146. ed across the other end of it, as in the figure; so that (i) their intersection may give a more precise line for determining the direction of any point. (216) When a telescope is substituted for this tube, and supported in such a way that it can turn over, so as to look both backwards and forwards, the instrument (with various other additions, which however do not affect the principle), is called the -Engineer's Transit. With the addition of a level, and a vertical circle, for measuring vertical angles, the instrument becomes a Theodolite; in which, however, the telescope does not usually admit of being turned over. (217) The Compass differs from the instruments which have been described, in the following respect. They all measure the angle which one line makes with another. The compass measures the angle which each of these lines makes with a third line, viz: that shown by the magnetic needle, which always points (approximately) in the same direction, i. e. North and South, Fig. 147. in the Magnetic Meridian. Thus, in the figure, the N line AB makes an angle of 30 degrees with the line AN, and the line AC makes an angle of 75 de- j@7So grees with AN. The difference of these angles, or 45 degrees, is the angle which AC makes with AB, agreeing with the result obtained in Art. (214). S [CHAP. I. Angular Surveying in general. 125 (218) Surveying with the compass is, therefore, a less direct operation than surveying with the Transit or Theodolite. But as the use of the compass is much more rapid and easy (only one sight and reading at each station being necessary, instead of two, as in the former case), for this reason, as well as for its smaller cost, it is the instrument most commonly employed in land surveying in this country, in spite of its imperfections and inaccuracies. As many may wish to learn " Surveying with the Compass," without being obliged to previously learn " Surveying with the Transit," (which properly, being more simple in principle, though less so in practice, should precede it, but which will be considered in Part IV), we will first take up COMPASS SURVEYING. (219) Angular Surveying in general, and therefore Compass Surveying, may employ either of the 3d, 4th and 5th methods of determining the position of a point, given in Part I; that is, any instrument which measures angles may be employed for Polar, Triangular, or Trilinear Surveying. The first of these, Polar Surveying, is the one most commonly adopted for the compass, and is therefore the one which will be specially explained in this part. The same method, as employed with the Transit and Theodolite, will be explained in the following part. The 4th and 5th methods will be explained in the next two parts. (220) The method of Polar Surveying embraces two minor methods. The most usual one consists in going around the field with the instrument, setting it at each corner and measuring there the angle which each side makes with its neighbor, as well as the length of each side. This method is called by the French the method of Chemninement. It has no special name in English, but may be called (from the American verb, To progress), the Method of Progression. The other system, the Method of Radiation, consists in setting the instrument at one point, and thence measuring the direction and distance of each corner of the field, or other object. The corresponding name of what we have called Triangular Surveying is the Method of Intersections; since it determines points by the intersections of straight lines. THE COMPASS. Fig. 198. JIM~~~~~~~~~~~~~~~~~~~I fil ~~~~~~~~~~~~~~~~~~~~~~~~~~~~, —— L 0~i _ - _'F W EL 0 CHAPTER II. THE COMPASS. (221) The Needle. The most essential part of the compass is the magnetic needle. It is a slender bar of steel, usually five or six inches long, strongly magnetized, and balanced on a pivot, so that it may turn freely, and thus be enabled to continue pointing in the same direction (that of the " Maynetic lVeridian," approximately North and South) however much the "Compass Box," to which the pivot is attached, may be turned around. As it is important that the needle should move with the least possible friction, the pivot should be of the hardest steel ground to a very sharp point; and in the centre of the needle, which is to rest on the pivot, should be inserted a cap of agate, or other hard material. Iridium for the pivot, and ruby for the cap, are still better. If the needle be balanced on its pivot before being magnetized, one end will sink, or " Dip," after the needle is magnetized. To bring it to a level, several coils of tire are wound around the needle so that they can be slid along it, to adjust the weight of its two ends, and balance it more perfectly. The North end of the needle is usually cut into a more ornamental form than the South end, for the sake of distinction. *The principal requisites of a compass needle are, intensity of directive force and susceptibility. "' Shear steel" was found by Capt. Kater to be the kind capable of receiving the greatest magnetic force. The best form is that of a rhomboid, Fig. 149. or lozenge, cut out in the middle, so as to dimi-, nish the extent of surface in proportion to the mass, as it is the latter on which the directive force depends. Beyond a certain limit, say five inches, no additional power is gained by increasing the length of the needle. On the contrary, longer ones are apt to have their strength diminished by several consecutive poles being formed. Short needles, made very hard, are therefore to be preferred. 128 COMPASS SURVEYING. [PART III The needle should not come to rest very quickly. If it does, it indicates either that it is weakly magnetized, or that the friction on the pivot is great. Its sensitiveness is indicated by the number of vibrations which it makes in a small space before coming to rest. A screw, with a milled head, on the under side of the plate which supports the pivot, is used to raise the needle off this pivot, when the instrument is carried about, to prevent the point being dulled by unnecessary friction. (222) The Sights. Next after the needle, which gives the direction of the fixed line, whose angles with the lines to be surveyed, are to be measured, should be noticed the Sights, which show the directions of these last lines. At each end of a line passing through the pivot is placed a "' Sight," consisting of an upright bar of brass, with openings in it of various forms; usually either slits, with a circular aperture at their top and bottom*; or of the form described in Art. (106); all these arrangements being intended to enable the line of sight to be directed to any desired object, with precision. (223) A Telescope which can move up and down in a vertical plane, i. e. a plunging telescope, or one which can turn completely over, is sometimes substituted for the sights. It has the great advantage of giving more distinct vision at long distances, and of admitting of sights up and down very steep slopes. Its accuracy of vision is however rendered nugatory by the want of precision in the readings of the needle. If a telescope is applied to the compass, a graduated scale and index should also be added, thus coenverting the compass into a "Transit." The Telescope will be found minutely described in Part IV, " Transit Surveying." (221) The divided circle. We now have the means of indicating the directions of the two lines whose angle is to be measured. The number of degrees contained in it is to be read from a circle, divided into degrees, in the centre of which is fixed the An inside and an outside view, or " Elevation," of such sights, are given onl each side of the figure of the Compass, on page 126. It is itself drawn in "' Military Perspective." CHeP. II.] The Compass. 129 pivot bearing the needle. The graduations are usually made to half a degree, and a quarter of a degree or less can then be " estimated." The pivot and needle are sunk in a circular box, so that its top may be on a level with the needle. The graduations are usually made on the top of the surrounding rim of the box, but' should also be continued down its inside circumference so that it may be easier to see with what division the ends of the needle coincide. The degrees are not numbered consecutively from 00 around to 3603; but run from 00 to 900, both ways from the two diametrically opposite points at which a line, passing through the slits in the middle of the sights, would meet the divided circle. The lettering of' the Surveyor's Compass has one important difference from that of the Mariner's Compass. When we stand facing the North, the East is on our right hand, and the West on our left. The graduated card of the Mariner's Compass which is fastened to the needle, and turns with it, is marked accordingly. But, in the Surveyor's compass, one of the 0 points being marked N, or North, (or indicated by a fleur-delis,) and the opposite one S, or South, the 90-degrees-point on the right of this line, as you stand at the S end and look towards the N, is marked W, or West; and the left hand 90-degrees-point is. marked E, or East. The reason of this will be seen when the method of using the compass comes to be explained in the following chapter. (225) The Points. In or- Fig. 150. dinary land surveying, only four points of the compass have names, viz: North, South, East E a and West; the direction of a,; line being described by the an- Q, b gle which it makes with a North.AW. and South line, to its East or to t bs its West. But for nautical pur- tod poses, the circle of the compass _ is divided into 32 points, the I/ names of which are shown in 9 130 COMPISS SURVEYING. [PART III. the figure. Two rules embrace all the cases. 10 When the letters indicating two points are joined together, the point half way between the two is meant; thus, N. E. is half way between North and East; and N. N. E. is half way between North and North East. 20 When the letters of two points are joined together with the intermediate word by, it indicates the point which comes next after the first, in going towards the second; thus, N. by E, is the point which follows North in going towards the East; S. E. by S. is the next point from South East, going towards the South. (226) Eccentricity. The centre-pin, or pivot of the needle, ought to be exactly in the centre of the graduated circle; the needle ought to be straight; and the line of the sights ought to pass exactly through this centre and through the 0 points of the circle. If this is not the case, there will be an error in every observation. This is called the error of eccentricity. When the maker of a compass is about to fix the pivot in place, he is in doubt of two things; whether the needle is perfectly straight, and whether the pivot is exactly in the cen- Fig. 151. tre. In figures 151 and 152, both of these are represented as being excessively in error. Firstly, to examine if the needle be straight. Fix the pivot temporarily, so that the ends of the needle may cut opposite degrees, i. e. degrees differing by 1800. The condition of things at this Fig15 stage of progress, will be represented by Fig. 151. Then turn the compass-box half way around. The error will now be doubled, as is shown by Fig. 152, in which the former position of the needle is indi- - cated by a dotted line.* Now bend the needle, as in Fig. 153, till it cuts divisions midway between those cut by it in * This is another example of the fruitful princiule of Reverszon, first noticed ill Art. (105). CHAP. II.] The Compass, 131 its present and in its former position Fig 153. This makes it certain that the needle is S straight, or that its two ends and its centre lie in the same straight line. Secondly, to put the pivot in the centre. Move it till the straightened needle cuts opposite divisions. It is then certain that the direction of the needle passes through the centre. Turn the compass box one-quarter around, and if the needle does not then cut opposite divisions, move the pivot till it does. Repeat the operation in various positions of the box. It will be a sufficient test if it cuts the opposite divisions of 00, 450 and 900. To fix the sights precisely in line, draw a hair through their slits and move them till the hair passes over the 0 points on the circle. The surveyor can also examine for himself, by the principle of Reversion, whether the line of the sights passes through the centre or not. Sight to any very near object. Read off the number of degrees indicated by one end of the needle. Then turn the compass half around, and sight to the same object. If the two readings do not agree, there is an error of eccentricity, and the arithmetical mean, or half sum of the two readings is the correct one. Fig. 154. Fig. 155. In Fig. 154, the line of sight AB is represented as passing to one side of the centre, and the needle as pointing to 46~. In Fig. 155, the compass is supposed to have been turned half around and the other end of the sights to be directed to the same object. Suppose that the needle would have pointed to 450, if the line of 132 COMPASS SURVEYING. [PART III. sight had passed through the centre. The needle will now point to 440, the error being doubled by the reversion, and the true reading being the mean. This does not, however, make it certain that the line of the sights passes through the 0 points, which can only be tested by the hair, as mentioned above. (227) Levels. On the compass plate are two small spirit levels. They consist of glass tubes, slightly curved upwards, and nearly filled with alcohol, leaving a bubble of air within them. They are so adjusted that when the bubbles are in the centres of the tubes, the plate of the compass shall be level. One of them lies in the direction of the sights, and the other at right angles to this direction. (228) Tangent Scale. This is a convenient, though not essential, addition to the compass, for the purpose of measuring the slopes of ground, so that the proper allowance in chaining may be made. In the figure of the compass, page 126, may be seen, on the edge of the left hand sight, a small projection of brass with a hole through it. On the edge of the other sight are engraved lines numbered from 00 to 20~, the 00 being of the same height above the compass plate that the eye-hole is. To use this, set the compass at the bottom of a slope, and at the top set a signal of exactly the height of the eye-hole from the ground. Level the compass very carefully, particularly by the level which lies lengthwise, and, with the eye at the eye-hole, look to the signal and note the number of the division on the farther sight which is cut by the visual ray. That will be the angle of the slope; the distances of the engraved lines from the 00 line being tangents (for the radius equal to the distance between the sights) of the angles corresponding to the numbers of the lines. (229) Vernier. The compass box is connected with the plate, which carries it and the sights, so that it can turn around on this plate. This motion is given to it by a screw, (called a slow-motion, or Tangent screw), the head of which is the nearest one in CHAP II.] The Compass. 133 the figure on page 126. If two marks be made opposite to each other, one on the projecting part of the compass box, and the other on the plate to which the sights are fastened, these marks will separate when the slow-motion screw is turned. Their distance apart (in angular measurement, i. e. fractions of a circle), in any position, is measured by a contrivance called a Vernier, which is the minutely divided arc of a circle seen between the left hand sight and the compass box. It will be better to defer explaining the mode of reading the vernier for the present, since it is rarely used with the compass, and an entire chapter will be given to it in Part IV. Its principle is similar to that of the Vernier Scale, described in Art. (50). Its applications in " Field-work" will be noticed under that head. (230) Tripod. The compass, like most surveying instruments, is usually supported on a Tripod, consisting of three legs, shod with iron, and so connected at top as to be movable in any direction. There are many forms of these. Lightness Fig.56. Fig. 157 and stiffness are the qualities desired. The most usual form is shewn in the figures of the Transit and the Theodolite, at the beginning of Part IV. Of the two represented in Figs. 156 and 157, the first has the advantage of being very easily and cheaply made; and the second that of being light and yet capable of very firmly resisting horizontal torsion. The joints, by which the instrument is connected with the tripod, are also various. Fig. 158 is the " Ball-and-socket joint," most usual in this country. It takes its name from the ball, in which 134 COMPASS SURVEYING. [PART III. Fig. 158. Fig. 159. Fig. 160. I'll' __________Illllnlnnl____rt,,,,,,,,,,,,,, _ _, 1111111i_111_ i 1i- T hfi l uihl l -IIIIIII 4!U _ terminates the covered spindle which enters a corresponding cavity under the compass plate, and the socket in which this ball turns. It admits of motion in any direction, and can be tightened or loosened by turning the upper half of the hollow piece enclosing it. which is screwed on the lower half. Fig. 159 is called the " Shelljoint." In it the two shell-shaped pieces enclosing the ball are tightened by a thumb-screw. Fig. 160, is "Cugnot's joint." It consists of two cylinders, placed at right angles to each other, and through the axes of which pass bolts, which turn freely in the cylinder and can be tightened or loosened by thumb-screws at their ends. The combination of the two motions which this joint permits, enables the instrument which it carries, to be placed in any desired position. This joint is much the most stable of the three. (231) Jacob's Staff. A single leg, called a "Jacob's Staff," has some advantages, as it is lighter to carry in the field, and can be made of any wood, on the spot where it is to be used, thus saving the expense of a tripod and the trouble of its transportation. Its upper end is fitted into the lower end of a brass head which has a ball and socket ioint, and axis above. Its lower end should be shod with iron, and a spike running through it is useful for pressing it into the ground with the foot. Of course it cannot be conveniently used on frozen ground, or on pavements. It may, however, be set before or behind the spot at which the angle is to be mea CHaP. II.] The Compass, 135 sured, provided that it is placed very precisely in the line of direction from that station to the one to which a sight is to be taken. (232) The Prismatic Compass. The peculiarity of this instrument (often called Schmalcalder's) is that a glass triangular prism is substituted for one of the sights. Such a prism has this peculiar property that at the same time, it can be seen through, so that a sight can be taken through it, and that its upper surface reflects like a mirror, so that the numbers of the degrees immediately under it, can be read off at the same time that a sight to any object is taken. Another peculiarity, necessary for profiting by the last one, is, that the divided circle is not fixed, but is a card fastened to the needle and moving around with it, as in the Mariner's Compass. The minute description, which follows, is condensed from Simms. In the figure, A repre- Fig. 161. sents the compass box, and B the card, which, being attached to the magnetic E needle, moves as it moves, around the agate centre, a, on which it is suspend- ed. The circumference of the card is usually di- D vided to ~ or - of a de- gree. C is a prism, which 1, the observer looks through. The perpendicular thread, of the sight-vane, E, and the divisions on the card, appear together on looking through the prism, and the division with which the thread coincides, when the needle is at rest, is the "' Bearing" of whatever object the thread may bisect, i. e. is the angle which the line of sight makes with the direction of the needle. The prism is mounted with a hinge joint, D. The sight-vane has a fine thread stretched along its opening, in the direction of its length, which is brought to bisect any object, by turning the box around horizontally. F is a mirror, made to 136 COMPASS SURVEYING. [PART III. slide on or off the sight-vane, E; and it may be reversed at pleasure, that is, turned face downwards; it can also be inclined at any angle, by means of its joint, d; and it will remain stationary on any part of the vane, by the friction of its slides. Its use is to reflect the image of an object to the eye of an observer when the object is much above or below the horizontal plane. The colored glasses represented at G, are intended for observing the sun. At e, is shown a spring, which being pressed by the finger at the time of observation, and then released, checks the vibrations of the card, and brings it more speedily to rest. A stop is likewise fixed to the other side of the box, by which the needle may be thrown off its centre. The method of using this instrument is very simple. First raise the prism in its socket, b, until you obtain a distinct view of the divisions on the card. Then, standing over the point where the angles are to be taken, hold the instrument to the eye, and, looking through the slit, C, turn around till the thread in the sight-vane bisects one of the objects whose bearing is required; then by touching the spring, e, bring the needle to rest, and the division on the card which coincides with the thread on the vane, will be the bearing of the object from the north or south points of the magnetic meridian. Then turn to any other object, and repeat the operation; the difference between the bearing of this object and that of the former, will be the angular distance of the objects in question. Thus, suppose the former bearing to be 400 30', and the latter 100 15', both east, or both west, Fig. 162. from the north or south, the angle will be 300 15'. The divisions are generally numbered 50, 10~, 15, &c. around the circle to 3600~. The figures on the compass card are reversed, or written upside down, as in the figure (in which only every fifteenth degree is mark- B ed), because they are again reversed by the prism, CHAP. II.] The Compass. 137 (233) The prismatic compass is generally held in the hand, the bearing being caught, as it were, in passing; but more accurate readings would of course be obtained if it rested on a support, such as a stake cut flat on its top. In the former mode, the needle never comes completely to rest, particularly in the wind. In such cases, observe the extreme divisions between which the needle vibrates, and take their arithmetical mean. (234) Defects of compass. The compass is deficient in both precision and correctness.* The former defect arises from the indefiniteness of its mode of indicating the part of the circle to which it points. The point of the needle has considerable thickness; it cannot quite touch the divided circle; and these divisions are made only to whole or half degrees, though a fraction of a division may be estimated, or guessed at. The Vernier does not much better this, as we shall see when explaining its use. Now an inaccuracy of one quarter of a degree in an angle, i. e. in the difference of the directions of two lines, causes them to separate from each other 5{ inches at the end of 100 feet; at the end of 1000 feet nearly 41 feet; and at the end of a mile, 23 feet. A difference of only one-tenth of a degree, or six minutes, would produce a difference of 13 feet at the end of 1000 feet; and 91 feet at the distance of a mile. Such are the differences which may result from the want of precision in the indications of the compass. But a more serious defect is the want of correctness in the compass. Its not pointing exactly to the true north does not indeed affect the correctness of the angles measured by it. But it does not point in the same or in a parallel direction, during even the same day, but changes its direction between sunrise and noon nearly a quarter of a degree, as will be fully explained in Chapter VIII. The effect of such a difference we have just- seen. This direction * The student must not confound these two qualities. To say that the sun appears to rise in the eastern quarter of the heavens and to set in the western, is correct, but not precise. A watch with a second hand indicates the time of day precisely, but not always correctly. The statement that two and two make five, is precise, but is not usually regarded as correct. 138 COMPASS SURVEYING. [PART III. may also be greatly altered in a moment, without the knowledge of the surveyor, by a piece of iron being brought near to the compass, or by some other local attraction, as will be noticed hereafter. This is the weak point in the compass. Notwithstanding these defects, the compass is a very valuable instrument, from its simplicity, rapidity and convenience in use; and though never precise, and seldom correct, it is generally not very wrong. CHAPTER III. THE FIELD WORK. (235) Taking Bearings. The " Bearing" of a line is the angle which it makes with the direction of the needle. Thus, in Fig. 147, page 124, the angle NAB is the Bearing of the line AB, and NAC is the Bearing of AC. The Bearing and-length of a line are named collectively the Course. To take the Bearing of any line, set the compass exactly over any point of it by a plumb-line suspended from beneath the centre of the compass, or, approximately, by dropping a stone. Level the compass by bringing the air bubbles to the middle of the level tubes. Direct the sights to a rod held truly vertical, or "' plumb," at another point of the line, the more distant the better. The two ends are usually taken. Sight to the lowest visible point of the rod. When the needle comes to rest, note what division on the circle it points to; taking the one indicated by the North end of the needle, if the North point on the circle is farthest from you, and vice versa. In reading the division to which one end of the needle points, the eye should be placed over the other end, to avoid the error which might result from the " parallax," or, apparent change of place, of the end read from, when looked at obliquely. CHAP. III.] The Field Work. 139 The bearing is read and recorded by noting between what letters the end of the needle comes, and to what number; naming, or writing down, firstly, that letter, N or S, which is at the 00 point nearest to that end of the needle from which you are reading; secondly, the number of degrees to which it points, and thirdly, the letter, E or W, of the 900 point which is nearest to the same end of the needle. Thus, in the figure, if when the sights were directed along a line, (the North Fig. 163. point of the compass being most distant from the observer), the A A North end of the needle was at the point A, the bearing of the line sighted on, would be North 450~ n East; if the end of the needle was at B, the bearing would be East; if at C, S. 300~ E; if at D, South; if at E, S. 60~ W; if at F, West; if at G, N. 60~ W; if at H, North. D (236) We can now understand why W is on the right hand of the compass-box, and E on the left. Let the direction from the centre of the compass to the point Fig. 164. B in the figure, be required, and suppose the sights in the first place to be pointing in the direction of the needle, S N, and the North sight to be ahead. When the sights (and -E........ the circle to which they are fastened) have been turned so as to point I in the direction of B, the point of the circle marked E, will have come round to the North end of the needle, (since the needle remains immovable,) and the reading will therefore be " East," as it should be. The effect on the reading is the same as if the needle had moved to the left the same quantity which the sights have moved to the right, and the left side is therefore properly marked " East," and vice versa. So, too, if the hearing of the line to C be desired, half-way between North and 140 COMPASS SURVEYING. [PART III East, i. e. N. 450 E.; when the sights and the circle have turned 45 degrees to the right, the needle, really standing still, has apparently arrived at the point half-way between N. and E., i. e. N. 450 E. -Some surveyors' compasses are marked the reverse of this, the E on the right and the W on the left. These letters must then be reversed in the mind before the bearing is noted down. (237) Reading with Vernier. When the needle does not point precisely to one of the division marks on the circle, the fractional part of the smallest space is usually estimated by the eye, as has been explained. But this fractional part may be measured by the Vernier, described in Art. (229), as follows. Suppose the needle to point between N. 310 E. and N. 31~0 E. Turn the tangent screw, which moves the compass-box, till the smaller division (in this case 310) has come round to the needle. The Vernier will then indicate through what space the compass-box has moved, and therefore how much must be added to the reading of the needle. Suppose it indicates 10 minutes of a degree. Then the bearing is N. 310 10' E. It is, however, so difficult to move the Vernier without disturbing the whole instrument, that this is seldom resorted to in practice. The chief use of the Vernier is to set the instrument for running lines and making an allowance for the variation of the needle, as will be explained in the proper place. A Vernierare is sometimes attached to one end of the needle and carried around by it. (238) Practical Hints. Mark every station, or spot, at which the compass is set, by driving a stake, or digging up a sod, or piling up stones, or otherwise, so that it can be found if any error, or other cause, makes it necessary to repeat the survey. Very often when the line of which the bearing is required, is a fence, &c., the compass cannot be set upon it. In such cases, set the compass so that its centre is a foot or two from the line, and set the flag-staff at precisely the same distance from the line at the other end of it. The bearing of the flag-staff from the compass will be the same as that of the fence, the two lines being parallel. CHAP. III.] The Field Work. 141 The distances should be measured on the real line. If more convenient the compass may be set at some point on the line prolonged, or at some intermediate point of the line, " in line " between its extremities. In setting the compass level, it is more important to have it level crossways of the sights than in their direction; since if it be not so, on looking up or down hill through the upper part of one sight and the lower part of the other, the line of sight will not be parallel to the N and S, or zero line, on the compass, and an incorrect bearing will therefore be obtained. The compass should not be levelled by the needle, as some books recommend, i. e. so levelled that the ends of the needle shall be at equal distances below the glass. The needle should be brought so originally by the maker, but if so adjusted in the morning, it will not be so at noon, owing to the daily variation in the dip. If then the compass be levelled by it, the lines of sight will generally be more or less oblique, and therefore erroneous. If the needle touches the glass, when the compass is levelled, balance it by sliding the coil of wire along it. The same end of the compass should always go ahead. The North end is preferable. The South end will then be nearest to the observer. Attention to this and to the caution in the next paragraph, will prevent any confusion in the bearings. Always take the readings from the same end of the needle; from the North end, if the North end of the compass goes ahead; and vice versa. This is necessary, because the two ends will not always cut opposite degrees. With this precaution, however, the angle of two meeting lines can be obtained correctly from either end, provided the same one is used in taking the bearings of both the lines. Guard against a very frequent source Fig. 165. of error with beginners, in reading from L I'j the wrong number of the two between which the needle points, such as reading 340 for 26~0, in a case like that in the figure. 142 COMPASS SURVEYING. [PART III. Check the vibrations of the needle by gently raising it off the pivot so as to touch the glass, and letting it down again, by the screw on the under side of the box. The compass should be smartly tapped after the needle has settled, to destroy the effect of any adhesion to the pivot, or friction of dust upon it. All iron, such as the chain, &c., must be kept at a distance from the compass, or it will attract the needle, and cause it to deviate from its proper direction. The surveyor is sometimes troubled by the needle refusing to traverse and adhering to the glass of the compass, after he has briskly wiped this off with a silk handkerchief, or it has been carried so as to rub against his clothes. The cause is the electricity excited by the friction. It is at once discharged by applying a wet finger to the glass. A compass should be carried with its face resting against the side of the surveyor, and one of the sights hooked over his arm. In distant surveys an extra centre pin should be carried, (as it is very liable to injury, and its perfection is most essential), and, also, an extra needle. When two such are carried, they should be placed so that the north pole of one rests against the south pole of the other. (239) When the magnetism of the needle is lessened or destroyed by time, it may be renewed as follows. Obtain two bar magnets. Provide a board with a hole to admit of the axis, so that its collar may fit fairly, and that the needle may rest flat on it, without bearing at the centre. Place the board before you, with the north end of the needle to your right. Take a magnet in each hand, the left holding the North end of the bar, or that which has the mark across, downwards; and the right holding the same mark upwards. Bring the bars over the axis, about a foot above it, without approaching each other within two inches:- bring them down vertically on the needle, (the marks as directed) about an inch on each side of its axis; slide them outwards to its ends with slight pressure; raise them up; bring them to their former position, and repeat this a number of times. CHAP. III.] The Field Work. 143 (210) Back Sights, To test the accuracy of the bearing of a line, taken at one end of it, set up the compass at the other end, or point sighted to, and look back to a rod held at the first station, or point where the compass had been placed originally. The reading of the needle should now be the same as before. If the position of the sights had been reversed, the reading would be the -Reverse Bearing; a former bearing of N. 300 E. would then be S. 300 XV., and so on. (241) Local attraction. If the Back-sight does not agree with the first or forward sight, this latter must be taken over again. If the same difference is again found, this shows that there is local attraction at one of the stations; i. e. some influence, such as a mass of iron ore, ferruginous rocks, &c., under the surface, which attracts the needle, and makes it deviate from its usual direction. Any high object, such as a house, a tree, &c., has recently been found to produce a similar effect. To discover at which station the attraction exists, set the compass at several intermediate points in the line which joins the two stations, and at points in the line prolonged, and take the bearing of the line at each of these points. The agreement of several of these bearings, taken at distant points, will prove their correctness. Otherwise, set the compass at a third station; sight to each of the two doubtful ones, and then from them back to this third station. This will show which is correct. When the difference occurs in a series of lines, such as around a field, or along a road, proceed Fig. 166. thus. Let C be the station at C D which the back-sight to B dif- A- / fers from the foresight from B to C. Since the back-sight from B to A is supposed to have agreed with the foresight from A to B, the local attraction must be at C, and the forward bearing must be corrected by the difference just found between the fore and back sights, adding or subtracting it, according to circumstances. An easy method is to draw a 144 COMPASS SURVEYING. [PART LI. figure for the case, as in Fig. 167. In Fig. 167. it, suppose the true bearing of BC, as N given by a fore-sight from B to C, to be // N. 400 E., but that there is local attraction at C, so that the needle is drawn aside 100, and points in the direction / S'N', instead of SN. The back-sight from C to B will then give a bearing / of N. 500 E.; a difference, or correc- 47 tion for the next fore-sight, of 10~. If the next fore-sight, from C to D, be N. 700 E, this 100 must be subtracted from it, making the true fore-sight N. 600 E. A general rule may also be given. When the back-sight is greater than the fore-sight, as in this case, subtract the difference from the next fore-sight, if that course and the preceding one have both their letters the same (as in this case, both being N. and E.), or both their letters different; or add the difference if either the first or last letters of the two courses are different. Wlen the back-sight is less than the fore-sight, add the difference in the case in which it has just been directed to subtract it, and subtract it where it was before directed to add it. (212) Angles of deflection, When the compass indicates much local attraction, the difference between the directions of two meeting lines, (or the " angle of deflection" of one from the other), can still be correctly measured, by taking the difference of the bearings of the two lines, as observed at the same point. For, the error caused by the local attraction, whatever it may be, affects both bearings equally, inasmuch as a "Bearing" is the angle which a line makes with the direction of the needle, and that here remains fixed in some one direction, no matter what, during the taking of the two bearings. Thus, in Fig. 167, let the true bearing of BC, i. e. the angle which it makes with the line SN, be, as before, N. 400 E., and that of CD N. 600 E. The true " angle of deflection" of these lines, or the angle B'CD, is therefore 200~. Now, if local attraction at C causes the needle to point in the direcS'N', 100 to the left of its proper direction, BC will bear N. 500 CHAP. III.] The Field Work, 145 E., and CD N. 700 E., and the difference of these bearings, i. e. the angle of deflection, will be the same as before. (243) Angles between Courses. To determine the angle of deflection of two courses meeting at any point, the following simple rules, the reasons of which will appear from the accompanying figures, are sufficient. Fig. 168. Case 1. When the first letters of the bearing are alike, (i. e. both N. or both o / S.), and the last letters also alike, (i. e. both E. or both W.), take the difference /,/ of the bearings. Example. If AB bears r —--— / - N. 30~ E. and BC bears N. 100 E., the angle of deflection CBB' is 20~. /0.Fig. 169. N Case 2. When the first letters are alike and the last letters different; take the sum of the bearings. Ex. If AB bears N. 400 E. and BC bears N. 200~... -E W.; the angle CBB' is 600. / Fig. 170. N Case 3. When the first letters are / different and the last letters alike, sub-, / tract the sum of the bearings from 180~. w ------------- --- - Ex. If AB bears N. 300 E. and BC bears S. 400 E.; the angle CBB' is 1100. 10 146 COMPASS SURVEYING. [PART III. Fig. 170. Case 4. When both the first and last letters are different, subtract the / difference of the bearings from 180~. /C Ex. If AB bears S. 300 W. and BC -. -.. —. bears N. 700 E.; the angle CBB' is / 1400. / If the angles included between the courses are desired, they will be at once found by reversing one bearing, and then applying the above rules; or by subtracting the results obtained as above from 1800; or an analogous set of rules could be formed for them. (244) To change Bearings. It is convenient in certain calculations to suppose one of the lines of a survey to change its direction so as to become due North and South; that is, to become a new Meridian line. It is then necessary to determine what the bearings of the other lines will be, supposing them to change with it. The subject may be made plain by supposing the survey to be platted in the usual way, with the North uppermost, and the plat to be then turned around, till the line to be changed is in the desired direction. The effect of this on the other lines will be readily seen. A General Rule can also be formed. Take the difference between the original bearing of the side which becomes a Meridian and each of those bearings which have both their letters the same as it, or both different from it. The changed bearings of these lines retain the same letters as before, if they were originally greater than the original bearing of the new Meridian line; but, if they were less, they are thrown on the other side of the N. and S. line, and their last letters are changed; E. being put for W. and W for E. Take the sum of the original bearing of the new Meridian line, and each of those bearings which have one letter the same as one letter of the former bearing, and one different. If this sum exceeds CHAP. III.] The Field Work. 147 90~, this shews that the line is thrown on the other side of the East or West point, and the difference between this sum and 1800 will be the new bearing and the first letter will be changed, N. being put for S. and S. for N. Example. Let the Bearings of the sides of a field be as follows: N. 320 E.; N. 800 E.; S. 480 E.; S. 180 W.; N. 73o0 W.; North. Suppose the first side to become due North; the changed bearings will then be as follows: North; N. 480 E.; S. 800 E.; S. 140 E.; S. 7410 W.; N. 320 W. To apply the rule to the " North" course, as above, it must be called N. 0~ W.; and then by the Rule, 320 must be added to it. The true bearings can of course be obtained from the changed bearings, by reversing the operation, taking the sum instead of the difference, and vice versa. (245) Line Surveying. This name may be given to surveys of lines, such as the windings of a brook, the curves of a road, &c., by way of distinction from Farm Surveying, in which the lines surveyed enclose a space. To survey a brook, or any similar line, set the compass at, or near, one end of it, and take the bearing of an imaginary or visual line, running in the general average direction of the brook, Fig. 172. uha inhefigure. Meas.urehislifseeB such as AB in the figure. Measure this line, taking offsets to the various bends of the brook, as to the fence explained in Art. (115). Then set the compass at B, and take a back-sight to A, and if they agree, take a fore-sight to C, and proceed as before, noting particularly the points where the line crosses the brook. To survey a road, take the bearings and lengths of the lines Fig. 173. ~c i I 1 r(~ —— ~-~ ~D 148 COMPASS SURVEYING. [PART III. which can be most conveniently measured in the road, and mea, sure offsets on each side, to the outside of the road. When the line of a new road is surveyed, the bearings and lengths of the various portions of its intended centre line should be measured, and the distance which it runs through each man's land should be noted. Stones should be set in the ground at recorded distances from each angle of the line, or in each line prolonged a known distance, so as not to be disturbed in making the road. In surveying a wide river, one bank may be surveyed by the method just given, and points on the opposite banks, as trees, &c., may be fixed by the method of intersections, founded on the Fourth Method of determining the position of a point; and fully explained in Part IV. (246) Checks by intersecting bearings. At each station at which the compass is set, take bearings to some remarkable object, such as a church steeple, a distant house, a high tree, &c. At least three bearings should be taken to each object to make it of any use: since two are necessary to determine it, (by our Fourth Method), and, till thus determined, it can be no check. When the line is platted, by the methods to be explained in the next chapter, plat also the lines given by these bearings. If those taken to the same object from three different stations, intersect in the same point, this proves that there has been no mistake in the survey or platting of those stations. If any bearing does not intersect a point fixed by previous bearings, it shows that there has been an error, either between the last station and one of those which fixed the point, or in the last bearing to the point. To discover which it was, plat the following line of the survey, and, at its extremity, set off the bearing from it to the point; and if the line thus platted passes through the point, it proves that there was no error in the line, but only in the bearing to the point. If otherwise, the error was somewhere in the line between the stations from which the bearings to that point were taken. caAP. III.] The Field Work. 149 (247) Keeping the Field-notes. The simplest and easiest method for a beginner is to make a rough sketch of the survey by eye, and write down on the lines their bearings and lengths. An improvement on this is to actually lay down the precise bearings and lengths of the lines in the field-book in the manner to be explained in the chapter on Platting, Art. (269). (248) A second method is to draw a straight line up the page of the field-book, and to write on it the bearings and lengths of the lines. The only advantage of this method is that the line will not run off the side of the page, as it is apt to do in the preceding method. (249) A third method is to represent the line surveyed, by a double column, as in Part II, Chapter I, Art. (95), which should be now referred to. The bearings are written obliquely up the columns. At the end of each course, its length is written in the column, and a line drawn across it. Dotted lines are drawn across the column at any intermediate measurement. Offsets are noted as explained in Art. (114). The intersection-bearings, described in Art. (246), should be entered in the field-book before the bearings of the line, in order to avoid mistakes of platting, in setting off the measured distances on the wrong line. (250) A fourth method is to write the Stations, Bearings, and Distances in three columns. This is compact, and has the advantage, when applied to farm surveying, of presenting a form suitable for the subsequent calculations of Content, but does not give facilities for noting offsets. Examples of these four methods are given in Art. (254); which contains the field-notes of the lines bounding a field. (251) New-York Canal Maps. The following is a description of the original maps of the survey of the line of the New-York Erie Canal, as published by the Canal Commissioners. The figure represents a portion of such a map; but, necessarily, with all its lines black; red and blue lines being used on the real map. 150 COMPASS SURVEYING. [PART III. Fig. 174. |SG WiDE S 2'E 30 ES 3* 4A0 ES 52~SJ " The RED LINE described along the inner edge of the towing path is the base line, upon which all the measurements in the direction of the length of the canal were made. The bearings refer to the magnetic meridian at the time of the survey. The lengths of the several portions are inserted at the end of each, in chains and links. The offsets at each station are represented by red lines drawn across the canal in such a direction as to bisect the angles formed by the two contiguous portions of the red or base line, upon the towing path. The intermediate offsets are set off at right angles to the base line; and the distances on both are given from it in links. The intermediate offsets are represented by red dotted lines, and the distances to them upon the base line are reckoned, in each case, from the last preceding station. The same is likewise done with the other distances upon the base line; those to the Bridges being taken to the lines joining the nearest angles, or corner posts of their abutments; those to the Locks extending to the lines passing through the centres of the two nearest quoin posts; and those to the Aqueducts, to the faces of their abutments. The space enclosed by the BLUE LINES represents the portion embraced within the limits of the survey as belonging to the state; and the names of the adjoining proprietors are given as they stood at the time of executing the survey. The distances are projected upon a scale of two chains to the inch." (252) Farm Surveying. A farm, or field, or other space included within known lines, is usually surveyed by the compass thus. Begin by walking around the boundary lines, and setting stakes at all the corners, which the flag-man should specially note, CHAP. III.] The Field Work. 151 so that he may readily find them again. Then set the compass at any corner, and send the flag-man to the next corner. Take the bearing of the bounding line, running from corner to corner, which is usually a fence. Measure its length, taking offsets if necessary. Note where any other fence, or road, or other line, crosses or meets it, and take their bearings. Take the compass to the end of this first bounding line; sight back, and if the back-sight agrees, take the bearing and distance of the next bounding line; and so proceed till you have got back to the point of starting. (253) Where speed is more important than accuracy in a survey, whether of a line or a farm, the compass need be set only at every other station, taking a forward sight, from the Ist station to the 2d; then setting the compass at the 3d station, taking a backsight to the 2d station (but with the north point of the compass always ahead), and a fore-sight to the 4th; then going to the 5th, and so on. This is, however, not to be recommended. (254) Field-notes. The Field-notes of a Farm survey may be kept by any of the methods which have been described with reference to a Line survey. Below are given the Field-notes of the same field recorded by each of the methods. F'irst Method. Fig. 175. N 83'0 _E 0~~~~~~i 4Z T ~`c Y~~~~i ~~~A 0z'd.5~ 152 COMPASS SURVEYING. [*PART III Second Third Fourth Method. Mlethod. Method.* 3.23 STATIONS.| BEARINGS. DISTANCES. IN.l 1 N. 350 E. 2.70 us C13 mic2 N. 8310 E. 1.29'0 i S. 570 E. 2.22 (5) 4 S. 34koW. 3.55 5. 56 W. 3.23 Fig. 176. 0(4) -0.90-1 2.22,zci~~ ~-0.25 —6 1.29 (2) 00 o -(2)2.70 0(1) m -i —-. 0_ (255) The Field-notes of a field, in which offsets occur, may be most easily recorded by the Third Method; as in Fig, 176. When the Field-notes are recorded by the Fourth Method, the offsets may be kept in a separate Table; in which the Ist column will contain the stations from which the measurements are made, the 2d column the distances at which they occur, the 3d * In the " Third Method," the bearings should be written obliquely upward, as directed in Art, (249), but are not so printed here, from typographical difficulties. CHAP. III.] The Field Work. 153 column the lengths of the offsets, and the 4th column the side of the line, " Right," or " Left," on which they lie. For calculation, four more columns may be added to the table, containing the intervals between the offsets; the sums of the adjoining pairs; and the products of the numbers in the two preceding columns, separated into Right and Left, one being additive to the field, and the other subtractive. (256) Tests of accuracy. Ist. The check of intersections described in Art. (246), may be employed to great advantage, when some conspicuous object near the centre of the farm can be seen from most of its corners. 2nd. When the survey is platted, if the last course meets the starting point, it proves the work, and the survey is then said to " close." 3d. Diagonal lines, running from corner to corner of the farm, like the " Proof-lines" in Chain Surveying, may be measured and their bearings taken. When these are laid down on the plat, their meeting the points to which they had been measured, proves the work. 4th. The only certain and precise test is, however, that by "Latitudes and Departures." This is fully explained in Chapter V, of this Part. (257) A very fallacious test is recommended by several writers on this subject. It is a well-known proposition of Geometry, that in any figure bounded by straight lines, the sum of all the interior angles is equal to twice as many right angles, as the figure has sides less two; since the figure can be divided into that number of triangles. Hence this common rule. "Calculate [by the last paragraph of Art. (243)] the interior angles of the field or farm surveyed; add them together, and if their sum equals twice as many right angles as the figure has sides less two, the angles have been correctly measured." This rule is not applicable to a compass survey; for, in Fig. 167, page 144, the interior angle BCD will contain the same number of degrees (in that case 1600) whether the bearings of the sides have been noted correctly, as being the 154 COMPASS SURVEYING. [PART III. angles which they make with NS-or incorrectly, as being the angles which they make with N'S'. This rule would therefore prove the work in either case. (258) Method of Radiation. A field may be surveyed from one station, either within it or without it, by taking the bearings and the distances from that point to each of' the corners of the field. These corners are then " determined," by the 3d method, Art. (7). This modification of that method, we named, in Art. (220), the Method of Radiation. All our preceding surveys with the compass have been by the Method of Progression. The compass may be set at one corner of the field, or at a point in one of its sides, and the same method of Radiation employed. This method is seldom used however, since, unlike the method of Progression, its operations are not checks upon each other. (259) Method of Intersection. A field. may also be surveyed by measuring a base line, either within it or without it, setting the compass at each end of the base line, and taking, from each end, the bearings of each corner of the field; which will then be fixed and determined, by the 4th method, Art. (8). This mode of surveying is the Method of Intersections, noticed in Art. (220). It will be fully treated of in Part V, under the title of Triangular Surveying. (260) Running out old lines. The original surveys of lands in the older States of the American Union, were exceedingly deficient in precision. This arose from two principal causes; the small value of land at the period of these surveys, and the want of skill in the surveyors. The effect at the present day is frequent dissatisfaction and litigation. Lots sometimes contain more acres than they were sold for, and sometimes less. Lines which are straight in the deed, and on the map, are found to be crooked on the ground. The recorded surveys of two adjoining farms often make one overlap the other, or leave a gore between them. The most difficult and delicate duty of the land-surveyor, is to run out these old boundary lines. In such cases, his first business is to find CHAP. III.] The Field Work. 155 monuments, stones, marked trees, stumps, or any other old " corners," or landmarks. These are his starting points. The owners whose lands join at these corners should agree on them. Old fences must generally be accepted by right of possession; though such questions belong rather to the lawyer than to the surveyor.' His business is to mark out on the ground the lines given in the deed. When the bounds are given by compass-bearings, the surveyor must be reminded that these bearings are very far from being the same now as originally, having been changing every year. The method of determining this important change, and of making the proper allowance, will be found in Chapter VIII, of this Part. (261) Town Surveying. Begin at the meeting of two or more of the principal streets, through which you can have the longest prospects. Having fixed the instrument at that point, and taken the bearings of all the streets issuing from it, measure all these lines with the chain, taking offsets to all the corners of streets, lanes, bendings, or windings; and to all remarkable objects, as churches, markets, public buildings, &c. Then remove the instrument to the next street, take its bearings, and measure along the street as before, taking offsets as you go along, with the offset-staff. Proceed in this manner from street to street, measuring the distances and offsets as you proceed. Fig. 177. In the description of land conveyed, the rule is, that known and fixed mon* " In the description of land conveyed, the rule is, that known and fixed monuments control courses and distances. So, the certainty of metes and bounds will include and pass all the lands within them, though they vary from the given quantity expressed in the deed. In New-York, to remove, deface or alter landmalrks maliciously) is ail indictable offence."-Kent's Commentaries, IV, 515. 156 COMPASS SURVEYING. [PART III. Thus, in the figure, fix the instrument at A, and measure lines in the direction of all the streets meeting there, noting their bearings; then measure AB, noting the streets at X, X. At the second station, B, take the bearings of all the streets which meet there; and measure from B to C, noting the places and the bearings of all the cross-streets as you pass them. Proceed in like manner from C to D, and from D to A, " closing" there, as in a farm survey. Having thus surveyed all the principal streets in a particular neighborhood, proceed then to survey the smaller intermediate streets, and last of all, the lanes, alleys, courts, yards, and every other place which it may be thought proper to represent in the plan. The several cross-streets answer as good check lines, to prove the accuracy of the work. In this manner you continue till you take in all the town or city. (262) Obstacles in Compass Surveying. The various obstacles which may be met with in Compass Surveying, such as woods, water, houses, &c., can be overcome much more easily than in Chain Surveying. But as some of the best methods for effecting this involve principles which have not yet been fully developed, it will be better to postpone giving any of them, till they can be all treated of together; which will be done in Part VII. CHAPTER IV. PLATTING THE SURVEY. (263) The platting of a survey made with the compass, consists in drawing on paper the lines and the angles which have been measured on the ground. The lines are drawn " to scale," as has been fully explained in Part I, Chapter III. The manner of platting angles was referred to in Art. (41), but its explanation has been reserved for this place. (264) With a Protractor. A Protractor is an instrument made for this object, and is usually a semicircle of brass, as in the figure, with its semi-circumference divided into 180 equal parts, or Fig. 178. degrees, and numbered in both directions. It is, in fact, a miniature of the instrument, (or of half of it), with which the angles have been measured. To lay off any angle at any point of a straight line, place the Protractor so that its straight side, the diameter of the semi-circle, is on the given line, and the middle of this diameter, which is marked by a notch, is at the given point. With a needle, or sharp pencil, make a mark on the paper at the required number of degrees, and draw a line from the mark to the given point. 158 COMPASS SURVEYING. [PART III. Sometimes the protractor has an arm turning on its centre, and extending beyond its circumference, so that a line can be at once drawn by it when.it is set to the desired angle. A Vernier scale is sometimes added to it to increase its precision. A Rectangular Protractor is sometimes used, the divisions of degrees being engraved along three edges of a plane scale. The semi-circular one is preferable. The objection to the rectangular protractor is that the division corresponding to a degree is very Fig. 179.,_ N\ \\\ 5 \00I/ /\/1 -7- 1/7/ / 7 - _i 5:_ iri.~ l&oPaon idii}l 0 S-W i unequal on different parts of the scale, being usually two or three times as great at its ends as at its middle. A Protractor embracing an entire circle, with arms carrying verniers, is also sometimes employed, for the sake of greater accuracy. (265) Platting Bearings, Since " Bearings" taken with the Compass are the angles which the various lines make with the M.Iagnetic Meridian, or the direction of the compass-needle, which, as we have seen, remains always (approximately) parallel to itself, it is necessary to draw these meridians through each station, before laying off the angles of the bearings. The T square, shown in Fig. 14, is the most convenient instrument for this purpose. The paper on which the plat is to be made is fastened on the board so that the intended direction of the North and South line may be parallel to one of the sides of the board. The inner side of the stock of the T square being pressed against one of the other sides of the board and slid along, the edge of the long blade of the square will always be parallel to itself and to the first named side of the board, and will thus represent the meridian passing through any station. CHAP. IV.] Platting the Survey. 159 If a straight-edged drawing Fig. 180. board or table cannot be pro- cured, nail down on a table of any shape a straight-edged ru- _ lert, and slide along against it the outside of the stock of a T square, one side of the stock being flush with the blade. A parallel ruler may also be used, one part of it being screwed down to the board in the proper position. If none of these means are at hand, approximately parallel meridians may be drawn by the edges of a common ruler, at distances apart equal to its width, and the diameter of the protractor made parallel to them by measuring equal distances between it and them. (266) To plat a survey with these instruments, mark, with a fine point enclosed in a circle, a convenient spot in the paper to represent the first station, 1 in the figure. Its place must be so chosen Fig. 181. 83' K me,~~~~ 160 COMPASS SURVEYING. [PART -III. that the plat may not "run off" the paper. With the T square draw a meridian through it. The top of the paper is usually, though not necessarily, called North. With the protractor lay off the angle of the first bearing, as directed in Art. (264). Set off the length of the first line, to the desired scale, by Art. (42), from 1 to 2. The line 1 --— 2 represents the first course. Through 2, draw another meridian, lay off the angle of the second course, and set off the length of this course, from 2 to 3. Proceed in like manner for each course. When the last course is platted, it should end precisely at the starting point, as the survey did, if it were a closed survey, as of a field. If the plat does not " close," or" come together," it shows some error or inaccuracy either in the original survey, if that have not been " tested" by Latitudes and Departures, or in the work of platting. A method of correction is explained in Art. (268). The plat here given is the same as that of Fig. 175, page 151. This manner of laying down the directions of lines, by the angles which they make with a meridian line, has a great advantage, in both accuracy and rapidity, over the method of platting lines by the angles which each makes with the line which comes before it. In the latter method, any error in the direction of one line makes all that follow it also wrong in their directions. In the former, the direction of each line is independent of the preceding line, though its position would be changed by a previous error. Instead of drawing a meridian through each station, sometimes only one is drawn, near the middle of the sheet, and all the bearings of the survey are laid off from some one point of it, as shown in the figure, and numbered to correspond with the stations from which these bearings were taken. The circular protractor is convenient for this. They are then transferred to the places where they are wanted, by a triangle or other parallel ruler, as explained on page 27. The figure at the top of the next page represents the same field platted by this method. A semi-circular protractor is sometimes attached to the stock end of the T square, so that its blade may be set at any desired angle with the meridian, and any bearing be thus protracted without drawing a meridian. It has some inconveniences. CHAP Iv.] Platting the Survey. 161 Fig. 182. 3 s 5 (267) The Compass itself may be used to plat bearings. For this purpose it must be attached to a square board so that the N and S line of the compass box may be parallel to two opposite edges of the board. This is placed on the paper, and the box is turned till the needle points as it did when the first bearing was taken. Then a line drawn by one edge of the board will be in a proper direction. Mark off its length, and plat the next and the succeeding bearings in the same manner. (268) When the plat of a survey does not " close," it may be corrected as follows. Let Fig. 183. ABCDE be the boundary A lines platted according to - O the given bearings and g/ B. C. distances, and suppose that, the last course comes to E, Ic~ / instead of ending at A, as /_ / it should. Suppose also that there is no reason to E/ suspect any single great n error, and that no one of the lines was measured over very rough 11 162 COMPASS SURVEYING. [PART11:In. ground, or was specially uncertain in its direction when observed. The inaccuracy must then be distributed among all the lines in proportion to their length. Each point in the fig re, B, C, D, E, must be moved in a direction parallel to EA, by a certain distance which is obtained thus. Multiply the distance EA by the distance AB, and divide by the sum of all the courses. The quotient will be the distance BB'. To get CC', multiply EA by AB + BC, and divide the product by the same sum of all the courses. To get DD', multiply EA by AB + BC + CD, and divide as before. So for any course, multiply by the sum of the lengths of that course and of all those preceding it, and divide as before. Join the points thus obtained, and the closed polygon AB'C'D'A will thus be formed, and will be the most probable plat of the given survey.* The method of Latitudes and Departures, to be explained hereafter, is, however, the best for effecting this object. (269) Field Platting. It is sometimes desirable to plat the courses of a survey in the field, as soon as they are taken, as was mentioned in Art. (247), under the head of " IKeeping the fieldnotes." One method of doing this is, is to have the paper of the Field-book ruled with parallel lines, at unequal distances apart, and to use a rectangular pro- Fig. 184. tractor (which may be made __ of Bristol-board, or other stout ] __ ~ ___ drawing paper,) with lines rul- ed across it at equal distances — / 0 of some fraction of an inch. A bearing having been taken and i-t noted, the protractor is laid on the paper and its centre placed at the station where the bearing is to be laid off. It is then turned till one of its cross-lines coincides with some one of the lines on the paper, which represent East and West lines. The long side of the protractor will then be on a meridian and the proper angle (40~ in the figure) can be at once marked off. The length of the course can also be set off by the equal spaces between the cross-lines, letting each space represent any convenient number of links. " This was demonstrated by Dr. BOWDITCH, in No. 4, Of " The Analyst." OHzAP. IV.] Platting the Survey, 163 (270) A common rectangular protractor without any cross-lines, or a semi-circular one, can also Fig. 185. be used for the same purpose. T The parallel lines on the paper (which, in this method, may be equi-distant, as in common o ruled writing paper) will now represent meridians. Place the centre of the protractor on the meridian nearest to the I. station at which the angle is to be laid off, and turn it till the given number of degrees is cut by the meridian. Slide the protractor up or down the meridian (which must continue to pass through the centre and the proper degree) till its edge passes through the station, and then draw by this edge a line, which will have the bearing required. (271) Paper ruled into squares, (as are sometimes the righthand pages of surveyors' field-books), may be used for platting bearings in the field. The lines running up the page may be called North and South lines, and those running across the page will then be East and West lines. Any course of the survey will be the hypothenuse of a right-angled triangle, and the ratio of its other two sides will determine the Fig.186. angle. Thus, if the ratio of _ B the two sides of the right-an- _ _ gled triangle, of which the line I AB in the figure is the hypoth- enuse, is 1, that line makes an angle of 450 with the meridian. - - If the ratio of the long to the short side of the right-angled - triangle of which the line AC - -i - - is the hypothenuse, is 4 to, l 1 7 I the line AC makes an angle -] _ 1 - of 14~ with the meridian. The line AD, the hypothenuse of an 164 COMPASS SURVEYING. [PART III. equal triangle, which has its long side lying East and West, makes likewise an angle of 14~ with that side, and therefore makes an angle of 760 with the meridian.' To facilitate the use of this method, the following table has been prepared. TABLE FOR PLATTING BY SQUARES. O of 0 Ratio of | Ratio of o long side to o long sideto t longfide to o 50 short side. short side. short side. side 10 57.3 to 1 89 16~0 3.49 to i 740 310 1.664 to 1 590 20 28.6 to 1 880 17~0 3.27 to 1 730 32~ 1.600 to 1 580 30 19.1 to 1 870 180 3.08 to 1 720 330 1.540 to 1 570 40 1.4.3 to 1 860 19~ 2.90 to 1 710 340 1.483 to 11 560 5~ 11.4 to 1 85~ 20~ 2.75 to 1 700 3501.428 to 11550 6~ 9.5 to 1 840 210 2.61 to 1 690 36C 1.376 to 1 540 70 8.1 to 1 830 220 2.48 to 1 680 370 1.327 to 1 530 80 7.1 to 1 82~ 230~ 2.36 to 1 67~ 380 1.280 to 11 520 90 6.3 to 1 81~ 24~ 2.25 to 1 660 390 1.235 to 1 510 100 5.7 to 1 800~ 250~ 2.14 to 1 651 400 1.192 to 1 500 110 5.1 to 1 790 260 2.05 to 1 640 410 1.150 to 11490 12~ 4.9 to 1 780~ 270~ 1.96 to 11630 42~ 1.111 to 11480 130 4.3 to 1 770 28~ 1.88 to 1 620 430 1.072 to 11 47 140 4.0 to 1 760 290 1.80 to 1 61 440 1.036 to 11460 150 3.7 to 1 750 300 1.73 to 1 600 451 1.000 to 11450 To use this table, find in it the ratio corresponding to the angle which you wish to plat. Then count, on the ruled paper, any number of squares to the right or to the left of the point which represents the station, according as your bearing was East or West; and count upward or downward according as your bearing was North or South, the number of squares given by multiplying the first number by the ratio of the Table. Thus; if the given bearing from A in the figure, was N. 200 E. and two squares were counted to the right, then 2 x 2.75 = 5-i squares, should be counted upward, to E, and AE would be the required course. (272) With a paper protractor. Engraved paper protractors may be obtained from the instrument-makers, and are very conve* This and all the following ratios may be obtained directly from Trigonometrical Tables; for the ratio of the long side to the short side, the latter being taken as unity, is the natural cotangent of the angle. CHAP. IV.] Platting the Survey. 165 nient. A circle of large size, divided into degrees and quarters, is engraved on copper, and impressions from it are taken on drawing paper. The divisions are not numbered. Draw a straight line to represent a meridian, through the centre of the circle, in any convenient direction. Number the degrees from 0 to 90~, each way from the ends of this meridian, as on the compass-plate. The protractor is now ready for Fig. 187. use. Choose a convenient N point for the first station. Suppose the first bearing to be N. 30~ E. The line passing through the centre of the 2 E circle and through the opposite points N. 30~ E. and S. / 300 W. has the bearing re- ld/ quired. But it does not pass s through the station 1. Transfer it thither by drawing through station 1 a line parallel to it, which will be the course required, its proper length being set off on it from 1 to 2. Now suppose the bearing from 2 to be S. 60~ E. Draw through 2 a line parallel to the line passing through the centre of the circle and through the opposite points S. 60~ E., and N. 600 W., and it will be the line desired, On it set off the proper length from 2 to 3, and so proceed. When the plat is completed, the engraved sheet is laid on a clean one, and the stations " pricked through," and the points thus obtained on the clean sheet are connected by straight lines. The pencilled plat is then rubbed off from the engraved sheet, which can be used for a great number of plates. If the central circle be cut out, the plat, if not too large, can be made directly on the paper where it is to remain. The surveyor can make such a paper protractor for himself, with great ease, by means of the Table of Chords at the end of this volume, the use of which is explained in Art. (275). The engraved ones may have shrunk after being printed. Such a circle is sometimes drawn on the map itself. This will be particularly convenient if the bearings of any lines on the map, 166 COMPASS SURVEYING. [PART III. not taken on the ground, are likely to be required. If the map be very long, more than one may be needed. (273) Drawing-Board Protractor. Such a divided circle, as has just been described, or a circular protractor, may be placed on a drawing board near its centre, and so that its 0~ and 900 lines are parallel to to the sides of the drawing board. Lines are then to be drawn, through the centre and opposite divisions, by a ruler long enough to reach the edges of the drawing board, on which they are to be cut in, and numbered. The drawing board thus becomes, in fact, a double rectangular protractor. A strip of white paper may have previously been pasted on the edges, or a narrow strip of white wood inlaid. When this is to be used for platting, a sheet of paper is put on the board as usual, and lines are drawn by a ruler laid across the 0~ points and the 90~ points, and the centre of the circle is at once found, and should be marked (D. The bearings are then platted as in the last method. (274) With a scale of chords. On the plane scale contained in cases of mathematical drawing instruments will be found a series of divisions numbered from 0 to 90, and marked C H, or C. This is a scale of chords, and gives the lengths of the chords of any arc for a radius equal in length to the chord of 60~ on the scale. To lay off an angle with this scale, as for Fig. 188. example, to draw a line making at A an angle B of 40~ with AB, take, in the dividers, the dis- C = tances from 0 to 60 on the scale of chords; with. this for radius and A for centre, describe an in- X definite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the are as a chord, from C to some point D. Join AD, and A prolong it. BAE is the angle required. The Sector, represented on page 36, supplies a modification of this method, sometimes more convenient. On each of its legs is a scale marked C, or CH. Open it at pleasure; extend the compass from 60 to 60, one on each leg, and with this radius describe an arc. Then extend the compasses from 40 to 40, and the dis caAP. Iv.] Platting the Survey. 167 tance will be the chord of 40~ to that radius. It can be set off as above. The smallness of the scale renders the method with a scale of chords practically deficient in exactness; but it serves to illustrate the next and best method. (2'75) With a Table of chords. At the end of this volume will be found a Table of the lengths of the chords of arcs for every degree and minute of the quadrant, calculated for a radius equal to 1. To use it, take in the compasses one inch, one foot, or any other convenient distance (the longer the better) divided into tenths and hundredths, by a diagonal scale, or otherwise. With this as radius describe an are as in the last case. Find in the table of chords the length of the chord of the desired angle. Take it from the scale just used, to the nearest decimal part which the scale will give. Set it off as a chord, as in the last figure, and join the point thus obtained to the starting point. This gives the angle desired. The superiority of this method to that which employs a protractor, is due to the greater precision with which a straight line can be divided than can a circle. A slight modification of this method is to take in the compasses 10 equal parts of any convenient length, inches, half inches, quarter inches, or any other at hand, and with this radius describe an arc as before, and set off a chord 10 times as great as the one found in the Table, i. e. imagine the decimal point moved one place to the right. If the radius be 100 or 1000 equal parts, imagine the decimal point moved two, or three, places to the right. Whatever radius may be taken or given, the product of that radius into a chord of the Table, will give the chord for that radius. This gives an easy and exact method of getting a right angle; by describing an arc with a radius of 1, and setting off a chord equal to 1.4142. If the angle to be constructed is more than 90~, construct on the other side of the given point, upon the given line prolonged, an angle equal to what the given angle wants of 180~; i. e. its Stupplement, in the language of Trigonometry. 168 COMPASS SURVEYING. [PART III. This same Table gives the means of measuring any angle. With the angular point for a centre, and 1, or 10, for a radius, describe an arc. Measure the length of the chord of the arc between the legs of the angle, find this length in the Table, and the angle corresponding to it is the one desired.* (276) With a Table of natural sines. In the absence of a Table of chords, heretofore rare, a table of natural sines, which can be found anywhere, may be used as a less convenient substitute. Since the chord of any angle equals twice the sine of half the angle, divide the given angle by two; find in the table the natural sine of this half angle; double it, and the product is the chord of the whole angle. This can then be used precisely as was the chord in the preceding article. An ingenious modification of this method has been much used. Describe an arc from the given point as centre, as in the last two articles, but with a radius of 5 equal parts. Take, from a Table, the length of the natural sine of half the given angle to a radius of 10. Set off this length as a chord on the arc just described, and join the point thus obtained to the given point.t (277) By Latitudes and Departures. When the Latitudes and Departures of a survey have been obtained and corrected, (as explained in Chapter V), either to test its accuracy, or to obtain its content, they afford the easiest and best means of platting it. The description of this method will be given in Art. (285). * This Table will also serve to find the natural sine, or cosine, of any angle. Multiply the given angle by two; find, in the Table, the chord of this double angle; and half of this chord will be the natural sine required. For, the chord of any angle is equal to twice the sine of half the angle. To find the cosine, proceed as above, with the angle which added to the given angle would make 90~. Another use of this Table is to inscribe regular polygons in a circle by setting off the chords of the arcs which their sides subtend. Still another use is to divide an arc or angle into any number of equal parts, by setting off the fractional arc or angle. Fig. 189. The reason of this is apparent from the figure. DE is the sine of half the angle BAc, to a radius of 10 equal parts, and BC is the chord directed to be set off, to a radius of 5 cqual parts. BO is equal to DE; for BC = 2.BF, by Trigonometry, and DE P / =2.BF, by similar triangles; hence BC = O DE. CHAPTER V. LATITUDES AND DEPARTURES. (278) Definitions. The LATITUDE of a point is its distance North or South of some " Parallel of Latitude," or line running East or West. The LONGITUDE of a point is its distance East or West of some "'Meridian," or line running North and South. In Compass-Surveying, the Magnetic Meridian, i. e. the direction in which the Magnetic Needle points, is the line from which the Longitudes of points are measured, or reckoned. The distance which one end of a line is due North or South of the other end, is called the Difference of Latitude of the two ends of the line; or its Naorthing or Southing; or simply its Latitude. The distance which one end of the line is due East or West of the other, is here called the Difference of Longitude of the two ends of the line; or its Easting or Westing; or its D)eparture. Latitudes and Departures are the most usual terms, and will be generally used hereafter, for the sake of brevity. This subject may be illustrated geographically, by noticing that a traveller in going from New-York to Buffalo in a straight line, would go about 150 miles due north, and 250 miles due west. These distances would be the differences of Latitude and of Longitude between the two places, or his Northing and Westing. Returning from Buffalo to New-York, the same distances would be his Southing and Easting.* In mathematical language, the operation of finding the Latitude and Longitude of a line from its Bearing and Length, would be called the transformation of Polar Co-ordinates into Rectangular Co-ordinates. It consists in determining, by our Second Principle, the position of a point which had originally been determined by the Third Principle. Thus, in the figure, (which is the same as * It should be remembered that the following discussions of the Latitudes and Longitudes of the points of a survey will not always be fully applicable to those of distant places, such as the cities just named, in consequence of the surface of the earth not being a plane. 170 COMPASS SURVEYING. [PART III. that of Art.(9)),the point S is determin- Fig. 190. ed by the angle SAC and by the dis- tance AS. It is also determined by the distances AC and CS, measured at right /-\ angles to each other; and then, supposing A - CS to run due North and South, CS will be the Latitude, and AC the _Departure of the line AS. (279) Calculation of Latitudes and Departures. Let AB be a given line, of which the length Fig. 191. AB, and the bearing (or angle, BAC, N which it makes with the Magnetic Meridian), are known. It is required to find the differences of Latitude and of Longitude between its two extremities A and B: that is, to find AC and CB; or, what is the same thing, BD --..-. —-- E and DA. It will be at once seen that AB is the hypothenuse of a right-angled tri- s angle, in which the " Latitude" and the " Departure" are the sides about the right angle. We therefore know, from the principles of trigonometry, that AC AB. cos. BAC, BC -- AB. sin. BAC. Hence, to find the Latitude of any course, multiply the natural cosine of the bearing by the length of the course; and to find the Departure of any course, multiply the natural sine of the bearing by the length of the course. If the course be Northerly, the Latitude will be North, and will be marked with the algebraic sign +, plus, or additive; if it be Southerly, the Latitude will be South, and will be marked with the algebraic sign -, minus, or subtractive. If the course be Easterly, the D)eparture will be East, and marked +, or additive; if the course be Westerly, the Departure will be West, and marked -, or subtractive. cHAP. V.] Latitudes and Departures. 171 (280) Formulas. The rules of the preceding article may be expressed thus; Latitude = Distance x cos. Bearing, Departure = Distance x sin. Bearing.* From these formulas may be obtained others, by which, when any two of the above four things are given, the remaining two can be found. When the Bearing and Latitude are given; Distance = coL j = Latitude x sec. of Bearing, Departure = Latitude x tang. of Bearing. When the Bearing and )Departure are given; Distance Departure = Departure x cosec. Bearing, sin. Bearing Latitude = Departure x cotang. Bearing. When the.Distance and Latitude are given; Cos. Bearing = Latitude COS. Bearmg-Distance) Departure = Latitude x tang. Bearing. 7When the -Distance and D)eparture are given; Sin. Bearing = Departure ~Sinm~. B eDistance Latitude = Departure x cotang. Bearing. When the Latitude and -Departure are given; Tang. of Bearing = Departure Latitude' Distance - Latitude x sec. Bearing. Still more simply, any two of these three-Distance, Latitude and Departure -being given, we have Distance =,/(Latitude2 + Departure2) Latitude = /(Distance2 - Departure2) Departure = V (Distance2 - Latitude2) (281) Traverse Tables. The Latitude and Departure of any distance, for any bearing, could be found by the method given in Art. (279), with the aid of a table of Natural Sines. But to W* henever sines, cosines, tangents, &c., are here named, they mean the natu ral sines, &c., of an arc described with a radius equal to one, or to the unit by which the sinies, &c., are measured. 172 COMPASS SURVEYING. [PART III. facilitate these calculations, which are of so frequent occurrence and of so great use, Traverse Tables have been prepared, originally for navigators, (whence the name Traverse), and subsequently for surveyors.' The Traverse Table at the end of this volume gives the Latitude and Departure for any bearing, to each quarter of a degree, and for distances from 1 to 9. To use it, find in it the number of degrees in the bearing, on the left hand side of the page, if it be less than 45~, or on the right hand side if it be more. The numbers on the same line running across the page,t are the Latitudes and Departures for that bearing, and for the respective distances —i, 2, 3, 4, 5, 6, 7, 8, 9,which are at the top and bottom of the page, and which may represent chains, links, rods, feet, or any other unit. Thus, if the bearing be 15~, and the distance 1, the Latitude would be 0.966 and the Departure 0.259. For the same bearing, but a distance of 8, the Latitude would be 7.727, and the Departure 2.071. Any distance, however great, can have its Latitude and Departure readily obtained from this table; since, for the same bearing, they are directly proportional to the distance, because of the similar triangles which they form. Therefore, to find the Latitude or Departure for 60, multiply that for 6 by 10, which merely moves the decimal point one place to the right; for 500, multiply the numbers found in the Table for 5, by 100, i. e. move the decimal point two places to the right, and so on. Merely moving the decimal point to the right, one, two, or more places, will therefore enable this Table to give the Latitude and Departure for any decimal multiple of the numbers in the Table. For compound numbers, such as 873, it is only necessary to find separately the Latitudes and Departures of 800, of 70, and of 3, and add them together. But this may be done, with scarcely any risk of error, by the following simple rule. * The first Traverse Table for Surveyors seems to have been published in 1791, by John Gale. The most extensive table is that of Capt. Boileau, of the British army, being calculated for every minute of bearing, and to five decimal places, for distances fiom 1 to 10. The Table in this volume was calculated for it, and then compared with the one just mentioned. t In using this or any similar Table, lay a ruler across the page, just above or below the line to be followed out. This is a very valuable mechanical assistance. CHAP. v.] Latitudes and Departures. 173 Write down the Latitude and Departure for the first figure of the given number, as found in the Table, neglecting the decimal point; write under them the Latitude and Departure of the second figure, setting them one place farther to the right; under them write the Latitude and Departure of the third figure, setting them one place farther to the right, and so proceed with all the figures of the given number. Add up these Latitudes and Departures, and cut off the three right hand figures. The remaining figures will be the Latitude and Departure of the given number in links, or chains, or feet, or whatever unit it was given in. For example; let the Latitude and Departure of a course having a distance of 873 links, and a bearing of 20~, be required. In the Table find 200, and then take out the Latitude and Departure for 8, 7 and 3, in turn, placing them as above directed, thus: Distances. Latitudes. Departures. 800 7518 2736 70 6578 2394 3 2819 1026 873 820.399 298.566 Taking the nearest whole numbers and rejecting the decimals, we find the desired Latitude and Departure to be 820 and 299.' When a 0 occurs in the given number, the next figure must be set two places to the right, the reason of which will appear from the following example, in which the 0 is treated like any other number. Given a bearing of 350, and a distance of 3048 links. Distances. Latitudes. Departures. 3000 2457 1721 000 0000 0000 40 3277 2294 8 6553 4589 3048 2496.323 1748.529 Here the Latitudes and Departures are 2496 and 1749 links. * It is frequently doubtful, in many calculations, when the final decimal is 5, whether to increase the preceding figure by one or not. Thus, 43.5 may be called 43 or 44 with equal correctness. Itiis better in such cases niot to increase the whole number, so as to escape the trouble of changing the original figure, and.the increased chance of error. If, however, more than one such a case occurs in the same column to be added up, the larger and smaller number should be taken alternately. 174 COMPASS SURVEYING. [PART III. When the bearing is over 45~, the names of the columns must be read from the bottom of the page, the Latitude of any bearing, as 50~, being the Departure of the complement of this bearing, or 400, and the Departure of 40~ being the Latitude of 500, &c. The reason of this will be at once seen on inspecting the last figure, (page 170), and imagining the East and West line to become a Meridian. For, if AC be the magnetic meridian, as before, and therefore BAC be the bearing of the course AB, then is AC the Latitude, and CB the Departure of that course. But if AE be the meridian and BAD (the complement of BAC) be the bearing, then is AD (which is equal to CB) the Latitude, and DB, (which is equal to AC), the Departure. As an example of this, let the bearing be 6.3 ~, and the distance 3469 links. Proceeding as before, we have -Distances. Latitudes. Departures. 3000 1350 2679 400 1800 3572 60 2701 5358 9 4051 8037 3469. 1561.061 3097.817 The required Latitude and Departure are 1561 and 3098 links. In the few cases occurring in Compass-Surveying, in which the bearing is recorded as somewhere between the fractions of a degree given in the Table, its Latitude and Departure may be found by interpolation. Thus, if the bearing be 100~, take the half sum of the Latitudes and Departures for 10 I and 1010. If it be 10~ 20', add one-third of the difference between the Lats. and Deps. for 10 and for 10 ~, to those opposite to 10~~; and so in any similar case. The uses of this table are very varied. The principal applications of it, which will now be explained, are to Testing the accuracy of surveys; to Supplying omissions in them; to Platting them, and to Calculating their content.' " The Traverse Table admits of many other minor uses. Thus, it may be used for solving, approximately, any right-angled triangle by mere inspection, the bearing being taken for one of the acute angles; the Latitude being the side adjacent, the Departure the side opposite, and the Distance the hypothenuse. Any two of these being given, the others are given by the Table. The Table will therefore serve to show the allowance to be made in chaining on slopes (see Art. CHAP. v.] Latitudes and Departures. 175 (282) Application to Testing a Survey. It is self-evident, that when the surveyor has gone completely around a field or farm, taking the bearings and distances of each boundary line, till he has got back to the starting point, that he has gone precisely as far South as North, and as far West as East. But the sum of the North Latitudes tells how far North he has gone, and the sum of the South Latitudes how far South he has gone. Hence these two sums will be equal to each other, if the survey has been correctly made. In like manner, the sums of the East and of the West Departures must also be equal to each other. We will apply this principle to testing the accuracy of the survey of which Fig. 175, page 151, is a plat. Prepare seven columns, and head them as below. Find the Latitude and Departure of each course to the nearest link, and write them in their appropriate columns. Add up these columns. Then will the difference between the sums of the North and South Latitudes, and between the sums of the East and West Departures, indicate the degree of accuracy of the survey. LATITUDE. DEPARTURE. STATION. BEARING. DISTANCE. EPI_ N. S. UIE. W. 1 N. 350 E. 2.70 2.21 1.55 2 N. 83-~0E. 1.29.15 1.28 3 S. 570 E. 2.22 1.21 1.86 4 S. 340 XW. 3.55 2.93 2.00 5 N. 56!0 W. 3.23 1.78 1 2.69 4.14 4.14 4.69 4.69 The entire work of the above example is given below. 350 1638 1147 34~o 2480 1688 57340 40150 4133 2814 ---— _ —- 4133 2814 270. 221.140 154:850..355. 293.463 199.754 (26)); for, look in the column of bearings for the slope of the ground, i. e. the angle it makes with the horizon, find the given distance, and the Latitude corresponding will be the desired horizontal measurement, and the difference between it and the Distance will be the allowance to be made. 176 COMPASS SIJRVEYING. [PART mII 8310 113 994 5610 1656 2502 226 1987 2 1104 1668 1019 8942 1656 2502 129. 14.579 128.212 323. 178.296 269.382 570 1089 1677 57~ 1089 1677 The nearest link is taken 1089 1677 1089 1677 to be inserted in the Table, and the remaining Decimals 222. 120.879 186.147 are neglected. Q In the preceding example the respective sums were found to be exactly equal. This, however, will rarely occur in an extensive survey. If the difference be great, it indicates some mistake, and the survey must be repeated with greater care; but if the difference be small it indicates, not absolute errors, but only inaccuracies, unavoidable in surveys with the compass, and the survey may be accepted. How great a difference in the sums of the columns may be allowed, as not necessitating a new survey, is a dubious point. Some surveyors would admit a difference of 1 link for every 3 chains in the sum of the courses: others only 1 link for every 10 chains. One writer puts the limit at 5 links for each station; another at 25 links in a survey of 100 acres. But every practical surveyor soon learns how near to an equality his instrument and his skill will enable him to come in ordinary cases, and can therefore establish a standard for himself, by which he can judge whether the difference, in any survey of his own, is probably the result of an error, or only of his customary degree of inaccuracy, two things to be very carefully distinguished.' (283) Application to supplying omissions. Any two omissions in the Field-notes can be supplied by a proper use of the method of Latitudes and Departures;. as will be explained in Part VII, which treats of " Obstacles to Measurement," under which head this subject most appropriately belongs. But a knowledge of the fact that any two omissions can be supplied, should not lead * A French writer fixes the allowable difference in chaining at 1-400 of level lines; 1-200 of lines on moderate slopes; 1-100 of lines on steep slopes. CHAPV.] Latitudes and Departures. 177 the young surveyor to be negligent in making every possible measurement, since an omission renders it necessary to assume all the notes taken to be correct, the means of testing them no longer existing. (284) Balancing a Survey. The subsequent applications of this method require the survey to be previously Balanced. This operation consists in correcting the Latitudes and Departures of the courses, so that their sums shall be equal, and thus " balance." This is usually done by distributing the differences of the sums among the courses in proportion to their length; saying, As the sum of the lengths of all the courses Is to the whole difference of the Latitudes, So is the length of each course To the correction of its Latitude. A similar proportion corrects the Departures.* It is not often necessary to make the exact proportion, as the correction can usually be made, with sufficient accuracy, by noting how much per chain it should be, and correcting accordingly. In the example given below, the differences have purposely been made considerable. The corrected Latitudes and Departures have been here inserted in four additional columns, but in practice they should be written in red ink over the original Latitudes and Departures, and the latter crossed out with red ink. CORRECTEDICORRECTED LATITUDES. DEPTIJRES. ATITUDES DEP TURES. STA. BEARING. DIST. _ATITUDS DPTURS. _______N.+ S.- E.+ W.- N.+I S.- E.+-\W.1 N. 52 E. 10.63 6.55 8.38 6.58 8.33 2 S. 291 E. 4.10 3.55 2.02 3.54 2.001 3 S. 311 W. 7.69 6.53j 4.04 6.51 1 4.07 4 N. 61 W. 7.13 3.45 6.23 3.47 6.26 29i5 10.00 10.08 10.40 10.27 10.05 10.05 10.33 10.33 The corrections are made by the following proportions. For the Latitudes. For the Departures. 29.65: 10.63:: 8 3 29.65: 10.63 13: 5 29.65: 4.10 8: 1 29.65: 4.10:: 13: 2 29.65: 7.69:: 8: 2 29.65: 7.69:: 13: 3 29.65: 7.13: 8: 2 29.65: 7.13:: 13: 3 * A demlonstration of this principle was given by Dr. Bowditch, in No. 4. of "The Analyst." 12 178 COMPASS SURVEYING. [PART III. This rule is not always to be strictly followed. If one line of a survey has been measured over very uneven and rough ground, or if its bearing has been taken with an indistinct sight, while the other lines have been measured over level and clear ground, it is probable that most of the error has occurred on that line, and the correction should be chiefly made on its Latitude and Departure. If a slight change of the bearing of a long course will favor the Balancing, it should be so changed, since the compass is much more subject to error than the chain. So, too, if shortening any doubtful line will favor the Balancing, it should be done, since distances are generally measured too long. (285) Application to Platting. Rule three columns; one for Stations; the next for total Latitudes; and the third for total Departures. Fill the last two columns by beginning at any convenient station (the extreme East or West is best) and adding up (algebraically) the Latitudes of the following stations, noticing that the South Latitudes are subtractive. Do the same for the Departures, observing thatthe Westerly ones are also subtractive. Taking the example given on page 175, Art. (282), and beginning with Station 1, the following will be the results: TOTAL LATITUDES TOTAL DEPARTURES STA.FROM STATION 1. FRO()I STATION 1. 1 0.00 0.00 2 +2.21 N. + 1.55 E. 3 +2.36 N. +2.83 E. 4 +1.15 N. +4.69 E. 5 -1.78 S. +2.69 E. 1 0.00 0.00 It will be seen that the work proves itself, by the total Latitudes and Departures for Station 1, again coming out equal to zero. To use this table, draw a meridian through the point taken for Station 1, as in the figure on the following page. Set off, upward from this, along the meridian, the Latitude, 221 links, to A, and from A, to the right perpendicularly, set off the Departure, 155 links.* This gives the point 2. Join 1....2. From 1 again, set This is most easily done with the aid of a right-augled trianl)gl, slidilg one of the sides acljacent to the right angle along the blade of the square, to wllicl the other side wilt then be perpendicular. CHAP. V.] Latitudes and Departures. 179 off, upward, 236 Fig. 192. links, to B, and from B, to the right, per- pendicularly, set off A 283 links, which will fix the point 3. Join 2....3; and so pro- c A- 4 ceed, setting off North Latitudes along the Meridian IL upwards, and South Latitudes along it downwards; East Departures perpendicularly to the right, n and West Depar-. tures perpendicularly to the left. The advantages of this method are its rapidity, ease and accuracy; the impossibility of any error in platting any one course affecting the following points; and the certainty of the plat "coming together," if the Latitudes and Departures have been "Balanced." CHAPTER VI. CALCULATING THE CONTENT. (286) Methods. WHEN a field has been platted, by whatever method it may have been surveyed, its content can be obtained from its plat by dividing it up into triangles, and measuring on the plat their bases and perpendiculars; or by any of the other means explained in Part I, Chapter IV. But these are only approximate methods; their degree of accuracy depending on the largeness of scale of the plat, and the skill of the draftsman. The invaluable method of Latitudes and Departures gives another means, perfectly accurate, and not requiring the previous preparation of a plat. It is sometimes called the IRectangular, or the Pennsylvania, or Rittenhouse's, method of calculation.' (287) Definitions. Imagine a Meridian line to pass through the extreme East or West corner of a field. According to the definitions established in Chapter V, Art. (278), (and here recapitulated for convenience of reference), the perpendicular distance of each Station from that Meridian, is the Longitude of that Station; additive, or plus, if East; subtractive, or minus, if West. The distance of the middle of any line, such as a side of the field, from the Meridian, is called the Longitude of that side.t The difference of the Longitudes of the two ends of a line is called the D)eparture of that line. The difference of the Latitudes of the two ends of a line is called the Latitude of the line. * It is, however, substantially the same as Mr. Thonmas Burgh's "Method to determine the areas of right lined figures universally," published nearly a century ago. t The phrase " Meridian Distance," is generally used for what is here called "Longitude"; but the analogy of" Differences of Longitude" with " Differences of Latitude," usually but anomalously united with the word " Departure," borrowed from Navigation, seems to put beyond all question the propriety of the innovation here introduced. CHAP. VI.] Calculating the Content. 181 (288) Longitudes. To give more definiteness to the develop ment of this subject, the figure in the margin will be referred to, and may be considered to represent any space enclosed by straight lines. Let NS be the Meridian passing through the extreme Westerly Station of the field ABCDE. From Fig. 193. the middle and ends of each side N C draw perpendiculars to the Meridi- --- an. These perpendiculars will be —: L —. the Longitudes and Departures of I the respective sides. The Longitude, FG, of the first course, AB, is evidently equal to half its Depar- a ture HB. The Longitude, JK, of the second course, BC, is equal to A JL + LM + MK, or equal to the Longitude of the preceding course, / plus half its Departure, plus half the Departure of the course itself. --- The Longitude, YZ, of some other course, as EA, taken anywhere, is S equal to WX- VX - UV, or equal to the Longitude of the preceding course, minus half its Departure, minus half the Departure of the course itself; i. e. equal to the Algebraic sum of these three parts, remembering that Westerly Departures are negative, and therefore to be subtracted when the directions are to make an Algebraic addition. To avoid fractions, it will be better to double each of the preceding expressions. We shall then have a GENERAL RULE FOR FINDING DOUBLE LONGITUDES. The Double longitude of the FIRST COURSE is equal to its Departure. The Double Longitude of the SECOND COURSE is equal to the Double Longitude of the first course, plus the Departure of that course, plus the Departure of the second course. The Double Longitude of the THIRD COURSE is equal to the Double Longitude of the second course, plus the Departure of that course, plus the Departure of the course itself. 182 COMPASS SURVEYING. [PART III. The Double Longitude of ANY course is equal to the -Double Longitude of the preceding course, plus the Departure of that course, plus the Departure of the course itself.* The Double Longitude of the last course (as well as of the first) is equal to its Departure. Its " coming out" so, when obtained by the above rule, proves the accuracy of the calculation of all the preceding Double Longitudes. (289) Areas. We will now proceed to find the Area, or Content of a field, by means of the " Double Longitudes" of its sides, which can be readily obtained by the preceding rule, whatever their number. (290) Beginning with a three-sided field, ABC in the figure, draw a Meridian through A, and draw perpendi- Fig. 194. culars to it as in the last figure. It is N plain that its content is equal to the differ- i ence of the areas of the Trapezoid DBCE, D B and of the Triangles ABD and ACE. \ The area of the Triangle ABD is equal A to the product of AD by half of DB, or to H —- ------— \ the product of AD by FG; i. e. equal to the product of the Latitude of the Ist course by its Longitude. The area of the Trapezoid DBCE is equal t C to the product of DE by half the sum of DB and CE, or by HJ; i. e. to the product of S the Latitude of the 2d course by its Longitude. The area of the Triangle ACE is equal to the product of AE by half EC, or by KL; i. e. to the product of the Latitude of the 3d course by its Longitude. Calling the products in which the Latitude was North, North Products, and the products in which the Latitude was South, South Products, we shall find the area of the Trapezoid to be a South Product, and the areas of the Triangles to be North Pro* The last course is a" preceding course" to the first course, as will appear on remembering that these two courses join each other on the ground. CHAP. VI.] Calculating the Content. 183 ducts. The lDifference of the North Products and the South Products is therefore the desired area of the three-sided field ABC. Using the -Double Longitudes, (in order to avoid fractions), in each of the preceding products, their difference will be the double area of the Triangle ABC. (291) Taking now a four-sided field, ABCD in the figure, and drawing a Meridian and Longitudes as be- Fig. 195. fore, it is seen, on inspection, that its area NX would be obtained by taking the two Trian- k B gles, ABE,ADG, from the figure EBCDGE, or from the sum of the two Trapezoids EBCF --- and FCDG. ----------- The area of the Triangle AEB will be found, as in the last article, to be equal to the product of the Latitude of the 1st course -- by its Longitude. The Product will be North. The area of the Trapezoid EBCF will be _, found to equal the Latitude of the 2d course by its Longitude. The product will be South. The area of the Trapezoid FCDG will be found to equal the product of the Latitude of the 3d course by its Longitude. The product swill be South. The area of the Triangle ADG will be found to equal the product of the Latitude of the 4th course by its Longitude. The product will be North. The difference of the North and South products will therefore be the desired area of the four-sidedfield ABCD. Using the Double Longitude as before, in each of the preceding products, their difference will be double the area of the field. (292) Whatever the number or directions of the sides of a field, or of any space enclosed by straight lines, its area will always be equal to half of the difference of the North and South Products 184 COMPASS SURVEYING. [PART III. arising from multiplying together the Latitude and Double Longitude of each course or side. We have therefore the following GENERAL RULE FOR FINDING AREAS. 1. Prepare ten columns, headed as in the example below, and in the first three write the Stations, Bearings and D)istances. 2. _Find the Latitudes and Departures of each course, by the Traverse Table, as directed in Art. (281), placing them in the four following columns. 3. Balance them, as in Art. (284), correcting them in red ink. 4. _Find the D)ouble Longitudes, as in Art. (288), with reference to a Meridian passing through the extreme East or West Station, and place them in the eighth column. 5. Multiply the Double Longitude of each course by the corrected Latitude of that course, placing the North Products in the ninth column, and the South Products in the tenth column. 6. Add up the last two columns, subtract the smaller sum from the larger, and divide the difference by two. The quotient will be the content desired. (293) To find the most Easterly or Westerly Station of a survey, without a plat, it is best to make a rough hand-sketch of the survey, drawing the lines in an approximation to their true directions, by drawing a North and South, and East and West lines, and considering the Bearings as fractional parts of a right angle, or 90~; a course N. 45~ E. for example, being drawn about half way between a North and an East direction; a course N. 28~ W. being not quite one-third of the way around from North to West; and so on, drawing them of approximately true proportional lengths. (294) Example 1, given below, refers to the five-sided field, of which a plat is given in Fig. 175, page 151, and the Latitudes and Departures of which were calculated in Art. (282), page 175. Station 1 is the most Westerly Station, and the Meridian will be supposed to pass through it. The Double Longitudes are best CHAP. VI.] Calculating the Content. 185 found by a continual addition and subtraction, STA. as in the margin, where they are marked D. L. 1 + 1.55 D. L. ~ 1.55 The Double Longitude of the last course comes + 1.28 out equal to its Departure, thus proving the 2 + 4.38 D. L. + 1.28 work. + 1.86 The Double Longitudes being thus obtained, 3 + 7.52 D. L. + 1.86 are multiplied by the corresponding Latitudes, - 2.00 and the content of the field obtained as directed 4 + 7.38 D. L. -2.00 in the General Rule. - 2.69 This example may serve as a pattern for the 5 +2.69 D. LI most compact manner of arranging the work. DIS LATITUDES. DEP'TURNS DOUBLE DOUBLE AREAS. STATION. BEARINGS. TANCEs. N. + S.- E.+ W.- LONGITUDES. N. + S. - 1 N. 350 E. 2.70 2.21 1.55 + 1.55 3.4255 2 N. 83io E. 1.29.15 1.28 + 4.38 0.6570 3 S. 570 )V. 2.22 1.21 1.86 + 7.52 9.0992 4 S. 3410 W. 3.55 2.93 2.00 + 7.38 21.6234 5 N. 56~O W. 3.23 1.78 2.69 ~- 2.69 4.7882 4.14 1 4.141 4.69 1 4.691 118.8707 130.7226 8.8707 C'ontent — 1A. OR. 15P. 2)21.8519 Square Chains, 10.9259 (295) The Meridian might equally well have STA, been supposed to pass through the most Easterly 4 - 2.00 D. L. ZD VIL ~MrrVNVII,V J~WNIV ~~~ILV — 2.00 station, 4 in the figure. The Double Longitudes - 2.69 could then have been calculated as in the mar- 5 - 6.69 D. L. -- 2.69 gin. They will of course be all West, or minus. + 1.55 The products being then calculated, the sum of 1 - 7.83 D. L. + 1.55 the North products will be found to be 29.9625, + 1.28 and of the South products 8.1106, and their 2 - 5.00 D. L. -+- 1.28 difference to be 21.8519, the same result as be- + 1.86 fore. 3 -1.86 (296) A number of examples, with and without answers, will now be given as exercises for the student, who should plat them by some of the methods given in the preceding chapter, using each of them at least once. He should then calculate their content by the method just given, and check it, by also calculating the area of the plat by some of the Geometrical or Instrumental methods given in Part I, Chapter IV; for no single calculation is ever reliable. 186 COMPASS SURVEYING. [PART III. All the examples (except the last) are from the author's actual surveys. 3 a Fig. 196. Example 2, given below, is also fully worked out, as anoth- / er pattern for the student, who need have no difficulty with any possible case if he strictly follows the directions which have been given. The plat is on a scale of 2 chains to 1 inch, (= 1: 1584). 6 DIS- LATITUDES. DEP'TURES. DOUBLE DOUBLE AREAS. STATION. BEARINGS. TANCES. N. + i S.- E.+ W. —LONGITUDES. N. + S. - ~1 N. 12~ E. 2.81 2.75.60 + 6.56 18.0400 2 N. 760 W. 3.20.77 3.11 + 4.05 3.1185 3 S. 24j~ W. 1.14 1.04.47 +.47.4888 4 S. 480 E. 1.53 1.02 1.14 + 1.14 1.1628 5 S. 12j~ E. 1.12 1.09.24 + 2.52 2.7468 6 S. 77~ E 1.64.i.37 1.60 + 4.36 1.6132 3.52 1 3.521 3.58 1 3.58 1 21.158516.0116 6.01 16 Content-OA. 3R. 1P. 2)15.1469 Square Chains, 7.5734.Example 3. -Example 4. STA. BEAAING. DISTANCE. STA. BEARING. DISTANCE. 1 N. 520 E. 10.64 1 S. 21~ W. 12.41 2 S. 2930 E. 4.09 2 N. 831~E. 5.86 3 S. 3130 W. 7.68 3 N. 120 E. 8.25 4 N. 610 W. 7.24 4 N. 470 W. 4.24 Ans. 4A. 3R. 28P. Ans. 4A. 2R. 37P. -Example 5. Example 6. STA. BEARING. DISTANCE. STA. BEARING: mDISTANCE. 1 N. 34 0E. 2.73 1 N. 350 E. 6.49 2 N. 850 E. 1.28 2 S. 561~ E. 14.15 3 -S. 5630 E. 2.20 3 S. 34~ W. 5.10 4 S. 341~ W. 3.53 4 N. 56~ W. 5.84 5 N. 5610 W. 3.20 5 S. 2910 W. 2.52 Ans.lA.OR.14P.6 N. 4810 W. 8,73 Ans. 1A. OR. 14P. CHAP. VI.] Calculating the Content. 187 Example 7. Example 8. ISTA. BEARING. DISTANCE. STA..BEARING.'DISTANCE. 1S.211 W. 17.62 1 S. 65-~0 E. 1 4.98 2 S. 34o W. 10.00 2 S. 58~ E. 8.56 3 N. 560 W. 14.15 -3 S. 141 W. 20.69 4 N. 340 E. 9.76 4 S. 470 W. 0.60 5 N. 67~ E. 2.30 5 S. 571j W. 8.98 6 N. 23~ E. 7.03 6 N. 560 W. 12.90 7 N. 18o0 E. 4.43 7 N. 340 E. 10.00 8 S. 76~0 E. 12.41 8 N. 211 E. 17.62 Example 9. Example 10. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 S. 570 E. 5.77 1 N. 63~ 51' W. 6.91 2 S. 3610 W. 2.25 2 N. 63~ 44' W. 7.26 3 S. 39. W. 1.00 3 N. 690 35' W. 3.34 4 S. 701 W. 1.04 4 N. 770 50' W. 6.54 5 N. 6830 W. 1.23 5 N. 310 24' E. 14.38 6 N. 560 W. 2.19 6 N. 310 18' E. 16.81 7 N. 33~0 E. 1.05 7 S. 680 55' E. 13.64 8 N.o 560 W. 1.54 8 S. 680 42' E. 11.54 9 N. 3310 E. 3.18 9 S. 330 45' W. 31.55 Ans. 2A. OR. 32P. Ans. 74 Acres. -Example 11. Example 12. STA. BEARING. DISTANCE. STA. BEARING. JDISTANCE. 1 N. 183~ E. 1.93 1 N.7230 E. 0.88 2 N. 9~ W. 1.29 2 S. 2002 E. 0.22 3 N. 140 W. 2.71 3 S. 630 E. 0.75 4 N. 740 E. 0.95 4 N. 51~ E. 2.35 5 S. 48~ E. 1.59 5 N. 440 E. 1.10 6 S. 140~ E. 1.14 6 N.250 W. 1.96 7S. 19-1 F. 2.15 7 N. 8S W. 1.05 S. 23?0 W. 1.22 8 S. 290 W. 1.63 9 S. 50 W. 1.40 9 N. 71~0 W. 0.81 10 S. 300 W. 1.02 10 N. 1310 W. 1.17 11 S. 81~0 W. 0.69 11 N. 630 W. 1.28 12 N. 321~ W. 1.98 12 West. 1.68 13 N. 490 W. 0.80 14 S. 191 E. 6.20 188 CORPASS SURVEYING. [PART III. xExample 13. A farm is described in an old Deed, as bounded thus. Beginning at a pile of stones, and running thence twentyseven chains and seventy links South-Easterly sixty-six and a half degrees to a white-oak stump; thence eleven chains and sixteen links North-Easterly twen- Fig. 197. ty and a half degrees to a hickory tree; thence two chains and thirty-five links North-Easterly thirty-six degrees to the South-Easterly corner of the homestead; thence nineteen chains and thirty-two links North-Easterly twenty-six degrees to a stone set in x1. the ground; thence twentyeight chains and eightylinks North-Westerly sixty-six degrees to a pine stump; thence thirty-three chains and nineteen links South-Westerly twenty-two degrees to the place of beginning, containing ninety-two acres,; be the same more or less. Required the exact content. (297) Mascheroni's Theorem. The surface of any polygon is equal to half the sum of the products of its sides (omitting any one side) taken two and two, into the sines of the angles which those sides make with each other. Fig. 198. Thus, take any polygon, such as the fivesided one in the figure. Express the argle which the directions of any two sides, as AB, CD, make with each other, thus (ABACD). Then will A<(GK —JFxBGxKH-. This shews,lt that the points G-, IH, J, lie in a straight line, which i.s a transversal to the triangle BFK. What Poinsot said, forty years ago, that "TThe simple and fruitful principles of this ingenious theory of tranTsversals deserve well to be admitted into the number of the elements (.!' geometry," is even more true and desirable at the present timCe. * By the theorem "If a straight line be drawn so as to cut any two tides of a triangle and the third side, one or all being prolonged, thus dividing them into six segments (the prolonged sides and the prolongations being taken as sc-ments) then will the product of any three of those segments whose extremities are not contiguous, be equal to the product of the other three segments.' t By the converse of the preceding theorem. PART IV. TRANSIT AND THEODOLITE SURVEYING: By the Third Method. CHAPTER I, THE INSTRUMENTS. (321) THE TRANSIT and THE THEODOLITE (figures of which are given on the next two pages) are Goniomneters, or Angle-Measurers. Each consists, essentially, of a circular plate of metal,supported in such a manner as to be horizontal, and divided on its outer circumference into degrees, and parts of degrees. Through the centre of this plate passes an upright axis, and on it is fixed a second circular plate, which nearly touches the first plate, and can turn freely around to the right and to the left. This second plate carries a Telescope, which rests on upright standards firmly fixed to the plate, and which can be pointed upwards and downwards. By the combination of this motion and that of the second plate around its axis, the Telescope can be directed to any object. The second plate has some mark on its edge, such as an arrow-head, which serves as a pointer or index for the divided circle, like the hand of a clock. When the Telescope is directed to one object, and then turned to the right or to the left, to some other object, this index which moves with it, and passes around the divided edge of the other plate, points out the arc passed over by this change of direction, and thus measures the angle made by the lines imagined to pass from the centre of the instrument to the two objects. 212 TRANSIT AND THEODOLITE SURVEYING. [PART IV. THE TRANSIT. o Fi. 212. V1,,~ ,T~,.'T~E EO D figc 213 214 TRANSIT AND THEODOLITE SURVEYING. [PART IV. (325) Distinction. The preceding description applies to both the Transit and the Theodolite. But an essential difference between them is, that in the Transit the Telescope can turn completely over, so as to look both forward and backward, while in the Theodolite it cannot do so. Hence the name of the Transit.* This capability of reversal enables a straight line to be prolonged from one end of it, or to be ranged out in both directions from any one point. The Telescope of the Theodolite can indeed be taken out of the y shaped supports in which it rests, and be replaced end for end, but this operation is an imperfect substitute for the revolution of the Telescope of the Transit. So also is the turning half way around of the upper plate which carries the Telescope. The Theodolite has a level attached to its Telescope, and a vertical circle for measuring vertical angles. The Transit does not usually have these, though they are sometimes added to it. The instrument may then be named a Transit-Theodolite. It then corresponds to the altitude and azimuth instrument of Astronomy. As the greater part of the points to be explained are common to both the Transit and the Theodolite, the descriptions to be given may be regarded as applicable to either of the instruments, except when the contrary is expressly stated, and some point peculiar to either is noticed. (326) The great value of these instruments, and the accuracy of their measurements of angles are due chiefly to two things; to the Telescope, by which great precision in sighting to a point is obtained; and to the bVernier Scale, which enables minute portions of any are to be read with ease and correctness. The former assists the eye in directing the line of sight, and the latter aids it in reading off the results. Arrangements for giving slow and steady motion to the movable parts of the instruments add to the value of the above. A contrivance for Repeating the observation of angles still farther lessens the unavoidable inaccuracies of these observations. * It is sometimes called the " Engineers' Transit, or " Railroad Transit," to distinguish it from the Astronomical Transit-instrument. CHAP. I.] The Instruments. 215 The inaccurate division of the limb of the instrument is also averaged and thus diminished by the last arrangement. Its want of true " centring," is remedied by reading off on opposite sides of the circle. Imperfections in the parallelism and perpendicularity of the parts of the instrument in which those qualities are required, are corrected by various " adjustments," made by the various screws whose heads appear in the engravings. The arrangements for attaining all these objects render necessary the numerous parts and apparent complication of the instrument. But this complication disappears when each part is examined in turn, and its uses and relations to the rest are distinctly indicated. This we now propose to do, after explaining the engravings. (327) In the figures of the instruments, given on pages 212 and 213, the same letters refer to both figures, so far as the parts are common to both.* L is the limb or divided circle. V is the index, or " Vernier," which moves around it. In the Transit, only a small portion of the divided limb is seen, the upper circle (which in it is the movable one) covering it completely, so that,,only a short piece of the are is visible through an opening in the upper plate. S, S, are standards, fastened to the upper plate and supporting the telescope, EO. G is a compass-box, also fastened to the upper plate. c is a clamp-screw, which presses together the two plates, and prevents one from moving over the other. t is a tangentscrew, or slow-motion screw, which gives a slow and gentle motion to one plate over the other. C is a clamp-screw which fastens the lower plate to the body of the instrument, and thus prevents it from moving on its own axis. T is the tangent-screw to give this part a slow-motion. P and PI are parallel plates through which pass four screws, Q, Q, Q, Q, by which the circular plate L is made level, * The arrangements of these instruments are differently made by almost every maker; but any form of them being thoroughly understood, any new one will cause no difficulty. The figure of the Transit was drawn from one made by W. & L. E. Gurley, of Troy, N. Y. to the latter of whom the Author is indebted for some valuable information respecting the details of the instrument. The Theodolite is of the favorite English form. 216 TRANSIT AND THIEODOLITE SURVEYING. [PART IV. as determined by the bubbles in the small spirit levels, B, B, of which there are two at right angles to each other. In the figure of the Tiheodolite, the large level b, and the semicircle NN are for the purposes of Levelling, and of measuring Vertical angles. They will therefore not be described in this place. (328) As the value of either of these instruments depends greatly on the accurate fitting and bearings of the two concentric -vertical axes, and as their connection ought to be thoroughly understood, a vertical section through the body of the instrument is given in Fig. 214, to half the real size. The tapering spindle or Fig.. 214. mnverted frustum of a cone, marked AA, supports the upper plate BB, which carries the index, or Verniers, V, V, and the Telescope. The whole bearing of this plate is at C, C, on the top of the hollow inverted cone EE, in which the spindle turns freely, but steadily. This interior position of the bearings preserves them from dust and injury. This hollow cone carries the lower or graduated plate, and it can itself turn around on the bearings D, D, carrying with it the lower circle, and also the upper one and all above it. The Vernier scales V, V, are attached to the upper plate, but lie in the same plane as the divisions L, L, of the lower plate, (so that the two can be viewed together, without parallax,) and are CHAP. I.] The Instruments. 217 covered with glass, to exclude dust and moisture. In Fig. 215. the figure the hatchings are drawn in different directions on the parts which move with the Vernier, and on those which move only with the limb. (329) The Telescope. This is a combination of lenses, placed in a tube, and so arranged, in accordance with the laws of optical science, that an image of any object to which the Telescope may be directed, is formed within the tube, (by thie rays of light coming from the object and bent in passing through the object-glass) and there magnified by an Eye-glass, or Eye-piece, composed of several lenses. The arrangement of these lenses are very various. Those two combinations, which are preferred for surveying instruments, will be here explained. Fig. 215 represents a Telescope which inverts objects. Any object is rendered visible by every point of it sending forth rays of light in every direction. In this figure, 9 the highest and lowvest points of the object, which here is an arrow, A, are alone considered. Those of the rays proceeding from them, which meet the object-glass, O, form a cone. The centre line of each cone, and its extreme upper and lower lines are alone shown in the figure. It will be seen that these rays, after passing through the object-glass, are refracted, or bent, by it, so as to cross one another, and thus to form at B an inverted image of the object. This would be rendered visible, if a piece of ground glass, or other semi-transparent substance, was placed at the point B, which is called the focus of the object-glass. The rays which form this image continue onward and pass through the two lenses i C and D, which act like one magnifying glass, so that j the rays, after being refracted by them, enter the eye at such angles as to form there a magnified and inverted image of the object. This combination of the two plano-convex lenses, C and D, is known as " Ramsden's Eye-piece." 218 TRANSIT AND THEODOLITE SURVEYING. [PART IV This Telescope, inverting objects, shows them upside Fig. 16. down, and the right side on the left. They can be shown erect by adding one or two more lenses as in the marginal figure. But as these lenses absorb light and lessen the distinctness of vision, the former arrangement is preferable for the glasses of a Transit or a Theodolite. A little practice makes it equally convenient for the observer, who soon becomes accustomed to seeing his flagmen standing on their heads, and soon learns to motion them to the right when he wishes them to go to the left, and vice versa. Figure 216 represents a Telescope which shows objects erect. Its eye-piece has four lenses. The eyepiece of the common terrestrial Telescope, or spy-glass, has three. Many other combinations may be used, all intended to show the object achromatically, or free from false coloring, but the one here shown is that most generally preferred at the present day. It will be seen that an inverted image of the object A, is formed at B, as before, but that the rays continuing onward are so refracted in passing through the lens C as to again cross, and thus, after farther refraction by the lenses D and E, to form, at F, an erect image, which is magnified by the lens G. In both these figures, the limits of the page render it necessary to draw the angles of the rays very much out of proportion. (330) Cross-hairs. Since a considerable field of view is seen in l6oking through the Telescope, it is necessary to provide means for directing the line of sight to the precise point which is to be observed. This could be effected by placing a very fine point, such as that of a needle, within the Telescope, at some place where it could be distinctly seen. In practice this fine point is obtained by the intersection of two very fine lines, placed in the common focus of the object-glass and CHAP. I.] The Instruments. 219 ot the eye-piece. These lines are called the cross-ha1irs, or crosswires. Their intersection can be seen through the eye-piece, at the same time, and apparently at the same place, as the image of the distant object. The magnifying powers of the eye-piece will then detect the slightest deviation from perfect coincidence. " This application of the Telescope may be considered as completely annihilating that part of the error of observation which might otherwise arise from an erroneous estimation of the direction in which an object lies from the observer's eye, or from the centre of the instrument. It is, in fact, the grand source of all the precision of modern Astronomy, without which all other refinements in instrumental workmanship would be thrown away." What Sir John Herschel here says of its utility to Astronomy, is equally applicable to Surveying. The imaginary line which passes through the intersection of the cross-hairs and the optical centre of the object-glass, is called the line qf collimation of the Telescope.* The cross-hairs are attached to a ring, or short thick tube of brass, placed within the Tele- Fig. 217. scope tube, through holes in which pass loosely four screws, (their heads being seen at a, a, a, in Figs. 212 and 213),, whose threads enter and take l / hold of the ring, behind or in front of the cross-hairs, as / shown (in front view and in section) in the two figures in'" the margin. Their movements will be explained in Chapter III. Usually, one cross-hair is horizontal, and the Fig. 218. other vertical, as in Fig. 217, but sometimes they are arranged as in Fig. 218, which is thought to / enable the object to be bisected with more precision. A horizontal hair is sometimes added. The cross-hairs are best made of platinum wire, drawn out very fine by being previously enclosed' From the Latin word Collimo, or Collineo, meaning to direct one thing towards another in a straight line, or to aim at. The line of aim would express the meaning. 220 TRANSIT AND THEODOLITE SURVEYING. [PART IV. in a larger wire of silver, and the silver then removed by nitric acid. Silk threads from a cocoon are sometimes used. Spiders' threads are, however, the most usual. If a cross-hair is broken, the ring must be taken out by removing two opposite screws, and inserting a wire with a screw cut on its end, or a stick of suitable size, into one of the holes thus left open in the ring, it being turned sideways for that purpose, and then removing the other screws. The spider's threads are then stretched across the notches seen in the end of the ring, and are fastened by gum, or varnish, or beeswax. The operation is a very delicate one. The following plan has been employed. A piece of wire is bent, as in the figure, so as to leave an opening a little wider than the Fig. 219. ring of the cross-hairs. A cobweb is chosen, at the end of which a spider is hanging, and it is wound around the bent wire, as in the figure, the weight of the insect keeping it tight and stretching it ready for use, each part being made fast by gum, &c. When a cross-hair is wanted, one of these is laid across the ring and there attached. Another method is to draw the thread out of the spider, persuading him to spin, if he sulks, by tossing him from hand to hand. A stock of such threads must be obtained in warm weather for the winter's wants. A piece of thin glass, with a horizontal and a vertical line etched on it, may be made a substitute. (331) Instrumental Parallax. This is an apparent movement of the cross-hairs about the object to which the line of sight is directed, taking place on any slight movement of the eye of the observer. It is caused by the image and the cross-hairs not being precisely in the common focus, or point of distinct vision of the eye-piece and the object glass. To correct it, move the eye-piece out or in till the cross-hairs are seen clearly and sharply defined against any white object. Then move the object glass in or out till the object is also distinctly seen. The cross-hairs will then seem to be fixed to the object, and no movement of the eye will cause them to appear to change their place. CHAP. I.] The Instruments. 221 (332) The milled-headed screw seen at M, passing into the telescope has a pinion at its other end entering a Fig. 220. toothed rack, and is used to move the object glass, s 0, out and in, according as the object looked at is nearer or farther than the one last observed. Short distances require a long tube: long distances a short tube. The eye-piece, E, is usually moved in and out by hand, but a similar arrangement to the preceding is a great improvement. This movement is necessary in order to obtain a distinct view of the cross-hairs. Short-sighted persons require the eye-piece to be pushed farther in than persons of ordinary sight, and old or longsighted persons to have it drawn further out. (333) Supports. The Telescope of the Transit is supported by a hollow axis at right angles to it, which itself rests at each end, on two upright pieces, or standards, spreading at their bases so as to increase their stability. In the Theodolite, the telescope rests at each end in forked supports, called ys, from their shape. These ys are themselves supported by a cross-bar, which is carried by an axis at right angles to it and to the telescope. This axis rests on standards similar to those of the Transit. The Telescope of the Theodolite can be taken out of the ys, and turned " end for end." This is not usual in the Transit. Either of the above arrangements enables the Telescope to be raised or depressed so as to suit the height of the object to which it is directed. A telescope so disposed is called a" plunging telescope." In some instruments there is an arrangement for raising or lowering one end of the axis. This is sometimes required for reasons to be given in connection with'" Adjustments." (331) The Indexes. The supports, or standards, of the telescope just described are attached to the upper, or index-carrying circle.' This, as has been stated, can turn freely on the lower or graduated circle, by means of its conical axis moving in the hollow conical axis of the latter circle. This upper circle carries the index, V, * In some instruments this circle is the under one. In our figures it is the upper one, and we will therefore always speak of it as such. 222 TRANSIT AND THEODOLITE SURVEYING. [PART IV. which is an arrow-head or other mark on its edge, or the zero-point of a Vernier scale. There are usually two of these, situated exactly opposite to each other, or at the extremities of a diameter of the upper circle, so that the readings on the graduated circle pointed out by them differ, if both are correct, exactly 180~. The object of this arrangement is to correct ally error of eccentricity, arising from the centre of the axis which barries the upper circle, (and with which it and its index pointers turn), not being precisely in the centre of the graduated circle. In the figure, let C Fig. 221. be the true centre of the graduated circle, but C' the centre on which the plate carrying the indexes turns. Let AC'B represent the direction of a sight taken a to one object, and D'C'E' the direction when turned to a second object. The / angle subtended by the two objects at / the centre of the instrument is required. Let DE be a line passing through C, and parallel to D'E'. The angle ACD equals the required angle, which is therefore truly measured by the arc AD or BE. But if the arc shown by the index is read, it will be AD' on one side, and BE' on the other; the first being too small by the arc DD' and the other too large by the equal arc EE'. If however the half-sum of the two arcs AD' and BE' be taken, it will equal the true arc, and therefore correctly measure the angle. Thus if AD' was 190, and BE' 210, their half sum, 200, would be the correct angle. Three indexes, 120~ apart, are sometimes used. They have the advantage of averaging the unavoidable inaccuracies and inequalities of graduation on different parts of the limb, and thus diminishing their effect on the resulting angle. Fig. 222. Four were used on the large Theodolite of A the English Ordnance Survey, two, A and B, opposite to each other, and two, C and D, 1200 from A and from each other. The half-sum or arithmetical mean, of A and B was taken, then c the mean of A, C, and D, and then the mean of these two means. But this was wrong, for B CHAP. I.] The Instruments. 223 it gave too great value to the reading of A, and also to B, though in a less degree; since the share of each Vernier in the final mean was as follows: A - 5, B = 3, C - 2, D = 2. This results from the expression for that mean, = (A + B + A- C F ) (5 A + 3 B + 2 C + 2 D). (335) The graduated circle. This is divided into three hundred and sixty equal parts, or Degrees, and each of these is subdivided into two or three parts or more, according to the size of the instrument. In the first case, the smallest division on the circle will of course be 30'; in the second case 20'. More precise reading, to single minutes or even less, is effected by means of the Vernier of the index, all the varieties of which will be fully explained in the next chapter. The numbers run from 0~ around to 3600, which number is necessarily at the same point as the 0, or zero-point.' Each tenth degree is usually numbered, each fifth degree is distinguished by a longer line of division, and each degree-division line is longer than those of the sub-divisions. A magnifying glass is needed for reading the divisions with ease. In the Theodolite engraving this is shown at m. It should be attached to each Vernier. (336) Movements. When the line of sight of the telescope is directed to a distant well-defined point, the unaided hand of the observer cqannot move it with sufficient delicacy and precision to make the intersection-ofthe- cross -irs- exactly -cover or "bisect" that point. To effect this, a clamp, and a Tangent, or slow-motion, screw are required. This arrangement, as applied to the movement of the upper, or Vernier plate, consists of a short piece of brass, D, which is attached to the Vernier plate, and through which passes a long and fine-threaded " Tangent-screw," t. The other end of this screw enters into and carries the clamp. This consists of two pieces of brass, which, by turning the clamp-screw c, which passes through them on the outside, can be made to take * In some instruments there is another concentric circle on which the degrees are also numbered from 00 to 90o as on the compass circle. 224 TRANSIT AND THEODOLITE SURVEYING. [PART IV. hold of and pinch tightly the edge of the lower circle, which lies between them on the inside. The upper circle is now prevented from moving on the lower one; for, the tangent-screw, passing through hollow screws in both the clamp and the piece D, keeps them at a fixed distance apart, so that they cannot move to or from one another, nor consequently the two circles to which they are respectively made fast. But when this tangent-screw is turned by its milled-head, it gives the clamp and with it the upper plate a smooth and slow motion, backward or forward, whence it is called the 1" Slow motion screw," as well as " Tangent-screw," from the direction in which it acts. It is always placed at the south end of the' compass-box. A little different arrangement is employed to give a similar motion to the lower circle (which we have hitherto regarded as immovable) on the body of the instrument. Its axis is embraced by a brass ring, into which enters another tangent-screw, which also passes through a piece fastened to the plate P. The clampscrew, C, causes the ring to pinch and hold immovably the axis of the lower circle, while a turn of the Tangent-screw, T, will slowly move the clamp ring itself, and therefore with it the lower circle. When the clamp is loosened, the lower circle, and with it every thing above it, has a perfectly free motion. A recent improvement is the employment for this purpose of two tangent screws, pressing against opposite sides of a pieae projecting from the clamp-ring. One is tightened as the other is loosened, and a very steady motion is thus obtained. (337) Levels. Since the object of the instrument is to measure horizontal angles, the circular plate on which they are measured must itself be made horizontal. Whether it is so or not is known by means of two small levels placed on the plate at right angles to each other. Each consists of a glass tube, slightly curved upward in its middle and so nearly filled with alcohol, that only a small bubble of air is left in the tube. This always rises to the hihllestpart of the tubes. They are so "adjusted" (as will be explained in chapter III) that when this bubble of air is in the middle of the tubes, or its ends equidistant from the central mark, the plate CHAP. I.] The Instruments. 225 on which they are fastened shall be level, which way soever it may be turned. The levels are represented in the figure of the Transit, on page 212, as being under the plate. They are sometimes placed above it. In that case, the Verniers are moved to one side, between the feet of the standards, and one of the levels is fixed between the standards above one of the Verniers, and the other on the plate at the south end of the compass-box. (338) Parallel Plates, To raise or lower either side of the circle, so as to bring the bubbles into the centres of the tubes, requires more gentle and steady movements than the unaided hands can give, and is attained by the Parallel Plates P, P', (so called because they are never parallel except by accident), and their four screws Q, Q, Q, Q, which hold the plates firmly apart, and, by being turned in or out, raise or lower one side or the other of the upper plate P', and thereby of the graduated circle.' The two plates are held together by a ball and socket joint. To level the instrument, loosen the lower clamp and turn the circle till each level is parallel to the vertical plane passing through a pair of opposite screws. Then take hold of two opposite screws and turn them simultaneously and equally, but in contrary directions, screwFig. 223. ing one in and the other out, as shown by the arrows in the figures. A rule easily remembered is that both thumbs must turn in, or both out. The movements represented in the first of these figures would raise the left-hand side of the circle and lower the right-hand side. The movements of the second figure would produce the reverse effect. Care is needed to turn the opposite screws equally, so that they shall not become so loose that the instrument will rock, or so tight as to be cramped. When this last occurs, one of the other pair should be loosened. 15 226 TRANSIT AND THEODOLITE SURVEYING. [PART IV. Sometimes one of each pair of the screws is replaced by a strong spring against which the remaining screws act. The French and German instruments are usually supported by only three screws. In such cases, one level is brought parallel to one pair of screws and levelled by them, and the other level has its bubble brought to its centre by the third screw. If there is only one level on the instrument, it is first brought parallel to one pair of screws and levelled, and is then turned one quarter around so as to be perpendicular to them and over the third screw, and the operation is repeated. (339) Watch Telescope. A second Telescope is sometimes attached to the lower part of the instrument. When a number of angles are to be observed from any one station, direct the upper and principal Telescope to the first object, and then direct the lower one to any other well-defined point. Then make all the desired observations with the upper Telescope, and when they are finished, look again through the lower one, to see that it and therefore the divided circle has not been moved by the movements of the Vernier plate. The French call this the Witness Telescope, (ILunette temoin). (340) The Compass. Upon the upper plate is fixed a compass. Its use has been fully explained in Part III. It is little used in connection with the Transit or Theodolite, which are so incomparably more accurate, except as a'" check," or rough test of the accuracy of the angles taken, which should about equal the difference of the magnetic bearings. Its use will be farther noticed in Chapter IV, on " Field Work." The name of " Surveyor's Transit" has been given to a modification of the " Engineer's Transit," enabling the compass to be set so as to run lines with an allowance for the magnetic variation, as was explained in Art. (312). This cannot be done in the instrument which has been described. (311) Theodolites. The distinctive peculiarities of the ordinary Theodolite have been explained in connection with those of the Transit. Some modifications will now be described. CHAP. I.] The Instruments. 227 Two Telescopes, one exactly under the other, so that their lines of collimation are in the same vertical plane, have been attached to a Theodolite, for the purpose of ranging a straight line forward and backward, as a substitute for the much more elegant operation of reversion as in the Transit. A remarkable arrangement, made by Gurley, of Troy, had an object-glass, an eye-glass, and a pair of cross-hairs at each end of the instrument, the eye glass being set in a skeleton frame so as to obstruct the light as little as possible. The focus of each glass was so adjusted that the Telescope could be looked through in either direction. Two Telescopes, mounted one above the other, and one attached to the graduated circle and the other to the Vernier plate, have also been used. With these the angle subtended at the instrument, can be measured at once, each object having a Telescope directed to it at the same moment, thus avoiding any danger of slipping, and rendering unnecessary a Watch Telescope, for which one of these Telescopes serves as a more perfect substitute. (342),Goniasmometre. A very compact in- Fi 4 strument to which the above name has been given in France, where it is much used, is shown,w 1 p in the figure. The upper half of the cylinder is movable on its lower half. The observations lli may be taken through the slits, as in the Survey- or's Cross, or a Telescope may be added to it. i Readings may be taken both from the compass, K I X and from the divided edge of the lower half of N the cylinder, by means of a Vernier on the upper half.* * The proper care of instruments must not be overlooked. If varnished, they should be wiped gently with fine and clean linen. If polished with oil, they should be rubbed more strongly. The parts neither varnished nor oiled, should be cleaned with Spanish white and alcohol. Varnished wood, when spotted, should be wiped with very soft linen, moistened with a little olive oil or alcohol. Unpainted wood is cleaned with sand-paper. Apply olive oil where steel rubs against brass; and wax softened by tallow where brass rubs against brass.Clean the glasses with kid or buck skin. Wash them, if dirtied, with alcohol. 228 [PART IV. CHAPTER II. VERNIERS. (343) Definition. A Vernier is a contrivance for measuring smaller portions of space than those into which a line is actually divided. It consists of a second line or scale, movable by the side of the first, and divided into equal parts, which are a very little shorter or longer than the parts into which the first line is divided. This small difference is the space which we are thus enabled to measure. The Vernier scale is usually constructed by taking a length equal to any number of parts on the divided line, and then dividing this length into a number of equal parts, one more or one less than the number into which the same length on the original line is divided. (344) Illustration. The figure represents (to twice the real size) a scale of inches divided into tenths, with a Vernier scale beside it, by which hundredths of an inch can be measured. The Fig. 225. Vernier is made by setting off on it 9 tenths of an inch, and dividing that length into 10 equal parts. Each space on the Vernier is therefore equal to a tenth of nine-tenths of an inch, or to ninehundredths of an inch, and is consequently one-hundredth of an inch shorter than one of the divisions of the original scale. The " The Vernier is so named from its inventor, in 1631. The name " Nonius," often improperly given to it, belongs to an entirely different contrivance for a similar object. CHAP. II.] Verniers, 229 first space of the Vernier will therefore fall short of, or be overlapped by, the first space on the scale by this one-hundredth of an inch'; the second space of the Vernier will fall short by two-hundredths of an inch; and so on. If then the Vernier be moved up by the side of the original scale, so that the line marked 1 coincides, or forms one straight line, with the line of the scale which was just above it, we know that the Vernier has been moved onehundredth of an inch. If the line marked 2 comes to coincide with a line of the scale, the Vernier has moved up two-hundredths of an inch; and so for other numbers. If the position of the Fig. 226.'1 I' t I I 1 1111o I 1' 1 1 1..... ~c O Vernier be as in this figure, the line marked 7 on the Vernier corresponding with some line on the scale, the zero line of the Vernier is 7 hundredths of an inch above the division of the scale next below this zero line. If this division be, as in the figure, 8 inches and 6 tenths, the reading will be 8.67 inches.* A Vernier like this is used on some levelling rods, being engraved on the sides of the opening in the part of the target above its middle line. The rod being divided into hundredths of a foot, this Vernier reads to thousandths of a foot. It is also used on some French Mountain Barometers, which are divided to hundredths of a metre, and thus read to thousandths of that unit. (345) General rules. To find what any Vernier reads to, i. e. to determine how small a distance it can measure, observe how many parts on the original line are equal to the same number increased or diminished by one on the Vernier, and divide the i The student will do well to draw such a scale and Vernier on two slips of thick paper, and move one beside the other till he can read them in any possible position; and so with the following Verniers. 230 TRANSIT AND THEODOLITE SURVEYING. [PART IV. length of a part on the original line by this last number. It will give the required distance.* To read any Vernier, firstly, look at the zero line of the Vernier, (which is sometimes marked by an arrow-head), and if it coincides with any division of the scale, that will be the correct reading, and the Vernier divisions are not needed. But if, as usually happens, the zero line of the Vernier comes between any two divisions of the scale, note the nearest next less division on the scale, and then look along the Vernier till you come to some line on it which exactly coincides, or forms a straight line, with some line (no matter which) on the fixed scale. The number of this line on the Vernier (the 7th in the last figure) tells that so many of the sub-divisions which the Vernier indicates, are to be added to the reading of the entire divisions on the scale. When several lines on the Vernier appear to coincide equally with lines of the scale, take the middle line. When no line coincides, but one line on the Vernier is on one side of a line on the scale, and the next line on the Vernier is as far on the other side of it, the true reading is midway between those indicated by these two lines. (346) Retrograde Verniers. The spaces of the Vernier in modern instruments, are usually each shorter than those on the scale, a certain number of parts on the scale being divided into a larger number of parts on the Vernier.f In the contrary case,: there is the inconvenience of being obliged to number the lines of the Vernier and to count their coincidences with the lines of the scale, in a retrograde or contrary direction to that in which the numbers on the scale run. We will call such arrangements retrograde Verniers. * In Algebraic language, let s equal the length of one part on the original line, and v the unknown length of one part on the Vernier. Let m of the former = m + 1 of the latter. Then?ns = (m + 1) v. v= -s. s- v = — -- m+i5s m- 1. If ms = (m- i) v, then v-s =. m m ii. e. Algebraically, v - s. 1 i.e. When v = m —-s. nt + 1 CHAP. II.] Verniers. 231 (347) Illustration. In this figure, the scale, as before, represents (to twice the real size) inches divided into tenths, but the Vernier is made by dividing 11 parts of the scale into 10 equal Fig. 227. I I IA lll Gl i V\ t l 1 Vl parts, each of which is therefore one-tenth of eleven-tenths of an inch, i. e. eleven-hundredths of an inch, or a tenth and a hundredth. Each space of the Vernier therefore overlaps a space on the scale by one-hundredth of an inch. The manner of reading this Vernier is the same as in the last one, except that the numbers run in a reverse direction. The reading of the figure is 30.16. This Vernier is the one generally applied to the common Barometer, the zero point of the Vernier being brought to the level of the top of the mercury, whose height it then measures. It is also employed for levelling rods which read downwards from the middle of the target. (348) The figure below represents (to double size) the usual scale of the English Mountain Barometer.* The scale is first divided into inches. These are subdivided into tenths by the Fig. 228. A__/~111 l I I I i I I l I i i /] * Tifigra in I t i I' This figure, and others in this chapter, are from Bree's " Present Practice." 232 TRANSIT AND THEODOLITE SURVEYING. [PART IV. longer lines, and the shorter lines again divide these into halftenths, or to 5 hundredths. 24 of these smaller parts are set off on the Vernier, and divided into 25 equal parts, each of which is 24 x.05 therefore 25 -=.048 inch, and is shorter than a division of the scale by.050.-.048 =.002, or two thousandths of an inch, a twenty-fifth part of a division on the scale, to which minuteness the Vernier can therefore read. The reading in the figure is 30.686, (30.65 by the scale and.036 by the Vernier), the dotted line marked D showing where the coincidence takes place.;,'~' bCircle divided into degrees. The following illustrations apply to the measurements of angles, the circle being variously divided. In this article,.the circle is supposed to be divided into degrees. If 6 spaces on the Vernier are found to be equal to 5 on the circle, the Vernier can read to one-sixth of a space on the circle, i. e. to 10'. If 10 spaces on the Vernier are equal to 9 on the circle, the Vernier can read to one-tenth of a space on the circle, i. e. to 6'. If 12 spaces on the Vernier are equal to 11 on the circle, the Vernier can read to one-twelfth of a space on the circle, i. e. to 5'. Fig. 229. I -mi go 45 30 I5 0 The above figure shows such an arrangement. The index, or zero, of the Vernier is at a point beyond 3580, a certain distance, which the coincidence of the third line of the Vernier (as indicated CHAP. II.] Verniers. 233 by the dotted and crossed line) shows to be 15'. The whole reading is therefore 358~ 15'. If 20 spaces on the Vernier are equal to 19 on the circle, the Vernier can read to one-twentieth of a division on the circle, i. e. to 3'. English compasses, or " Circumferentors," are sometimes thus arranged. If 60 spaces on the Vernier are equal to 59 on the circle, the Vernier can read to one-sixtieth of a division on the circle, i. e. to 1'. (350) Circle divided to 30'. Such a graduation is a very common one. The Vernier may be variously constructed. Suppose 30 spaces on the Vernier to be equal to 29 on the 29 x 30' circle. Each space on the Vernier will be = 30 = 29', and will therefore be less than a space of the circle by 1', to which the Vernier will then read. Fig. 230. The above figure shows this arrangement. The reading is 0~, or 360~. In the following figure, the dotted and crossed line shows what divisions coincide, and the reading is 20~ 10'; the Vernier being the same as in the preceding figure, and its zero being at a point of the circle 10' beyond 20~. 234 TRANSIT AND THEODOLITE SURVEYING. [PARr IV. Fig. 231. I-i J t- 1,./ f 31ll It Alp In the following figure, the reading is 20~ 40', the index being at a point beyond 20~ 30', and the additional space being shown by the Vernier to be 10'. Fig. 232. 2. I 30 )_h"< CHAP. II.] Verniers. 235 Sometimes 30 spaces on the Vernier are equal to 31 on the circle. 31. x 30' Each space on the Vernier will therefore be -- 30 - 31', and will be longer than a space on the circle by 1', to which it will therefore read, as in the last case, but the Vernier will be " retrograde." This is the Vernier of the compass, Fig. 148. The peculiar manner in which it is there applied is shown in Fig. 239. If 15 spaces on the Vernier are equal to 16 on the circle, each 16 x 30' space on the Vernier will be -- 15 - 32', and the Vernier will therefore read to 2'. (351) Circle divided to 20'. If 20 spaces on the Vernier are equal to 19 on the circle, each space of the latter will be = 9 x 20 - 19', and the Vernier will read to 20'- 19' 1'. If 40 spaces on the Vernier are equal to 41 on the circle, each 41 x 20" space on the Vernier will be - 0 20; and the Vernier will therefore read to 20k' - 20' = 30". It will be retrograde. In the following figure the reading is 360~, or 0~; and it will be seen that the 40 spaces on the Vernier (numbered to whole minutes) are equal to 130 40' on the limb, i. e. to 41 spaces, each of 20'. Fig.233. 360 10 If 60 spaces on the Vernier are equal to 59 on the circle, each 59 x 201 of the former will be = = 19' 40", and the Vernier 236 TRANSIT AND THEODOLITE SURVEYING. [PART IV. will therefore read to 20' - 19' 40" = 20". The following figure shows such an arrangement. The reading in that position would be 400 46' 20". Fig. 234. 50 45 40 I / -I F _ I X I I 1it I0987i6 5432 L4 (352) Circle divided to 15', If 60 spaces on the Vernier are equal to 59 on the circle, each space on the Vernier will be = 69 x 15' 14' 45", and the Vernier will read to 15". In the 60 following figure the reading is 10~ 20' 45", the index pointing to 100 15', and something more, which the Vernier shows to be 5' 45". Fig. 235. 20 "1>1 10 114S1/,/1 T!1l l II Ii11 iiiii_ 1098 76 5H i 4 13 "I' CHAH. II.] Verniers. 237 (353) Circle divided to 10'. If 60 spaces on the Vernier be equal to 59 on the limb, the Vernier will read to 10". In the following figure, the reading is 70 25' 40", the reading on the circle being 7~ 20', and the Vernier showing the remaining space to be 5' 40". Fig. 236. 15' 0 I K (354) Reading backward s. When an index carrying a Vernier is moved backwards, or in a contrary direction to that in which the numbers on the circle run, if we wish to read the space which it has passed over in this direction from the zero point, the Vernier must be read backwards, (i. e. the highest number be called 0), or its actual reading must be subtracted from the value of the smallest space on the circle. The reason is plain; for, since the Vernier shows how far the index, moving in one direction, has gone past one division line, the distance which it is from the next division line (which it may be supposed to have passed, moving in a contrary direction), will be the difference between the reading and the value of one space. Thus, in Fig. 229, page 232, the reading is 358~ 15'. But, counting backwards.from the 360~, or zero point, it is 1~ 45'. Caution on this point is particularly necessary in using small angles of deflection for railroad curves. 238 TRANSIT AND THIEODOLITE SURVEYING. [PART IVo (355) Arc of excess. On the sextant and similar instruments, the divisions of the limb are carried onward a short distance beyond the zero point. This portion of the limb is called the "Arc of excess." When the index of the Vernier points to this arc, the reading must be made as explained in the last article. Thus, in the figure, the reading on the arc from the zero of the limb to the Fig. 235 0 _ il ~ i illIllll I -l 1)1' 1111'1I1111//11qST/ zero of the Vernier is 40 20', and something more, and the reading of the Vernier firo- 10 towards to the right, where the lines coincide, is 3' 20", (or it is 10' - 6' 40" = 3' 20"), and the entire reading is therefore 40 23' 20". (356) Double Verniers. To avoid the inconveniences of reading backwards, double Verniers are sometimes used. The figure below shows one applied to a Transit. Each of the Verniers is Fig. 238. 30 o0 200 X0 2,30 IrMAi; i1 II Al','[ CHAP. II.] Verniers. 239 like the one described in Art. (350), Figs. 230, 231, and 232. When the degrees are counted to the left, or as the numbers run, as is usual, the left-hand Vernier is to be read, as in Art. (350); but when the degrees are counted to the right, from the 360~ line, the right-hand Vernier is to be used. (357) Compass-Vernier. Another form of double Vernier, often applied to the compass, is shown in the following figure. The Fig. 239. ~9 295 310 2,15 21 5 5 m, 5 o limb is divided to half degrees, and the Vernier reads to minutes, 30 parts on it being equal to 31 on the limb. But the Vernier is only half as long as in the previous case, going only to 15', the upper figures on one half being a sort of continuation of the lower figures on the other half. Thus in moving the index to the right, read the lower figures on the left hand Vernier (it being retrograde) at any coincidence, when the space passed over is less than 15'; but if it be more, read the upper figures on the right hand Vernier: and vice versa. 240 [PART IV. CHAPTER III. ADJUSTMENTS. (358) THE purposes for which the Transit and Theodolite (as well as most surveying and astronomical instruments) are to be used, require and presuppose certain parts and lines of the instrument to be placed in certain directions with respect to others; these respective directions being usually parallel or perpendicular. Such arrangements of their parts are called their Adjustments. The same word is also applied to placing these lines in these directions. In the following explanations the operations which determine whether these adjustments are correct, will be called their Verifications; and the making them right, if they are not so, their BRectifieations.' (359) In observations of horizontal angles with the Transit or the Theodolite,t it is required, 1~ That the circular plates shall be horizontal in whatever way they may be turned around. 2~ That the Telescope, when pointed forward, shall look in precisely the reverse of its direction when pointed backward, i. e. that its two lines of sight (or lines of collimation) forward and backward shall lie in the same plane. 3~ That the Telescope in turning upward or downward, shall move in a truly vertical plane, in order that the angle measured between a low object and a high one, may be precisely the same as would be the angle measured between the low object and a point exactly under the high object, and in the same horizontal plane as the low one. i It has been well said, that " in the present state of science it may be laid down as a maxim, that every instrument should be so contrived, that the observer may easily examine and rectify the principal parts; for, however careful the instrument-maker may be, however perfect the execution thereof, it is not possible that any instrument should long remain accurately fixed in the position in which it came out of the maker's hands."-Adams' " Geometrical and Graphical Essays," 1791. t The Theodolite adjustments, which relate only to levelling, or to measuring vertical angles, will not be hcre discussed. CHAP. III.] Adjustments. 241 We shall see that all these adjustments are finally resolvable into these; 1st. Making the vertical axis of the instrument perpendicular to the plane of the levels; 2d. Making the line of collimatlon perpendicular to its axis; and 3d. Making this axis parallel to the plane of the levels. They are all best tested by the invaluable principle of " Reversion." We have now, firstly, to examine whether these things are so, that is, to " verify" the adjustments; and, secondly, if we find that they are not so, to make them so, i. e. to " rectify," or " adjust" them correctly. The above three requirements produce as many corresponding adjustments. (360) First adjustment. To cause the circle to be horizontal in every position.Verification.-Turn the Vernier plate which carries the levels, till one of them is parallel to one pair of the parallel plate screws. The other will then be parallel to the other pair. Bring each bubble to the middle of its tube, by that pair of screws to which it is parallel. Then turn the vernier plate half way around, i. e. till the index has passed over 180~. If the bubbles remain in the centres of the tubes, they are in adjustment. If either of them runs to one end of the tube, it requires rectification. IRectification.-The fault which is to be rectified is that the plane of the level (i. e. the plane tangent to the highest point of the level tube) is not perpendicular to the vertical axis, AA in figure 214, on which the plate turns. For, let AB represent this Fig. 240. Fig. 241. C —---— A *s;,Z&\XW\\\.......\\.... B —-. 1 i D!D plane, seen edgeways, and CD the centre line of the vertical axis, * This applies equally to the Transit and the Theodolite. 16 242 TRANSIT AND THEODOLITE SURVEYING. [PART IV. which is here drawn as making an acute angle with this plane on the right hand side. The first figure represents the bubble brought to the centre of the tube. The second figure represents the plate turned half around. The centre line of the axis is supposed to remain unmoved. The acute angle will now be on the left hand side, and the plate will no longer be horizontal. Consequently the bubble will run to the higher end of the tube. The rectification necessary is evidently to raise one end of the tube and lower the other. The real error has been doubled to the eye by the reversion. Half of the motion of the bubble was caused by the tangent plane not being perpendicular to the axis, and half by this axis not being vertical. Therefore raise or lower one end of the level by the screws which fasten it to the plate, till the bubble comes about half way back to the centre, and then bring it quite back by turning its pair of parallel plate screws. Then again reverse the vernier plate 180~. The bubble should now remain in the centre. If not, the operation should be repeated. The same must be done with the other level if required. Then the bubbles will remain in the centre during a complete revolution. This proves that the axis of the vernier plate is then vertical; and as it has been fixed by the maker perpendicular to the plate, the latter must then be horizontal. It is also necessary to examine whether the bubbles remain in the centre, when the divided circle is turned round on its axis. If not, the axes of the two plates are not parallel to each other. The defect can be remedied only by the maker; for if the bubbles be altered so as to be right for this reversal, they will be wrong for the vernier plate reversal. (361) Second adjustment. To cause the line of collimation to revolve in a plane.* Verification. Set up the Transit in the middle of a level piece of ground, as at A in the figure. Level it carefully. Set a stake, with a nail driven into its head, or a chain pin, as far from the instrument as it is distinctly visible, as at B. Direct the telescope * This adjustment is not the same in the Transit and illn the Theodolite. That for the Transit will be first given, and that for the Theodolite in the next article. c01P; III.] Adjustments. 243 Fig. 242. A F D to it, and fix the intersection of the cross-hairs very precisely upon it. Clamp the instrument. Measure from A to B. Then turn over the telescope, and set another stake at an equal distance from the Transit, and also precisely in the line of sight. If the line of collimation has not continued in the same plane during its half-revolution, this stake will not be at E, but to one side, as at C. To discover the truth, loosen the clamp and turn the vernier plate half around without touching the telescope. Sight to B, as at first, and again clamp it. Then turn over the telescope, and the line of sight will strike, as at D in the figure, as far to the right of the point, as it did before to its left. Rectification. The fault which is to be rectified, is that the line of collimation of the telescope is not perpendicular to the horizontal axis on which the telescope revolves. This will be seen by the figures, which represent the position of the lines in each of the four A A. A Fig. 2465. B..1.. Fig. 246In each of the figures the observations which have been made. In each of the figures the long thick line represents the telescope, and the short one the axis on which it turns. In Fig. 243 the line of sight is directed to P 244 TRANSIT AND TIIEODOLITE SURVEYING. [PART IV. In Fig. 244 the telescope has been turned over, and with it the axis, so that the obtuse angle, marked 0 in the first figure, has taken the place, 0', of the acute angle, and the telescope points to C instead of to E. In Fig. 245 the vernier plate has been turned half around so as to point to B again, and the same obtuse angle has got around to 0". In Fig. 246 the telescope has been turned over, the obtuse angle is at 0"', and the telescope now points to D. To make the line of collimation perpendicular to the axis, the former must have its direction changed. This is effected by moving the vertical hair the proper distance to one side. As was explained in Art. (330), and represented in Fig. 217, the crosshairs are on a ring held by four screws. By loosening the lefthand screw and tightening the right-hand one, the ring, and with it the cross-hairs, will be drawn to the right; and vice versa. Two holes at right angles to each other pass through the outer heads of the screws. Into these holes a stout steel wire is inserted, and the screws can thus be turned around. Screws so made are called is capstan-headed." One of the other pair of screws may need to be loosened to avoid straimng the threads. In some French instruments, one of each pair of screws is replaced by a spring. To find how much to move this vertical hair, measure from C to D, Fig. 242, page 243. Set a stake at the middle point E, and set another at the point F, midway between D and E. Move the vertical hair till the line of sight strikes F. Then the instrument is adjusted; and if the line of sight be now directed to E, it will strike B, when the telescope is turned over; since the hair is moved half of the doubled error, DE. The operation will generally require to be repeated, not being quite perfected at first. generally require to be repeated, not being quite perfected at first. It should be remembered, that if the Telescope used does not invert objects, its eye-piece will do so. Consequently, with such a telescope, if it seems that the vertical hair should be moved to the left, it must be moved to the right, and vice versa. An inverting telescope does not invert the cross-hairs. If the young surveyor has any doubts as to the perfection of his rectification, he may set another stake exactly under the instrument by means of a plumb-line suspended from its centre; and then, in like manner, set his Transit over B or E. He will find that the CHAP. III.] Adjustments, 245 other two stakes, A and the extreme one, are in the same straight line with his instrument. In some instruments, the horizontal axis of the telescope can be taken out of its supports, and turned over, end for end. In such a case, the line of sight may be directed to any well defined point, and the axis then taken out and turned over. If the line of sight again strikes the same point, this line is perpendicular to the axis. If not, the apparent error is double the real error, as appears from the figures, the obtuse angle O coming to 0', and the desired perFig. 247. ----------— 4 Fig. 248. B pendicular line falling at C midway between B and B'. The rectification may be made as before; or, in some large instruments, in which the telescope is supported on Ys, by moving one of the ys laterally. (362) The Theodolite must be treated differently, since its telescope does not reverse. One substitute for this reversal, when it is desired to range out a line forward and backward from one station, is, after sighting in one direction, to take the telescope out of the ys and turn it end for end, to sight in the reverse direction. This it can be made to do by adjusting its line of collimation as explained in the last article. Another substitute is, after sighting in one direction, and noting the reading, to turn the vernier plate around exactly 180~. But this supposes not only that the graduation is perfectly accurate, but also that the line of collimation is exactly over the centre of the circle. To test this, after sighting to a point, and noting the reading, take the telescope out of the ys and turn it end for end, and then turn the vernier plate around exactly 180~. If the line of sight again strikes the same point, the latter condition exists. If not, the maker must remedy 246 TRANSIT AND THEODOLITE SURVEYIN'G [PART IV. the defect. This error of eccentricity is similar to that explained with respect to the compass, in the latter part of Art. (226). (363) Third adjustment. To cause the line of collimation to rezolve in a vertical plane.* Verification. Suspend a long plumb-line from some high point. Set the instrument near this line, and level it carefully. Direct the telescope to the plumb-line, and see if the intersection of the cross-hairs follows and remains upon this line, when the telescope is turned up and down. If it does, it moves in a vertical plane. The angle of a new and well-built house will form an imperfect substitute for the plumb-line. Otherwise; the instrument being set up and levelled as above, place a basin of some reflecting liquid (quicksilver being the best, though molasses, or oil, or even water, will answer, though less perfectly,) so that the top of a steeple, or other point of a high object, can be seen in it through the telescope by reflection. Make the intersection of the -cross-hairs cover it. Then turn up the telescope, and if the intersection of the cross-hairs bisects also the object seen directly, the line of sight has moved in a vertical plane. If a star be taken as the object, the star and its reflection will be equivalent (if it be nearly over head) to a plumb-line at least fifty million million miles long. Otherwise; set the instrument as close as possible to the base of a steeple, or other high object; level it, and direct Fig. 249 it to the top of the steeple, or to some other elevated and well defined point. Clamp the plates. Turn down the telescope, and set up a pin in the ground precisely "in line." Then loosen the clamp, turn over the telescope, and turn it half-way around, or so far as to again sight to the high point. Clamp the plates, and again turn down the telescope. If the line of sight again strikes the pin, the telescope has moved in a vertical plane. If not, the apparent p' P 1' error is double the real error. For, let S be the top of the steeple, * This applies to both the Transit and the Theodolite, with the exception of the method of verification by the steeple and pin, which applies only to the Transit. CHAP. III.] Adjustments. 247 (Fig.249) and P' the pin; then the plane in which Fig. 250. the telescope moves, seen edgewise, is SP'; and, after being turned around, the line of sight moves in the plane SP", as far to one side of the vertical plane SP, as SP' was on the other side of it. Rectification. Since the second adjustment causes the line of sight to move in a plane perpendicular to the axis on which it turns, it will move in a vertical plane if that axis be horizontal. It may be made so by filing off the feet of the standards which support the higher end of the axis. This will be best done by the maker. In some instruments one end of the axis can be raised or lowered. (364) Centring eye-piece. In some instruments, such as that of which a longitudinal \ section is shown in the margin, the inner end of the eye-piece may be moved so that the BS cross-hairs shall be seen precisely in the centre of its field of view. This is done by means of four screws, arranged in pairs, like those of the cross-hair-ring screws, and capable of moving the eye-piece up and down, and to right or left, by loosening one and tightening the opposite one. Two of them are shown at A, A, in the figure; in which B, B, are two of the cross-hair screws. (365) Centring object-glass. In some instruments four screws, similarly arranged, two of which are shown at C, C, can move, in any direction, the inner end of the slide which carries the object-glass. The necessity for such an arrangement arises from the impossi- v 248 TRANSIT AND THEODOLITE SURVEYING. [PART IV. bility of drawing a tube perfectly straight. Consequently, the line of collimation, when the tube is drawn in, will not coincide with the same line when the tube is pushed out. If adjusted for one position, it will therefore be wrong for the other. These screws, however, can make it right in both positions. They are used as follows. Sight to some well defined point as far off as it can be distinctly seen. Then revolve the telescope half around in its supports; i. e. turn it upside down.' If the line of collimation was not in the imaginary axis of the rings or collars on which the telescope rests, it will now no longer bisect the object sighted to. Thus, if the horizontal hair was too high, as in Fig. 251, this line of Fig. 251.' —-:'-: —:: -- - ---- - - -o-c ~ collimation would point at first to A, and after being turned over, it would point to B. The error is doubled by the reversion, and it should point to C, midway between A and B. Make it do so, by unscrewing the upper capstan-headed screw, and screwing in the lower one, till the horizontal hair is brought half way back to the point. Remember that in an erecting telescope, the cross-hairs are reversed, and vice versa. Bring it the rest of the way by means of the parallel plate screws. Then revolve it in the Ys back to its original position, and see if the intersection of the cross-hairs now bisects the point, as it should. If not, again revolve, and repeat the operation till it is perfected. If the vertical hair passes to the right or to the left of the point when the telescope is turned half around, it must be adjusted in the same manner by the other pair of cross-hairs screws. One of these adjustments may disturb the other, and they should be repeated alternately. When they are perfected, the intersection of the cross-hairs, when once fixed on a point, will not move from it when the telescope is revolved in its i In Theodolites, the Telescope is revolved in the Ys. In Transits, the maker, by whom this adjustment is usually performed, revolves the Telescope, in the same manner, before it is fixed in its cross-bar. CHAP. III.] Adjustments. 249 supports. This double operation is called adjusting the line of collimation.* This line is now adjusted for distant objects. It would be so for near ones also, if the tube were perfectly straight. To test this, sight to some point, as near as is distinctly visible. Then turn the telescope half over. If the intersection does not now bisect the point, bring it half way there by the screws C, C, of Fig. 250, moving only one of the hairs at a time, as before. Then repeat the former adjustment on the distant object. If this is not quite perfect, repeat the operation. This adjustment, in instruments thus arranged, should precede the first one which we have explained. It is usually performed by the maker, and its screws are not visible in the Transit, being enclosed in the ball seen where the telescope is connected with the cross-bar.i All the adjustments should be meddled with as little as possible, lest the screws should get loose; and when once made right they should be kept so by careful usage. * This "adjustment of the line of collimation" has merely brought the intersection of the cross-hairs (which fixes the line of sight) into the line joining the centres of the collars on which the telescope turns in the Ys; but the maker is supposed to have originally fixed the optical axis of thetelescope, (i. e. the line joining the optical centres of the glasses), in the same line. tThe adjustment of " Centrilg the ol.'ect-glass is the invention of Messrs. Gualey, of Troy. 250 [PART 1V. CHAPTER IV. THE FIELD-WORK. (366) To measure a horizontal angle. Set up the instrument so that its centre shall be Fig. 252. exactly over the angular point, or in the intersection of the two lines whose difference of A - direction is to be measured; as at B in the figure. A plumb line must be suspended from under the centre. Dropping a stone is an imperfect substitute for this. Set the instrument so that its lower parallel plate may be as nearly horizontal as possible. The levels will serve as guides, if the four parallel-plate screws be first so screwed up or down that equal lengths of them shall be above the upper plate. Then level the instrument carefully, as in Art. (338). Direct the telescope to a rod, stake, or other object, A in the figure, on one of the lines which form the angle. Tighten the clamps, and by the tangent-screw, (see Art. (336)), move the telescope so that the intersection of the crosshairs shall very precisely bisect this object. Note the reading of the vernier, as explained in. the preceding chapter. Then loosen the clamp of the vernier, and direct the telescope'on the other line (as to C) precisely as before, and again read. The difference of the two readings will be the desired angle, ABC. Thus, if the first reading had been 40~ and the last 1900, the angle would be 150~. If the vernier had passed 360~ in turning to the second object, 360~ should be added to the last reading before subtracting. Thus, if the first reading had been 300~, and the last reading 90~, the angle would be found by calling the last reading, as it really is, 3600 + 900 = 450~, and then subtracting 300~. It is best to sight first to the left hand object and then to the right hand one, turning " with the sun," or like the hands of a watch, since the numbering of the degrees usually runs in that direction. CHAP. IV.] The Field-work., 251 It is convenient, though not necessary, to begin by setting the vernier at zero, by the upper movement (that of the vernier plate on the circle) and then, by means of the lower motion, (that of the whole instrument on its axis), to direct the telescope to the first object. Then fasten the lower clamp, and sight to the second object as before. The reading will then be the angle desired. An objection to this is that the two verniers seldom read alike.* After one or more angles have been observed from one point, the telescope must be directed back to the first object, and the reading to it noted, so as to make sure that it has not slipped. A watch-telescope (see Art. 339) renders this unnecessary. The error arising from the instrument not being set precisely over the centre of the station, will be greater the nearer the object sighted to. Thus a difference of one inch would cause an error of only 3" in the apparent direction of an object a mile distant, but one of nearly 3' at a distance of a hundred feet. (367) Reduction of high and low objects. When one of the objects sighted to is higher than the other, the "plunging telescope" of these instruments causes the angle measured to be the true horizontal angle desired; i. e. the same angle as if a point exactly under the high object and on a level with the low object (or vice versa) had been sighted to. For, the telescope has been caused to move in a vertical plane by the 3d adjustment of Chapter II, and the angle measured is therefore the angle between the vertical planes which pass through the two objects, and which " project" the two lines of sight on the same horizontal plane. This constitutes the great practical advantage of these instruments over those which are held in the planes of the two objects observed, such as the sextant, and the " circle" much used by the French. X The learner will do well to gauge his own precision and that of the instrument and he may rest assured that his own will be the one chiefly in fault) by measuring, from any station, the angles between successive points all around him, till he gets back to the first point, beginning at different parts of the circle for each angle. The sum of all these angles should exactly equal 3600. He will probably find quite a difference from that. 252 TRANSIT AND THEODOLITE SURVEYING. [PART IV. (368) Notation of angles. The angles observed may be noted in various ways. Thus, the observation of the angle ABC, in Fig. 252, may be noted " At B, from A to C, 150~," or better, " At B, between A and C, 150~." In column form, this becomes Between A 150~land C. At B When the vernier had been set at zero before sighting to the first object, and other objects were then sighted to, those objects, the readings to which were less than 180~, will be on the left of the first line, and those to which the readings were more than 1800, will be on iid right, looking in the direction in which thesurvey is proceeding, from A to B, and so on.* (369) Probable error. When a number of separate observations of an angle have been made, the mean or average of them all, (obtained by dividing the sum of the readings by their number,) is taken- as the true reading. The " Probable error" of this mean, is the quantity, (minutes or seconds) which is such that there is an even chance of the real error being more or less than it. Thus, if ten measurements of an angle gave a mean of 350 18', and it was an equal wager that the error of this result, too much or too little, was half a minute, then half a minute would be the " Probable error" of this determination. This probable error is equal to the square root of the sunm of the squares of the errors (i. e. the differences of each observation from the mean) divided by the number of observations, and multiplied by the decimal 0.674489. The same result would be obtained by using what is called " The weight" of the observation. It is equal to the square of the number of observations divided by twice the sum of the squares of the errors. The " Probable error" is equal to 0.476936 divided by the square root of the weight. These rules are proved by the "Theory of Probabilities." (370) To repeat an angle. Begin as in Art. (366), and measure the angle as there directed. Then unclamp below, and turn the circle around till the telescope is again directed to the first object, and made to bisect it precisely by the lower tan* This is very useful in preventing any ambiguity in the field-notes. CHAP. IV.] The Field-work. 253 gent screw. Then unclamp above and turn the vernier plate till the telescope again points to the second object, the first reading remaining unchanged. The angle will now have been measured a second time, but on a part of the circle adjoining that on which it was first measured, the second arc beginning where the first ended. The difference between the first and last readings will therefore be twice the angle. This operation may be repeated a third, a fourth, or any number of times, always turning the telescope back to the first object by the lower movement, (so as to start with the reading at which the preceding observation left off) and turning it to the second object by the upper movement. Take the difference of the first and last readings and divide by the number of observations. The advantage of this method is that the errors of observation (i. e. sighting sometimes to the right and sometimes to the left of the true point) balance each other in a number of repetitions; while the constant error of graduation is reduced in proportion to this number. This beautiful principle has some imperfections in practice, probably arising from the slipping and straining of the clamps. (371) Angles of deflection. The angle of deflection of one line from another, is the Fig. 253. angle which one line ce makes with the other line produced. Thus, in _ the figure, the angle of deflection of BC from AB, is B'BC. It is evidently the supplement of the angle ABC. To measure it with the Transit, set the instrument at B, direct the telescope to A, and then turn it over. It will now point in the direction of AB produced, or to B', if the 2d adjustment of Chapter II, has been performed. Note the reading. Then direct the telescope to C. Note the new reading, and their difference will be the required angle of deflection, B'BC. If the vernier be set at zero, before taking the first observation, the readings for objects on the right of the first line will be less than 254 TRANSIT AND THEODOLITE SURVEYING. [PART IV. 180~, and more than 180~ for objects on the left; conversely to Art. (368). (372) Line surveying. The survey of a line, such as a road, &c., can be made by the Theodolite or Transit, with great precision; measuring the angle which each line makes with the preceding line, an'd noting their lengths, and the necessary offsets on each side. Short lines of sight should be avoided, since a slight inaccuracy in setting the centre of the instrument exactly over or under the point previously sighted to, would then much affect the angle, as noticed at close of Art. (366). Very great accuracy can be obtained by using three tripods. One would be set at the first station and sighted back to from the instrument placed at the second station, and a forward sight be then taken to the third tripod placed at the third station. The instrument would then be set on this third tripod, a back sight taken to the tripod remaining on the second station, and a foresight taken to the tripod brought from the first station to the fourth station; to which the instrument is next taken: and so on. This is especially valuable in surveys of mines. The field-notes may be taken as directed in Chapter III of Compass Surveying, pages 149, &c., the angles taking the place of the Bearings. The " Checks by intersecting Bearings," explained in Art. (246), should also be employed. The angles made on each side of the stations may both be measured, and the equality of their sum to 3600, would at once prove the accuracy of the work. If the magnetic Bearing of any one of the lines be given, and that of any of the other lines of the series be required, it can be deduced by constructing a diagram, or by modifications of the rules given for the reverse object, in Art. (243). (373) Traversing: Or Surveying by the back-angle. This is a method of observing and recording the different directions of successive portions of a line, (such as a road, the boundaries of a farm, &c.,) so as to read off on the instrument, at each station, the angle which each line makes-not with the preceding line, but-with the first line observed. This line is, therefore, called the meridian of that survey. CHAP. IV.] The Field-work. 255 Fig. 254. A B C - 4 —-- - Set up the instrument at the first angle, or second station, (B, in the figure), of the line to be surveyed. Sight to A and then to C. Clamp the vernier, and take the instrument to C. Loosen the lower clamp, and direct the telescope to B, the reading remaining as it was at B. Clamp below, loosen above, and sight to D. The reading of the instrument will be the angle which the line CD makes with the first line, or Meridian, AB. Take the instrument to D. Sight back to C, and then forward to E, as before directed, and the reading of the instrument will be the angle which DE makes with AB. So proceed-for any number of lines. When the Transit is used, the angles of deflection of each line from the first, obtained by reversing the telescope, may be used in "Traversing," and with much advantage when the successive lines do not differ greatly in their directions. A 00 A 0~ B 2000 The survey represented in the figure, B 20~ C t50( is recorded in the first of the accompa- C 500 D 1800 nyino' Tables, as observed with the The- D 00 E 3000' E 3000 F 2100 odolite; and in the second Table, as F 300 G 2500 observed with the Transit. G 2500 The chief advantage of this method is its greater rapidity in the field and in platting, the angles being all laid down from one meridian, as in Compass-surveying. This also increases the accuracy of the plat, since any error in the direction of one line does not affect the directions of the following lines.' (371) Use of the Compass. The chief use of the Compass attached to a Transit or Theodolite, is as a check on the observations; for the difference between the magnetic Bearings of any * If there are two verniers; take care always to read the degrees from the same vernier. Mark it A. 256 TRANSIT AND THEODOLITE SURVEYING. [PART IV. two lines should be the same, approximately, as the angle between them, measured by the more accurate instruments. The Bearing also prevents any ambiguity, as to whether an angle was taken to the right or to the left. The instrument may also be used like a simple compass, the telescope taking the place of the sights, and requiring similar tests of accuracy. A more precise way of taking a Bearing is to turn the plate to which the compass box is attached, till the needle points to zero, and note the reading of the vernier; then sight to the object, and again read the vernier. The Bearing will thus be obtained more minutely than the divisions on the compass box could give it. (375) Measuring distances with a telescope and rod. On the cross-hair ring, described in Art. (330), stretch two more horizontal spider-threads at equal distances above and below the original one; or all may be replaced by a plate of thin Fig. 255. glass, placed precisely in the focus, with the necessary lines, as in the figure, etched by fluoric acid. Let a rod, 10 or 15 feet long, be heldup at 1000 feet off, and let there be marked on it precisely the length which the distance between two of these lines covers. Let this be subdivided as minutely as the spaces, painted alternately white and red and numbered, can be seen. If ten subdivisions are made, each will represent a distance of 100 feet off, and so on. Continue these divisions over the whole length of the rod. It is now ready for use. The French call it a stadia. When it is held up at any unknown distance, the number of divisions on it intercepted between the two lines, will indicate the distance with considerable precision. It should be tested at various distances. A "Levelling-rod," divided into feet, tenths and hundredths, may be used as a stadia, with less convenience but more precision. Experiments must previously determine at what distances the space between the lines in the telescope covers one foot, &c. Then, at any unknown distance, let the sliding " target" of the rod be moved till one line bisects it, and its, place on the rod be read off; let-the target be then moved so that the other line bisects it and let CHAP. IV.] The Feld-work. 257 its place be again noted. Then the required distance will be equal to the difference of the readings on the rod, in feet, multiplied by the distance at which a foot was intercepted between the lines. One of the horizontal hairs may be made movable, and its distance from the other, when the space between them exactly covers an object of known height, can be very precisely measured by counting the number of turns and fractions of a turn, of a screw by which this movable hair is raised or lowered. A simple proportion will then give the distance. On sloping ground a double correction is necessary to reduce the slope to the horizon and to correct the oblique view of the rod. The horizontal distance is, in consequence, approximately equal to the observed distance multiplied by the square of the cosine of the slope of the ground. The latter of the above two corrections will be dispensed with by holding the rod perpendicular to the line of sight, with the aid of a right angled triangle, one side of which coincides with the rod at the height of the telescope, and the other side of which adjoining the right angle, is caused, by leaning the rod, to point to the telescope. Other contrivances have been used for the same object, such as a Binocular Telescope with two eye-pieces inclined at a certain angle; a Telescope with an object-glass cut into two movable parts; &c. (376) Ranging out lines. This is the converse of Surveying lines. The instrument is fixed over the first station with great precision, its telescope being very carefully adjusted to move in a vertical plane. A series of stakes, with nails driven in their tops, or otherwise well defined, are then set in the desired line as far as the power of the instrument extends. It is then taken forward to a stake three or four from the last one set, and is fixed over it, first by the plumb and then by sighting backward and forward to the first and last stake. The line is then continued as before. A good object for a long sight is a board painted like a target, with black and white concentric rings, and made to slide in grooves cut in the tops of two stakes set in the ground about in the line. It 17 258 TRANSIT AND THEODOLITE SURVEYING. [PART IV. is moved till the vertical hair bisects the. circles (which the eye can determine with great precision) and a plumb-line dropped from their centre, gives the place of the stake. "M ason & Dixon's Line" was thus ranged. If a Transit be used for ranging, its " Second Adjustment" is most important to ensure the accuracy of the reversal of its Telescope. If a Theodolite be used, the line is continued by turning the vernier 180~, or by reversing the telescope in its ys, as noticed in Arts. (325) and (362). (377) Farm Surveying, &c. A large farm can be most easily and accurately surveyed, by measuring the angles of its main boundaries (and a few main diagonals, if it be very large,) with a Theodolite or Transit, as in Arts. (366) or (371), and filling up the interior details, as fences, &c., with the Compass and Chain. If the Theodolite be used, Fig. 25G. keep the field on the left hand, as in following the or- B der of the letters in this figure, and turn the telescope' 200~, around "with the sun," and C0,' a E A the angles measured as in D Art. (366), will be the interior angles of the field, as noted in the figure. The accuracy of the work will be proved. as alluded to in Art. (257), if the sum of all the interior angles be equal to the product of 180~ by the number of sides of the figure less two. Thus in the figure, the sum of all the interior angles = 540~ = 1800 x (5 - 2). The sum of the exterior angles would of course equal 1800 x (5 + 2) = 1260~. If the Transit be used, the farm should be kept on the right hand, and then the angles measured will be the supplements of the interior angles. If the angles to the right be called'positive, and those to the left negative, their algebraic sum should equal 360~. If the boundary lines be surveyed by " Traversing," as in Art. (373), the reading, on getting back to the last station and looking back to the first line, should be 360~, or 0~. CHAP. IV.] The Field-work. 259 The content of any surface surveyed by " Traversing " with the Transit can be calculated by the Traverse Table, as in Chapter VI, of Part III, by the following modification. When the angle of deflection of any side from the first side, or Meridian, is less than 900, call this angle the Bearing, find its Latitude and Departure, mand call them both plus. When the angle is between 90~ and 1800, call the difference between the angle and 180~ the Bearing, and call its Latitude minus and its Departure plus. When the angle is between 1800 and 270~, call its difference from 1800 the Bearing, and call its Latitude minus and its Departure mninus. When the angle is more than 2700, call its difference from 3600 the Bearing, and call its Latitude plus and its Departure minus. Then use these as in getting the content of a Compass-survey. The signs of the Latitudes and Departures follow those of the cosines and sines in the successive quadrants. Town-Surveying would be performed as directed in Art. (261), substituting" angles" for "Bearings." "Traversing" is the best method in all these cases. Inaccessible areas would be surveyed nearly as in Art. (134), except that the angles of the lines enclosing the space would be measured with the instrument, instead of with the chain. (378) Platting. Any of these surveys can be platted by any of the methods explained and characterized in Chapter IV, of the preceding Part. A circular Protractor, Art. (264), may be regarded as a Theodolite placed on the paper. "Platting Bearings," Art. (265), can be employed when the survey has been made by " Traversing." But the method of " Latitudes and Departures," Art. (285), is by far the most accurate. PART V. TRIANGULAR SURVEYING; OR By the Fourth Method. (379) TRIANGULAR SURVEYING is founded on the Fourth Method of determining the position of a point, by the intersection of two known lines, as given in Art. (8). By an extension of the principle, a field, a farm, or a country, can be surveyed by measuring only one line, and calculating all the other desired distances, which are made sides of a connected series of imaginary Triangles, whose angles are carefully measured. The district surveyed is covered with a sort of net-work of such triangles, whence the name given to this kind of Surveying. It is more commonly called " Trigonometrical Surveying;" and sometimes " Geodesic Surveying," but inmproperly, since it does not necessarily take into account the curvature of the earth, though always adopted in the great surveys in which that is considered. (380) Outline of operations. A base line, as long as possible, (5 or 10 miles in surveys of countries), is measured with extreme accuracy. From its extremities, angles are taken to the most distant objects visible, such as steeples, signals on mountain tops, &c. The distances to these and between these are then calculated by the rules of Trigonometry. The instrument is then placed at each of these new stations, and angles are taken from them to still more distant stations, the calcu lated lines being used as new base lines. This process is repeated and extended till the whole district is embraced by these " primary triangles " of as large sides as possible. PART v.] TRIANGULAR SURVEYING. 261 One side of the last triangle is so located that its length can be obtained by measurement as well as by calculation, and the agreement of the two proves the accuracy of the whole work. Within these primary triangles, secondary or smaller triangles are formed, to fix the position of the minor local details, and to serve as starting points for common surveys with chain and compass, &c. Tertiary triangles may also be required. The larger triangles are first formed, and the smaller ones based on them, in accordance with the important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. Each of these steps will now be considered in turn, in the following order: 1. The Base; articles (381), (382). 2. The Triangulation; articles (383) to (390). 3. Modifications of the method; articles (391) to (395). (381) Measuring a Base. Extreme accuracy in this is necessary, because any error in it will be multiplied in the subsequent work. The ground on which it is located must be smooth and nearly level, and its extremities must be in sight of the chief points in the neighborhood. Its point of beginning must be marked by a stone set in the ground with a bolt let into it. Over this a Theodolite or Transit is to be set, and the line'" ranged out" as directed in Art. (376). The measurement may be made with chains, (which should be formed like that of a watch,) &c. but best with rods. We will notice in turn their Materials, Supports, Alinement, Levelling, and Contact. As to Materials, iron, brass and other metals have been used, but are greatly lengthened and shortened by changes of temperature. Wood is affected by moisture. Glass rods and.tubes are preferable on both these accounts. But wood is the most convenient. Wooden rods should be straight-grained white pine, &c.; well seasoned, baked, soaked in boiling oil, painted and varnished. They may be trussed, or framed like a mason's plumb-line level, to prevent their bending. Ten or fifteen feet is a convenient length. Three are required, which may be of different colors, to prevent 262 TRIANGULAR SURVEYING. [PART v. mistakes in recording. They must be very carefully compared with a standard measure. Supports must be provided for the rods, in accurate work. Posts set in line at distances equal to the length of the rods, may be driven or sawed to a uniform line, and the rods laid on them, either directly, or on beams a little shorter. Tripods, or trestles, with screws in their tops to raise or lower the ends of the rods resting on them, or blocks with three long screws passing through them and serving as legs, may also be used. Staves, or legs, for the rods have been used, up and down which sockets carrying the rods could slide. The Alinement of the rods can be effected, if they are laid on the ground, by strings, two or three hundred feet long, stretched between the stakes set in the line, a notched peg being driven when the measurement has reached the end of one string, which is then taken on to the next pair of stakes; or, if the rods rest on supports, by projecting points on the rods being alined by the instrument. The Levelling of the rods can be performed with a common mason's level; or their angle measured, if not horizontal, by a " slope-level." The Contacts of the rods may be effected by bringing them end to end. The third rod must be applied to the second before the first has been removed, to detect any movement. The ends must be protected by metal, and should be rounded (with radius equal to length of rod) so as to touch in only one point. Round-headed nails will answer tolerably. Better are small steel cylinders, horizontal on one end and vertical on the other. Sliding ends, with verniers, have been used. If one rod be higher than the next one, one must be brought to touch a plumb-line which touches the other, and its thickness be added. To prevent a shock from contact, the rods may be brought not quite in contact, and a wedge be let down between them till it touches both at known points on its graduated edges. The rods may be laid side by side, and lines drawn across the end of each be made to coincide or form one line. This is more accurate. Still better is a " visual contact," a double microscope with cross-hairs being used, so placed that one tube bisects a dot at the end of one rod, and the other tube bisects a dot at the end PART v.] TRIANGULAR SURVEYING, 263 of the next rod. The rods thus never touch. The distance between the two sets of cross-hairs is of course to be added. The combination of this principle with an arrangement by which bars of two metals preserve a length invariable at all temperatures, as used on the U. S. Coast Survey, leaves nothing to desire in precision. A Base could be measured over very uneven ground, or even water, by suspending a series of rods from a stretched rope by rings in which they can move, and levelling them and bringing them into contact as above. (382) Corrections of Base, If the rods were not level, their length must be reduced to its horizontal projection. This would be the square root of the difference of the squares of the length of the rod (or of the base) and of the height of one end above the other; or the product of the same length by the cosine of the angle which- it makes with the horizon.* If the rods were metallic, they would need to be corrected for temperature. Thus, if an iron bar expands 7 of its length for 10 Fahrenheit, and had been tested at 32~, and a Base had been measured at 72~ with such a bar 10 feet long, and found to contain 3000 of them, its apparent length would be 30,000 feet, but its real length would be 8.4 feet more. (383) Choice of Stations. The stations, or " Trigonometrical points," which are to form the vertices of the triangles, and to be observed to and from, must be so selected that the resulting triangles may be'" well-conditioned," i. e. may have such sides and angles that a small error in any of the measured quantities will cause the least possible errors in the quantities calculated from them. The higher Calculus shows that the triangles should be as nearly equilateral as possible. This is seldom attainable, but no angle should be admitted less than 30~, or more than 120~.t * More precisely, A being this angle, and not more than 20 or 30, the difference between the inclined and horizontal lengths, equals the inclined or real length multiplied by the square of the minutes in A, and that by the decimal 0.00000004231; as shewn in Appendix B. In a Geodesic survey, the base would also be required to be reduced to the level of the sea. t When two angles only are observed, as is often the case in the secondary triangulation, the unobserved angle ought to be nearly a right angle. 264 TRIANGULAR SURVEYING. [PART V. To extend the triangulation, by continually increasing the sides of the triangles, without introducing "ill-conditioned" triangles, may be effected as in the figure. AB is the measured base. Fig. 257. -------- --—' —-I C and D are the nearest stations. In the triangles ABC and ABD, all the angles being observed and the side AB known, the other sides can be readily calculated. Then in each of the triangles DAC and DBC, two sides and the contained angles are given to find DC, one calculation checking the other. DC then becomes a base to calculate EF; which is then used to find GH; and so on. The fewer primary stations used, the better; both to prevent confusion and because the smaller number of triangles makes the correctness of the results more "' probable." The United States Coast Survey, under the superintendence of Prof. A. D. Bache, displays some fine illustrations of these principles, and of the modifications they may undergo to suit various localities. The figure on the opposite page represents part of the scheme of the primary triangulation resting on the Massachusetts base and including some remarkably well-conditioned triangles, as well as the system of quadrilaterals which is a valuable feature of the scheme when the sides of the triangles are extended to considerable lengths, and quadrilaterals, with both -:,n'nals determined, take the place of simple triangles. The engraving is on a scale of 1:1200,000. PART v.] TRIANGULAR SURVEYING. 265 Fig. 258. ZI 0 0 * / / k~Bton i57'L / 266 TRIANGULAR SURVEYING. [PART V. (381) Signals. They must be high, conspicuous, and so made that the instrument can be placed precisely under them. Three or four timbers framed into a Fig. 259. pyramid, as in the figure, with a long mast projecting above, fulfil the first and last conditions. The mast may be made vertical by directing two theodolites to it and adjusting it so that their telescopes follow it up and down, their lines of sight being at right angles to each other. Guy ropes may be used to keep it vertical. A very excellent signal, used on the Massachusetts State Survey, by Mr. Borden, is represented in the three following figures. It Fig. 260. Fig. 261. Fig. 262. consists merely of three stout sticks, which form a tripod, framed with the signal staff, by a bolt passing through their ends and its middle. Fig. 260 represents the signal as framed on the ground; Fig. 261 shews it erected and ready for observation, its base being steadied with stones; and Fig. 262 shews it with the staff turned aside, to make room for the Theodolite and its pro- Fig. 263 tecting tent. The heights of these signals varied between 15 and 80 feet. Another good signal consists of a stout post let into the ground, with a mast fastened to it by a bolt below and a collar above. By opening the collar, the mast lI can be turned down and the Theodolite set exactly under the former summit of the signal, i. e. in its vertical axis. / Signals should have a height equal to at least T- of their dis PART V.] TRIANGULAR SURVEYING. 267 tance, so as to subtend an angle of half a minute, which experience has shown to be the least allowable To make the tops of the signal-masts conspicuous, flags may be attached to them; white and red, if to be seen against the ground, and red and green if to be seen against the sky.' The motion of flags renders them visible, when much larger motionless objects are not. But they are useless in calm weather. A disc of sheetiron, with a hole in it, is very conspicuous. It should be arranged so as to be turned to face each station. A barrel, formed of muslin sewn together four or five feet long, with two hoops in it two feet apart, and its loose ends sewn to the signal-staff, which passes through it, is a cheap and good arrangement. A tuft of pine boughs fastened to the top of the staff, will be well seen against the sky. In sunshine, a number of pieces of tin nailed to the staff at different angles, will be very conspicuous. A truncated cone of burnished tin will reflect the sun's rays to the eye in almost every situation. But a " heliotrope," which is a piece of looking-glass, so adjusted as to reflect the sun directly to any desired point, is the most perfect arrangement. For night signals, an Argand lamp is used; or, best of all, Drummond's light, produced by a stream of oxygen gas directed through a flame of alcohol upon a ball of lime. Its distinctness is exceedingly increased by a parabolic reflector behind it, or a lens in front of it. Such' a light was brilliantly visible at 66 miles distance. (385) Observations of the Angles. These should be repeated as often as possible. In extended surveys, three sets, of ten each, are recommended. They should be taken on different parts of the circle. In ordinary surveys, it is well to employ the method of'" Traversing," Art. (373). In long sights, thestate of the atmosFig. 264. * To determine at a station A, c whether its signal can be seen C from B, projected against the sky or not, measure the vertical angles BAZ and ZAC. If their A!,, sum equals or exceeds 1800, A. will be thus seen from B. If not, the signal at A must be rais. ed till this sum equals 180~. 268 TRIANGULAR SURVEYING. [PART V. phere has a very remarkable effect on both the visibility of the signals, and on the correctness of the observations. When many angles are taken from one station, it is important to record them by some uniform system. The form given below is convenient. It will be noticed that only the minutes and seconds of the second vernier are employed, the degrees being all taken from the first. Observations at STATION READINGS. MIEAN RIGHT OR LEFT OF OBSERVED TO VERNIER A. VERNIER B. READING. PRECED G OBJ' T. REMARKS. A 700 19' 0" 18' 40" 700 18' 50" B 1030 32' 20" 32' 40" 1030 32' 30" R. C 1150 14' 20'I 14' 50" 1150 14' 35" R. When the angles are "repeated," Art. (370), the multiple arcs will be registered under each other, and the mean of the seconds shewn by all the verniers at the first and last readings be adopted. (386) Reduction to the centre. It is often impossible to set the instrument precisely at or under the signal which has been observed. In such cases pro- Fig. 265. ceed thus. Let C be the cen-. tre of the signal, and RCL the -- desired angle, R being the right hand object and L the left hand D one. Set the instrument atD, AO R as near as possible to C, and measure the angle RDL. It may be less than RCL, or greater than it, or equal to it, according as D lies without the circle passing through C, L and- R, or within it, or in its circumference. The instrument should be set as nearly as possible in this last position. To find the proper correction for the observed angle, observe also the angle LDC, (called the angle of direction), counting it from 0~ to 3600, going from the left-hand object toward the left; and measure the distance DC. Calculate the distances CR and CL with the angle RDL instead of RCL, since they are sufficiently nearly equal. Then PART V.] TRIANGULAR SURVEYING. 269 RCL = RDL + CD. sin. (RDL + LDC) CD. sin. LDC, CR. sin. 1" CL. sin. 1" The last two terms will be the number of seconds to be added or subtracted. The Trigonometrical signs of the sines must be attended to. The log. sin. 1" =4.6855749. Instead of dividing by sin. 1", the correction without it, which will be a very small fraction, may be reduced to seconds by multiplying it by 206265. Example. Let RDL = 32~ 20'18".06; LDC = 101~ 15' 32".4; CD =0.9; CR = 35845.12; CL = 29783.1. The first term of the correction will be + 3".750, and the second term - 6".113. Therefore, the observed angle RDL must be diminished by 2".363, to reduce it to the desired angle RCL. Much calculation may be saved by taking the station D so that all the signals to be observed can be seen from it. Then only a single distance and angle of direction need be measured. It may also happen that the centre, C, of the Fig. 261 signal cannot be seen from D. Thus, if the signal be a solid circular tower, set the Theodolite at D, and turn its telescope so that its line of sight be- TT comes tangent to the tower at T, T'; measure on these tangents equal distances DE, DF, and direct the telescope to the middle, G, of the line EF. It will then point to the centre, C; and the distance DC will equal the distance from D to the tower plus the radius obtained by measuring the circumference. If the signal be rectangular, measure DE, DF. Fig 267 Take any point G on DE, and on DF set off DH DF =DGDE. Then is GH parallel to EF, (since E F DG: DH:: DE: DF) and the telescope directed GQ to its middle, K, will point to the middle of the diagonal EF. We shall also have DC = DK DE. _ Any such case may be solved by similar methods. * For the investigation, see Appendix B. 270 TRIINGULIR SURVEYING. [PART V. The "Phase" of objects is the effect produced by the sun shining on only one side of them, so that the telescope will be directed from a distant station to the middle of that bright side instead of to the true centre. It is a source of error to be guarded against. Its effect may however be calculated. (387) Correction of the angles. When all the angles of any triangle can be observed, their sum should equal 180.' If not they must be corrected. If all the observations are considered equally accurate, one-third of the difference of their sum from 180~, is to be added to, or subtracted from, each of them. But if the angles are the means of unequal numbers of observations, their errors may be considered to be inversely as those numbers, and they may be corrected by this proportion; As the sum of the reciprocals of each of the three numbers of observations Is to the whole error, So is the reciprocal of the number of observations of one of the angles To its correction. Thus if one angle was the mean of three observations, another of four, and the third of ten, and the sum of all the angles was 180~ 3', the first named angle must be diminished by the fourth term of this proportion; +: 3'::: 1' 27".8. The second angle must in like manner be diminished by 1' 5".9; and the third by 26".3. Their corrected sum will then be 180~. It is still more accurate but laborious, to apportion the total error, or difference from 180~, among the angles inversely as the " Weights," explained in Art. (369). On the U. S. Coast Survey, in six triangles measured in 1844 by Prof. Bache, the greatest error was six-tenths of a second. (388) Calculation and platting. The lengths of the sides of the triangles should be calculated with extreme accuracy, in two ways if possible, and by at least two persons. Plane Trigonometry may be used for even large surveys; for, though these sides are really arcs and not straight lines, the difference will be only one* If the triangles were very large, they would have to be regarded as spherical, and the sum of their angles would be more than 1800; but this " spherical excess" would be only 1" for a triangle containing 76 square miles, 1' for 4500 square miles, &c.; and may therefore be neglected in all ordinary surveying operations. PART v.] TRIANGULAR SURVEYING. 271 twentieth of a foot in a distance of 111 miles; half a foot in 23 miles; a foot in 341 miles, &c. The platting is most correctly done by constructing the triangles, as in Art. (90), by means of the calculated lengths of their sides. If the measured angles are platted, the best method is that of chords, Art. (275). If many triangles are successively based on one another, they will be platted most accurately, by referring all their sides to some one meridian line by means of" iRectangular Coordinates," the Method of Art. (6), and platting as in Art. (277.) In the survey of a country, this Meridian would be the true North and South line passing through some well determined point. (389) Base of Verification. As mentioned in Art. (380), a side of the last triangle is so located that it can be measured, as was the first base. If the measured and calculated lengths agree, this proves the accuracy of all the previous work of measurement and calculation, since the whole is a chain of which this is the last link, and any error in any previous part would affect the very last line, except by some improbable compensation. How near the agreement should be, will depend on the nicety desired and attained in the previous operations. Two bases 60 miles distant differed on one great English survey 28 inches; on another one inch; and on a French triangulation extending over 500 miles, the difference was less than two feet. Results of equal or greater accuracy are obtained on the U. S. Coast Survey. (390) Interior filling ap. The stations whose positions have been determined by the triangulation are so many fixed points, from which more minute surveys may start and interpolate any other points. The Trigonometrical points are like the observed Latitudes and Longitudes which the mariner obtains at every opportunity, so as to take a new departure from them and determine his course in the intervals by the less precise methods of his compass and log. The chief interior points may be obtained by " Secondary Triangulation," and the minor details be then filled in by any of the methods of surveying, with Chain, Compass, or Transit, already explained, or by the Plane Table, described in Part VIII. 272 TRIANGULAR SURVEYING. [PART V. With the Transit, or Theodolite, "Traversing" is the best mode of surveying, the instrument being set at zero, and being then directed from one of the Trigonometrical points to another, which line therefore becomes the "Meridian" of that survey. On reaching this second point, in the course of the survey, and sighting back to the first, the reading should of course be 0~0, as explained in Art. (377). (391) Radiating Triangulation. This name may be given to a method shown in the figure. Choose Fig. 268. a conspicuous point, 0, nearly in the F centre of the field or farm to be sur- B veyed. Find other points, A, B, C, / D, &c. such that the signal at 0 can be G x.X seen from all of them, and that the tri-.. angles ABO, BCO, &c, shall be as A / \ nearly equilateral as possible. Mea- / sure one side, AB for example. At A / measure the angles OAB, and OAG; at B B measure the angles OBA and OBC; and so on, around the polygon. The correctness of these measurements may be tested by the sum of the angles, as in Art. (377). It may also be tested by the Trigonometrical principle that the product of the sines of every alternate angle, or the odd numbers in the figure, should equal the product of the sines of the remaining angles, the even numbers in the figure.' The calculations of the unknown sides are readily made. In the triangle ABO, one side and all the angles are given to find AO and BO. In the triangle BCO, BO and all the angles are given to find BC and CO; and so with the rest. Another proof of the accuracy of the work will be given by the calculation of the length of the side AO in the last triangle, agreeing with its length as obtained in the first triangle. (392) Farm Triangulation. A Farm or Field may be surveyed by the previous methods, but the following plan will often be more * For the demonstration, see Appendix B. PART v.] TRIANGULAR SURVEYING. 273 convenient. Choose abase, as XY, within Fig. 269. the field, and from its ends measure the A angles between it and the direction of each corner of the field, if the Theodo- p c lite or Transit be used, or take the bearing of each, if the Compass be used. Consider first the triangles which have D XY for a base, and the corners of the field, A, B, C, &c., for vertices. In each of them one side and the angles will be known to find the other sides, XA, XB, &c. Then consider the field as made up of triangles which have their vertices at X. In each of them two sides and the included angle will be given to find its content, as in Art. (65). If Y be then taken for the common vertex, a test of the former work will be obtained. The operation will be somewhat simplified by taking for the base line a diagonal of the field, or one of its sides. (393) Inaccessible Areas. A field or farm may be surveyed, by this " Fourth Method," without entering Fig, 270. it. Choose a base line XY, from which all the corners of the field can be seen. Take / their Bearings, or the angles between the, / Base line and their directions. The dis- /', tances from X and Y to each of them can I!, < / be calculated as in the last article. The 11/'' / figure will then shew in what manner the,' content of the field is the difference between x the contents of the triangles, having X (or Y) for a vortex, whicht lie outside of it, and those which lie partly within the field and partly outside of it. Their contents can be calculated as in the last article. and their difference will be the desired content. If the figure be regarded as generated by the revolution of a line one end of which is at X, while its other end passes along'the boundaries of the field, shortening and lengthening accordingly, and if those triangles generated by its movement in one direction be called plus and those generated by the contrary movement be called minus, their algebraic sum will be the content. 18 274 TRIINGULAR SURVEYING. [PART V. (394) Inversion of the Fourth Method. In all the operations which have been explained, the position of a point has been determined, as in Art. (8), by taking the angles, or bearings, of two lines passing from the two ends of a Base line to the unknown point. But the same determination may be effected inversely, by taking from the point the bearings, by compass, of the two ends of the Base line, or of any two known points. The unknown point will then be fixed by platting from the two known points the opposite bearings, for it will be at the intersection of the lines thus determined. (395) Defects of the Method of Intersection. The determination of a point by the Fourth Method (enunciated in Art. (8), and developed in this Part) founded on the intersection of lines, has the serious defect that the point sighted to will be very indefinitely determined if the lines which fix it meet at a very acute or a very obtuse angle, which the relative positions of the points observed from and to, often render unavoidable. Intersections at right angles should therefore be sought for, so far as other considerations will permit. PART VI. TRILINEAR SURVEYING; By the Eifth Method. (396) TRILINEAR SURVEYING is founded on the Fifth Method of determining the position of a point, by measuring the angles betwen three lines conceived to pass from the required point to three known points, as illustrated in Art. (10). To fix the place of the point from these data is much more difficult than in the preceding methods, and is known as the " Problem of the three points." It will be here solved Geometrically, Instrumentally and Analytically. (397) Geometrical Solution. Let A, B and C be the known Fig. 271. objects observed from S, the angles ASB and BSC being there measured. To fix this point, S, on the plat containing A, B and C, draw lines from A and B, making angles with AB each equal 276 TRILINEAR SURVEYING. [PART VI. to 900 -ASB. The intersection of these lines at 0 will be the centre of a circle passing through A and B, in the circumference of which the point S will be situated.* Describe this circle. Also, draw lines from B and C, making angles with BC, each equal to 90o-BSC. Their intersection, 0', will be the centre of a circle passing through B and C. The point S will lie somewhere in its circumference, and therefore in its intersection with the former circumference. The point is thus determined. In the figure the observed angles, ASB and BSC, are supposed to have been respectively 40~ and 60~. The angles set off are therefore 500 and 30~. The central angles are consequently 80~ and 120~, twice the observed angles. The dotted lines refer to the checks explained in the latter part of this article. When one of the angles is obtuse, set off its difference from 900 on the opposite side of the line joining the two objects to that on which the point of observation lies. When the angle ABC is equal to the supplement of the sum of the observed angles, the position of the point will be indeterminate; for the two centres obtained will coincide, and the circle described from this common centre will pass through the three points, and any point of the circumference will fulfil the conditions of the problem. A third angle, between one of the three points and a fourth point, should always be observed if possible, and used like the others, to serve as a check. Many tests of the correctness of the position of the point determined may be employed. The simplest one is that the centres of the circles, O and 0', should lie in the perpendiculars drawn through the middle points of the lines AB and BC. Another is that the line BS should be bisected perpendicularly by the line 00'. A third check is obtained by drawing at A and C perpendiculars to AB and CB, and producing them to meet BO and BO' produced, For, the arc AB measures the angle AOB at the centre, which angle = 1800 -2 (90o - ASB) = 2 ASB. Therefore, any angle inscribed in the circumference and measured by the same arc is equal to ASB. PART VI.] TRILINEAR SURVEYING. 277 in D and E. The line DE should pass through S; for, the angles BSD and BSE being right angles, the lines DS and SE form one straight line. The figure shews these three checks by its dotted lines. (398) Instrumental Solution. The preceding process is tedious where many stations are to be determined. They can be more readily found by an instrument called a Station-pointer, or Chorograph. It consists of three arms, or straight-edges, turning about a common centre, and capable of being set so as to make with each other any angles desired. This is effected by means of graduated arcs carried on their ends, or by taking off with their points (as with a pair of dividers) the proper distance from a scale of chords (see Art. (274)) constructed to a radius of their length. Being thus set so as to make the two observed angles, the instrument is laid on a map containing the three given points, and is turned about till the three edges pass through these points. Then their centre is at the place of the station, for the three points there subtend on the paper the angles observed in the field. A simple and useful substitute is a piece of transparent paper, or ground glass, on which three lines may be drawn at the proper angles and moved about on the paper as before. (399) Analytical Solution. The distances of the required point from each of the known points may be obtained analytically. Let AB = e; BO = a; ABC = B; ASB= S; BSO = S'. Also, make T = 360'~- S —S'- B. Let BAS = U; BCS =V. Then we shall have (as will be shewn in Appendix B) Cot. U = cot. T ( c. sin. S + ) ia. si. cos. T + V=T-U c.sin. U a. sin. V sin. S sin. SI sin. S sin. C' SA~c~s~~BS. SC~a~l B 278 TRILINEAR SURVEYING. [PART vI. Attention must be given to the algebraic signs of the trigonometrical functions. Example. ASB - 330 45'; BSC = 22~ 30'; AB = 600 feet; BC - 400 feet; AC = 800 feet. Required the distances and directions of the point S from each of the stations. In the triangle ABC, the three sides being known, the angle ABC is found to be 104~ 28' 39". The formula then gives the angle BAS -= U = 105 8' 10"; whence BCS is found to be 940 8' 11"; and SB = 1042.51; SA = 710.193; and SC = 934.291. (400) Maritime Surveying. The chief application of the Trilinear Method is to Miaritime or H[ydrographical Surveying, the object of which is to fix the positions of the deep and shallow points in harbors, rivers, &c., and thus to discover and record the shoals, rocks, channels and other important features of the locality. To effect this, a series of signals are established on the neighboring shore, any three of which may be represented by our points A, B, C. They are observed to from a boat, by means of a sextant, and the position of the boat is thus fixed as just shewn. The boat is then rowed in any desired direction, and soundings are taken at regular intervals, till it is found convenient to fix the new position of the boat as before. The precise point where each sounding was taken can now be platted on the map or chart. A repetition of this process will determine the depths and the places of each point of the bottom. PART VII. OBSTACLES IN ANGULAR SURVEYING. (401) THE obstacles, such as trees, houses, hills, vallies, rivers, &c., which prevent the direct alinement or measurement of any desired course, can be overcome much more easily and precisely with any angular instrument, than with the chain, methods for using which were explained in Part II, Chapter V. They will however be taken up in the same order.* As before, the given and measured lines are drawn with fine full lines; the visual lines with broken lines; and the lines of the result with heavy full lines. CHAPTER I. PERPENDICULARS AND PARALLELS. (402) Erecting Perpendiculars. To erect a perpendicular to a line at a given point, set the instrument at the given point, and, if it be a Cormpass, direct its sights on the line, and then turn them till the new Bearing differs 90~ from the original one, as explained in Art. (243). A convenient approximation is to file notches in the Compass-plate, at the 90~ points, and stretch over them a thread, sighting across which will give a perpendicular to the direction of the sights. The Transit or Theodolite being set as above, note the reading of the vernier and then turn it till the new reading is 90~ more or less than the former one. * The Demonstrations of the Problems which require them, and from which they can conveniently be separated, will be found in Appendix B. 280 OBSTICLES IN INGULAR SURVEYING [PART VII. (403) To erect a perpendicular to an inaccessible line, at a given point of it. Let AB be the line Fig. 272. and A the point. Calculate the distance Ar, B from A to any point C, and the angle CAB, by the method of Art. (430). Set -— = —the instrument at C, sight to A, turn an angle = CAB, and measure in the direc- - tion thus obtained a distance CP = CA. cos. CAB. PA will be the required perpendicular. (404) Letting fall perpendiculars. To let fall a perpendicular to a line from a given point. With the Compass, take the Bearing of the given line and then from the given point run a line, with a Bearing differing 90~ from the original Bearing, till it reaches the given line. With the Transit or Theodolite, set it at any point of the given line, as A, and observe the angle between this. Fig. 273. line and a line thence to the given point, A B P. Then set at P, sight to the former position of the instrument, and turn a number of degrees equal to what the observed angle at A wanted of 90~. The instrument will then P point in the direction of the required perpendicular PB. (405) To let fall a perpendicular to a line from an inaccessible point. Let AB be the line and P the Fig. 274. point. Measure the angles PAB, and -P PBA. Measure AB. The angles APC and BPC are known, being the comple- F —-1 7 ments of the angles measured. Then is A, C B tan. APC AC = AB. tan. APC +- tan. BPC' CHAP. I.] Perpendiculars and Parallels. 281 (406) To let fall a perpendicular to an inaccessible line from a given point. Let C be the point and Fig. 275. AB the line. Calculate the angle CAB A- E by the method of Art. (430). Set the = instrument at C, sight to A, and turn an =. _angle = 90 — CAB. It will then point in the direction of the required perpendicular CE. (407) Running Parallels, To trace a line through a given point parallel to a given line. With the Compass, take the Bearing of the given line, and then, from the given point, run a line with the same Bearing. With the Transit or Theodolite, set it at any convenient roint of the given line, as A, direct Fig. 276. it on this line, and note the read- X 3B ing. Then turn the vernier till the cross-hairs bisect the given \ point, P. Take the instrument to P Q this point and sight back to the former station, by the lower motion, without changing the reading. Then move the vernier till the reading is the supplement of what it was when the telescope was directed on the given line. It' -will then be directed on PQ, a parallel to AB, since equal angles have been measured at A and P. The manner of reading them is similar to the method of " Traversing," Art. (373). (408) To trace a line through a given point parallel to an inaccessible line. Let C be the given Fig. 277. point, and AB the inaccessible line. A\ B Find the angle CAB, as in Art. (430). = _ _ _ Set the instrument at C, direct it to A, = and then turn it so as to make an angle C E with CA equal to the supplement of the angle CAB. It will then point in a direction, CE, parallel to AB. 282 OBSTACLES IN ANGULAR SURVEYING, [PART VII. CHAPTER II. OBSTACLES TO ALINEMENT. A. To PROLONG A LINE. (409) The instrument being set at the farther end of'a line, and directed back to its beginning, the sights of the Compass, if that be used, will at once give the forward direction of the line. They serve the purpose of the rods described in Art. (169). A distant point being thus obtained, the Compass is taken to it and the process repeated. The use of the Transit or Theodolite, for this purpose, was fully explained in Art. (376). (410) By perpendiculars. When a tree, or house, obstructing the line, is met with, place the instru- Fig. 278. ment at a point B of the line, and set A B off there a perpendicular, to C; set off another at C to D, a third at D to E, a D and a fourth at E, which last will be in the direction of AB prolonged. If perpendiculars cannot be conveniently used, let BC and DE make any equal angles with the line AB, so as to make CD parallel to it. (411) By an equilateral triangle. Fig. 279. At B, turn aside from the line at an A B D 1 angle of 60~, and measure some convenient distance BC. At C, turn 60~ in the contrary direction, and mea- c sure a distance CD = BC. Then will D be a point in the line AB prolonged. At D, turn 60~ from CD prolonged, and the new direction will be in the line of AB prolonged. This method re quires the measurement of one angle less than the preceding. CIHAP. II.] Obstacles to Alinement. 283 (412) By triangulation. Let Fig. 280. AB be the line to be prolonged. AI B < E TChoose some station C, whence d <' can be seen A, B, and a point beyond the obstacle. Measure, AB and the angles A and B, of c the triangle ABC, and thence calculate the side AC. Set the instrument at C, and measure the angle ACD, CD being any line which will clear the obstacle. Let E be the desired point in the lines AB and CD prolonged. Then in the triangle ACE, will be known the side AC and its including angles, whence CE can be calculated. Measure the resulting distance on the ground, and its extremity will be the desired point E. Set the instrument at E, sight to C, and turn an angle equal to the supplement of the angle AEC, and you will have the direction, EF, of AB prolonged. (413) When the line to be prolonged is inaccessible. In this case, before the preceding method can be applied, it will be necessary to determine the lengths of the lines AB and AC, and the angle A, by the method given in Art. (430). (414) To prolong a line with only an angular instrument. This may be done when no means of measuring any distance can be obtained. Let AB be the line Fig. 281. to be prolonged. Set the in- C strument at B and deflect angles of 45~ in the directions C A and D. Set at some point, C, on one of these lines and deflect from CB 45~, and mark the D point D where this direction intersects the direction BD. Also, at C, deflect 90~ from B. Then, at D, deflect 90~ from I DB. The intersections of these last directions will fix a point E. At E deflect 135~ from EC or ED,.and a line EF, in the direction of AB will be obtained and may be continued. * This ingenious contrivance is due to a former student, Mr. R. Hood, in whose practice, while running an air line for a railroad, the necessity occurred. 284 OBSTACLES IN ANGULAR SURVEYING. [PART VII. B. To INTERPOLATE POINTS IN A LINE. (415) The instrument being set at one end of a line and directed to the other, intermediate points can be found as in Art. (177), &c. If a valley intervenes, the sights of the Compass, (if the Compass-plate be very carefully kept level cross-ways), or the telescope of the Transit or Theodolite, answer as substitutes for the plumb-line of Art. (179). (416) By a random line. When a wood, hill, or other obstacle, prevents one end of the line, Z, Fig. 282. from being seen from the other, A, run, a random line AB with the Compass or A V-~Z,TC'~[ Transit, &c., as nearly in the desired B direction as can be guessed, till you arrive opposite the point Z. Measure the error, BZ, at right angles to AB, as an offset. Multiply this error by 57Z-, and divide the product by the distance AB. The quotient will be the degrees and decimal parts of a degree, contained in the angle BAZ. Add or subtract this angle to or from the Bearing or reading with which AB was run, according to the side on which the error was, and start from A, with this corrected Bearing or reading, to run another line, which will come out at Z, if no error has been committed.'.Example. A random line was run, by compass, with a Bearing of S. 80~ E. At 20 chains' distance a point was reached opposite to the desired point, and 10 links distant from it on its right. Required the correct Bearing. Ans. By the rule, 10 x 57.3 0.2865 1= 7'. The cor2000 ~ rect Bearing is therefore S. 800 17' E. If the Transit had been used, its reading would have been changed for the new line by the same 17'. A simple diagram of the case will at once shew whether the correction is to be added to the original Bearing or angle, or subtracted from it. * This rule is substantially identical with that of.Art. (319), where its reason is given. CHAP. II.] Obstacles to Alinement. 285 If Trigonometrical Tables are at hand, the correction will be more precisely obtained from this equation; Tan. BAZ = BZ AB' In this example, B.005 = tan. 17'. AB 2000 The 57~.3 rule, as it is sometimes called, may be variously modified. Thus, multiply the error by 86~, and divide by one and a half times the distance; or, to get the correction in minutes, multiply by 3438 and divide by the distance; or, if the error is given in feet and the distance in four-rod chains, multiply the former by 52 and divide by the distance, to get the correction in minutes. The correct line may be run with the Bearing of the random line, by turning the vernier for the correction, as in Art. (312). (417) By Latitudes and Departures. When Fig 283. a single line, such as AB, cannot be run so as to come opposite to the given point Z, proceed thus, __ Z with the Compass. Run any number of zig-zag courses, AB, BC, CD, DZ, in any convenient ]j direction, so as at last to arrive at the desired point. Calculate the Latitude and Departure of each of these courses and take their algebraic sums. The sum of the Latitudes will be equal to AX, and that of the Departures to XZ. Then is Tan. ZAX; XA i. e. the algebraic sum of the Departures divided by the algebraic sum of the Latitudes is 9qual to the tangent of the Bearing.' (418) When the Transit or Theodolite is used, any line may be taken as a Meridian, i. e. as the line to which the following lines are referred; as in " Traversing," Art. (373), page 254, all the successive lines were referred to the first line. In the figure, on the next page, the same lines as in the preceding figure are repre* The length of the line AZ can also be at once obtained, since it is equal to the square root of the sum of the squares of AX and XZ; or to the Latitude divided by the cosine of the Bearing. 286 OBSTICLES IN INGULAR SURVEYING. [PART VII. sented, but they are referred to the first course, Fig. 284. AB, instead of to the Magnetic Meridian as Z before, and their Latitudes are measured along its produced line, and its Departures perpendicular to it. As before, a right-angled triangle ~,>?/ will be formed, and the angle ZAY will be "iy the angle at A between the first line AB and the desired line AZ. This method of operation has many useful&: applications, such as in obtaining data for running Railroad Curves, &c., and the student should master it thoroughly. The desired angle (and at the same time the distance) can be obtained, approximately, in this and the preceding case, by finding in a Traverse Table, the final Latitude and Departure of the desired line (or a Latitude and Departure having the same ratio) and the Bearing and Distance corresponding to these will be the angle and distance desired. (419) By similar triangles. Fig. 285. Through A measure any line CD. H e _ Take a point E, on the line CB, | beyond the obstacle, and from"it A hT_ P UB set off a parallel to CD, to some / C point, F, in the line DB. Measure EF, CD, and CA. Then this proportion, CD: CA:: EF: EG, will give the distance EG, from E to a point in the line AB. So for other points. (420) By triangulation. When Fi. 286. obstacles prevent the preceding me- - thods beinog used, if a point, C, can be found, from which A and B are accessible, measure the distances CA, CB, c and, the angle ACB, and thence calculate the angle CAB. Then observe any angle ACD, beyond the obstacle. In the triangle ACD, a side and its including angles are known, to find CD. Measure it, and a point, D, in the desired line, will be obtained. CHAP. III.] Obstacles to Measurement. 287 CHAPTER III. OBSTACLES TO MEASUREMENT. A. WHEN BOTH ENDS OF THE LINE ARE ACCESSIBLE. (421) The methods given in the preceding Chapter for prolonging a line and for interpolating points in it, will generally give the length of the line by the same operation. Thus, in Fig. 278, the inaccessible distance BE is equal to CD; in Fig. 279, BD = BC - CD; in Fig. 280, the distance BE can be calculated from the same data as CE; in Fig. 282, AZ -= V(AB2 ~ BZ2); in Fig. 283, AZ = V(AX2 + XZ2); in Fig. 284, AZ- = (AY2 + YZ2); iFi. 285,AG GB (CA in Fig. 286, the triangle ACD will give the distance AD. The method of Latitudes and Departures, Arts. (417) and (418), is very generally applicable. So is the following. (422) By triangulation. Let AB Fig. 287. be the inaccessible distance. From AX~ \, 2. -/ B any point, C, from which both A and B are accessible, measure CA, CB, and the angle ACB. Then in the triangle ABC two sides and the included angle are known to find the side AB. If all the angles can be measured, they-may be corrected, as in Art. (387).' (423) A broken Base, When the angle C is very obtuse, the preceding problem may be modified as follows. Naming the lines as is usual in Trigonometry, by small letters corresponding to the * In this figure, and the followingones,the angular point enclosed in a circle indicates the place at which the instrument is set. 288 OBSTACLES IN ANGULAR SURVEYING. [PART VII. capital letters at the angles to which they are opposite, and letting K - the number of minutes in the supplement of the angle C, we Fig. 288. shall have AB - c = a + b — 0.000000042308 x "bK2 a+ b' This formula is chiefly used in the case of what is called in Triangular Surveying "' A broken Base;" such as above; AC and CB being measured and forming very nearly a straight line, and the length of AB being required. Log. 0.000000042308 = 2.6264222 —10. (421) By angles to known points. The length of a line, both ends of which are accessible, may also be determined by angles measured at its extremities between it and the directions of two or more known points. But as the methods of calculation involve subsequent problems, they will be postponed to Articles (435), (436) and (437). B. WHEN ONE END OF THE LINE IS INACCESSIBLE. (425) By Perpendiculars. Many of the methods given for the chain, in Part II, Chapter V, may be still more advantageously employed with angular instruments, which can so much more easily and precisely set off the Perpendiculars required in Articles (191), (192), (193), &c. (426) By equal angles. Let AB Fig. 289. be the inaccessible line. At A set off'd) A (Lll I/ i AC, perpendicular to AB, and as' nearly equal to it, by estimation, as G the ground will permit. At C, measure the angle ACB, and turn the c sights, or vernier, till ACD = ACB. Find the point, D, at the intersections of the lines CD and BA produced. Then is AD - AB. CHAP.:III.] Obstacles to Measurement, 289 (427) By triangulation, Measure a distance Fig. 290. AC, about equal to AB. Measure the angles at 4(a l A and C. Then in the triangle ABC, two angles and the included side are known, to find another side AB AC sin. ACB B sin. ABC When the compass is used, the angles between 8 j the lines will be deduced from their respective Bearings, by the principles of Art. (243). If the angle at A is 900, AB = AC. tang. ACB. If the angle ACB: 45~, then AC = AB; but this position could not easily be obtained, except by the use of the Sextant,,a reflecting instrument, not described in this volume. (428) When one point cannot be seen from the other,Choose two points, C and D, in the line Fig. 291. of A, and such that from C, A and B can!D) ii be seen, and from D, A and B. Measure AC, AD, and the angles C and D. Then, -... in the triangle BCD, are known two an-, gles and the included side, to find CB. Then, in the triangle ABC, are known two sides and the included angle, to find D the third side, AB. (429) To find the distance from a given point to an inaccessible line, In Fig.' 275, Art. (406), the required distance is CE. The operations therein directed give the line CA and the angle CAB, or CAE. The required distance CE - CA. sin. CAE. 19 290 OBSTACLES IN ANGULAR SURVEYING. [PART VIL C. WHEN BOTH ENDS OF THE LINE ARE INACCESSIBLE. (430) General Method. Let Fig. 292. AB be the inaccessible line. -, —;,B Measure any convenient distance,,,, / CD, and the angles ACD, BCD,' ADC, BDC. Then, in the triangle CDA, two angles and the included side are given, to find CA. In the -,, - triangle CDB, two angles and the included side are given, to find _ / 7 CB. Then, in the triangle ABC, -— Atwo sides and the included angle C" D are given, to find AB. The work may be verified by taking another set of triangles, and finding AB from the triangle ABD instead of ABC. The following formulas will however give the desired distance with less labor. Find an angle K, such that tang. K = sin. ADC sin. CBD sin. CAD. sin. BDO Then find the difference of the unknown angles in the triangle CAB from the formula Tang. - (CAB-ABC) = tang. (450 - K). cot. 2 ACB. Then is CAB = - (CAB -ABC) + ~ (CAB + ABC). sin. BDC. sin. ACB Finally, AB = CD iBD s. AOB Finally, AB CD sin. CD. sin. CAB Example. Let CD = 7106.25 feet; ACD = 950 17' 20"; BCD = 61~ 41' 50"; ADC = 39~ 38' 40"; BDC = 780 35' 10"; required AB. The figure is constructed with these data on a scale of 5000 feet to 1 inch = 1:60000. By the above formulas, K is found to be 30~ 26' 5"; CAB = 113~ 55' 37"; and lastly AB = 6598.32. Both the methods may be used as mutual checks in any im portant case. CItAP. III. Obstacles to Measurement. 291 Fig. 293. If the lines AB and CD crossed 4 each other, as in Fig. 293, instead of >, /{" being situated as in the preceding_ figure, the same method of calculation would apply. (431) Problems To measure an inaccessible distance, AB, when a point, C, in its line can be obtained. Set the instrument at a point, D, from which A, B Fig. 294. and C can be seen, and measure c_' A B the angles CDA and ADB. " Measure also the line DC and the angle C. Then in the triangle ACD two angles and the included side are given to find AD. In the triangle DAB, the D angle DAB is known, (being equal to ACD + CDA), and AD having been found, we again have two angles and the included side to find AB. (432) Problem, To measure an inaccessible distance, AB, when only one point, C, can be foundfrom which both ends of the line can be seen. Consider CA Fig. 295. and CB as distances to be deter- Ar B mined, having one end accessible., Determine them, as in Art. (427), - - =-__-_ _ by choosing a point D, from which C and A are visble, and a point E /-,! from which C and B are visible. c At C observe the angles DCA,ACB D and BCE. Measure the distances CD and CE. Observe the angles ADC and BEC. Then in the triangle ADC, two angles and the included side are given, to find CA; and the same in the triangle CBE, to find CB. Lastly, in the triangle ACB two sides and the included angle are known, to find AB. 292 OBSTACLES IN INGULAR SURVEYING. [PART VII. (433) Problem. To measure an inaccessible distance, AB, when no point can be found from which the two ends can be seen. Let C be a point from which A is Fig. 296. visible, and D a point from which A __ B is visible, and also C. Measure I2 CD. Find the distances CA and / \ ~ / DB, as in the preceding problem; /' -,:' i. e. choose a point E, from which A / 6 and C are visible, and another E D point, F, from which D and B are visible. Measure EC and DF. Observe the angles AEC, ECA, BDF and DFB; and at the same time the angles ACD and CDB, for the subsequent work. Then CA and DB will be found, as were CA and CB in the last problem. Then in the triangle CDB, two sides and the included angle are known to find CB and the angle DCB; and, lastly, in the triangle ACB, two sides and the included angle (the difference of ACD and DCB) to find AB. (434) Problem. To interpolate a Base. Four inaccessible objects, A, B, C, D, being in a right Fig. 297. line, and visiblefrom only one point, A ___:4'\ (( c D E, it is required to determine the dis- __ I tance between the middle points, B -___/ and C, the exterior distances, AB and CD, being known. Let AB -a, CD =b, BC = x; AEB - P, AEC Q, AED - R. Calculate an auxiliary angle, K, such that tang.2 4ab sin. Q. sin. (R- P) (a - b) sin. P.sin. (R —Q) Then is x= a +-A a-b 2 2 2.cos. K' Of the two values of x, the positive one is alone to be taken. This problem is used in Triangular Surveying when a portion of Base line passes over water, &c. CHAP. III.] Obstacles to Measurement. 293 (435) Problem. Gaiven the angles observed, at the ends of a line which cannot be measuredc, between it and the ends of a line of known length but inaccessible, required the length of the former line. This Problem is the converse of that given in Art. (430). Its figure, 292, may represent the case, if the distance AB be regarded as known and CD as that to be found. Use the first and second formulas as before, and invert the last formula, obtaining sin. CBD. sin. CAB sin. BDC. sin. ACB' This problem may also be solved, indirectly, by assuming any length for CD, and thence calculating as in the first part of Art. (430), the length of AB on this hypothesis. The imaginary figure thus calculated is similar to the true one; and the true length of CD will be given by this proportion; calculated length of AB: true length of AB:: assumed length of CD: true length of CD. The length of AB can also be obtained Fig. 298. graphically. Take a line of any length, a B' as C'D', and from C' and D' lay off angles equal to those observed at C and D, and, thus fix points A, B'. Produce AB' till it,C equals the given distance AB, on any de- / " sired scale. From B draw a parallel to. B'D', meeting AD' produced in D; and from D draw a parallel to D'C' meeting AC' produced in C. Then CD will be the required distance to the same scale as AB. (436) Problem. Three points, A, B, C, being given by their distances from each other, and two other points, P and Q, being so situated that from each of them two of the three points can be seen and the angles APQ, BPQ, CQP, BQP, be measured, it is required to determine the positions of P and Q. 294 OBSTACLES IN ANGULAR SURVEYING. [PART VII. CONSTRUCTION. Begin, Fig. 299. as in Art. (397), by describ-. ing a circle passing through / A and B, and having the tcen- A<. tral angle subtended by AB,,. —. -.equal to twice the given an- p gle APB, and thus contain- ", ing that angle. The point P will lie somewhere in its circumference. Describe another circle passing through B and C, and having a central angle subtended by BC equal to twice the given angle BQC. The point Q will lie somewhere in its circumference. From A draw a line making with AB an angleBPQ, and meeting at X the circle first drawn. From C draw a line making with CB an angle BQP, and meeting the second circle in Y. Join XY and produce it till it cuts the circles in points P and Q, which will be those required; since BPX = BAX = BPQ; and BQY 13BCY -BQP. CALCULATION. In the triangle ABC, the sides being given, the angle ABC is known. In the triangle ABX, a side and all the angles are known, to find BX. In the triangle CBY, BY is similarly found. By subtracting the angle ABC from the sum of the angles ABX and CBY, the angle XBY can be obtained. Then in the triangle XBY, the sides BX, BY, and the included angle are given to find the other angles. Then in the triangle BPX are knbwn all the angles and the side BX to find BP. In the triangle BQY, BQ is found in like manner. Finally, in the triangle BPQ, PQ can then be found. If desired, we can also obtain AP in the triangle APB; and CQ in the triangle CBQ. (437) Problem. Four points, A, B, C, D, being given in position, by their mutual distances and directions, and two other points, P and Q, being. so situated that from each of them two of the four points can be seen and the angles APB, APQ, PQC and PQD measured, it is required to determine the position of P and Q. CHAP. III.] Obstacles to Measurement. 295 Fig. 300. \.;;.......... - - - --- - E,, C CONSTRUCTION. Begin as in the last article, by describing on AB the segment of a circle to contain an angle equal to APB. From B draw a chord BE, making an angle with BA equal to the supplement of the angle APQ. On CD describe another segment to contain an angle equal to CQD. From C draw a chord CF, making an angle with CD equal to the supplement of the angle DQP. Draw the line EF, and it will cut the two circles in the required points P and Q.* CALCULATION. To obtain PQ — EF- EP — QF, we proceed to find those three lines thus. In the triangle ABE, we know the side AB, the angle ABE, and the angle AEB = APB; whence to find EB. In the same way, the triangle CFD gives FC. In the triangle EBC are known EB and BC, and the angle EBC= — ABC - ABE; whence EC and the angle ECB are found. In the triangle ECF are known EC, FC, and the angle ECF = BCD — ECB —FCD; whence we find EF, and the angles CEF and CFE. In the triangle BEP, we have EB, the angle BEP =-BEC + CEP, and the angle BPE = BPA + APE; to find EP and PB. In the triangle QCF, we have CF, and the angles CQF and CFQ, to find QC and QF. Then we know PQ EF - EP -- QF. * For, the angle APQ in the figure equals the measured angle APQ, because the supplement of the former, EPA, equals the supplement of the latter, since it is measured by the same arc as the angle ABE, equal to that supplement by construction. So too with the angle DQP. 296 OBSTACLES IN INGULAR SURVEYING. [PART VII. The other distances, if desired, can be easily found from the above data, some of the calculations, not needed for PQ, being made with reference to them. In the triangle ABP, we know AB, BP, and the angle BAP, to find the angle ABP and AP. In the triangle QDC we know QC, CD, and the angle CQD, to find the angle QCD and QD. In the triangle PBC, we know PB, 13C, and the angle PBC - ABC - ABP, to find PC. Lastly, in the triangle QCB, we know QC, CB, and the angle QCB = DCB- DCQ, to find QB. The solution of this problem includes the two preceding; for, let the line BC be reduced to a point so that its two ends come together and the three lines become two, and we have the problem of Art. (436); and let the line AB be reduced to a point, B, and CD to a point, C, and we have but one line, and the problem becomes that of Art. (435). In these three problems, if the two stations lie in a right line with one of the given points, the problem is indeterminate. (435) Problem of the eight points. Eour points, A, B, C, D, are inaccessible, but visible Fig. 301. from four other points, E, II F, G, H; it is required to A find the respective distances of these eight points; the = only data being the observation, from each of the - points of the second systemn, of the angles under ( which are seen the points j of thefirst system. 0 This problem can be solved, but the great length and complication of the investigation and resulting formulas render it more a matter of curiosity than of utility. It may be found in Puissant's " Topographie," page 55; Lefevre's " Trigonometrie," p. 90, and Lefevre's "Arpentage," No. 387. CHAP. IV.] To Supply Omissions. 297 CHAPTER, IV. TO SUPPLY OMISSIONS. (439) Any two omissions in a closed survey, whether of the direction or of the length, or of both, of one or more of the sides bounding the area surveyed, can always be supplied by a suitable application of the principle of Latitudes and Departures, as was stated in Art. (283); although this means should be resorted to only in cases of absolute necessity, since any omission renders it impossible to " Test the survey," as directed in Art. (282). In the following articles the survey will be considered to have been made with the Compass. All the rules will however apply to a Transit or Theodolite survey, the angles being referred to any line as a meridian, as in " Traversing." To save unnecessary labor, the examples in the various cases now to be examined, will all be taken Fig. 302. from the same survey, a plat of which A - is given in the margin on the scale of 40 chains to 1 inch (1:31,680), and 3 the Field-notes of which, with the Latitudes and Departures carried out A to five decimal places, are given on the following page.*' The teacher can make any number of examples for his own use by taking a tolerably accurate survey, striking out the bearing and distance of any one course, and calculating it precisely as in Case 1, given below. He can then omit any two quantities at will, to be supplied by the student by means of the rules now to be given. 298 OBSTACLES IN ANGULAR SURVEYING. [PART VII. DIST. LATITUDES. DEPARTURES..STA. B ARING. IIN LINKS. N. E. W. A North. | 1284 1284.00000 0 0 B N. 32~0 E. 1782 1511.22171 944.31619 C N. 800 E. 2400 416.75568 2363.53872 D S. 48~ E: 2700 1806.65262 2006.49096 E S. 18~ W. 2860 2720.02159 883.78862 F N. 730 28' 21" W. 4621k 1314.69682 | 4430.55725 4526.67421 4526.67421 5314.34587 5314.34587 CASE 1. When the length and the Bearing of any one side are wanting. (440) Find the Latitudes and the Departures of the remaining sides. The difference of the North and South Latitudes of these lines, is the Latitude of the omitted line, and the difference of their Departures is its Departure. This Latitude and Departure are two sides of a right angled triangle of which the omitted line is the hypothenuse. Its length is therefore equal to the square root of the sum of their squares, and the quotient of the Departure divided by the Latitude is the tangent of its Bearing; as in Art. (417). In the above survey, suppose the course from F to A to have been omitted or lost. The difference of the Latitudes of the remaining courses will be found to be 1314.69682, and the difference of the Departures to be 4430.55725. The square root of the sum of their squares is 4621.5; and the quotient of the Departure divided by the Latitude is the tangent of 730 28' 21". The deficiencies were in North Latitude and West Departure; and the omitted course is therefore N. 73~ 28' 21" W., 4621.5 CASE 2. When the length of one side and the Bearing of another are wanting. (441) When the deficient sides adjoin each other. Find, as in Case 1, the length and Bearing of the line joining the ends of the remaining courses. This line and the deficient lines will form a triangle, in which two sides will be known, and the angle between the calculated side and the side whose Bearing is given can be found by Art. (243). The parts wanting can then be obtained by the common rules of Trigonometry. CHAP. Iv.] To Supply Omissions. 299 In the figure, let the length of EF, Fig. 303. and the Bearing of FA be the omitted D parts. The difference of the sums of the N. and S. Latitudes, and the E. and W. Departures of the complete courses from A to E, are respectively 1405.32477 North Latitude, and A m 5314.34587 East Departure. The course, EA, corresponding to this de- r ficiency we find, by proceeding as in case 1, to be S. 75~ 11' 15" W., 5497.026. The angle AEF is therefore 75~ 11' 15" - 18~ =57~ 11' 15". Then in the triangle AEF are given the sides AE, AF, and the angle AEF to find the remaining parts; viz. the angle AFE — 91 28' 21", whence the Bearing of FA 91~ 28' 21" — 180 - N. 730 28' 21" W.; and the side EF- 28.60. (442) When the deficient sides are separated from each other. A modification of the preceding method will still apply. In this figure let the omissions be the Bearing Fig. 304. of FA and the length of CD. Imagine b the courses to change places without changing Bearings or lengths, so as to / bring the deficient lines next to each B" E other, by transferring CD to AG, AB to GH, and BC to HD. This will not A / affect their Latitudes or Departures. Join GF. Then in the figure DEFGH, F the Latitudes and Departures of all the sides but FG are known, whence its length and Bearing can be found as in Case 1. Then the triangle AGF may be treated like the triangle AEF in the last article, to obtain the length of AG = CD, and the Bearing of FA. (443) Otherwise, by changing the Meridian. Imagine the field to turn around, till the side of which the distance is unknown, becomes the Meridian, i. e. comes to be due North and South; 300 OBSTACLES IN ANGULAR SURVEYING. [PART VII. all the other sides retaining their relative positions, and continuing to make the same angles with each other. Change their Bearings, accordingly, as directed in Art. (244). Find the Latitudes and Departures of the sides in their new positions. Since the side whose length was unknown has been made the Meridian, it has no Departure, whatever may be its unknown length; and the difference of the columns of Departure will therefore be the Departure of the side whose Bearing is unknown. The length of this side is given. It is the hypothenuse of a right angled triangle, of which the De. parture is one side. Hence the other side, which is the Latitude, can be at once found; and also the unknown Bearing. Put this Latitude in the Table in the blank where it belongs. Then add up the columns of Latitude, and the difference of their sums will be the unknown length of the side which had been made a Meridian.' Let the omitted quantities be, as in the last article, the length of CD and the Bearing of FA. STA.OLD BEARING. NEW BEARING. e BEARING. Make CD the Meridian. The changA North. N. 50~ W. B N. 32 E. N. 4 W. ed Bearings will then be found by C N. 800 E. North. Art. (244) to be as in the margin. D S. 48~ E. N. 520 E. To aid the imagination, turn the E S. 18~ W. jS. 62~ E. book around till CD points up and down, as North lines are usually placed on a map. Then obtain the Latitudes of the courses with their new Bearings and old distances, and proceed as has been directed. CASE 3. When the lengths of two sides are wanting. (444) When the deficient sides adjoin each other, Find the Latitudes and Departures of the other courses, and then, by Case 1, find the length and Bearing of the line joining the extremities of the deficient courses. Then, in the triangle thus formed, are known one side and all the angles (deduced from the Bearings) to find the lengths of the other two sides.' This conception of thus changing the Bearings is stated to be due to Prof. Robert Patterson, of Philadelphia, by whom it was communicated to Mr. John Gummere, and published by him, in 1814, in his " Treatise on Surveying." CHAP. IV.] To Supply Omissions. 301 Thus, in Fig. 303, page 299, let EF and FA be the sides whose lengths are unknown. EA is then to be calculated, and its length will be found, as in Art. (441), to be 5497.026, and its bearing S. 75~ 11' 15" W., whence the angle AEF- 75~ 11' 15" - 18 -- 57o 11' 15"; AFE = 18~ + 730 28' 21" = 910 28' 21"; and EAF = 310 20' 24"; whence can be obtained EF = 28.60 and FA -- 46.215. (445) When the deficient sides are separated from each other. Let the lengths of BC and DE be those Fig. 305. omitted. Again imagine the courses to change places, so as to bring the /\ deficient lines together, DE being,/ transferred to CG, and CD to GE. B.'\", Join BG. Then in the figure ABGEFA, are known the Latitudes A and Departures of all the courses except BG, whence its length and Bearing F can be found as in Case 1. Then in the triangle BCG, the angle CBG can be found from the Bearings of CB and BG, and the angle CGB from the Bearings of BG and GC. Then all the angles of the triangle are known and one side, BG, whence to find the required sides, BC = 178:2, and CG = DE = 2700. (446) Otherwise, by changing the Meridian. As in Art. (443), imagine the field to turn around, till one of the sides whose length is wanting, becomes a Meridian or due North and South. Change all the Bearings correspondingly.: Find the Latitudes and Departures of the changed courses. The difference of the columns of Departure will be the Departure of the second course of unknown length, since the course made Meridian has now no Departure. The new Bearing of this second course being given, in the right angled triangle formed by this course, as an hypothenuse, and its Departure and Latitude, we know one side, the Departure, and the acute angles, which are the Bearing and its complement. The length of the course is then readily calculated; and also its Latitude. This Latitude being inserted in its proper place, the differ 302 OBSTACLES IN ANGULAR SURVEYING. [PART VII. ence of the columns of Latitude will be the length of that wanting side which had been made a Meridian. Thus, let the lengths of BC and DE be wanting, as in the pre- -LD BEAG.ceding example. Make BC STA.OLD BEARNG. EW BEARING. Ma LMeridian. The other BearA North. N.o320 IV. ings are then changed as in B NT. 800 E. N.orthE. the margin. Calculate new D S. 480 E. S. 800 E. Latitudes and Departures. E S. 180 W. S. 14~ E. The difference of the DeparF N. 730 28' 21" W. S. 74~ 31' 39 W. tures will be the Departure of DE, since BC, being a Meridian, has no Departure. Hence the length and Latitude of DE are readily bbtained. This Latitude being put in the table, and the columns of Latitude then added up, their difference will be the length of BC. CASE 4. When the Bearings of two sides are wanting. (417) ihea the deficient sides adjoin each othler, Find the Latitudes and Departures of the other sides, and then, as in Case 1, find the length and bearing of the line joining the extremities of the deficient sides. Then in the triangle thus formed we have the three sides to find the angles and thence the Bearings. (448) When the deficient sides are separated from each other. Change the places of the sides so as to bring the deficient ones next to each other. Thus, in the Fig. 306. figure, supposing the Bearings of CD, - and EF to be wanting, transfer EF to DG, and DE to GF. Then calculate, \\ as in Case 1, the length and Bearing \ / of the line joining the extremities of / the deficient sides, CG in the figure. A/ This line and the deficient sides form a triangle in which the three sides are F given to determine the angles and thence the required Bearings.* * The fullest investigation of this subject, developing many curious points, will be found in Mascheroni's " Problemes de Geomnetrie pour les Arpenteurs," and Lhuillier's " Polyg-onom.etrie." PART VIII. PLANE TABLE SURVEYING. (449) THE Plane Table is in substance merely a drawing board fixed on a tripod, so that lines may be drawn on it by a ruler placed so as to point to any object in sight. All its parts are mere additions to render this operation more convenient and precise. Such an arrangement may be applied to any kind of " Angular Surveying"; such as the Third Mcthod, " Polar Surveying," in its two modifications of Radiation and Progression, (characterized in Art. (220)), and the Fourth Method, by Intersections. Each of these will be successively explained. The instrument is very convenient for filling in the details of a survey, when the principal points have been determined by the more precise method of " Triangular Surveying" and can then be platted on the paper in advance. It has the great advantage of dispensing with all notes and records of the measurements, since they are platted as they are made. It thus saves time and lessens mistakes, but is wanting in precision. (450) The Table. It is usually a rectangular board of well seasoned pine, about 20 inches wide and 30 long. The paper to be drawn upon may be attached to it by drawing-pins, or by clamping plates fixed on its sides for that purpose, or by springs pressed upon it, or it may be held between rollers at opposite sides of the table. Tinted paper is less dazzling in the sun. Cugnot's joint, described on page 134, is the best for connecting it with its tripod, though a pair of parallel plates, like those of the Theodolite, are often used. A detached level is placed on the board to test its horizontality; though a smooth ball, as a marble, will answer the same purpose approximately. 304 PLANE TABLE SIURVEYING. [PART VIII. A pair of sights, like those of the compass, are sometimes placed under the board, serving, like a " Watch Telescope," (Art. (339), to detect any movement of the instrument. To find what point on the lower side of the board is exactly under a point on the upper side, so that by suspending a plumb-line from the former the latter may be exactly over any desired point of ground, a large pair of" callipers," or dividers with curved legs, may be used, one of their points being placed on the upper point of the board, and their other point then determining the corresponding under point; or a frame forming three sides of a rectangle, like a slate frame, may be placed so that one end of one side of it touches the upper point, and the end of the corresponding side is under the table precisely below the given point, so that from this end a plumb-line can be dropped. A compass is sometimes attached to the table, or a detached compass, consisting of a needle in a narrow box, (called a Declinator), is placed upon it, as desired. The edges of the table are sometimes divided into degrees, like the " Drawing board Protractor," Art. (273). It then becomes a sort of Goniometer, like that of Art. (213). (451) The Alidade. The ruler has a fiducial or feather edge, which may be divided into inches, tenths, &c. At each end it carries a sight like those of the compass. Two needles would be tolerable substitutes. The sights project beyond its edge so that their centre lines shall be precisely in the same vertical plane as this edge, in order that the lines drawn by it may correspond to the lines sighted on by them. To test this, fix a needle in the board, place the alidade against it, sight to some near point, draw a line by the ruler, turn it end for end, again place it against the needle, again sight to the same point, and draw a new line. If it coincides with the former line, the above condition is satisfied. The ruler and sights together take the name of Altdade. If a point should be too high or too low to be seen with the alidade, a plumb-line, held between the eye and the object, will remove the difficulty. A telescope is sometimes substituted for the sights, being supported above the ruler by a standard, and capable of pointing upward or downward. It admits of adjustments similar in principle PART vIII.] PLANE TABLE SURVEYING. 305 to the 2d and 3d adjustments of the Transit, Part IV, Chapter 3, pages 242 and 246. But even without these adjustments, whether of the sights or of the telescope, a survey could be made which wvould be perfectly correct as to the relative position of its parts, however far the line of sight might be from lying in the same vertical plane as the edge of the ruler, or even from being parallel to it; just as in the Transit or Theodolite the index or vernier need not to be exactly under the vertical hair of the telescope, since the angular deviation affects all the observed directions equally. (452) Method of Radiation, This is the simplest, though not the best, method of surveying with the Plane-table. It is especially applicable to survey- Fig. 307. ing a field, as in the figure. B In it and the following fi- A gures, the size of the Table C is much exaggerated. Set the instrument at any convenient point, as O; level it, G / and fix a needle (having a head of sealing-wax) in the board to represent the sta- F D tion. Direct the alidade to any corner of the field, as A, the fiducial edge of the ruler touching the needle, and draw an indefinite line by it. Measure OA, and set off the distance, to any desired scale, from the needle point, along the line just drawn, to a. The line OA is thus platted on the paper of the table as soon as determined in the field. Determine and plat in the same way, OB, OC, &c., to b, c, &c. Join ab, be, &c., and a complete plat of the field is obtained. Trees, houses, hills, bends of rivers, &c., may be determnied in the same manner. The corresponding method with the Compass or Transit, was described in Articles (258) and (391). The table may be set at one of the angles of the field, if more convenient. If the alidade has a telescope, the method of measuring distances with a stadia, described in Art. (375), may be here applied with great advantage. 20 306 PLANE TIBLE SURVEYING. [PART VIII. (453) MIethod of Progression. Let ABCD, &c., be the line to be surveyed. Fig. 308. Fix a needle at a F convenient point E of the Plane-table, A -a near a corner so as to leave room for the plat, and -— D set up the table at! B, the second an- B 6 gle of the line., so that the needle, whose point repre- c sents B, and which should be named b, shall be exactly over that station. Sight to A, pressing the fiducial edge of the ruler against the needle, and draw a line by it. Measure BA, and set off its length, to the desired scale, on the line just drawn, from b to a point a, representing A. Then sight to C, draw an indefinite line by the ruler, and on it set off the length of BC from b to c. Fix the needle at c. Set up at C, the point c being over this station, and make the line eb of the plat coincide in direction with CB on the ground, by placing the edge of the ruler on cb, and turning the table till the sights point to B. The compass, if the table have one, will facilitate this. Then sight forward from C to D, and fix CD, cd on the plat, as be was fixed. Set up at D, make dc coincide with DC, and proceed as before. The figure shews the lines drawn at each successive station. The Table drawn at A shews how the survey might be commenced there. In going around a field, the work would be proved by the last line " closing" at the starting point, and, during the progress of the survey, by any direction, as from C to A on the ground, coinciding with the corresponding line, ea, on the plat. This method is substantially the same as the method of surveying a line with the Transit, explained in Art. (372). It requires all the points to be accessible. It is especially suited to the survey of a road, a brook, a winding path through woods, &c. The offsets required may often be sketched in by eye with sufficient preclsion. PART VIII.] PLANE TABLE SURVEYING. 307 When the paper is filled, put on a new sheet, and begin by fixing on it two points, such as C and DI, which were on the former sheet, and from them proceed as before. The sheets can then be afterwards united, so that all the points on both shall be in their true relative positions. (451) Method of Intersection. This is the most usual and the most rapid method of using the Plane-table. The principle was referred to in Articles (259) and (392). Set up the instrument at any convenient point, as X in the figure, and sight to all Fig. 309. B C D X'Y the desired points A, B. C, &c., which are visible, and draw indefinite lines in their directions. Measure any line XY, Y being one of the points sighted to, and set off this line on" the paper to any scale. Set up at Y. and turn the table till the line XY on the paper lies in the direction of XT, on the ground, as at C in the last method. Sight to all the former points and draw lines in their directions, and the intersections of the two lines of sight to each point will determine them, by the Fourth Method, Art. (8). Points on the other side of the line XY could be determined at the same time. In. surveying a field, one side of it may be taken for the base XY. Very acute or obtuse intersections should be avoided. 300 and 150~ should be the extreme limits. The impossibility of always doing this, renders this method often deficient in precision. When the paper is filled, put on a new sheet, by fixing on it two known points, as in the preceding method. on it two known points, as in the preceding method. 308 PLANE TABLE SURVEYING. [PART VIII. (455) Method of Resection. This method (called by the French Recoupement) is a modification of the preceding method of InterFig. 310. E a D section. It requires the measurement of only one distance, but all the points must be accessible. Let AB be the measured distance. Lay it off on the paper as ab. Set the table up at B, and turn it till the line ba on the paper coincides with BA on the ground, as in the Method of Progression. Then sight to C, and draw an indefinite line by the ruler. Set up at C, and turn the line last drawn so as to point to B. Fix a needle at a on the table, place the alidade against the needle and turn it till it sights to A. Then the point in which the edge of the ruler cuts the line drawn from B will be the point c on the table. Next sight to D, and draw an indefinite line. Set up at D, and make the line last drawn point to C. Then fix the needle at a or b, and by the alidade, as at the last station, get a new line back from either of them, to cut the last drawn line at a point which will be d. So proceed as far as desired. (456) To orient the table,* The operation of orientation consists in placing the table at any point so that its lines shall have the same directions as when it was at previous stations in the same survey. * The French phrase, To orient one's self, meaning to determine one's position, usually with respect to the four quarters of the heavens, of which the Orient is the leading one, well deserves naturalization in our language. PART VIII.] PLANE TABLE SURVEYING. 309 With a compass, this is very easily effected by turning the table till the needle of the attached compass, or that of the Declinator, placed in a fixed position, points to the same degree as when at the previous station. Without a compass the table is oriented, when set at one end of a line previously determined, by sighting back on this line, as at C in the Method of Progression, Art. (453). To orient the table, when at a station unconnected with others, is more difficult. It may be Fig. 311. effected thus. Let ab on the table represent a line AB on the L[_ ground. Set up at A, make ab coincide with AB, and draw a - line from a directed towards a steeple, or other conspicuous object, as S. Do the same at ]B. Draw a line cd, parallel to ab, and intercepted between aS, and bS. Divide ab and ed into the same number of equal parts. The table is then prepared. Now let there be a station, P, p on the table, at which the table is to be oriented. Set the table, so that p is over P, apply the edge of the ruler to p, and turn it till this edge cuts ed in the division corresponding to that in which it cuts ab. Then turn the table till the sights point to S, and the table will be oriented. (457) To find one's place on the ground. This problem may be otherwise expressed as Interpolating a point in a plat. It is most easily performed by reversing the Method of Intersection. Set up the table over the station, Fig. 312. 0 in the figure, whose place on the plat already on the table is 0 desired, and orient it, by one of A 1 the means described in the last i article. Make the edge of the \ ruler pass through some point, a on the table, and turn it till the sights point to the corresponding station, A on the ground. Draw a line by the ruler. The desired 310 PLANE TABLE SURVEYING. [PART VIII. point is somewhere in this line. Make the ruler pass through another point, b on the table, and make the sights point to B on the ground. Draw a second line, and its intersection with the first will be the point desired. Using C in the same way would give a third line to prove the work. This operation may be used as a new method of surveying with the plane-table, since any number of points can have their places fixed in the same manner. This problem may also be executed on the principle of Trilinear Surveying. Three points being given on the table, lay on it a piece of transparent paper, fix a needle any where on this, and with the alidade sight and draw lines towards each of these three points on the ground. Then use this paper to find the desired point, precisely as directed in the last sentence of Art. (398), page 277. (158) Inaccessible distances. 3Many of the problems in Part VII. can be at once solved on the ground by the plane-table, since it is at the same time a Goniometer and a Protractor. Thus, the Problem of Art. (4135) may be solved as follows, on the principle of the construction in the last paragraph of that article. Set the table at C. Mark on it a point, c', to represent C, placing c' vertically over C. Sight to A, B and D, and draw corresponding lines from c'. Set up at D, mark any point on the line drawn from c' towards D, and call it d'. Let d' be exactly over D, and direct d'c' toward C. Then sight to A and B, and draw corresponding lines, and their intersections with the lines before drawn towards A and B will fix points a' and 6'. Then on the line joining a and b, given on the paper to represent A and B, ab being equal to AB on any scale, construct a figure, abed, similar to a'b'e'd', and the line od thus determined will represent CD on the same scale as AB. PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) THE Principles which were established in Part I, and subsequently applied to surveying with various instruments, may also be employed, with tolerable correctness, for determining and representing the relative positions of larger or smaller portions of the earth's surface without any Instruments but such as can be extemporized. The prominent objects on the ground, such as houses, trees, the summits of hills, the bends of rivers, the crossings of roads, &c., are regarded as "points" to be " determined." Distances and angles are consequently required. Approximate methods of obtaining these will therefore be first given. (460) Distances iy pacing. Quite an accurate measurement of a line of ground may be made by walking over it at a uniform pace, and counting the steps taken. But the art of walking in a straight line must first be acquired. To do this, fix the eye on two objects in the desired line, such as two trees, or bushes, or stones, or tufts of grass. Walk forward, keeping the nearest of these objects steadily covering the other. Before getting up to the nearest object, choose a new one in line farther ahead, and then proceed as before, and so on. It is better not to attempt to make each of the paces three feet, but to take steps of the natural length, and to ascertain the value of each by walking over a known distance, and dividing it by the number of paces required to traverse it. Every person should thus determine the usual length of his own steps, repeating the experiment sufficiently often. The French "Geographical Engineers" accustom themselves to take regular 312 SURVEYING WITHOUT INSTRUMENTS. [PART IX. steps of eight-tenths of a metre, equal to two feet seven and a half inches. The English military pace is two feet and six inches. This is regarded as a usual average. 108 such paces per minute give 3.07 English miles per hour. Quick pacing of 120 such paces per minute gives 3.41 miles per hour. Slow paces, of three feet each and 60 per minute, give 2.04 miles per hour.' An instrument, called a Pedometer, has been contrived, which counts the steps taken by one wearing it, without any attention on his part. It is attached to the body, and a cord, passing from it to the foot, at each step moves a toothed wheel one division, and some intermediate wheelwork records the whole number upon a dial. (461) Distances by visual angles. Prepare a scale, by marking off on a pencil what length of it, when it is held off at arm's length, a man's height appears to cover at different distances (previously measured with accuracy) of 100, 500, 1000 feet, &c. To apply this, when a man is seen at any unknown distance, hold up the pencil at arm's length, making the top of it come in the line from the eye to his head, and placing the thumb nail in the line from Fig. 313. the eye to his feet, as in Fig. 313. The pencil having been previously graduated by the method above explained, the portion of it now intercepted between these two lines will indicate the corresponding distance. If no previous scale have been prepared, and the distance of a man be required, take a foot-rule, or any measure minutely divided, hold it off at arm's length as before, and see how much a man's height covers. Then knowing the distance from the eye to the rule, a statement by the Rule of Three (on the principle of similar triangles) will give the distance required. Suppose a man's height, of 70 inches, covers 1 inch of the rule. He is then 70 times as far * A horse, on a walk, averages 330 feet per minute, on a trot 650, and on a comrn. mon gallop 1040. For longer times, the difference in horses is more apparent. PART IX.] ISURVEYING WITHOUT INSTRUMENTS. 313 from the eye as the rule; and if its distance be 2 feet, that of the man is 140 feet. Instead of a man's height, that of an ordinary house, of an apple-tree, the length of a fence-rail, &c., may be be taken as the standard of comparison. To keep the arm immovable, tie a string of known length to the pencil, and hold between the teeth a knot tied at the other end of the string. (462) Distances by visibility. The degree of visibility of various well-known objects will indicate approximately how far distant they are. Thus, by ordinary eyes, the windows of a large house can be counted at a distance of about 13000 feet, or 2- miles; men and horses will be perceived as points at about half that distance, or 1 miles; a horse can be clearly distinguished at about 4000 feet; the movements of men at 2600 feet, or half a mile; and the head of a man, occasionally, at 2300 feet, and very plainly at 1300 feet, or a quarter of a mile. The Arabs of Algeria define a mile as " the distance at which you can no longer distinguish a man from a woman." These distances of visibility will of course vary somewhat with the state of the atmosphere, and still more with individual acuteness of sight, but each person should make a corresponding scale for himself. (463) Distances by sound. Sound passes through the air with a moderate and known velocity; light passes almost instantaneously. If, then, two distant points be visible from each other, and a gun be fired at night from one of them, an observer at the other, noting by a stop-watch the time at which the flash is seen, and then that at which the report is heard, can tell by the intervening number of seconds how far apart the points are, knowing how far sound travels in a second. Sound moves about 1090 feet per second in dry air, with the temperature at the freezing point, 320 Fahrenheit. For higher or lower temperatures add or subtract 11- foot for each degree of Fahrenheit. If a wind blows with or against the movement of the sound, its velocity must be added or subtracted. If it blows obliquely, the correction will evidently equal its velocity multiplied by the cosine of the angle which the direction of the wind makes 314 SURVEYING WITHOUT INSTRUMENTS. [PART IX. with the direction of the sound.* If the gun be fired at each end of the base in turn, and the means of the times taken, the effect of the wind will be eliminated. If a watch is not at hand, suspend a pebble to a string (such as a thread drawn from a handkerchief) and count its vibrations. If it be 391 inches long, it will vibrate in one second; if 93 inches long, in half a second, &c. If its length is unknown at the time, still count its vibrations; measure it subsequently; and then will /~length of string the time of its vibration, in seconds, = /lentho. (464) Angles. Right angles are those most frequently required in this kind of survey, and they can be estimated by the eye with much accuracy. If other angles are desired, they will be determined by measuring equal distances along the lines which make the angle, and then the line, or chord, joining the ends of these distances, thus forming chain angles, explained in Art. (100). (465) Methods of operation. The " First Method" of determining the position of a point, Art. (5), is the one most generally applicable. Some line, as AB in Fig. 1, is paced, or otherwise measured, and then the lines AS and BS; the point S is thus determined. The " Second Method," Art. (6), is also much employed, the right angles being obtained by eye, or by the easy methods given in Part II, Chapter V, Arts. (140), &c. It is used for offsets, as in Part II, Chapter III, Arts. (114), &c. The " Third Method," Art. (7), may also be used, the angles being determined as in Art. (164). The "Fourth Method," Art. (8), may also be employed, the angles being similarly determined. The " Fifth Method," Art. (10), wouid seldom be used, unless by making an extempore plane-table, and proceeding as directed in the last paragraph of Art. (157). * A gentle, pleasant wind has a velocity of 10 feet per second; a brisk gale 20 feet per second; a very brisk gale 30 feet; a high wind 50 feet; a very high wind 70 feet; a storm or tempest 80 feet; a great storm 100 feet; a hurricane 120 feet; and a violent hurricane, that tears up trees, &c., 150 feet per second PART Ix.] SURVEYING WITHOUT INSTRUMENTS. 315 The method referred to in Art. (11) may also be employed. When a sketch has made some progress, new points may be fixed on it by their being in line with others already determined. All these methods of operation are shown in the following figure. AB is a line paced, or otherwise measured approximately. Fig. 314. / ___. A._H/ L__ I i,! I,.! |, 7 - / A N The hill C is determined by the first method. The river on the other side of AB is determined by offsets according to the Second Method. The house D is determined by the Third Method, EBF being a chain angle. The house G is determined by the Fourth Method, chain angles being measured at B and H, a point in AB prolonged. The pond IK is determined, as in Art. (11), by the intersection of the alinements CD and GH prolonged. The bend of the river at L is determined by its distance from H in the line of AH prolonged. A new base line, HM, is fixed by a chain angle at It, and employed like the former one so as to fix the hill at N, &c. All these methods may thus be used collectively and successively. The necessary lines may always be ranged with rods, as directed in Art. (169), and very many of the instrumental methods already explained, may be practiced with extempore contrivances. The use of the Plane-table is an admirable preparation for this style of surveying or sketching, which is most frequently employed by Military Engineers, though they generally use a prismatic Compass, or pocket Sextant, and a sketching case, which may serve as a Plane-table. PART X. MAPPIN G. CHAPTER I. COPYING PLATS. (466) THE Plat of a survey necessarily has many lines of construction drawn upon it, which are not needed in the finished map. These lines, and the marks of instruments, so disfigure the paper that a fair copy of the plat is usually made before the map is finished. The various methods of copying plats, &c., whether on the same scale, or reduced or enlarged, will therefore now be described. (467) Stretching the paper. If the map is to be colored, the paper must first be wetted and stretched, or the application of the wet colors will cause its surface to swell or blister and become uneven. Therefore, with a soft sponge and clean water wet the back of the paper, working from the centre outward in all directions. The " water-mark" reads correctly only when looked at from the front side, which it thus distinguishes. When the paper is thoroughly wet and thus greatly expanded, glue its edges to the drawing board, for half an inch in width, turning them up against a ruler, passing the glue along them, and then turning them down and pressing them with the ruler. Some prefer gluing down opposite edges in succession, and others adjoining edges. The paper must be moderately stretched smooth during the process. Hot glue is best. Paste or gum may be used, if the paper be kept wet by a damp cloth, so that the edges may dry first. "Mouth-glue " may be used CHAP. I.] Copying Plats. 317 by rubbing it (moistened in the mouth or in boiling water) along the turned up edges, and then rubbing them dry by an ivory folder, a piece of dry paper being interposed. As this is a slower process, the middle of each side should first be fastened down, then the four angles, and lastly the intermediate portions. When the paper becomes dry, the creases and puckerings will have disappeared, and it will be as smooth and tight as a drum-head. (468) Copying by tracing. Fix a large pane of clear glass in a frame, so that it can be supported at any angle before a window, or, at night, in front of a lamp. Place the plat to be copied on this glass, and the clean paper upon it. Connect them by pins, &c. Trace all the desired lines of the original with a sharp pencil, as lightly as they can be easily seen. Take care that the paper does not slip. If the plat is larger than the glass, copy its parts successively, being very careful to fix each part in its true relative position. Ink the lines with India ink, making them very fine and pale, if the map is to be afterwards colored. (469) Copying on tracing paper. A thin transparent paper is prepared expressly for the purpose of making copies of maps and drawings, but it is too delicate for much handling. It may be prepared by soaking tissue paper in a mixture of turpentine and Canada balsam or balsam of fir (two parts of the former to one of the latter), and drying very slowly. Cold drawn linseed oil will answer tolerably, the sheets being hung up for some weeks to dry. Linen is also similarly prepared, and sold under the name of "Vellum tracing paper." It is less transparent than the tracing paper, but is very strong and durable. Both of these are used rather for preserving duplicates than for finished maps. (470) Copying by transfer paper. This is thin paper, one side of which is rubbed with blacklead, &c., smoothly spread by cotton. It is laid on the clean paper, the blackened side downward, and the plat is placed upon it. All the lines of the plat are then gone over with moderate pressure by a blunt point, such as the eye-end of a small needle. A faint tracing of these lines will then be found 318 MAPPING. [PART X. on the clean paper, and can be inked at leisure. If the original cannot be thus treated, it may first be copied on tracing paper, and this copy be thus transferred. If the transfer paper be prepared by rubbing it with lampblack ground up with hard soap, its lines will be ineffaceable. It is then called " Camp-paper." (471) Copying by punctures. Fix the clean paper on a drawing board and the plat over it. Prepare a fine needle with a sealing-wax head. Hold it very truly perpendicular to the board, and prick through every angle of the plat, and every corner and intersection of its other lines, such as houses, fences, &c., or at least the two ends of every line. For circles, the centre and one point of the circumference are sufficient. For irregular curves, such as rivers, &c., enough points must be pricked to indicate all their sinuosities. Work with system, finishing up one strip at a time, so as not to omit any necessary points nor to prick through any twice, though the latter is safer. When completed, remove the plat. The copy will present a wilderness of fine points. Select those which determine the leading lines, and then the rest will be easily recognized. A beginner should first pencil the lines lightly, and then ink them. An experienced draftsman will omit the pencilling. Two or three copies may be thus pricked through at once. The holes in the original plat may be made nearly invisible by rubbing them on the back of the sheet with a paper-folder, or the thumb nail. (472) Copying by intersections. Draw a line on the clean paper equal in length to some important line of the original. Two starting points are thus obtained. Take in the dividers the distance from one end of the line on the original to a third point. From the corresponding end on the copy, describe an arc with this distance for radius and about where the point will come. Take the distance on the original from the other end of the line to the point, and describe a corresponding are on the copy to intersect the former arc in a point which will be that desired. The principle of the operation is that of our " First Method," Art. (5). Two pairs of dividers may be used as explained in Art. (90). " Tri CHAP. I.] Copying Plats. 319 angular compasses," having three legs, are used by fixing two of their legs on the two given points of the original, and the third leg on the point to be copied, and then transferring them to the copy. All the points of the original can thus be accurately reproduced. The operation is however very slow. Only the chief points of a plat may be thus transferred, and the details filled in by the following method. (473) Copying by squares. On the original plat draw a series of parallel and equidistant lines. The T square does this most readily. Draw a similar series at right angles to these. The plat will then be covered with squares, as in Fig. 38, page 48. On the clean paper draw a similar series of squares. The important points may now be fixed as in the last article, and the rest copied by eye, all the points in each square of the original being properly placed in the corresponding square of the copy, noticing whether they are near the top or bottom of each square, on its right or left side, &c. This method is rapid, and in skilful hands quite accurate. Instead of drawing lines on the original, a sheet of transparent paper containing them may be placed over it; or an open frame with threads stretched across it at equal distances and at right angles. This method supplies a transition to the Redutetion and _Enlargement of plats in any desired ratio; under which head copying by the Pantagraph and Camera Lucida will be noticed. (474) Reducing by squares. Begin, as in the preceding article, by drawving squares on the original, or placing them over it. Then on the clean paper draw a similar set of squares, but with their sides one-half, one third, &c., (according to the desired reduction), of those of the original plat. Then proceed as before to copy into each small square all the points and lines found in the large square of the plat in their true positions relative to the sides and corners of the square, observing to reduce each distance by eye, or as directed in the following article, in the given ratio. 320 MIAPPING. [PART X. (475) Reducing by proportional scales. Many graphical methods of finding the proportionate length on the copy, of any line of the original, may be used. The "Angle of reduction" is constructed thus. Draw any line AB. With it for radius and A Fig. 315. for centre, describe an indefinite are. With B for centre and a,\ \\ \ radius equal to one-half, one-third,\ \\ \ \\ &c., of AB according to the de-. K\' \\. \ \ 1 1ll 1 sired reduction describe another A D are intersecting the former are in C. Join AC. From A as centre describe a series of arcs. Now to reduce any distance, take it in the dividers, and set it off from A on AB, as to D. Then the distance from D to E, the other end of the arc passing through D, will be the proportionate length to be set off on the copy, in the manner directed in Art. (472). The Sector, or " Compass of proportion," described in Art. (52), presents such an " Angle of reduction," always ready to be used in this manner. The I" Angle of reduction" may be simplified Fig. 316. thus. Draw a line, AB, parallel to one side FB — iC of the drawing board, and another, BC, at right angles to it, and one-half, &c., of it, as desired. Join AC. Then let AD be the distance required to be reduced. Apply a T square so as to pass through D. It will meet AC in some point E, and DE will be the reduced A length required. Another arrangement for the same object is shown in Fig. 317. Draw two lines, AB, AC, at any angle, and de- Fig. 317. scribe a series of arcs from their intersection, A, as in the figure. Suppose the reduced scale is to be half the original scale. Divide the outermost are into three equal parts, and draw a line from A to one of the points of division, as D. Then each arc will be divided into parts, one of which is twice the other. Take any distance on the original scale, and find by trial which of the arcs on CHAP. I.] Copying Plats. 321 the right hand side of the figure it corresponds to. The other part of that are will be half of it, as desired. "-Proportional compasses," being properly set, reduce lines in any desired ratio. A simple form of them, known as " Wholes and halves," is often useful. It consists of two slender bars, pointed at each end, and united by a pivot which is twice as far from one pair of the points as from the other pair. The long ends being set to any distance, the short ends will give precisely half that distance. (476) Reducing by a pantagraph. This instrument consists of two long and two short rulers, connected so as to form a parallelogram, and capable of being so adjusted that when a tracing point attached to it is moved over the lines of a map, &c., a pencil attached to another part of it will mark on paper a precise copy, reduced on any scale desired. It is made in various forms. It is troublesome to use, though rapid in its work. (477) Reducing by a camera lucida, This is used in the Coast Survey Office. It cannot reduce smaller than one-fourth, without losing distinctness, and is very trying to the eyes. Squares drawn on the original are brought to apparently coincide with squares on the reduction, and the details are then filled in with the pencil, as seen through the prism of the instrument. (478) Enlarging plats. Plats may be enlarged by the principal methods which have been given for reducing them, but this should be done as seldom as possible, since every inaccuracy in the original becomes magnified in the copy. It is better to make a new plat from the original data. 21 322 MAPPING. [PART X. CHAPTER II. CONVENTIONAL SIGNS. (479) Various conventional signs or marks have been adopted, more or less generally, to represent on maps the inequalities of the surface of the ground, its different kinds of culture or natural products, and the objects upon it, so as not to encumber and disfigure it with much writing or many descriptive legends. This is the purpose of what is called Topogrcaphieal Maapping. (480) The relief of ground. The inequalities of the surface of the earth, its elevations and depressions, its hills and hollows, constitute its " Relief."* The representation of this is sometimes called " Hill drawing." Its difficulty arises from our being accustomed to see hills sideways, or " in elevation," while they must be represented as they would be seen from above, or " in plan." Various modes of thus drawing them are used; their positions being laid down in pencil as previously sketched by eye or measured. If light be supposed to fall vertically, the slopes of the ground will receive less light in proportion to their steepness. The relief of ground will be indicated on this principle by making the steep slopes very dark, the gentler inclinations less so, and leaving the level surfaces white. The shades may be produced by tints of India ink applied with a brush, their edges, at the top and bottom of a hill or ridge, being softened off with a clean brush. If light be supposed to fall obliquely, the slopes facing it will be light, and those turned from it dark. This mode is effective, but not precise. In it the light is usually supposed to come from the upper left hand corner of the map. Horizontal contour lines are however the best convention for this purpose. Imagine a hill to be sliced off by a number of equidistant horizontal planes, and their intersections with it to be drawn as they would be seen from above, or horizontally projected on the CHAP II.] Conventional Signs, 323 map. These are'" Contour lines." Theyare the same lines as would be formed by water surrounding the hill, and rising one foot at a time (or any other height) till it reached the top of the hill. The edge of the water, or its shore, at each successive rise, would be one of these horizontal contour lines. It is plain that their nearness or distance on the map would indicate the steepness or gentleness of the slopes. A right cone would thus be repreFig. 318. Fig. 319. Fig. 320. sented by a series of concentric circles, as in Fig. 318; an oblique cone by circles not concentric, but nearer to each other on the steep side than on the other, as in Fig. 319; and a half-egg, somewhat as in Fig. 320. Vertical sections, perpendicular to these contour lines, are usually combined with them. They are the " Lines of greatest slope," and may be supposed to represent water running down the sides of the hill. They are also made thicker and nearer together on the steeper slopes, to produce the effect required by the convention of vertical light Fig. 321. already referred to. \ The marginal figure ~.. shews an elongated " ~ x half-egg, or oval hill, ______'~ thus represented.; The spaces between I' the rows of vertical //l' Hatchings" indicate the contour lines, which are not actually drawn. The beauty of the graphical execution of this work depends on the uniformity of the strokes representing uniform slopes, on their perfectly regular gradation in thickness and nearness, for varying slopes, and on their being made precisely at right angles to the contour lines between which they are situated. 324 MAPPING, [PART X The methods of determining the contour lines are applications of Levelling, and will therefore be postponed, together with the farther details of " Hill-drawing," to the volume treating of that subject, which is announced in the Preface. (481) Signs for natural surface. Sand is represented by fine dots made with the point of the pen; gravel by coarser dots. Bocks are drawn in their proper places in irregular angular forms, imitating their true appearance as seen from above. The nature of the rocks, or the Geology of the country, may be shown by applying the proper colors, as agreed on by geologists, to the back of the map, so that they may be seen by holding it up against the light, while they will thus not confuse the usual details. (482) Signs for vegetation. Woods are represented by scolloped circles, irregularly disposed, Fig: _2. imitating trees seen "in plan," and A closer or farther apart according to - a the thickness of the forest. It is VI P" j - usual to shade their lower and right /1 hand sides and to represent their By t' "' shadows, as in the figure, though, in strictness, this is inconsistent with the hypothesis of vertical light, adopted for " hill-drawing." For pine and similar forests, the signs may have a star-like form, as on the right hand side of the figure. Trees are sometimes drawn " in elevation," or sideways, as usually seen. This makes them more easily recognized, but is in utter violation of the principles of mapping in horizontal projection, though it may be defended as a pure convention. Orchards are represented by trees arranged in rows. Bushes may be drawn like trees, but smaller. Grass-land is drawn with irregularly Fig. 323. scattered groups of short lines, as in the..L..k..l.,, v figure, the lines being arranged in odd.r\~k'.... ^x1JJ, ~""~~ "V V 1v numbers, and so that the top of each group is -,,Vj,,,,.;ll:: \., convex and its bottom horizontal or parallel ~,-s" vi to the base of the drawing. Meadows are V,~ J, U,-, a.s rIo sometimes represented by pairs of diverging lines, (as on the right CHAP. II.] Conventional Signs. 325 of the figure), which may be regarded as tall blades of grass. Uncultivated land is indicated by appropriately intermingling the signs for grass land, bushes, sand and rocks. Cultivated land is shown by parallel rows of broken and dotted Fig. 324. lines, as in the figure, representing furrows.'lilil'! Crops are so temporary that signs for them are i i! unnecessary, though often used. They are usu- L i: I_ I,1 _t ally imitative, as for cotton, sugar, tobacco, rice, vines, hops, &c. Gardens are drawn with circular and other beds and walks. (483) Signs for water. The Sea-coast is represented by drawing a line parallel to the shore, following all its windings and indentations, and as close to it as possible, then another parallel line a little more distant, then a third still more distant, and so on. Examples are seen in figures 287, &c. If these lines are drawn from the low tide mark, a similar set may be drawn between that and the high tide mark, and dots, for sand, be made over the included space. Rivers have each shore treated like the sea shore, as in the figures of Part VII.* Brooks would be shown by only two lines, or one, according to their magnitude. Ponds may be drawn like sea shores, or represented by Fig. 325. parallel horizontal lines ruled across them. -I==. 3cfarshes and Swamps are represented by an:_ _.,,,__ irregular intermingling of the preceding,-,.... sign with that for grass and bushes, as in the -,..., figure. -''",' (484) Colored Topography. The conventional signs which have been described, as made with the pen require much time and labor. Colors are generally used by the French as substitutes for them, and combine the advantages of great rapidity and effectiveness. Only three colors (besides India ink) are required; viz. Gamboge (yellow), Indigo (blue), and Lake (pink). Sepia, Burnt Sienna, Yellow ochre, Red lead, and Vermillion, are also sometimes used. The last three are difficult to work with. To f Those in Part II, Chapter V, have the lines too ciose together in the middle. 326 MAPPING. [PART X. use these paints, moisten the end of a cake and rub it up with a drop of water, afterwards diluting this to the proper tint, which should always be light and delicate. To cover any surface with a uniform flat tint, use a large camel's hair or sable brush, keep it always moderately full, incline the board towards you, previously moisten the paper with clean water if the outline is very irregular, begin at the top of the surface, apply a tint across the upper part, and continue it downwards, never letting the edge dry. This last is the secret of a smooth tint. It requires rapidity in returning to the beginning of a tint to continue it, and dexterity in following the outline. Marbling, or variegation, is produced by having a brush at each end of a stick, one for each color, and applying first one, and then the other beside it before it dries, so that they may blend but not mix, and produce an irregularly cloud&d appearance. Scratched parts of the paper may be painted over by first applying strong alum water to the place. The conventions for colored Topography, adopted by the French Military Engineers, are as follows. WOODS, yellow; using gamboge and a very little indigo. GRASS-LAND, green; made of gamboge and indigo. CULTIVATED LAND, brown; lake, gamboge, and a little India ink. "Burnt Sienna" will answer. Adjoining fields should be slightly varied in tint. Sometimes furrows are indicated by strips of various colors. GARDENS are represented by small rectangular patches of brighter green and brown. UNCULTIVATED LAND, marbled green and light brown. BRUSH, BRAMBLES, &c., marbled green and yellow. IHIEATH, FURZE, &C., marbled green and pink. VINEYARDS, pourple; lake and indigo. SANDS, a light brown; gamboge and lake. " Yellow ochre" will do. LAKES and RIVERS, light blue, with a darker tint on their upper and left hand sides. SEAS, dark blue, with a little yellow added. MARSHES, the blue of water, with spots of grass green, the touches all lying horizontally. ROADS, brown; between the tints for sand and cultivated ground, with more India ink. HILLS, greenish brown; gamboge, indigo, lake and India ink, instead of the pure India ink, directed in Art. (480). WOODS may be finished up by drawing the trees as in Art. (482) and coloring them green, with touches of gamboge towards the light (the upper and left hand side) and of indigo on the opposite side. CHAP. II.] Conventional Signs, 327 (485) Signs for detached objects. Too great a number of these will cause confusion. A few leading ones will be given, the meanings of which are apparent. Figs. Figs. Court house,.326. Wind mill, 334. Post office, ia 327. Steam mill, t 335. Tavern, F328. _Furnace,, 336. Blacksmith's shop, ~ 329. Woollen factory, a 337. Guide board, t 330. Cotton factory, X 338. Quarry, 331. Glass works, Grist mill, i 332. Church, ) 340. Saw mill, 333. Grave yard, -.U42341. An ordinary house is drawn in its true position and size, and the ridge of its roof shown if the scale of the map is large enough. On a very small scale, a small shaded rectangle represents it. If colors are used, buildings of masonry are tinted a deep crimson, (with lake), and those of wood with India ink. Their lower and right hand sides are drawn with heavier lines. Fences of stone or wood, and hedges, may be drawn in imnitation of the realities; and, if desired, colored appropriately. Minzes may be represented by the signs of the planets which were anciently associated with the various metals. The signs here given represent respectively, Gold, Silver, Iron, Copper, Tin, Lead, Quicksilver. Q } 3? ~24 1 A large black circle, *O, may be used for Coal. Boundary lines, of private properties, of townships, of counties, and of states, may be indicated by lines formed of various combinations of short lines, dots and crosses, as below.' ~"" e *-........ *...... -. * Very minute directions for the execution of the details described in this chapter, are given in Lieut. R. S. Smith's "Topographical Drawing." Wiley, N. Y. 328 MIPPING. [PART X. CHAPTER III. FINISHING THIE MAP. (486) Orientation. The map is usually so drawn that the top of the paper may represent the North. A Meridian line should also be drawn, both True and Magnetic, as in Fig. 199, page 189. The number of degrees and minutes in the Variation, if known, should also be placed between the two North points. Sometimes a compass-star is drawn and made very ornamental. (487) Lettering. The style in which this is done very much affects the general appearance of the map. The young surveyor should give it much attention and careful practice. It must all be in imitation of the best printed models. No writing, however beautiful, is admissible. The usual letters are the ordinary ROMAN CAPITALS, Small Roman, ITALIC CAPITALS, Small ltalic, and G O T HIC O R E G Y PT I A N. This last, when well done, is very effective. For the Titles of maps, various fancy letters may be used. For very large letters, those formed only of the shades of the letters regarded as blocks (the body being rubbed out after being pencilled as a guide to the placing of the shades) are most easily made to look well. The simplest lettering is generally the best. The sizes of the names of places, &c., should be proportional to their importance. Elaborate tables for various scales have been published. It is better to make the letters too small than too large. They should not be crowded. Pencil lines should always be ruled as guides. The lettering should be in lines parallel to the bottom of the map, except the names of rivers, roads, &c., whose general course should be followed. (488) Borders. The Border may be a single heavy line, enclosing the map in a rectangle, or such a line mxy be relieved by a finer line drawn parallel and near to it. Time should not be wasted in ornamenting the border. The simplest is the best. CHAP. III.] Finishing the Mapo 329 (489) Joining paper. If the map is larger than the sheets of paper at hand, they should be joined with a feather-edge, by proceeding thus. Cut, with a knife guided by a ruler, about onethird through the thickness of the paper, and tear off on the under side, a strip of the remaining thickness, so as to leave a thin sharp edge. Treat the other sheet in the same way on the other side of it. When these two feather edges are then put together, (with paste, glue or varnish), they will make a neat and strong joint. The sheet which rests upon the other must be on the right hand side, if the sheets are joined lengthways, or below if they are joined in that direction, so that the thickness of the edge may not cast a shadow, when properly placed as to the light. The sheets must be joined before lines are, drawn across them, or the lines will become distorted. Drawing paper is now made in rolls of great length, so as to render this operation unnecessary. (4190) Mounting maps. A map is sometimes required to be mounted, i. e. backed with canvas or muslin. To do this, wet the. muslin and stretch it strongly on a board by tacks driven very near together. Cover it with strong paste, beating this in with a brush to fill up the pores of the muslin. Then spread paste over the back of the paper, and when it has soaked into it, apply it to the muslin, inclining the board, and pasting first a strip, about two inches wide, along the upper side of the paper, pressing it down with clean linen in order to drive out all air bubbles. Press down another strip in like manner, and so proceed till all is pasted. Let it dry very gradually and thoroughly before cutting the muslin from the board. Maps may be varnished with picture varnish; or by applying four or five coats of isinglass size, letting each dry well before applying the next, and giving a full flowing coat of Canada balsam diluted with the best oil of turpentine. PART XI. LAYING OUT, PARTING OFF, AND DIVIDING UP LAND.S CHAPTER I. LAYING OUT LAND. (491) Its nature. This operation is precisely the reverse of those of Surveying properly so called. The latter measures certain lines as they are; the former marks them out in the ground where they are required to be, in order to satisfy certain conditions. The same instruments, however, are used as in Surveying. Perpendiculars and parallels are the lines most often employed. The Perpendiculars may be set out either with the chain alone, Arts. (140) to (159); still more easily with the Cross-staff, Art. (104), or the Optical-square, Art. (107); and most precisely with a Transit or Theodolite, Arts. (402) to (406). Parallels may also be set out with the chain alone, Arts. (160) to (166); or with Transit, &c., Arts. (407) and (408). The ranging out of lines by rods is described in Arts. (169) and (178), and with an Angular instrument, in Arts. (376), (409) and (415). (492) To lay out squares. Reduce the desired content to square chains, and extract its square root. This will be the length of the required side, which is to be set out by one of the methods indicated in the preceding article. An Acre, laid out in the form of a square, is frequently desired by farmers. Its side must be made 3161 links of a Gunter's * The Demonstrations of the Problems in this part, when required, will be found in Appendix B. CHAP. I.] Laying. out Land. 331 chain; or 208-17- feet; or 69"7~ yards. It is often taken at 70 paces. The number of plants, hills of corn, loads of manure, &c., which an acre will contain at any uniform distance apart, can be at once found by dividing 209 by this distance in feet, and multiplying the quotient by itself; or by dividing 43560 by the square of the distance in feet. Thus, at 3 feet apart, an acre would contain 4840 plants, &c.; at 10 feet apart, 436; at a rod apart, 160; and so on. If the distances apart be unequal, divide 43560 by the product of these distances in feet; thus, if the plants were in rows 6 feet apart, and the plants in the rows were 3 feet apart, 2420 of them would grow on one acre. (493) To lay out rectangles. The content and length being given, both as measured by the same unit, divide the former by the latter, and the quotient will be the required breadth. Thus, 1 acre or 10 square chains, if 5 chains long, must be 2 chains wide. The content being given and the length to be a certain number of tines the breadth. Divide the content in square chains, &c., by the ratio of the length to the breadth, and the square root of the quotient will be the shorter side desired, whence the longer side is also known. Thus, let it be required to lay out 30 acres in the form of a rectangle 3 times as long as broad. 30 acres 300 square chains. The desired rectangle will contain 3 squares, each of 100 sq. chs., having sides of 10 chs. The rectangle will therefore be 10 chs. wide and 30 long. An Acre laid out in a rectangle twice as long as broad, will be 224 links by 448 links, nearly; or 1471 feet by 295 feet; or 49k yards by 98] yards. 50 paces by 100 is often used as an approximation, easy to be remembered. The content being given, and the difference between the length and breadth. Let c represent this content, and d this difference. Then the longer side =- d ~+ /(d2 + 4 c). Example. Let the content be 6.4 acres, and the difference 12 chains. Then the sides of the rectangle will be respectively 16 chains and 4 chains. 332 LAYING OUT AND DIVIDING UP LAND. [PART XI. The content being given, and the sum of the length and breadth. Let c represent this content, and s this sum. Then the longer side-s + /(s2 — 4 c)..Example. Let the content be 6.4 acres, and the sum 20 chains. The above formula gives the sides of the rectangle 16 chains and 4 chains as before. (494) To lay out triangles. The content and the base being given, divide the former by half the latter to get the height. At any point of the base erect a perpendicular of the length thus obtained, and it will be the vertex of the required triangle. The content being given and the base having to be m times the height, the height will equal the square root of the quotient obtained by dividing twice the given area by m. The content being given and the triangle to be equilateral, take the square root of the content and multiply it by 1.520. The product will be the length of the side required. This rule makes the sides of an equilateral triangle containing one acre to be 480 2 links. A quarter of an acre laid out in the same form would have each side 240 links long. An equilateral triangle is very easily set out on the ground, as directed in Art. (90), under " Platting," using a rope or chain for compasses. (495) The content and base being given, and one side having to make a given angle, as B, with the base Fig. 342. AB, the length of the side BC-2 ABC. AB. sin. B Example. Eighty acres are to be laid out in the form of a triangle, on a base, AB, of sixty chains, bearing N. 800 W. A the bearing of the side BC being N. 700~ ]. Here the angle B is found from the Bearings (by Art. (243), reversing one of them) to be 30~. Hence BC =53.33. The figure is on a scale of 50 chains to 1 inch = 1: 39600. Any right-line figure may be laid out by analogous methods. (496) To lay out circles. Multiply the given content by 7, divide the product by 22, and take the square root of the quotient. CHAP. I.] Laying out Land. 333 This will give the radius, with which the circle can be described on the ground with a rope or chain. A circle containing one acre has a radius of 178k links. A circle containing a quarter of an acre will have a radius of 89 links. (497) Town lots. House lots in cities are usually laid off as rectangles of 25 feet front and 100 feet depth, variously combined in blocks. Part of New-York is laid out in blocks 200 feet by 800, each containing 64 lots, and separated by streets, 60 feet wide, running along their long sides, and avenues, 100 feet wide, on their short sides. The eight lots on each short side of the block, front on the avenues, and the remaining forty-eight lots front on the streets. Such a block covers almost precisely 32 acres, and 17- such lots about make an acre. But, allowing for the streets, land laid out into lots, 25 by 100, arranged as above, would contain only 11.9, or not quite 12 lots per acre. Lots in small towns and villages are laid out of greater size and less uniformity. 50 feet by 100 is a frequent size for new villages, the blocks being 200 feet by 500, each therefore containing 20 lots. (498) Land sold for taxes. A case occurring in the State of New-York will serve as an application of the modes of laying out squares and rectangles. Land Fig. 343. on which taxes are unpaid is C sold at auction to the lowest bidder; i. e. to him who will accept the smallest portion of it in return for paying the taxes on the whole. The lot in question was originally the east half of the square lot ABCD, containing 500 acres. At a sale for taxes in 1830, 70 acres were bid off, and this area was D set off to the purchaser in a square lot, from the north-east corner. Required the side of the square in links. Again, in 1834, 29 acres more were thus sold, to be set off in a strip of equal width 3B4 LAYING OUT AND DIVIDING UP LAND. [PART XI. around the square previously sold. Required the width of this strip. Once more, in 1839, 42 acres more were sold, to be set off around the preceding piece. Required the dimensions of this third portion. The answer can be proved by calculating if the dimensions of the remaining rectangle will give the content which it should have, viz. 250 - (70 + 29 + 42) -= 109 Acres. The figure is on a scale of 40 chains to 1 inch = 1: 31680. (499) New countries. The operations of laying out land for the purposes of settlers, are required on a large scale in new countries, in combination with their survey. There is great difficulty in uniting the necessary precision, rapidity and cheapness. "Triangular Surveying" will ensure the first of these qualities, but is deficient in the last two, and leaves the laying out of lots to be subsequently executed. "Compass Surveying" possesses the last two qualities, but not the first. The United States system for surveying and laying out the Public Lands admirably combines an accurate determination of standard lines (Meridians and Parallels) with a cheap and rapid subdivision by compass. The subject is so important and extensive that it will be explained by itself in Part XII. CHAPTER II. PARTING OFF LAND. (500) It is often required to part off from a field, or from an indefinite space, a certain number of acres by a fence or other boundary line, which is also required to run in a particular direction, to start from a certain point, or to fulfil some other condition. The various cases most likely to occur will be here arranged according to these conditions. Both graphical and numerical methods will generally be given.* * The given lines will be represented by fine full lines; the lines of construction by broken lines, and the lines of the result by heavy full lines. CAP. II.] Parting off Land. 335 The given content is always supposed to be reduced to square chains and decimal parts, and the lines to be in chains and decimals. A. BY A LINE PARALLEL TO A SIDE. (501) To part off a rectangle. If the sides of the field adjacent to the given side make right angles with it, the figure parted off by a parallel to the given side will be a rectangle, and its breadth will equal the required content divided by that side, as in Art. (493). If the field be bounded by a curved or zigzag line outside of the given side, find the content between these irregular lines and the given straight side, by the method of offsets, subtract it from the content required to be parted off, and proceed with the remainder as above. The same directions apply to the subsequent problems. (502) To part off a parallelogram. If the sides adjacent to the given side be parallel, the Fig. 344. figure parted off will be a parallel- D ogram, and its perpendicular width, CE, will be obtained as above. The length of one of the parallel A E B CE ABDC sides, as AC - sin. A AB. sin. A (503) To part off a trapezoid. When the sides of the field adjacent to the given side are not parallel, the figure parted off will be a trapezoid. When the field or figure is given on the ground, or on a plat, begin as if the sides were parallel, Fig. 345. dividing the given content by the base AB. The quotient will be c -D an approximate breadth, CE, or DF; too small if the sides converge, as in the figure, and vice versa. Measure CD. Calculate E F the content of ABDC. Divide the difference of it and the required 336 LAYING OUT AND DIVIDING UP LAND. [PART XI. content by CD. Set off the quotient perpendicular to CD, (in this figure, outside of it,) and it will give a new line, GH, a still nearer approximation to that desired. The operation may be repeated, if found necessary. (504) When the field is given by Bearings, de- Fig. 346. duce from them, as in Art. (243), the angles at A P and B. The required sides will then be given by C these formulas: sin. B sin. AD= (AB-CD) si. + B). sin. (A + B) sin. A BC = (AB- CD) sin.A sin. (A+ B)T D When the sides AD and BC diverge, instead of converging, as in the figure, the negative term, in the expression for CD, becomes positive; and in the expressions for both AD and BC, the first factor becomes (CD- AB). The perpendicular breadth of the trapezoid =AD. sin. A; or, BC. sin. B. Example. Let AB run North, six chains; AD, N. 80~ E.; BC, S. 60~ E. Let it be required to part off one acre by a fence parallel to AB. Here AB = 6.00, ABCD = 10 square chains, A = 800, B - 60~. Ans. CD - 4.57, AD = 1.92, BC = 2.18, and the breadth = 1.89. The figure is on a scale of 4 chains to 1 inch = 1: 3168. B. BY A LINE PERPENDICULAR TO A SIDE. (505) To part off a triangle. Let FG be the required line. When the field is given on the Fig. 347. ground, or on a plat, at any point, as E C D, of the given side AB, set out a "guess line," DE, perpendicular to AB, and calculate the content of -B D A DEB. Then the required distance BF, from the angular point to the foot of the desired perpendicular, = BD./(BFG. W V~ IVV VI ~loU~~UL~U ~lr~lUIUILU) UU R liE-E CHAP. II.] Parting off Land. 337 Example.. Let BD = 30 chains; ED 12 chains; and the desired area = 24.8 acres. Then BF -- 35.22 chains. The scale of the figure is 30 chains to 1 inch =1: 23760. (506) When the field is given by Bearings, Fig. 348. find the angle B from the Bearings; then is A c C BE = J(2x BFG) Example. Let BA bear S. 75~ E., and BC N. 60~ E., and let five acres be required to be parted off from the field by a perpendicular to BA. Here the angle B = 450, and BF = 10.00 chains. The scale of the figure is 20 chains to 1 inch = 1:15840. (507) To part off a quadrilateral. Produce the converging sides to meet at B. Calculate the Fig. 369. content of the triangle HKB, whe- C ther on the ground or plat, or from.. Bearings. Add it to the content of the quadrilateral required to be B " 1 F A parted off, and it will give that of the triangle FGB, and the method of the preceding case can then be applied. (508) To part off any figure. If the field be very irregularly shaped, find by trial any line which will part off a little less than the required area. This trial line will represent HK in the preceding figure, and the problem is reduced to parting off, according to the required condition, a quadrilateral, comprised between the trial line, two sides of the field, and the required line, and containing the difference between the required content and that parted off by the trial-line. C. BY A LINE RUNNING IN ANY GIVEN DIRECTION. (509) To part off a triangle. By construction, on the ground or the plat, proceed nearly as in Art. (505), setting out a line in the required direction, calculating the triangle thus formed, and obtaining BF by the same formula as in that Article. 22 338 LAYING OUT AND DIVIDING UP LAND [PART xI. (510) If the field be given by Bearings, find Fig. 350. from them the angles CBA and GFB; then is A BF__J(2 x BFG sin. (B + F)) B V/ sin. B. sin.F \ Example. Let BA bear S. 30~ E.; BC, N. 80~ E.; and a fence be required to run from some point in BA, a due North course, and to part off one acre. Required the distance from B to the point F, whence it must start. Ans. \ The angle B = 70~, and F =30~. Then BF- 6.47. The scale of Fig. 350 is 6 chains to 1 inch = 1: 4752. (511) To part off a quadrilateral. Let it be required to part off, by a line running in a Fig. 351. given direction, a quadrilateral from a field in which are given the side AB, and X the directions of the two B other sides running from A and from B. On the ground or plat produce the two converging, sides to meet at some point.$C E. Calculate the content A of the triangle ABE. Measure the side AE. From ABE subtract the area to be cut off, and the remainder will be the content of the triangle CDE. From A set out a line AF parallel to the given direction. Find the content of ABF. Take it from ABE, and thus obtain AFE. Then this formula, ED =-AE CFAE' will fix the point D, since AD = AE - ED. (512) When the field and the dividing line are given by Bearings, produce the sides as in the last article. Find all the angles from the Bearings. Calculate the content of the triangle ABE, by the formula for one side and its including angles. Take the mHAP. II.] Parting off Land, 339 desired content from this to obtain CDE. Calculate the side sin. B. 1/2 X CDE. sin.DCE\* AE-AB.-. ThenisAD =AE- /2E' sin. E V' sin. E. sin. POE Example. Let IDA bear S. 20 0 W.; AB, N. 514~ Mr.,'8.19; BC, N. 73x~ E.; and let it be required to part off two acres by a fence, DC, running N. 450 W. Ans. ABE - 32.50 sq. chains; whence CDE- =12.50 sq. chs. Also, AE 8.37; and finally AD - 8.37 - 5.49 - 2.88 chains. The scale of Fig. 351 is 5 chains to 1 inch = 1: 3960. If the sum of the angles at A and B was more than two right angles, the point E would lie on the other side of AB. The necessary modifications are apparent. (513) To part off any figure, Proceed in a similar manner to that described in Art. (508), by getting a suitable trial-line, producing the sides it intersects, and then applying the method just given. D. BY A LINE STARTING FROM A GIVEN POINT IN A SIDE. (511) To part off a triangle. Let it be required to cut off from a corner of a field a triangu- Fig. 352. lar space of given content, by a line starting from a given point on one of the sides, A in the figure, the base, AB, of the desired tri- _ i angle being thus given. If the field be given on the ground or on A a plat, divide the given content by half the base, and the quotient will be the height of the triangle. Set off this distance from any point of AB, perpendicular to it, as from A to C; from C set out a parallel to AB, and its intersection with the second side, as at D, will be the vertex of the required triangle. Otherwise, divide the required content by half of the perpendicular distance from A to BD, and the quotient will be BD. This original formula is very convenient for logarithmic computation. 340 LAYING OUT AND DIVIDING UP LAND. [PART Xr. (515) If the field be given by the Bearings of two sides and the length of one of them, deduce the angle B (Fig. 352) from the 2 x ABD Bearings, as in Art. (243). Then is BD ABD AB. sin. B If it is more convenient to fix the point D, by the Second Method, Art. (6), that of rectangular co-ordinates, we shall have BE BD. cos. B; and ED = BD. sin. B. The Bearing of AD is obtained from the angle BAD; which is known, sinceD E tang. BAD. nEA AB — BE Example. Eighty acres are to be set off from a corner of a field, the course AB being N. 800 W., sixty chains; and the Bearing of BD being N. 70~ E. Ans. BD 53.33; BE —- 46.19; ED =- 26.67; and the Bearing of AD, N.?o $1 W. The scale of Fig. 352 is 40 chains to 1 inch = 1: 31680. 2 ABD If the field were right angled at B, of course BD = ABD (516) To part off a quadrilateral, Imagine the two converging sides of the field produced to meet, as in Art. (511). Calculate the content of the triangle thus formed, and the question will then be reduced to the one explained in the last two articles. (517) To part off any figure. Proceed as directed in Art. (513). Otherwise, proceed as follows. The field being given on the ground or on a plat, find on which side of it the required line will end, by drawing or running " guess lines" from the given point to various angles, and roughly measuring the content thus parted off. Fig. 353. If, as in the figure, A,beinog the i D given point, the guess line AD c/ parts off less than the required con- / / tent, and AE parts off more, then the desired division line AZ will B I3 end in the side DE. Subtract the -- - area parted off by AD from the required content, and the difference will be the content of the tri CHAP. II.] Parting off Land. 341 angle ADZ. Divide this by half the perpendicular let fall from the given point A to the side DE, and the quotient will be the base, or distance from D to Z. Or, find the content of ADE and make this proportion; ADE: ADZ:: DE: DZ. (518) The field being given by Bearings and distances, find as before, by approximate trials on the plat, or otherwise, which side the desired line of division will terminate in, as DE in the last figure. Draw AD. Find the Latitude and Departure of this line, and thence its length and Bearing, as in Art. (440). Then calculate the area of the space this line parts off, ABCD in the figure, by the usual method, explained in Part III, Chapter VI. Subtract this area from that required to be cut off, and the remainder will be the area of the triangle ADZ. Then, as in Art. (515), 2 ADZ DZ -- AD. sin. ADZ This problem may be executed without any other Table than that of Latitudes and Departures, thus. Find the Latitude and Departure of DA, as before, the area of the space ABCD, and thence the content of ADZ.: Then find the Latitude and Departure of EA, and the content of ADE. Lastly, make this proportion: ADE: ADZ:: DE: DZ.* Examfple. In the field ABCDE, &c., part of which is shown in Fig. 353, (on a scale of 4 chains to 1 inch = 1: 3168), one acre is to be parted off on the west side, by a line starting from the angle A. Required the distance from D to Z, the other end of this dividing line.t The only courses needed are these. AB, N. 530 W., 1.55; BC, N. 200 E., 2.00; CD, N. 53~0 E., 1.32; DE, S. 570 E., 5.79. A rough measurement will at once shew that ABCD is less than an acre, and that ABCDE is more; hence the desired line will fall * The problem may also be performed by making the side on which the division line is to fall, a Meridian, and changing the Bearings as in Art. (244). The difference of the new Departures will be the Departure of the Division line. Its position, can then be easily determined, by calculations resembling those in Part VII, Chapter IV, Arts. (443), &c. t If the whole field has been surveyed and balanced, the balanced Latitudes and Departures should be used. We will here suppose the survey to have proved perfectly correct. 342 LAYING OUT AND DIVIDING UP LAND. [PART XI. on DE. The Latitudes and Departures of AB, BC and CD are then found. From them the course AD is found to be N. 8~ E., 3.63. The content of ABCD will be 3.19 square chains. Subtracting this from one acre, the remainder, 6.81 sq. chs., is the content of ADZ. AP 3.63 x sin. 650 - 3.29. Dividing ADZ by half of this, we obtain DZ = 4.14 chains. By the Second Method, the Latitude and Departure of DA, the area of ABCD, and of ADZ, being found as before, we next find the Latitude and Departure of EA, from those of AD and DE, and thence the area of ADE = 9.53. Lastly, we have the proportion 9.53: 6.81:: 5.79: DZ = 4.14, as before. E. BY A LINE PASSING THROUGH A GIVEN POINT WITHIN THE FIELD. (519) To part off a triangle. Let P be a point within a field through which it is required to 1f B run a line so as to part off from Fig. 354. the field, a given area in the form of a triangle. /// When the field is given on the /7/ ground or on a plat, the division,/ / I can be made by construction, / thus. From P draw PE, parallel to the side BC. Divide the / given area by half of the perpen- /, /. \I dicular distance from P to AC, / / 7,d and set off the quotient from C,, /:'" /\ to G. Bisect GC in H. On AI/ G A L C HE describe a semi-circle. On'- " it set off EK - EC. Join KH. Set off HL = HK. The line LM, drawn from L through P, will be the division line required.' If iK be set off in the contrary direction, it will fix another line L'PN', meeting CB produced, and thus parting off another triangle of the required content. Example. Let it be required to part off 31.175 acres by a fence passing through a point P, the distance PD of P from the * As some lines in the figure are not used in the construction, though needed for the Demonstration, the student should draw it himself to a large scale. 0HAP. II.] Parting off Land. 343 side BC, measured parallel to AC, being 6 chains, and DC 18 chains. The angle at C is fixed by a " tie-line" AB =48.00, BC being 42.00, and CA being 30.00. Ans. CL- 27.31 chains, or CL' -= 7.69 chains. The figure is on a scale of 20 chains to 1 inch =- 1: 15840. (520) If the angle of the field and the position of the point P are Fig. 355. given by Bearings or angles, proceed 1?A~ thus. Find the perpendicular distances, PQ and PR, from the given point to the sides, by the formulas PQ PC.sin. PCQ; and PR= 1 PC. sin. PCR. Let PQ= q, PR =p, and the required content c. zL' C Then CL 2 p V p2 sin. LCMI ]Ezxample. Let the angle LCM - 820~. Let it be required to part off the same area as in the preceding example. Let PC = 19.75, PCQ-17~ 301', PCR - 640 29k'. Required CL. Ans. PQ 5.94, PR _ 17.82, and therefore, by the formula, CL-= 27.31, or CL' 7.69; corresponding to the graphical solution. The figure is on the same scale. If the given point were without the field, the division line could be determined in a similar manner. (521) To part off a quadrilaterals Conceive the two sides of the field which the division line will intersect, Fig. 356. DA and CB, produced till they meet at a /1 point G, not shown in the figure. Calculate F the triangle thus formed outside of the field. Its area increased by the required area, will be that of the triangle EFG. Then the problem is identical with that in the last article. The following example is that given in Gummere's Surveying. The figure A E D represents it on a scale of 20 chains to 1 inch = 1 15840. 344 LAYING OUT AND DIVIDING UP LAND. [PART xI. Example. A field is bounded thus: N. 140 W., 15.20; N. 701~ E., 20.43; S. 60 E., 22.79; N. 86~0 W., 18.00. A spring within it bears from the second corner S. 750 E., 7.90. It is required to cut off 10 acres from the West side of the field by a straight fence through the spring. How far will it be from the first corner to the point at which the division fence meets the fourth side? Ans. 4.6357 chains. (522) To part off any figure. Let it be required to part off from a field a certain area by Fig. 357. B D a line passing through a given point P within the field. Run a guess-line AB through P. Calculate the area which it C' parts off. Call the difference between it and the required area —d. Let CD be the desired line of division, and. let P represent the angle, APC or BPD, which it makes with the given line. Obtain the angles PAC = A, and PBD = B, either by measurement, or by deduction from Bearings. Measure PA and PB. Then the desired angle P will be given by the following formula. Cot. P =- (cot. + cot. B - 2 -BP) j[ AP2 cot. B- BP2. cot. A cot. B cot. A. cot. B + 2 d X (cot. A + cot. B- AP2-BP2) )2] If the guess line be run so as to be perpendicular to one of the sides of the field, at A, for example, the preceding expression reduces to the following simpler form. Cot. P- (cot. B - AP2 BP2) v/[AP2 cot. B- -2 B e 2d 2d cEa. II.] Parting off Land. 345 Example. It was required to cut off from a field twelve acres by a line passing through a spring, P. A guess-line, AB, was run making an angle with one side of the field, at A, of 550, and with the opposite side, at B, of 810. The area thus cut off was found to be 13.10 acres. From the spring to A was 9.30 chains, and to B 3.30 chains. Required the angle which the required line, CD, must make with the guess line, AB, at P. Ans. 20~ 45'; or 86~ 25'. The heavy broken line, C'D', shows the latter. The scale of the figure is 10 chains to 1 inch = 1: 7920. If the given point were outside of the field, the calculations would be similar. F. BY THE SHORTEST POSSIBLE LINE. (523) To part off a triangle. Let it be required to part off a triangular space, BDE, of given content, from the Fig. 358. corner of a field, ABC, by the shortest possible line, DE. From B set off BD' and BE each equal to / /12 BDE E/( sin —-B) The line DE thus obtained will be v / perpendicular to the line, BF, which bisects the angle B. The length of DE V (2 DBE. sin. B) _ CoS.IB A.. Example. Let it be required to part off 1.3 acre from the corner of a field, the angle, B, being 30~. Ans. BD = BE= 7.21; and DE=- 3.73. The scale of the figure is 10 chains to 1 inch =1: 7920. G. LAND OF VARIABLE VALUE. (521) Let the figure represent a field in which Fig. 359. the land is of two qualities and values, divided by B c the "quality line" EF. It is required to part off from it a quantity of land worth a certain sum, by E a straight fence parallel to AB. Multiply the value per acre of each part by its length (in chains) on the line AB, add the products, multiply the value to be set off by 10, divide A D 346 LAYING OUT AND DIVIDING UP LAND. [PART XI. by the above sum, and the quotient will be the desired breadth, BC or AD, in chains. Example. Lqt the land on one side of EF be worth $200 per acre, and on the other sidle $100. Let the length of the former, BE, be 10 chains, and EA be 30 chains. It is required to part off a quantity of land worth $7500. Ans. The width of the desired strip will be 15 chains. The scale of the figure is 40 chains to 1 inch = 1: 31680. If the "' quality line" be not perpendicular to AB, it may be made so by'" giving and taking," as in Art. (124), or as in the article following this one. The same method may be applied to land of any number of different qualities; and a combination of this method with the preceding problems will solve any case which may occur. H. STRAIGHTENING CROOKED FENCES. (525) It is often required to substitute a straight fence for a crooked one, so that the former shall part off precisely the same quantity of land as did the latter. This can be done on a plat by the method given in Art. (83), by which the irregular figure Fig. 360. 2 t 1...2...3...4...5 is reduced to the equivalent triangle 1...5...3', and the straight line 5...3' therefore parts off the same quantity of land on either side as did the crooked one. The distance from 1 to 3', as found on the plat, can then be set out on the ground and the straight fence be then ranged from 3' to 5 The work may be done on the ground more accurately by running a guess line, AC, Fig. 361, across the bends of the fence which crooks from A to B, measuring offsets to the bends on each side of the guess line, and calculating their content. If the sums of these areas on each side of AC chanced to be equal, that would be the line desired; but if, as in the figure, it passes too far on one CHAP. III.] Dividing up Land. 347 Fi,. 361. side, divide the difference of the areas by half of AC, and set it off at right angles to AC, from A to D. DC will then be a line parting off the same quantity of land as did the crooked fence. If the fence at A was not perpendicular to AC, but oblique, as AE, then from D run a parallel to AC, meeting the fence at E, and EC will be the required line. CHAPTER III. DIVIDING UP LAND. (526) MosT of the problems for "' Dividing up" land may be brought under the cases in the preceding chapter, by regarding one of the portions into which the figure is to be divided, as an area to be " Parted off" from it. Many of them, however, can be most neatly executed by considering them as independent problems, and this will be here done. They will be arranged, firstly, according to the simplicity of the figure to'be divided up, and then sub-arranged, as in the leading arrangement of Chapter II, according to the manner of the division. DIVISION OF TRIANGLES. (527) By lines parallel to a side. Sup- Fig. 362. pose that the triangle ABC is to be divided into two equivalent parts by a line parallel to AC. The desired point, D, from which this line is to start, will be obtained by measuring BDAB / 1. So, too, E is fixed by BE = BC V/-. 348 LAYING OUT AND DIVIDING UP LAND. [PART XI. Generally, to divide the triangle into two parts, BDE and ACED, which shall have to each other a ratio =m: n, we have BDAB em-' This may be constructed thus. Describe a Fig. 363. semicircle on AB as a diameter. From B set / off BF +.BA. At F erectaperpendi- I\ cular meeting the semicircle at G. Set off BG, from B to D. D is the starting point of the divi- A e' sion line required. In the figure, the two parts are as 2 to 3, and BF is therefore= — 2 BA. To divide the triangle ABC into five Fig. 364. equivalent parts, we should have, similarly, BD=- AB V'.; BD' —AB VI'; BD" -AB V3*; BD"' -ABV. D/ E The same method will divide the trian- D / E gle into any desired number of parts hav- / - 1"' ing any ratios to each other. A D (528) By lines perpendicular to a sides Suppose that ABC is to be divided into two parts having Fig. 365. a ratio = m: n, by a line perpendicular to AC. Let EF be the dividing line whose position is required. Let BD be a perpendicular let fall from B to -. F —-l AC. ThenisAF=j.(AC x AD x + -). In this figure, AFE: EFBC:: m: n:: 1: 2. If the triangle had to be divided into two equivalent parts, the above expression would become AF = / ( AC x AD). (529)' By lines running in any given direction. Let a triangle, ABC, be given to be divided into two parts, having a ratio = n: n, by a line making a given angle with a side. Part off, as in Art. (509) or (510), Fig. 350, an area BFG = n ~ ABC. m-+ n CHAP. III.] Dividing up Land. 349 (530) By lines starting from an angle, Divide the side opposite to the given angle into the required num- Fig. 366. ber of parts, and draw lines from the angle to the points of division. In the figure the triangle is represented as being thus divided into two equivalent parts. A C If the triangle were required to be divided into two parts, having to each other a ratio m: n, we should have AD = ACm m n and DC = AC n m + n If. the triangle had to be divided into three Fig. 367. parts which should be to each other m n:::, we should have AD -= AC m + - DE m + n + p =AC n and EC=AC P A D E mn~rn+p mrn'p m+ M + n -- p' Suppose that a triangular field ABC, had to be divided among five men, two of them to have a quarter each, and three of them each a sixth. Divide AC into two equal parts, one of these again into two equal parts, and the other one into three equal parts. Run the lines from the four points thus obtained to the angle B. (531) By lines starting from a point in a side: Suppose that the triangle ABC is to be divided into two Fig. 368 equivalent parts by a line starting from a point D in the side AC. Take a point E in the middle of AC. Join BD, and from E draw a parallel to it, meeting AB in F. DF will be A.. E D the dividing line required. The point F will be most easily obtained on the ground by the proportion AD: AB:: AE = ~ AC: AF. The altitude of AFD of course equals.1 ABC ~ ~ AD. If the triangle is to be divided into two parts having any other ratio to each other, divide AC in that ratio, and then proceed as AB X AC m before. Let this ratio = m: n, then AF = A x AC AD m + n 350 LAYING OUT AND DIVIDING UP LAND. [PART XI. (532) Next suppose that the trian- Fig. 369. gle ABC is to be divided into three equivalent parts, meeting at D. The altitudes, EF and GHI, of the parts ADE and DCG, will be obtained by ~ l) D B c dividing ~ ABC, by half of the respective bases AID and DC. If one of these quotients gives an altitude greater than that of the triangle ABC, it will shew that the two lines DE and DG would both cut the same side, as in Fig. 370, in Fig. 370. which EF is obtained as above, and GH = ABC - 2 AD. In practice it is more convenient to de- E termine the points F and G, by these I proportions; F } K D BK: A:: EF: AF; and BIK: AK:: GH: AH. - The division of a triangle into a greater number of parts, having any ratios, may be effected in a similar manner. (533) This problem admits of a more elegant solution, analogous to that given for the division into two Fig. 371. parts, graphically. Divide AC into G three equal parts at L and M. Join BD, and from L and M draw parallels to it, meeting AB and BC in E Ai and G. Draw ED and GD, which will be the desired lines of division. The figure is the same triangle as Fig. 369. The points E and G can be obtained on the ground by measuring AD and AB, and making the proportion AD: AB:: AC: AE. The point G is similarly obtained. The same method will divide a triangle into a greater number of parts. (531) To divide a triangle into four equivalent triangles by lines terminating in the sides, is very Fig. 372. easy. From D, the middle point of AB, draw DE parallel to AC, and from F, the middle of AC, draw FD and FE. / The problem is now solved. it - -1, CRAP. III.] Dividing up Land. 351 (535) By lines passing through a point within the triangle. Let D be a given point (such as a well, Fig. 373. &c.) within a triangular field ABC, from which fences are to run so as to divide /) the triangle into two equivalent parts. Join AD. Take E in the middle of BC, AC F if' and from it draw a parallel to DA, meeting AC in F. EDF is the fence required. (536) If it be required to di- i B vide a triangle into two equiva- Fig. 374. lent parts by a straight line passing through a point within it, pro- / ceed thus. Let P be the given/ / point. From P draw PD paral- // lel to AC, and PE parallel to BC. / / Bisect AC at F. Join FB. From D/ lP' B draw BG parallel to DF. Then / /, /' bisect GC in H. On HE de- / / scribe a semicircle. On it set off -, EK = EC. Join KH. Set off/ / HL = HK. The line LM drawn -'- //'/ A L from L, through P, will be the, division line required. > K This figure is the same as that of Art. (519). The triangle ABC contains 62.35 acres, and the distance CL = 27.31 chains, as in the example in that article. (537) Next suppose that the trian- Fig. 375. gle ABC is to be divided into three equivalent parts by lines starting from a point D, within the triangle, given by r the rectangular co-ordinates AE and A c and ED. Let ED be one of the lines K E of division, and F and G the other points required. The point F will be determined if AH is known; AllH and HF being its rectangular co-ordinates. From B let fall the perpendicular BK on AC. 362 LAYING OUT AND DIVIDING iUP LAND. [PART XI. Then is AH =- AK ( ABC - AE x ED) The position of the AE x B -- ED x AK other point, G, is determined in a similar manner. (538) Let DB, instead of DE, Fig. 376. be one of the required lines of division. Divide 1 ABC by half of the perpendicular DH, let fall from D to AB, and the quotient F will be the distance BF. To find - G, if, as in this figure, the trian- h-E L i c gle BDC (=- BC x } DK) is less than ~ ABC, divide the excess of the latter (which will be CDG) by 2 DE, and the quotient will be CG. Example. Let AB = 30.00; BC = 45.00; CA = 50.00. Let the perpendiculars from D to the sides be these; DE = 10.00; DH = 20.00; DK = 5.171. The content of the triangle ABC will be 666.6 square chains. Each of the small triangles must therefore contain 222.2 sq. chs., BD being one division line. We shall therefore have BF- = 222.2. I DH = 22.2 chains. BDC 45 x } x 5.17- = 116.4 sq. chs., not enough for a second portipn, but leaving 105.8 sq. chs. for CDG; whence CG = 21.16 chs. To prove the work, calculate the content of the remaining portion, GDFA. We shall find DGA = 144.2 sq. chs., and ADF = 78.0 sq. chs., making together 222.2 sq. chs., as required. The scale of Fig. 376 is 30 chains to 1 inch = 1: 23760. (539) The preceding case may Fig. 377. be also solved graphically, thus. Take CL = AC. Join DL, and from B draw BG parallel to DL. Join DG. It will be a second line of division. Then take a point, / - M, in the middle of BG, and from A....- a.-Lo Iit draw a line, MF, parallel to DA. DF will be the third line of division. This method is neater on paper than the preceding; but less convenient on the ground. CHAP. III.] Dividing up Land. 353 (540) Let it be required to divide Fig. 378. the triangle ABC into three equivalent triangles, by lines drawn from the three angular points to some unknown point within the triangle. This F point is now to be found. On any A c side, as AB. take ADi= AB. From D draw DE parallel to AC. The middle, F, of DE, is the point required. If the three small triangles are not to be equivalent, but are to have to each other the ratios:: m: n:p, Fig. 379. divide a side, AB, into parts having these ratios, and through each point of division, D, E, draw a parallel to the side nearest to it. The intersection of these parallels, in F, is the A. C point required. In the figure the parts ACF, ABF, BCF, are as 2: 3: 4. (511) Let it be required to find Fig. 380. the position of a point, D, situated within a given triangle, ABC, and equally distant from the points A, B, C; and to determine the ratios to each other of the three triangles into A which the given triangle is divided. By construction, find the centre of the circle passing through A, B, C. This will be the required point. AB x BC x CA By calculation, the distance DA = DB = DC = ABC 4 x area ABC The three small triangles will be to each other as the sines of their angles at D; i. e. ADB: ADC: BDC:: sin. ADB: sin. ADC: sin. BDC. These angles are readily found, since the sine of half of each of them equals the opposite side divided by twice one of the equal distances. 23 354 LAYING OUT AND DIVIDING UP LAND. [PART XI. (512) By the shortest possible line. Let it be Fig. 381. required to divide the triangle ABC by the short. est possible line, DE, into two parts, which shall / be to each other:: me: n; or DBE: ABC:: m /: m + n. D From the smallest angle, B, of the triangle, measure along the sides, BA and BC, a distance BD=BE ( ABxBC) DEisthe A O line required. It is perpendicular to the line BF which bisects the angle ABC; and itis= n. B XAB x BC) cos. _ B \ -m + DIVISION OF RECTANGLES. (543) By lines parallel to a side. Divide two opposite sides into the required number of parts, either equal or in any given ratio to each other, and the lines joining the points of division will be the lines desired. The same method is applicable to any parallelogram. Exampfle. A rectangular field Fig. 382. ABCD, measuring 15.00 chains B_ 1 C by 8.00, is bought by three men, who pay respectively $300, $400 and $500. It is to be divided among them in that proportion. Ans. The portion of the first, A E ~ AEE'B, is obtained by making the proportion 300 + 400 + 500: 300:: 15.00: AE = 3.75. EF is in like manner found to be 5.00; and FD = 6.25. BE' is made equal to AE; E'F' to EF; and F'C to FD. Fences from E to E', and from F to F', will divide the land as required. The scale of the figure is 10 chains to 1 inch = 1: 7920. The other modes of dividing up rectangles will be given under the head of " Quadrilaterals," Art. (548), &c. CHAP. III.] Dividing up Land. 355 DIVISION OF TRAPEZOIDS. (544) By lines parallel to the bases. Given the bases and a third side of the trapezoid, ABCD, to be Fig. 383. divided into two parts, such that BCFE: C EFDA:: m: n. The length of: the desired dividing line, EF V-(m x AD + n x BC2). The distance BE- AB (EF — B0) AD-BC Example. Let AD = 30 chains; BC = 20 chs.; and AB = 541 chs.; and the parts to be as 1 to 2; required EF and BE. Ans. EF = 23.80; and BE=20.65. The figure is on a scale of 30 chains to 1 L D inch-= 1: 23760. (545) Given the bases of a trapezoid, and the perpendicular distance, BH, between them; it is required to divide it as before, and to find EF, and the altitude, BG, of one of the parts. Let BCxBH BCFE: EFDA:: mn: n. ThenBG= - B + AD-BC / m 2 x ABCD x BH BC x BH21 \/MI+ n AD-BC (AD- B-C] AD -BC EF = BC + BGx B BC Example. Let AD = 30.00; BC = 20.00; BH = 54.00; and the two parts to be to each other:: 46: 89. The above data give the content of ABCD = 1350 square chains. Substituting these numbers in the above formula, we obtain BG = 20.96, and EF = 23.88. (546) By lines starting from points in a side. To divide a trapezoid into parts equivalent, or having any ratios, divide its parallel sides in the same ratios, and join the corresponding points. 356 LAYING OUT AND DIVIDING UP LAND. [PART xI. If it be also required that the division lines shall start from given points on a side, proceed Fig. 384. thus. Let it be required to B R divide the trapezoid ABCD /: into three equivalent parts by i fences starting from P andQ / i Divide the trapezoid, as above directed, into three equivalent A F P if trapezoids by the lines EF and GH. These three trapezoids muns now be transformed, thus. Join EP, and from F draw FR paral lel to it. Join PR, and it will be one of the division lines required. The other division line, QS, is obtained similarly. (547) Other cases. For other cases of dividing trapezoids, apply those for quadrilaterals in general, given in the following articles.' DIVISION OF QUADRILATERALS. (548) By lines parallel to a side. Let ABCD be a quadrila teral which it is required to Fig. 385. divide, by a line EF, paral- a lel to AD, into two parts, BEFC and EFDA, which shall be to each other as. m: n. Prolong AB and CD to intersect in G. Let a be the area of the triangle I:' ADG, obtained by any me- E thod, graphical or trigono- / \ metrical, and a' = the area A' at of the triangle BCG, obtained by subtracting the area of the given quadrilateral from that of the triangle ADG. Then GK = GH (ma + a. Having measured this length of GK from G on GH, set off at K a perpendicular to GK, and it will be the required line of division. * If a line be drawn joining the middle points of the parallel bases of a trapezoid, any line drawn through the middle of the first line will divide the trapezoid into two equivalent parts. CHAP. III.] Dividing up Land. 357 Otherwise, take G:E = GA((m' ) a ); and from E run a parallel to AD. If the two parts of the quadrilateral were to be equivalent, m =n, and we have GK = GH a a-); and consequently GE to GA in the same ratio. -Example. Let a quadrilateral, ABCD, be required to be thus divided, and let its angles, B and C, be given by rectangular co-ordinates, viz: AB' = 6.00; B'B = 9.00; DC' = 8.00; C'C = 13.00; B'C' = 24.00. Here GH is readily found to be 29.64; ADG = 563.16 square chains; and BGC = 220.16 square chains. Hence, by the formula, GK = 24.72; whence KH = GH - GK = 4.92; and the abscissas for the points E and F can be obtained by a simple proportion. The scale of the figure is 20 chains to 1 inch = 1: 15840. If the quadrilateral be given by Bearings, part off the desired area=. ABCD, by the formulas of Art. (504). m +- n Suppose now that a quad- Fig. 386. rilateral, ABCD, is to be di- A vided into p equivalent parts, by lines parallel to AD. Measure, or calculate by Trigonometry, AG. Let Qbe' the quadrilateral ABCD, and, as before, a' = BCG. Then A D GE=AG _ GL=AG.Ja+ p| at + Q a' + Q a' 3Q If the quadrilateral be given by Bearings, part off, by Art. (504),. ABCD, then part off 2. ABCD; &c.; so in any similar case. p P 358 LAYING OUT AND DIVIDING UP LAND. [PART XI. (549) By lines perpendicular to a side. Let ABCD be a quadrilateral which is to be divided, by Fig. 387. a line perpendicular to AD, into two 1. E parts having a ratio = min: n. By hypothesis, ABEF = m. ABCD. Taking away the triangle ABG, the A G P i P remainder, GBEF, will be to the rest of the figure in a known ratio, and the position of EF, parallel to BG, will be found as in the last article. (550) By lines running in any given direction. To divide a quadrilateral ABCD into two parts:: m: n, part off from it an area = 7. ABCD, by the methods of Arts. (509) or (510), if the area parted off is to be a triangle, or Arts. (511) or (512), if the area parted off is to be a quadrilateral. (551) By lines starting from an angle. ABCD is to be divided, by the line CE, into two Fig. 388. parts having the ratio m: n. Since the area of the triangle 3 CDE-= n. ABCD, DEwill m + n be obtained by dividing this area A E r D by half of the altitude CF. (552) By lines starting from points in a side. Let it be required to divide ABCD into two Fig. 389. parts:: m: n, by a line starting from the point E. The area ABFE is /. known, (being = m. ABD) as / // also ABE; AB, BE, and EA be- Al E ing given on the ground. BEF will then be known = ABFE — BEF ABE. Then GF = ~, and the point F is obtained by running p el to BE a parallel to BE, at a perpendicular distance from it = GF. CH&P. III.] Dividing up Land. 359 To divide a quadrilateral, ABCD, Fig. 390. graphically, into two equivalent parts by a line from a point, E, on a' B side, proceed thus. Draw the diago- / nal CA, and from B draw a parallel I to it, meeting DA prolonged in F. \/\ Mark the middle point, G, of FD. f A -- Join GE. From C draw a parallel to EG, meeting DA in H. EH is the required line. The quadrilateral could also be divided in any ratio = m: n, by dividing FD in that ratio. If' the quadrilateral be given by Bearings, proceed to part off the desired area, as in Art. (515) or (516). (553) Let it be required to divide a quadrilateral, ABCD, into three equivalent parts. Fig. 391. From any angle, as C, B draw CE, parallel to DA. Divide AD and EC, each E/ — 1C into three equal parts, at F, F', and G, G'. Draw BF, BF'. From G draw GH, parallel to FB, and from G' draw G'H', pa- rallel to F'B. FH and F'H' are the required lines of division. Let it be required to make Fig. 392. the above division by lines C starting from two given L points, P and Q. ReduceJ D the quadrilateral to an equi- 7. /' valent triangle CBE, as in,/, - Art. (87). Divide EB into' / three equal parts at F and I4; / G. Join CQ, and, from G, A draw GK parallel to it. Join CP, and from F draw FL parallel to it. Join PL and QK, and they will be the division lines required. (554) By lines passing through a point within the figure, Proceed to part off the desired area as in Arts. (519), (520), or (521), according to the circumstances of the case. 360 LAYING OUT AND DIVIDING UP LAND. [PART XI. DIVISION OF POLYGONS. (555) By lines running in any direction. Let ABCDEFG be a given polygon, and BH the di- Fig. 393. rection parallel to which is to be R drawn a line PQ, dividing the polygon into two parts in any desired ratio = m: n. The area PCDEQ=. AB3CDEFG. - \Q Taking it from the area BCDEH, the remainder will be the area F BPQH. The quadrilateral A - F BCEH, CE being supposed to be drawn, can then be divided by the method of Art. (548), into two parts, BPQH arid PQEC, having to each other a known relation. If DK were the given direction, at right angles to the former, the position of a dividing line RS could be similarly obtained. (556) By lines starting from an angle. Produce one side, AB, Fig. 394. z ti\ \/ of the given polygon, both ways, and reduce the polygon to a single equivalent triangle, XYZ, by the method of Art. (82). Then divide the base, XY, in the required ratio, as at W, and draw ZW, which will be the division line desired. In this figure the polygon is divided into two equivalent parts. CHAP. III.] Dividing up Land. 361 If the division line should pass outside of the polygon, as does ZP, through P draw a parallel to BZ, meeting the adjacent side of the polygon in Q, and ZQ will be the division line desired. (557) By lines starting from a point on a side. See Articles (517) and (518) in the preceding chapter. (558) By lines passing through a point within the figure. Part off, as in Arts. (519) or (522) in the preceding chapter, if a straight line be required; or by guess lines and the addition of triangles, as in Art. (538) of this chapter, if the lines have merely to start from the point, such as a spring or well. (559) Other problems. The following is from Gummere's Surveying. Question. A tract of land is Fig. 395. bounded thus: N. 35o0 E., 23.00; N. 750~ E., 30.50; S. 310 E., 46.49; N. 661;~ W., 49.64. It is to be divided into four equivalent parts by two straight lines, H one of which is to run parallel to the third A side; required the distance of the parallel division line from the first corner, mea-w sured on the fourth side; also the Bearing of the other division line, and its distance from the same corner measured on the first -side. Ans. Distance of the parallel division line from the first corner, 32.50; the Bearing of the other, S. 88~ 22' E.; and its distance from the same corner 5.99. The scale of the figure is 40 chains to 1 inch = 1: 31680. An indefinite number of problems on this subject might be proposed, but they would be matters of curiosity rather than of utility, and exercises in Geometry and Trigonometry rather than in Surveying; and the youngest student will find his life too short for even the hastiest survey of merely the most fruitful parts of the boundless field of Mathematics. 362 U. S. PUBLIC LANDS. [PART XII. Fig. 396. I 7-~_:::~ I _ //t VE=S S-r- a SWm. X. r t.hI Y ~,I ~, I i 1/ rl IY PART XII. THE PUBLIC LANDS OF THE UNITED STATES.* (560) General system, The Public Lands of the United States of America are generally divided and laid out into squares, the sides of which run truly North and South, or East and West. This is effected by means of Meridian lines and Parallels of Latitude, established six miles apart. The principal meridians and base lines are established astronomically, and the intermediate ones are run with chain and eompass. The squares thus formed are called TOWNSHIPS. They contain 36 square miles, or 23040 acres, "' as nearly as may be." The map on the opposite page represents a portion of the Territory of Oregon thus laid out. The scale is 10 miles to 1 inch = 1: 633600. On it will be seen the "Willamette Meridian," running truly North and South, and a " Base line," which is a " Parallel of Latitude," running truly East and West. Parallel to these, and six miles from' them, are other lines, forming Townships. All the Townships, situated North or South of each other, form a RANGE. The Ranges are named by their number East or West of the principal Meridian. In the figure are seen three Ranges East and West of the Willamette Meridian. They are noted as R. I. E., R. I. W., &c. The Townships in each Range are named by their number North or South of the Base line. In' The substance of this Part is mainly taken from " Instructions to the Surveyor General of Oregon, being a Manual for Field Operations," prepared, in March, 1851, by John M. Moore, " Principal Clerk of Surveys," by direction of Hon. J. Butterfield, " Commissioner of the General Land Office," and communicated to the author by Hon. John Wilson, the present Commissioner. The aim of the " Instructions" is stated to be " simplicity, uniformity and permanency." They seem admirably adapted for these objects, and the lasting importance of the subject in this country has led the author to reproduce about half of them in this place. They were subsequently directed to be adopted for the Surveying service in Minnesota and California. 364 U. S. PUBLIC LANDS. [PART XII. the figure along the principal Meridian are seen four North and five South of the Base line. They are noted as T. 1 N., T. 2 N., T. 1 S., &c.* Each Township is divided into 36 SEC- N TIONS, each 1 mile square, and therefore 6 5 4 3 2 1 containing, " as nearly as may be," 640 7 8 9 10 11 12 acres. The sections in each Township are 18 17 16 15 14 13 E numbered, as in the margin, from 1 to 36, 19 20 21 22 23 24 beginning at the North-east angle of the 30129 281 276 25 Township, and going West from 1 to 6, 3l132 33134135 36 then East from 7 to 12, and so on alter- 8 nately to Section 36, which will be in the South-east angle of the Township. The Sections are sub-divided into Quarter-sections, half-a-mile square, and containing 160 acres, and sometimes into halfquarter-sections of 80 acres, and quarter-quarter-sections of 40 acres. By this beautiful system, the smallest subdivision of land can be at once designated; such as the North-east quarter of Section 31, in Township two South, in range two East of Willamette Meridian. (561) Difficulty. "The law requires that the lines of the public surveys shall be governed by the true meridian, and that the townships shall be six miles square, —two things involving in connection a mathematical impossibility-for, strictly to conform to the meridian, necessarily throws the township out of square, by reason of the convergency of meridians; hence, adhering to the true meridian renders it necessary to depart from the strict requirements of law as respects the precise area of townships, and the subdivisional parts thereof, the township assuming something of a trapezoidal form, which inequality developes itself, more and more as such, the higher the latitude of the surveys. In view of these circumstances, the law provides that the sections of a mile.square shall contain the quantity of 640 acres, as nearly as may be; and, moreover, provides that'In all cases where the exterior lines of the townships, thus to be subdivided into sections or half-sections, shall exceed, or shall not extend, six miles, the excess or deficiency * The marks 0, + and A, merely refer to the dates of the surveys. They are sometimes used to point out lands offered for sale, or reserved, &c. PART XII.] Difficulty. 365 shall be specially noted, and added to or deducted from the western or northern ranges of sections or half-sections in such township, according as the error may be in running the lines from east to west, or from south to north."' " In order to throw the excesses or deficiencies, as the case may be, on the north and on the west sides of a township, according to law, it is necessary to survey the section lines from south to north on a true meridian, leaving the result in the northern line of the township to be governed by the convexity of the earth and the convergency of meridians." Thus, suppose the land to be surveyed lies between 460 and 47~0 of North Latitude. The length of a degree of Longitude in Lat. 46~ N. is taken as 48.0705 statute miles, and in Lat. 47~0 N. as 47.1944. The difference, or convergency per square degree = 0.8761 = 70.08 chains. The convergency per Range (8 per degree of Longitude) equals one-eighth of this, or 8.76 chains; and per Township (11t per degree of Latitude) equals the above divided by 11i, i. e. 0.76 chain. We therefore know that the width of the Townships along their Northern line is 76 links less than on their, Southern line. The townships North of the base line therefore become narrower and narrower than the six mile width with which they start, by that amount; and those South of it as much wider than six miles. "STANDARD PARALLELS (usually called correction lines), are established at stated intervals (24 or 30 miles) to provide for or counteract the error that otherwise would result from the convergency of meridians; and, because the public surveys have to be governed by the true meridian, such lines-serve also to arrest error arising from inaccuracies in measurements. Such lines, when lying north of the principal base, themselves constitute a base to the surveys on the north of them; and where lying south of the principal base, they constitute the base for the surveys south of them." The convergency or divergency above noticed is taken up on these Correction lines, from which the townships start again with their proper widths. On these therefore there are found D)ouble Corners, both for Townships and Sections, one set being the Closing Corners of the surveys ending there, and the other set being the Standard Corners for the surveys starting there. 366 U. S. PUBLIC LANDS. [PAnT XII. (562) Running Township lines. " The principal meridian, the base line, and the standard parallels having been first astronomically run, measured and marked, according to instructions, on true meridians, and true parallels of latitude, the process of running, measuring, and marking the exterior lines of townships will be as follows. Townships situated NORTH of the base line, and WEST of the principal meridian.* Commence at No. 1, being the southwest corner of T. 1 N. —R. 1 W., as established on the base line; thence run north, on a true meridian line, four hundred and eighty chains, establishing the mile and half-mile corners thereon, as per instructions, to No. 2, (the northwest corner of the same township), whereat establish the corner of Tps. 1 and 2 N.-Rs. 1 and 2 W.; thence east, on a random or trial line, setting temporary mile and half-mile stakes to No. 3, (the northeast corner of the same township), where measure and note the distance at which the line intersects the eastern boundary, north or south of the true or established corner. Run and measure westward, on the true line, (taking care to note all the land and water crossings, &c., as per instructions), to No. 4, which is identical with No. 2, establishing the mile and half-mile PERMANENT CORNERS on said line, the last half-mile of which will fall short of being forty chains, by about the amount of the calculated convergency per township, 76 links in the case above supposed. Should it ever happen, however, that such random line materially falls short, or overruns in length, or intersects the eastern boundary of the township at any considerable distance from the true corner thereon, (either of which would indicate an important error in the surveying), the lines must be retraced, even if found necessary to remeasure the meridional boundaries of the township (especially the western boundary), so as to discover and correct the error; in doing which, the true corners must be established and marked, and the false ones destroyed and obliterated, to prevent confusion in future; and all the facts must be distinctly set forth in the notes. Thence proceed in a similar manner north, from No. 4 to No. 5, (the N. W. corner of T. 2 N. -R. 1 W.), east from No. 5 to No. 6, (the N. E. corner of the same township), west from No. 6 to No. 7, (the same as No. 5), north from No. 7 to No. 8, (the N. W. corner of T. 3 N., R. 1 W.), east from No. 8 to No. 9, (the N. E. corner of same township), and thence west to No. 10, (the same as No. 8), or the southwest corner T. 4 N.-R. 1 W. Thence north, still on a true meridian line, establishing the mile and half-mile corners, until reaching the STANDARD PARALLEL or correction line, (which is here four townThe Surveyor shoulld prepare a diagram of the townships, with the numbers here referred to,. in their proper places, as here indicated PART XII.] Running Township Lines,.67 ships north of the base line); throwing the excess over, or deficiency under, four hundred and eighty chains, on the last half-mile, according to law, and at the intersection establishing the " CLOSING CORNER," the distance of which from the standard corner must be measured and noted as required by the instructions. But should it ever so happen that some impassable barrier will have prevented or delayed the extension of the standard parallel along and above the field of present survey, then the surveyor will plant, in place, the corner for the township, subject to correction thereafter, should such parallel be extended. Townships situated NORTH of the base line, and EAST of the principal meridian. Commence at No. 1, being the southeast corner of T. I N. —R. 1 E., and proceed as with townships situated'" north and west," except that the random or trial lines will be run and measured west, and the true lines, east, throwing the excess over or deficiency under four hundred and eighty chains on the west end of the line, as required by law; wherefore, the surveyor will commence his measurement with' the length of the deficient or excessive half-section boundary on the west of the township, and thus the remaining measurements will all be even miles and half-miles. Townships situated SOUTH of the base line, and WEST of the principal meridian. Commence at No. 1, the northwest corner of township 1 S., range 1 W', and proceed due south in running and measuring line, establishing and marking the mile, half-mile, and township corners thereon, precisely in the method prescribed for running NORTH and WEST, with the exception that, in order to throw the excess or deficiency (over or under four hundred and eighty chains) of the western boundaries of such of those townships as close on the standard parallel on the south, upon the most northern half-mile of the townships, according to law, the proceeding will be as follows. The western (meridional) boundary line of every township, closing on the standard parallel, (being every fifth one in this case), will be carefully run south, on a true meridian, until it intersects the standard, planting stakes and making distinctive marks on line trees, in sufficient number to serve as guides in afterwards retracing the line north with ease and certainty. At the point of the line's intersection of the standard, the surveyor will establish the " closing" (southwest) corner of the township, noting in his field-book its distance and direction from the " standard corner." Then starting from such " closing corner," he will proceed north on the line identified by the guide stakes and marks, measuring such line, and establishing thereon the mile and half-mile stations, and noting, as he goes, all the land and water crossings, &c. 368 U. S. PUBLIC LANDS. [PART XII. Townships situated SOUTH of the base line, and EAST of the principal meridian. Commence at No. 1, at the northeast corner of township 1 S., range 1 E., and proceed precisely as with the townships situated "south and west," except that the random lines will be run and measured west, and the true lines east; the deficiency or excess of the measurements being, as in all other cases, thrown upon the most western half-mile of line." (563) Running Section lines. The interior or sectional lines of all townships, however situated in reference to the BASE and MERIDIAN lines, are laid off and surveyed as below. 31 32 33 34 35 36 97 71 53 35 17 I 6 5 4 3 2 I 6 99 98 96 72 70 54152 36 34 18 16 100 94 95 68 69 50 51 32 33 14 15 12 7 8 9 10 11 12 7 92 93 91 67 49 31 13 89 90 65 66 47 48 29 30 11 12 13 18 17 16 15 14 13 18 87 86 64 46 28 10 88 84 85 62 63 44 45 26 27 8 9 24 19 20 21 22 23 24 19 82 81 61 43 25 7 83 79 80 59 60 41 42 23 24 5 6 25 30 29 28 27 26 25 30 77 76 58 40 22 4 78 7 4 75 56 57 38 39 20 21'2 3 36 31 32 33 84 35 36 31 73 55 37 19 1 6 5 4 3 2 1 In the above Diagram, the squares and large figures represent sections, and the small figures at their corners are those referred to in the following directions. " Commence at No. 1, (see small figures on diagram), the corner established on the township boundary for sections 1, 2, 35, and PART XII.] Running Section Lines. 369 36; thence run north on a true meridian; at 40 chains setting the half-mile or quarter-section post, and at 80 chains (No. 2) establishing and marking the corner of sections 25, 26, 35, and 36. Thence east, on a random line, to No. 3, setting the temporary quarter-section post at 40 chains, noting the measurement to No. 3. and the measured distance of the randomn's intersection north or south of the true or established corner of sections 25, 36, 30, and 31, on the township boundary. Thence correct, west, on the true line to No. 4, setting the quarter-section post on this line exactly at the equidistant point, now known, between the section corners indicated by the small figures Nos. 3 and 4. Proceed, in like manner, from No. 4 to No. 5, 5 to 6, 6 to 7, and so on to No. 16, the corner to sections 1, 2, 11, and 12. Thence north/, on a ranldom line, to No. 17, setting a temporary quarter-section post at 40 chains, noting the length of the whole line, and the measured distance of the random's intersection east or west of the true corner of sections 1, 2, 35, and 36, established on the township boundary; thence southluardly from the latter, on a true line, noting the course and distance to No. 18, the established corner to sections 1, 2, 11, and 12, taking care to establish the quarter-section corner on the true line, at the distance of 40 chains from said section corner, so as to throw the excess or deficiency on the northern half-mile, according to law. Proceed in like manner throuigh all the intervening tiers of sections to No. 73, the corner to sections 31, 32, 5, and 6; thence north, on a true meridian line, to No. 74, establishing the quarter-section corner at 40 chains, and at 80 chains the corner to sections 29, 30, 31, and 32; thence east, on a random line to No. 75, setting a temporary quarter-section plost at 40 chains, noting the measurement to No. 75, and the distance of the random's intersection north or south of the established corner of sections 28, 29, 32, and 33; thence west from said corner, on the true line, setting the quarter-section post at the equidistant point, to No. 76, which is identical with 74; thence west, on a random line, to No. 77, setting a temporary quarter-section post at 40 chains, noting the measurement to No. 77, and the distance of the random's intersection with the western boundary, north or south of the established corner of sections 25, 36, 30, and 31; and from No. 77, correct, eastward, on the true line, giving its course, but establishing the quarter-section post, on this line, so as to retain the distance of 40 chains from the corner of sections 29, 30, 31, and 32; thereby throwing the excess or deficiency of measurement on the most western half-mile. Proceed north, in a similar manner, from'No. 78 to 79, 79 to 80, 80 to 81, and so on to 96, the south-east corner of section 6, where having established the corner for sections, 5, 6, 7, and 8, run thence, successively, on 24 370 U. S. PUBLIC LANDS. [PART XII. random line east to 95, north to 97, and west to 99; and by reverse courses correct on true lines back to said south-east corner of section 6, establishing the quarter-section corners, and noting the courses, distances, &c., as before described. In townships contiguous to standard parallels, the above method will be varied as follows. In every township SOUTH of the principal base line, which closes on a standard parallel, the surveyor will begin at the south-east corner of the township, and measure west on the standard, establishing thereon the mile and half-mile corners, and noting their distances from the pre-established corners. He then will proceed to subdivide, as directed under the above head. In the townships NORTI Of the principal base line, which close on the standard parallel, the sectional lines must be closed on the standard by true meridians, instead of by course lines, as directed under the above head for townships otherwise situated; and the connexions of the closing corners with the pre-established standard corners are to be ascertained and noted. Such procedure does away with any necessity for running the randoms. But in case he is unable to close the lines on account of the standard not having been run, from some inevitable necessity, as heretofore mentiond, he will plant a temporary stake, or mound, at the end of the sixth mile, thus leaving the lines and their connexions to be finished, and the permanent corners to be planted, at such time as the standard shall be extended." (561) Exceptional methods. Departures from the general system of subdividing public lands have been authorized by law in certain cases, particularly on water-fronts. Thus, an act of Congress, March 3, 1811, authorized the surveyors of Lousiana, "' in surveying and dividing such of the public lands in the said territory, which are or may be authorized to be surveyed and divided, as are adjacent to any river, lake, creek, bayou, or water course, to lay out the same into tracts, as far as practicable, of fifty-eight poles in front, and four hundred and sixty-five poles in depth, of such shape, and bounded by such lines, as the nature of the country will render practicable and most convenient." Another act, of May 24, 1824, authorizes lands similarly situated " to be surveyed in tracts of two acres in width, fronting on any river, bayou, lake, or water course, and running back the depth of forty acres; which tracts of land, so surveyed, shall be offered for sale entire, instead of in half-quarter-sections." The " Instructions" from which we have quoted say,:" In those localities where it would best subserve the interests of the people( to have fronts on the navigable streams, and to run back into thil PART XII.] Exceptional Methods. 371 uplands for quantity and timber, the principles of the act of May 24th, 1824, may be adopted, and you are authorized to enlarge the quantity, so as to embrace four acres front by forty in depth, forming tracts of one hundred and sixty acres. But in so doing it is designed only to survey the lines between every four lots, (or 640 acres), but to establish the boundary posts, or mounds, in froilnt and in rear, at the distances requisite to secure the quantity of lti;i acres to each lot, either rectangularly, when practicable, or at oblique angles, when otherwise. The angle is not important, so that the principle be maintained, as far as practicable, of making the work to square in the rear with the regular sectioning. The numbering of all anomalous lots will commence with No. 37, to avoid the possibility of conflict with the numbering of the regular sections." The act of Sept. 27, 1850, authorized the Department, should it deem expedient, to cause the Oregon surveys to be executed according to the principles of what is called the "Geodetic Method." The complete adoption of this has not been thought to be expedient; but " it was deemed useful to institute on the principal base and meridian lines of the public surveys in Oregon, ordered to be established by the act referred to, a system of triangulations front the recognized legal stations, to all prominent objects within the range of the theodolite; by means of which the relative distances of such objects, in respect to those main lines, and also to each other, might be observed, calculated, and protracted, with the view of contributing to the knowledge of the topography of the country in advance of the progressing linear surveys, and to obtain the elements for estimating areas of valleys intervening between the spurs of the mountains." "Meandering" is a name given to the usual mode of surveying with the compass, particularly as applied to navigable streams. The " Instructions" for this are, in part,. as follows. " Both banks of navigable rivers are to be meandered by taking the courses and distances of their sinuosities, and the same are to be entered in the'Meander field-book.' At those points where either the township or section lines intersect the banks of a navigable stream, POSTS, or, where necessary, MOUNDS of earth or stone, (as noted in Art. (566,)) are to be established at the time of running these lines. These are called " meander corners;" and in meandering you are to commence at one of those corners on the township line, coursing the banks, and measuring the distance of each course from your commencing corner to the next'meandle. 372 Ui. S. PUBLIC LANDS. [PART XI1. corner,' upon the same or another boundary of the same township; carefully noting your intersection with all intermediate meander corners. By the same method you are to meander the opposite bank of the same river. The crossing distance between the MEANDER CORNERS, on same line, is to be ascertained by triangulation, in order that the river may be protracted with entire accuracy. The particulars to be given in the field-notes. The courses and distances on meandered navigable streams, govern the calculations wherefrom are ascertained the true areas of the tracts of land (sections, quarter sections, &c.) known to the law as fractional, and bounding on such streams." You are also to meander, in manner aforesaid, all lakes and deep ponds of the area of twenty-five acres and upwards; also navigable bayous. The precise relative position of islands, in a township made fractional by the river in which the same are situated, is to be determined trigonometrically. Sighting to a flag or other fixed object on the island, from a special and carefully measured base line, connected with the surveyed lines, on or near the river bank, you are to form connexion between the meander corners on the river to points corresponding thereto, in direct line, on the bank of the island, and there establish the proper meander corners, and calculate the distance across." (565) Marking Lines, "All lines on which are to be established the legal corner boundaries, are to be marked after this method, viz: Those trees which may intercept your line, must have two chops or notches cut on each side of them without any other marks whatever. These are called' sighlt trees,' or' line trees.' A sufficient number of other trees standing nearest to your line, on either side of it, are to be blazed on two sides, diagonally or quartering towards the line, in order to render the line conspicuous, and readily to be traced, the blazes to be opposite each other, coinciding in direction with the line where the trees stand very near it, and to approach nearer each other, the further the line passes from the blazed trees. Due care must ever be taken to have the lines so well marked as to be readily followed." (566) Marking Corners,' After a true coursing, and most exact measurements, the corner boundary is the consummation of the work, for which all the previous pains and expenditure have been incurred. A boundary corner, in a timbered country, is to be a tree, if one be found at the precise spot; and if not, a post is to be planted thereat; and the position of the corner post is to be PART XII.] arking t6rPers. 373 indicated by trees adjacent, (called Bearing trees) the angular bearings and distances of which from the corner are facts to be ascertained and registered in your field book. In a region where stone abounds, the corner boundary will be a small momnunent of stones along side of a single marked stone, for a township corner —and a single stone for all other corners. In a region where timber is not near, nor stone, the corner will be a mound of earth, of prescribed size, varying to suit the case. Corners are to be fixed, for township boundaries at intervals of every six miles; for section boundaries at intervals of every mile, or 80 chains; and, for quarter section boundaries at intervals of every half mile, or 40 chains. MEANDER CORNER POSTS are to be planted at all those points where the township or section lines intersect the banks of such rivers, lakes, or islands, as are by law directed to be meandered," as explained in Art. (561). When posts are used, their length and size must be proportioned to the importance of the corner, whether township, section, or quarter-section, the first being at least 24 inches above ground, and 3 inches square. Where a township post is a corner common to four townships, N it is to be set in the earth diayonally, thus: w'~E, and the cardis nal points of the compass are to be indicated thereon by a cross line, or wedge, (one-eighth of an inch deep at least), cut or sawed out of its top, as in the figure. On each surface of the post is to be marked the number of the particular township, and its range, which it faces. Thus, if the post be a common boundary to foure townships, say one and two, south of the base line, of range one, west of the meridian; also to townships one and two, south of the base line, of range two, west of the meridian, it is to be marked thus: 1. 1 W. ) The position of the post which From N. to E. T. 1 S. 5 is here taken as an example, is S. 31 shewn in the following diagram. 2 2W.) from N. to W. 1 S.. 2 W. R. 1 W. 36 T. 1 S. T.1S. from E. to S. 2 S. 6 2 W.) 6 from W. to. T 2S. R. 2 W. I T 2 T. 2 S.T.2. 374 JU. S. PUBLIC LANDS. [PART XII. These marks are to be distinctly and neatly chiselled into the wood, at least the eighth of an inch deep.; and to be also marked with red chalk. The number of the sections which they respectively face, will also be marked on the township post. Section or mile posts, being corners of sections, when they are common to four sections, are to be set diagonally in the earth, (in the manner provided for township corner posts), and with a similar cross cut in the top, to indicate the cardinal points of the compass; and on each side of the squared surfaces is to be marked the appropriate number of the particular one of the four sections, respectively, which such side faces; also on one side thereof are to be marked the numbers of its township and range; and to make such marks yet more conspicuous, (in manner aforesaid), a streak of red chalk is to be applied. In the case of an isolated township, subdivided into thirty-six sections, there are twenty-five interior sections, the south-west corner boundary of each of which will be common to four sections. On all the extreme sides of an isolated township, the outer tiers of sections have corners common only to two sections then surveyed. The posts, however, must be planted precisely like the former, but presenting two vacant surfaces to receive the appropriate marks when the adjacent survey may be made. A quarter-section or half-mile post is to have no other mark on it than X S., to indicate what it stands for. Township corner posts are to be NOTCHED with six notches on each of the four angles of the squared part set to the cardinal points. All mile posts on township lines must have as many notches on them, on two opposite angles thereof, as they are miles distan' from the township corners, respectively. Each of the posts at the corners of sections in the interior of a township must indicate, by a number of notches on each of its four corners directed to the cardinal points, the corresponding number of miles that it stands from the outlines of the township. The four sides of the post will indicate the number of the section they respectivelyface. Should a tree be found at the place of any corner, it will be marked and notched, as aforesaid, and answer for the corner in lieu of a post; the kind of tree and its diameter being given in the field-notes. The position of all corner posts, or corner trees of whatever description, which may be established, is to be perpetuated in the following manner, viz: From such post or tree the courses shall be taken, and the distances measured, to two or more adjacent trees, in opposite directions, as nearly as may be, which are called'Bearing trees,' and are to be blazed near the ground, with a large l)aze facing the post, and having one notch in it, neatly and plainly PART XII.] Marking Corners. 375 made with an axe, square across, and a little below the middle ot the blaze. The kind of tree and the diameter of each are facts to be distinctly set forth in the field-book. On each bearing tree the letters B. T., must be distinctly cut into the wood, in the blaze, a little above the notch, or on the bark, with the number of the range, township, and section. At all township corners, and at all section corners, on range or township lines, four bearing trees are to be marked in this manner, one in each of the adjoining sections. At interior section corners four trees, one to stand within each of the four sections to which such corner is common, are to be marked in manner aforesaid, if such be found. From quarter section and meander corners two bearing trees are to be marked, one within each of the adjoining sections. Stones at township corners (a small monument of stones being alongside thereof) must have six notches cut with a pick or chisel on each edge or side towards the cardinal points; and where used as section corners on the range and township lines, or as section corners in the interior of a township, they will also be notched by a pick or chisel, to correspond with the directions given for notching posts similarly situated. Stones, when used as quarter-v:cion corners, will have X cut on them; on the west side on north and south lines, and on the north side on east and west lines. Whenever bearing trees are not found, MOUNDS of earth, o., stone, are to' be raised around posts on which the corners are to be marked in the manner aforesaid. Wherever a mound of earth is adopted, the same will present a conical shape; but at its base, on the earth's surface, a quadrangular trench will be dug; a spade deep of earth being thrown up from the four sides of the line, outside the trench, so as to form a continuous elevation along its outer edge. In mounds of earth, common to four townships or to four sections, they will present the angles of the quadrangular trench (diagonally) towards the cardinal points. In mounds common only to two townships or two sections, the sides of the quadrangular trench willface the cardinal points. Prior to piling up the earth to construct a mound, in a cavity formed at the corner boundary point'is to be deposited a stone, or a portion of charcoal, or a charred stake is to be driven twelve inches down into such centre point, to be a witness for the future. The surveyor is farther specially enjoined to plant nmidway between each pit and the trench, seeds of some tree, those of fruit trees adapted to the climate being always to be preferred. DOUBLE CORNERS are to be found nowhere except on the Standard Parallels or Correction lines, whereon are to appear both the cor 376 U. S. PUBLIC LANDS0 [IPAIT XI. ners which mark the intersections of the lines which close thereon, and those from which the surveys start in the opposite direction. The corners which are established on the standard parallel, at the time of running it, are to be known as' Standard Corners,' and, in addition to all the ordinary marks, (as herein prescribed), they will be marked with the letters S. C. The' closing corners' will be marked C. C." (567) Field Books. There should be several distinct and separate field-books; viz.: " 1. Field-notes of the iMERIDIAN and EASE lines, showing the establishment of the townshi2p, section or mile, and quarter-section or half-mile, boundary. corners thereon; with the crossings of streams, ravines, hills, and mountains; character of soil, timber, minerals, &c. These notes will be arranged, in series, by mile stations, from number one to number -. 2. Field-notes of the'STANDARD PARALLELS, or correction lines,' showing the establishment of the township, section, and quarter-section corners, besides exhibiting the topography of the country on line, as required on the base and meridian lines. 3. Field-notes of the EXTERIOR lines of TOWNSHIIPS, showing the establishment of the corners on line, and the topography, as aforesaid. 4. Field notes of the SUBDIVISIONS of TOWNSHIPS into sections and quarter-sections; at the close whereof will follow the notes of the MEANDERS of navigable streams. These notes will also show, b/ ocular observation, the estimated rise and fall of the land on the line. A description of the timber, undergrowth, surface, soil, and minerals, upon each section line, is to follow the notes thereof, and not to be mixed up with them." 5. The "' Geodetic Field-book," comprising all triangulations, angles of elevation and depression, levelling, &c. The examples on the next two pages, taken from the' Ill - tions" which we have followed throughout, will shew what is required. The ascents and descents are recorded in the right-hand columns, ATInT xII.] Field-N ots, 377 FIELD NOTES OF THE EXTERIOR LINES OF AN ISOLATED TOWNSHIP. Field no/e.' c' /e, Survey of township 25 north, of range 2 west, of the lWillanette mertzdiain, in the Terrilory of OREGON, by Robert Acres, deputy surveyor, under his contract JNo. 1, beari~n date the 2d day of January, 185 1.!2 Chs. iks. Feet. 5.~ Tow.NSH-IPe LINES commenced January 20, 1851. Southern boundary variation 18o 41' E. East. On a random line on the south boundaries of sections 31, 32,. 33, 34, 35, and 36. Set temporary mile and half-mile posts, and inltersected the eastern boundary 2 chains 20 links 110oth1 P, of the true corner 5 rmiles 74 chaitns 53 links. Therefore thle correction w;ill be 5 chains 47 1ilkS W. 37.1 O links S. per mile. TitUE SOUTHERN BOUNDARtY variation 18~ 41' E. West On the southern boundary of sec. 36, Jan. 24, 1851. 40.00 Set qi. sec. post from which a 10 a beech 124 in. dia. bears N. 11 E. 38 Iks. dist. a do 9 do do S. 9 E. 17 do o 62.50 a brook 8 1. wide, course NW.............................. d 10 80.00 Set post cor. of secs. 35 & 36, 1 & 2, fiom which........ a: a~ a beech 9 in. din,. bears S. 46 E. 8 1. (list. a a do 8 do do S. 62 W. 7 do a W. oak 10 (lo do N. 19 W. 14 do a B. oak 14 do do N. 29 E. 16 do Land level, parit et ct aInd swampy; timber beech, oak, ash, hicklory, &c. Wd Xest. On the S. boundary of sec. 35'40.00 Set qr. sec. post, with trench, friom which a 1i) a beechL 6 in. clis. bears N. 80 E. 8 1. dist. Co )IEplanted SW. a yellow locust seed. - 6 5.00 To beginning of hill..........................a 5 X 80.00 Set post, with trench, cor. of' sees. 34 & 35. 2 & 3, from wvic. a 20 a beech 10 in. dia. bears S. 51 E. 13 1. dist. ~E do 10 do do N. 56 W. 9 do pianted SW. a white oak acorn, ~: NE. a beech Int..[21'to Land level, rich, and good frou farming; timber same. PC West. On tlhe S. boundary of sec. 3440.00 Set qr. sec. post, with trench, from which a 5 a B;. oak i in. dia. bears N. 2 E. 635 1. dist. Planted SW. a beech nut. 80.00 To corner of sections 33, 34, 3 and 4, drove charred stakes a 10 raised mlound with trench as pe instructions, and Planted NE. a W. oalk ac'n-; NW. a yel. locust seed. SE. a butternut; SW. a beech nut. ~ CLanud level, rich and good for farming, smrne scattering oak = allcd walnut. &c., &c., &c. 37U. S. PUBLIC LANDS. PART XII.1 FIELD NOTES OF THE SUBDIVISIONAL OR SECTIONAL LINES, AND MEANDERS. Township 25 N., Range 2 W., Willamette Mer. ohs. lks. Feet. SUBDIVISIONS. Commenced February 1, 1851. North. Between sees. 35 and 369.19 A beech 30 in. dia........................................... d 10 29.97 A beech 30 in. dia................d 5 40.00 Set qr. sec. post, from which d 5 a beech 15 in. dia. bears S. 48 E. 12 1. dist. a do 8 do do N. 23 W. 45 do X 51.90 A beech 18 in. dia.......................................... d 5 76.73 A sugar 30 in. dia......................................... d 8 80.00 Set a post cor. of secs. 25, 26, 35, 36, fiom which d 2 a beech 24 in. dia. bears N. 62 W. 17 1. dist. a poplar 36 do do S. 66 E. 34 do. a do 20 do do S. 70 W. 50 do. a beech 28 do do N. 60 E. 45 do. Land level, second rate; timber beech, poplar, sugar, and und'gr. spice, &c. East. On random line between secs. 25 and 369.00 A brook 30 1. wide, course N. —.. —---—.-.-. —............. d 10 15.00 To foot of hill.............................................. d 10 40.00 Set temporary qr. sec. post...-..............-............... a 60 a 55.00 To opposite foot of hill.......-... ——. ——....... d 40 72.00 A brook 15 1. wide, course N........................ d 20 80.00 Intersect E. boundary at post......... a 10 Land level, second rate; timber, beech, oak, ash, &c. &c.,. &c., &c. MEANDERS OF CHICKEELES RIVER. Beginning at a meander post in the northern township boundary, and thonce on the left bank down stream. Commenced February 11, 1851. lDist. Courses. Dist. REMARKS. Ohs. Iks. S. 76 W. 18.46 In section 4 bearing to corner sec. 4 on right bank N. 700 W. S. 61 W. 10.00 Bearing to cor. sec. 4 and 5, right bank N. 520 W. S. 61 W. 8.18 To post in line between sections 4 and 5, breadth of river by triangulation 9 chains 51 links. S. 54 W. 10.69 In section 5. S. 40 W. 5.59 S. 50 W. 8.46 S. 37 W. 16.50 To upper corner of John Smith's claim, course E. S. 44 W. 21.96 S. 36 W. 27.53 To post in line between sections 5 and 8, breadth of river by triangulation 8 chains 78 links. &c., &c., &c. APPENDIX. APPENDIX A. SYNOPSIS OF PLANE TRIGONOMETRY.* (1) Definition. Plane Trigonometry is that branch of Mathematical Science which treats of the relations between the sides and angles of plane triangles. It teaches how to find any three of these six parts, when the other three are given and one of them, at least, is a side. (2) Angles and Arcs. The angles of a triangle are measured by the arcs described, with any radius, from the angular points as centres, and intercepted between the legs of the angles. These arcs are measured by comparing them with an entire circumference, described with the same radius. Every circumference is regarded as being divided into 360 equal parts, called degrees. Each degree is divided into 60 equal parts, called minutes, and each minute into 60 seconds. These divisions are indicated by the marks o' ". Thus 28 degrees, 17 minutes, and 49 seconds, are written 280 17' 49". Fractions of a second are best expressed decimally. An arce, including a quarter of a circumference and measuring a right angle, is therefore 90~. A semicircumference comprises 180~. It is often represented by r, which equals 3.14159, &c., or 31 approximately, the radius being unity. The length of 1~ in parts of radius -= 0.01745329; that of 1'= 0.00029089; and that of 1"=- 0.00000485. The length of the radius of a circle in degrees, or 360ths of the circumference -- 50~.29518 = 570 17' 24".8 = 3437'.747 = 206264".8.f An arc may be regarded as generated by a point, M, Fig. 39 moving from an origin, A, around a circle, in the direction of the arrow. The point may thus describe arcs of any lengths, such as AM; AB = 900~ = 7r; ABC = 1800 = r; / ABCD = 2'70~ %- 7r; ABCDA = 360~ = 2 r. The point may still continue its motion, and generate C o A arcs greater than a circumference, or than two circumferences, or than three; or even infinite in length. While the point, M, describes these arcs, the radius, OM, indefinitely produced, generates corresponding angles. * For merely solving triangles, only Articles (1), (2), (3), (5), (6), (10), (11), and (12), are needed. t The number of seconds in any arc which is given in parts of radius, radius being unity, equals the length of the arc so given divided by the length of the arc of one second; or multiplied by the number of seconds in radius. 380 TRIGONOMETRY. [APP. A. If the point, M, should move from the origin, A, in the contrary direction to its former movement, the arcs generated by it are regarded as negative, or minus; and so too, of necessity, the angles measured by the arcs. Arcs and angles may therefore vary in length from 0 to ~- c in one direction, and from 0 to- oo in the contrary direction. The Complement of an arc is the are which would remain after subtracting the are from a quarter of the circumference, or from 90~. If the are be more than 900, its complement is necessarily negative. The Supplement of an are is what would remain after subtracting it from half the circumference, or from 180~. If the are be more than 1800, its supplement is necessarily negative. (3) Trigooinetarical Lines. The relations of the sides of a triangle to its angles are what is required; but it is more convenient to replace the angles by arcs; and, once more, to replace the arcs by certain straight lines depending upon them, and increasing and decreasing with them, or conversely, in such a way that the length of the lines can be found from that of the arcs, and vice versa. It is with these lines that the sides of a triangle are compared.* These lines are called T'rigonometrical Lines; or Circular Functions, because their length is a function of that of the circular arcs. The principal Trigonometrical lines are Sines, Tangents, and Seccants. Chords and versed sines are also used. The SINE of an arc, AM, is the perpendicular, MP, let fall, from one extremity of the arc, upon Fig. 398. the diameter which passes through the other extremity. The TANGENT of an arc, AM, is the distance, T AT, intercepted, on the tangent drawn at one extremity of the arc, between that extremity and the prolongation of the radius which passes through the other extremity. The SECANT of an arc, AM, is the part, OT, of the prolonged radius, comprised between the 0L 1 A centre and the tangent. The sine, tangent, and secant of the complement of an arc are called the CoSINE, CO-TANGENT, and CO-SECANT of that arc. Thus, MIQ is the cosine of AM, BS its cotangent, and OS its cosecant. The cosine MIQ is equal to OP, the part of the radius comprised between the centre and the foot of the sine. The chord of an are is equal to twice the sine of half that arc. The versed-sine of an arc, AM, is the distance, AP, comprised between the origin of the arc and the foot of the sine. It is consequently equal to the difference between the radius and the sine. The Trigonometrical lines are usually written in an abbreviated form. Calling, the arc AM = a, we write, MP = sin. a. AT -= tan. a. OT = sec. a. MQ = cos. a. BS = cot. a. OS = cosec. a. The period after sin., tan., &c., indicating abbreviation, is frequently omitted. The arcs whose sines, tangents, &c., are equal to a line =a, are written, sin.-' a, or arc (sin. = a); tan.-' a, or arc (tan.= a); &c. * For the great value of this indirect mode of comparing the sides and angles of triangles, see Comte's "' Philosophy of Mathematics," (Harpers', 1851,) page 225. APP. A.] T'' O rI METRY. 381 (4) Tiae lei nes a raatio. The ratios Fig. 309. between the trigonometrical lines and the radius B are the same for the same angles, or number of degrees in an are, whatever the length of the radius or arc. Consequently, radius being unity, \ \ these lines may be expressed as simple ratios. Thus, in the right-angled triangle A B C, we /. would have. A C BC opposite side AC adjacent side sin. A-= -~= hphs cos. A= =O- - AB hypothenuse' AB hypothenuse' BC opposite side AC adjacent side tan.A= cot. A=AC adjacent side' -AB hypothenuse AB hypothenuse sec. A- - cosec- A= AC adjacent side' BC opposite side' When the radius of the arcs which measure the angles is unity, these ratios may be used for the lines. If the radius be any other length, the results which have been obtained by the above supposition, must be modified by dividing each of the trigonometrical lines in the result by radius, and thus rendering the equations of the results "hlomogeneous." The same effect would be produced by multiplying each term in the expression by such a power of radius as would make it contain a number of linear factors equal to the greatest number in any term. The radius is usually represented by r, or R. (5) T~ eir vsyaralooasi ini tletgafla. As the point M moves around the circle, and the are thus increases, the sines, tangents, and secants, starting from zero, also increase; till, -when the point M has arrived at B, and the are has Fig. 400. become 90~, the sine has become equal to S' _B S radius, or unity, and the tangent and se-... cant have become infinite. The comple- _ mentary lines have decreased; the cosine being equal to radius or unity at starting' and becomingl zero, and the cotangent and cocscant passing from infirl- P _A ity to zero. Vhen the point M has passed the first quadrantl at B and is proceeding towards C, the sines, tan-. N gents, and secants begin to decrease, till, \ / when the point has reached C, they have D the same values as at A. They then begin to increase again, and so on. The Table on page 382 indicates these variations. The sines and tangents of very small arcs may be regarded as sensibly proportional to the arcs theimselves; so that for sin. a", we may write a. sin. 1"; and similarly, though less accurately, for sin. a', we may write a. sin. 1'. The sines and tangents of very small arcs may similarly be regarded as sensibly of the same length as the arcs themselves.* * Consequently, the note on page 3T9 may read thus: The number of seconds in any very small are given in parts of radius, radius being unity, is equal to the length of the are so given divided by sin. 1". 382 TRIGONOMETRY, [APP. A. a being the length of any are expressed in parts of radius, the lengths of its sine and cosine may be obtained by the following series: a a5 a7 sin, a=-a -- 23 527 +, etc. 2 3. 4.5 2. 3....7 2 a a4 a cos. a= 1 — - 2 6 -+, etc. 2 2.3.4 2....6 Let it be required to find cos. 300, by the above series. 30~ =- = X 3.1416 =.5236. 0 Substituting this number for a, the series becomes, talking only three terms of it, 1 (.5236)2+ ('5236),etc. =1 - 0.137078 + 0.003130 —.866052; 2 24 which is the correct value of cos. 300 for the first four places of decimals. The lengths of the other lines can be obtained from the mutual relations given in Art. (7.) Some particular values are given below. sin. 300~ =. sin. 450 = -,- V2. sin. 60~ =- 3. tan. 0 = 3 tan. 45 = 1. tan. 600~ 3S. sec. 300 = 2\/3. sec. 45~ = /2. sec. 600 - 2. (6) Their changes of sign. Lines measured in contrary directions from a common origin, usually receive contrary algebraic signs. If then all the lines in the first quadrant are called positive, their signs will change in some of the other quadrants. Thus the sines in the first quadrant being all measured upward, when they are measured downward, as they are in the third and fourth quadrants, they will be negative. The cosines in the first quadrant are measured from left to right, and when they are measured from right to left, as in the second and third quadrants, they will be negative. The tangents and secants follow similar rules. The variations in length and the changes of sign are all indicated in the following table, radius being unity. The terms "increasing" and "decreasing" apply to the lengths of the lines without any reference to their signs. Lengths and Signs of the Trigonometrical Lines for Arcsfrom 00 to 360'. Arcs. 0o Between 0c and 900. 900 Between 900 and 1800. 1800 Sine... 0 +, and increasing, +1, and decreasing, 0 Tangent. 0 +, and increasing, oo -, and decreasing, 0 Secant.. +1 +, and increasing, i cO -, and decreasing, -1 Cosine.. -1 +, and decreasing, 0 -, and increasing, -1 Cotangent 4 co +-, and decreasing, 0 -, and increasing, cx Cosecant.. ~o +, and decreasing, +1 -, and increasing, ~ o Arcs. 180o Between 180 and 2700. 2700 Between 2700 and 3600. 3600 Sine... 0 -, and increasing, -1 -, and decreasing, 0 Tangent.. +, and increasing, c -, and decreasing, O Secant.. -1 -, and increasing, To +-, and decreasing, +1 Cosine... -1 -, and decreasing, 0 +, and increasing, -1 Cotangent =Fx +-, and decreasing, 0 -, and increasing, oo Cosecant.:.c: -, and decreasing, -1 -, and increasing, qoo APP. A.] TRIGONOMETRY.:383 From this table, and Fig. 400, we see that an arc and its supplement have the same sine; and that their tangents, secants, cosines, and cotangents are of equal length but of contrary signs; while the cosecants are the same in both length and sign. We also deduce from the figure the following consequences: sin. (a~+ 180~) =-sin. a~. cos. (a~+ 180~) =-cos. a~. tan. (a~+ 180~) = tan. a~. cot. (a0+ 180~) = cot. a~. sec. (a~+ 180) = —sec. a~. cosec. (a~+ 1800) =-cosec. a~. sin. (-ao) =-sin. a'. cos. (-a')= cos. ao. tan. (-a~) — tan. a~. cot. (-a~) = —cot. a0. sec. (-ao)= sec. a0. cosec. (-a~) =-cosec. ao. An infinite number of arcs have the same trigonometrical lines; for, an are a, the same are plus a circumference, the same are plus two circumferences, and so on, would have the same sine, &c. "To bring back to the first quadrant" the trigonometrical lines of any large arc, proceed thus: Let 1029~ be an arc the sine of which is desired. Take from it as many times 3600 as possible. The remainder will be 309~. Then we shall have sin.3090=sin.(1800- 3090)=sin.-1290= —sin.1290=- sin.(180 — 1290)=-sin.510. (7) Their mutual relations. Radius being unity, sin. a~ cos. ao tan. a - — 0 cot. a'~COS. a sin. a0 0 1 0 1 sec. a = 0 cosec. a~ =. cos. a sin. a~ tan. a~ Xcot. a~ = 1. (sin. a0)2 + (cos. a0)2 = 1 * 1 + (tan. a~)- = (sec. a0)2. 1 + (cot. a0)2 = (cosec. a0)2. Hence, any one of the trigonometrical lines being given, the rest can be found from some of these equations. (8) Two ares. Let a and b represent any two arcs, a being the greater. Then the following formulas apply: sin. (a + b) = sin. a. cos. b + cos. a. sin. b. sin. (a - b) = sin. a. cos. b - cos. a. sin. b. cos. (a - b) = cos. a. cos. b - sin. a. sin. b. cos. (a - b) = cos. a. cos. b + sin. a. sin. b. tan. a +- tan. b tan. (a J b) 1 - tan a. tan. 6b tan. a- tan. b tan. (a - b) 1+ tan. a. tan. b' cot. a. cot. b —1 cot. b + cot. a cot. a. cot. b6 1 cot. (a - b) = cot. b- cot. a * The square, &c., of the sine, &c., of an are, is often expressed by placing the exponent between the abbreviation of the name of the trigonometrical line and the number of the degrees in the arc; chus, sin.2 aO, tan.2 ao, &c. But the notation given above, places the index as used by Gauss, Delambre, Arbogast, &c., though the first two omit the parentheses. 384 TRIGONOMETRY. [APP. A. sin. a. sin. b =. cos. (a - b) - cos. (a + b). cos. a. cos. b = e. cos. (a + b) + i cos. (a - b). sin. a. cos. b = ~. sin. (a + b) +- ~ sin. (a - b). cos. a. sin. b = ~. sin. (a +- b) - sin. (a - 6b). sin. a + sin. b = 2 sin. (a +t- b) cqs. ~ (a - b). cos. a + cos. b = 2 cos. ~ (6 -+- ) cos. (a - 6). sin. a - sin. b = 2 sin. - (a - b) cos. - (a + b). cos. b - cos. a = 2 sin. ~ (a - b) sin. - (a + b). tan. a + tan. b= sin. (a +- b) cos. a. cos. b tan. a- tan. b sin. (a- b6) cos. a. cos. b sin. (a q- b) cot. b + cot. a sin. b sin. a. sin. b cot. b - cot. a = in ) sin. a. sin. b' (9) Double and Inalf arcs. Letting a represent any arc, as before, we have the following formulas: sin. 2 a — = 2 sin. a.cos. a. cos. 2 a = (cos. a)' - (Sin. a)2 = 2 (C0o. a)2 - 1 = 1 - 2 (sin. a)2 2 tan. a 2 cot. a 2 tan. 2 a ---- 1- (tan. a)2 (cot. a)2- 1 cot, a - tan. a (cot. a)2_- 1 cot. 2 a = 2 cot. a - (cot. a - tan. a). sin. ~ a V/ [ (1 - cos a)]. cos. ~a= V [ (1 -cos. a)]. sinl. a 1 - cos. a l - cos. a tan. a == =_ig. 1 + cos. a sin, a + cos. a c 1 +t cos. a sin. e 1 - cos. a) sin. a -os. a C - cos. ac (19) r_' agouomeltrcal Teb. e, In the usual tables of the natural Trigonometrical lines, the degrees from 0~ to 45~ are found at the top of the table, and those from 450 to 900 at the bottorm; the latter being complements of the former. Consequently, the columns which have Sine cand Tangent at top have Cosine and Cotanzgent at bottom, since the cosine or cotangent of any a:c is the same thilig as the sine or tangent of its complement. The minutes to eo added to the degrees are found in the left-hand column, when the number of degrees at the top of the page are used, and in the right-hand'column for the degrees when at the bottom of the page. The lines for arcs intermediate between those in the tables are found by Troportion. The lines are calculated for a radius equal unity. Hence, the values of the sines and cosines are decimal fractions, though the point is usually omitted. So too are the tangents from 0~ to 450, and the cotangents from 900 to 450. Beyond those points they are integers and decimals. The calculations, like all others involving large numbers, are shortened by the use of logarithms, which substitute addition and subtraction for multiplication and division; but the young student should avoid the frequent error of regarding loga rithms as a necessary part of trigonometry. APP. A.] TRIGONOMETRY. 385 SOLUTION OF TRIANGLES. Fig. 401 (11) Right-angled Triangles. Let B ABC be any right-angled triangle. Denote the sides opposite the angles by the corresponding small / letters. Then any one side and one acute angle, or t any two sides being given, the other parts can be obtained by one of the following equations: A -- C Given. Required. Formulas. a, b c, A, B c =e(a2 + 62); tan. A = cot. B=. 6 a, c b, A, B b =(c -a2); sin. A=-; cos. B =-. c C a, A b, c, B b =a.cot. A; c= i A B=90~- A. b, A a, c, B a= b.tan. A; c= A; B —90~-A. COS. A' c, A a, b, B a = c. sin. A; b= c cos. A; B = 90' - A. (12) Oblique-angled Trian- Fig. 402. gles. Let ABC be any oblique-angled B triangle, the angles and sides being noted as in the figure. Then any three of its six parts being given, and one of them being a side, the other parts can be obtained by one of the following methods, which are found- A C ed on these three theorems. THEOREM I.-In every plane triangle, the sines of the angles are to each other as the opposite sides. THEOREM II.-I-n every plane triangle, the sum of two sides is to their differene as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference. THEOREM III. —In every plane triangle, the cosine of any angle is equal to afraction whose numerator is the sum of the squares of the sides adjacent to the angle, minus the square of the side opposite to the angle, and whose denominator is twice the product of the sides adjacent to the angle. All the cases for solution which can occur, may be reduced to four. CASE 1. —Given a side and two angles. The third angle is obtained by subtracting the sum of the two given angles from 180~. Then either unknown side can be obtained by Theorem I. Calling the given side a, we have b a. si-.; and ca sin. C sin. A sin. A: 25 386 TRIGONOMETRY. [APP, A. CASE 2.-Given two sides and an angle opposite one of thenm. The angle opposite the other given side is found by Theorem I. Thie third angle is obtained by subtracting the sum of the other two from 180~. The remaining side is then obtained by Theorem I. Calling the given sides a and b, and the given angle A, we have sin. B = sin. A. -. Since an angle and its supplement have the same sine, the result is ambiguous; for the angle B may have either of the two supplementary values indicated by the sine, if b > a, and A is an acute angle. C = 180~ - (A + B). c = sin. C. sin. A' CASE 3.-Given two sides and their included angle. Applying Theorem II. (obtaining the sum of the angles opposite the given sides by subtracting the given included angle from 1800), we obtain the difference of the unknown angles. Adding this to their sum we obtain the greater angle, and subtracting it from their sum we get the less. Then Theorem I. will give the remaining side. Calling the given sides a and b, and the included angle C, we have A + B 180~-C. Then a-b tan. I (A - B) = tan. (A +- B). a b a+- b' sin. C (A +- B) +- (A —B) = A. (A+- B)- (A-B)=B. c=asin. A In the first equation cot. -} C may be used in the place of tan. I (A + B). CASE 4. —Given the three sides. Let s represent half the sum of the three sides = (a + b + c). Then any angle, as A, may be obtained from either of the following formulas, founded on Theorem III.: sin. A =/[(s — b) (s - c) tan. I A,J/ [(s b) (s c)] sin. A=-2A s (s- a) (s-b) (s —c) be COS. A = -- 2 bc The first formula should be used when A < 900, and the second when A > 900~. The third should not be used when A is nearly 180~; nor the fourth when A is nearly 90~; nor the fifth when A is very small. The third is the most convenient when all the angles are required. APPENDIX B. DEMONSTRATIONS OF PROBLEMS, ETC. MANY of the problems, &c., contained in the preceding pages, require Demonstrations. These will be given here, and will be designated by the same numbers as those of the Articles to which they refer. As many of these Demonstrations involve the beautiful Theory of Transversals, &c., which has not yet found its way into our Geometries, a condensed summary of its principal Theorems will first be given. TRANSVERSALS. THEOREM I.-If a straight line be drawn so as to cut any two sides of a triangle, and the third side prolonged, thus dividing them into six parts (the prolonged side and its prolongation being two of the parts), then will the product of any three of those parts, whose extremities are not contiguous, equal the product of the other three parts. That is, in Fig. 403, ABC being the triangle, and Fig. 403. DF the Transversal, BE X AD X CF=EA X DO X BF. A To prove this, from B draw BG, parallel to CA. From the similar triangles BEG and AED, we have BG BE:: AD: AE. From the similar triangles E BFG and CFD, we have CD: CF:: BG: BF. Multiplying these proportions together, we have BGXCD: BEXCF:: ADXBG: AEXBF. Multi- B plying extremes and means, and suppressing the common factor BG, we have. BEXADXCF= EAXDCXBF. These six parts are sometimes said to be in involution. If the Transversal passes entirely out- Fig. 404 side of the triangle, and cuts the prolonga- A tions of all three sides, as in Fig. 404, the theorem still holds good. The same demonstration applies without any change.* THEOREM II.-Conversely: If threepoints C be taken on two sides of a triangle, and on, the third side prolonged, or on the prolon, - gations of the three sides, dividing them into six parts, such that the product of f three non-consecutive parts equals the product of the other three parts; then will these three points lie in the same straight line, This Theorem is proved by a Reductio ad absurdum. * This Theorem may be extended to polygons. 388 TRANSVERSALS. [APP. B. THEOREM III.-If from the summits of a triangle, lines Fig. 405. be drawn, to a point situated either withisz or without the A triangle, and prolonged to meet the sides of the triangle, or their prolongations, thus dividing them into six parts; / then will the product of any three non-consecutive parts be E P equal to the product of the other three parts. B F F That is, in Fig. 405, or Fig. 406, AE X BF X CD=EB X FC X DA Fig. 406. For, the triangle ABF being cut by A the transversal EC, gives the relation (Theorem I.), AE X BC X FP = EB X FC X PA. The triangle ACF, being cut by the transversal DB, gives,P j _ DC X FB X PA = AD X CB X FP.' Multiplying these equations together, E and suppressing the common factors PA, CB, and FP, we have AE X BF X CD EB X FO X DA. THEOREM IV.-Conversely: If three points are situated on the three sides of a triangle, or on their prolongations (either one, or three, of these points being on the sides), so that they divide these lines in such a way that the product of any three non-consecutive parts equals the product of the other three parts, then will lines drawn from these points to the opposite angles meet in the same point. This Theorem can be demonstrated by a Reductio ad absurdum. COROLLARIES OF THE PRECEDING THEOREMS. Co. 1. —The MEDIANS of a triangle (i e., the lines drawn from its summits to the middles of the'6pposite sides) meet in the same point. For, supposing, ixnFig. 405, the points D, E, and F to be the middles of the sides, the products of th eon-con 6utive parts will be equal, i. e., AE X BF X CD= DAXEB XFC; sinpaeAE -RB, BF = FC, CD = DA. Then Theorem IV. applies. COR. 2.-The BISSECTRIOE'S of a triangle (i. e., the lines bisecting its angles) meet in the same point. For, in Fig. 40j/4upposing. the line4 AF, BD, CE to be Bissectrices, we have (Legendre IV. 17): BF: FC:: AB:AC) BFXACFXAB, CD: PA:: BC BA, whence CD X BA= DA X BC, AE:DEB:: CA: CB, AE X CB =EB X CA. Multiplying these equations together,:and omitting the common factors, we have BF X CD X AE = FC X DA X EB. Then Theorem IV. applies APP. B.] TRANSVERSALS. 389 CoR. 3.-The ALTITUDES of a triangle (i. e., the lines drawn from its summits perpendicular to the opposite sides) meet in the same point. For, in Fig. 405, supposing the lines AF, BD, and CE, to be Altitudes, we have three pairs of similar triangles, BCD -and FCA, CAE and DAB, ABF and EBC, by comparing which we obtain relations from which it is easy to deduce BF X CD xAE =EB XFC X DA; and then Theorem IV. again applies. CoR. 4.-If, in Fig. 405, or Fig. 406, the point F be taken in the middle of BC, then will the line ED be parallel to DC. For, since BF = FC, the equation of Theorem III. reduces to AE X CD=EB XDA; whence AE: EB:: AD: DC; consequently ED is parallel to BC. Con. 5.-Conversely: If ED be parallel to BC, then is BF = FC. For, since AE: EB:: AD:) DC, we have AE X DC = EB X AD; whence, in the equation of Theorem III., we must have BF = FC. Con. 6.-From the preceding Corollary, we derive the following: If two sides of a triangle are divided proportionally, Fig. 407. starting from the same summit, as A, and lines are drawn A from the extremities of the third side to the points of division, the intersections of the corresponding lines will all lie in the same straight line joining the summit A, and the middle of the base. Con.'. —A particular case of the preceding corollary is this: B F C In any trapezoid, the straight line which joins the intersection of the diagonals and the point of meeting of the non-parallel sides produced, passes through the middle of the two parallel bases. CoR. 8. —f the three lines drawn through the corresponding summits of two triangles cut each other in the same point, then the three points in which the corresponding sides, produced if necessary, will meet, are situated in the same straight line. This corollary may be otherwise enunciated, thus: If two triangles have their summits situated, two and two, on three lines which meet in the same point, then, &c. This is proved by obtaining by Theorem I. three equations, which, being multiplied together, and the six common factors cancelled, give an equation to which Theorem II. applies. Triangles thus situated are called homologic; the common point of meeting of the lines passing through their summits is called the centre of homology; and the line on which the sides meet, the axis of homology. 390 HARMONIC DIVISION. [APP. B. HARMONIC DIVISION. DEFINITIONS.-A straight line, AB, is said to Fig. 408. be harmonically divided at the points C and D, I I I 1 when these points determine two additive seg- A C B D ments, AC, BC, and two subtractive segments, AD, BD, proportional to one another; so that AC: BC:: AD: BD. It will be seen that AC must be more than BC, since AD is more than BD.* This relation may be otherwise expressed, thus: the product of the whole line by the middle part equals the product of the extreme parts. Reciprocally, the line DC is harmonically divided at the points B and A; since the preceding proportion may be written DB: CB:DA: CA. The four points, A, B, C, D, are called harmonics. The points C and D are called harmonic conjugates. So are the points A and B. When a straight line, as AB, is divided harmonically, its half is a mean propQrtional between the distance from the middle of the line to the two points, C and D, which divide it harmonically. If, from any point, 0, lines be drawn so as to Fig. 409. divide a line harmonically, these lines are called 0 an harmonic pencil. The four lines which compose it, OA, OC, OB, OD, in the figure, are called its radii, and the pairs which pass through the conjugate points are called conjugate radii. A C B D THEOREM V.-In any harmonic pencil, a line drawn parallel to any one of the radii, is divided by the three other radii into two equal parts. Let EF be the line, drawn parallel to Fig. 410. OA. Through B draw GiH, also parallel 0 to OA. We have, GB: OA:: BD: AD; and BH: OA:: BC: AC. But, by hypothesis, AC: B:: AD: BD.\ /E Hence, the first two proportions reduce to A GB = BH; and consequently, EK - KF. A A The Reciprocal is also true; i. e., If four lines radiating from a point are such that a line drawn parallel to one of them is divided into two equal parts by the other three, the four lines form an harmonic pencil. * Three numbers, m, n, p, arranged in decreasing order of size, form an harmonic proportion, when the difference of the first and the second is to the difference of the second and the third, as the first is to the third. Such are the numbers 6, 4, and 8;, or 6, 8, and 2; or 15, 12, and 10; &c. So, in Fig. 408, are the lines AD, AB, and AC, which thus give BD: CB:: AD: AC; or AC: CB:: AD: BD. The series of fractions, G1 Ad -1 11 1 &c., is called an harmoni progression, because any consecutive three of its terms form an harmonic proportion. APP. 13.] ITE COMPLETE QUADRILATERAL. 391 THEOREM VI.-If any transversal to an harmonic pencil be drawn, it will be divided harmonically. Let LM be the transversal. Through K, where LM intersects OB, draw EF parallel to OA. It is bisected at K by the preceding theorem; and the similar triangles, FMK and LMO, EKN and LNO, give the proportions LM: KM:: OL: FK, and LN: NK:: OL: EK; whence, since FK = EK, we have LN: NK:: LM: KM. COROLLARY.-The two sides of any angle, together with the bissectrices of the angle and of its supplement, form an harmonic pencil. THEOREM VILI-If, from the summits of any Fig. 411. triangle, ABC, through any point, P, there be C drawn the transversals AD, BE, OF, and the trans- Ed versal ED be drawn to nmeet AB prolonged, in Fr', the points F and F' will divide the base AB har- /.. —.\ mtonically. A A F B F' This may be otherwise expressed, thus.: The line, CP, which joins the intersection of the diagonals of any quadrilateral, ABDE, with the point of meeting, C, of two opposite sides prolonged, cuts the side AB in a point F, which is the harmonic conjugate of the point of meeting, F', of the other two sides, ED and AB, prolonged. For, by Theorem I., AF' X BD X CE = F'B X DC X EA; and by Theorem III., AF X BD X CE = FB X DC X EA; whence AF: FB:: AF': F'B. THE COMPLETE QUADRILATERAL. A Complete Quadrilateral is formed by Fig. 412. drawing any four straight lines, so that each of them shall cut each of the other three, so./ as to give six different points of intersection. L / It is so called because in the figure thus formed are found three quadrilaterals; viz.,. in Fig. 412, ABCD, a common convex quadri-' / lateral; EAFO, a uni-concave quadrilateral; F \ and EBAFD, a bi-concave quadrilateral, corn- posed of two opposite triangles. The complete quadrilateral, AEBCDF, has A three diagonals; viz., two interior, AC, BD; and one exterior, EF. B THEOREM VIII.-In every COMPLETE QUADRILATERAL the middle points of its three diagonals lie in the same straight line. AEBCDF is the quadrilateral, and LMN the middle points of its three diagonals. From A and D draw parallels to BC, and from B and C draw parallels to. 392 THE COMPLETE QUIDRILATERAL. [APP. B. -AD. The triangle EDO being cut by the transversal BF, we have (Theorem I.), DF X CB X EA = CF X EB X DA. From the equality of parallels between parallels, we have GB = E'B', EA -= CA', EB - DB', DA = EA'. Hence, the above equation becomes DF X E'B' X CA' = CF X DB' X E'A'; therefore, by Theorem II., the points, F', B', A', lie in the same straight line. Now, since the diagonals of the parallelogram IECA'A bisect each other at N, and those of the parallelogram EBB'D at N, we have EN: NA':: EM: MB'. Then MN is parallel to FA'; and we have EN: NA':: EL: LF, or EL = LF, so that L is the middle of EF, and the same straight line passes through L, M, and N. THEOREM IX.-In every complete qguadrilateral each of the three diagonals is divided harmonically by the two others. OEBADF is the complete quadrilateral. Fig. 413. The diagonal EF is divided harmonically at A G and H by DB and AC produced; since AH, DE, and FB are three transversals drawn from the summits of the triangle ( AEF through the same point C; and there- B fore, by Theorem VII., DBG and ACH di- \ vide EF harmonically. E So too, in the triangle ABD, GB, CA, CD, are the three transversals passing through C; and G and K therefore divide the diagonal BD harmonically. So too, in the triangle, ABC, DA, DB, DC are the transversals, and H and K the points which divide the diagonal AC harmonically. THEoREMr X.-If from a point, A, any num- Fig. 414. ber of lines be drawn, cutting the sides of an A angle POQ, the intersections of the diagonals of the quadrilaterals thus formed will all lie in the same straight line passing through the summit of the angle. By the preceding Theorem, the diagonal BC' of the complete quadrilateral, BAB'C'CO, o C C' C" Q is divided harmonically at D and E. Hence, OA, OP, OD, and OQ, form an harmonic pencil. So do OA, OP, OD', and OQ. Therefore, the lines OD, OD' coincide. So for the other intersections. If the point A moves on OA, the line OD is not displaced. If, on the contrary, OA is displaced, OD turns around the point O. Hence, the point A is said to be a pole with respect to the line OD, which is itself called the polar of the point A. Similarly, D is a pole of OA, which is the polar of D. OD is likewise the polar of any other point on the line OA; and this property is necessarily reciprocal for the two conjugate radii OA, OD, with respect to the lines OP, OQ, which are also conjugate radii. Hence; In every harmonic pencil, each of the radii is a polar with respect to each point of its conjugate; and each point of this latter line is a pole with respect to the former. D E M ON S T R AT I O N S.* PART II.; CHAPTER V. (140), (141) The equality of the triangles formed in these methods proves their correctness. (143), (144) These methods depend on the principle of the square of the hypothenuse. (145) CAD is an angle inscribed in a semicircle. (146) Let fall a perpendicular from B to AC, meeting it at a point E, not marked in Fig. 91. Then (Legendre, IV. 12), AB2 = AC'+ -BC- 2 AC. CE; whence CE = A2 +BC - 2 AC BC2 When AC =AB, this becomes CE = 2 A' The similar triangles, BCE and DCA, give EC CB:: AC: CD; whence CB X AC BCO2 2 AC2 0D - - = CB X AC__ -—.. CE 2 AC BC (147) Mark a point, G, in the middle of DF, and join GA. The triangle AGD will then be isosceles, since it is equal to the isosceles triangle ABC, having two sides and the included angle equal. Then AG - GD = AB = GF. The triangle AGF is then also isosceles. Now the angle FAG = ~ AGD; and GAD = ~ FGA. Therefore FAG + GAD = FAD = A (AGD + FGA) = ~ (1S0~) = 90~. (]149) See Part VII., Art. (403). (][50) The proof follows from the equal triangles formed. (151) The proof is found in the first half of the proof of Art. (146). (153) ACP is an angle inscribed in a semicircle. (154) Draw from C a perpendicular to the given line, meeting it at a point E. As in the proof of Art. (146), changing the letters suitably, we have AE -- AB. The similar triangles AEC and ADP give AP AP AC2 AP X AC AC AE AP: AD A X AE - X =' X AC AC 2AB 2AB (155) Similar triangles prove this. (156) The equal triangles which are formed give BP- = CF. Hence FP is parallel to BC, and consequently perpendicular to the given line DG. (157') The proof of this is found in the " Theory of Transversals," corollary 3. (158) The proof of this is the same as the last. (161]) The lines are parallel because of the equal angles formed. * Additional lines to the figures in the text will sometimes be employed. The student should draw them on the figures, as directed. 394 DEMONSTRATIONS [APP. B. (162) The equal triangles give equal angles, and therefore parallels. (163) AB is parallel to PF, since it cuts the sides of the triangle proportionally. (164) The proof is found in corollary 4 of " Transversals." (165) From the similar triangles, CAD and CEP, we have CE: CD:: CP: CA. From the similar triangles, CEF and CBD, we have CE: CD:: CF: CB. These two proportions give the following; CP: CA:: CF: CB. Therefore PF is parallel to AB. (166) Draw PE. The similar triangles PCE and ACD give PE: CE:: AD: CD. The similar triangles CEF and CDB give EF: CE:: DB: CD. These proportions produce PE: EF:: AD: DB. Hence PEF is similar to ADB, and PF is parallel to AB. (173) The equality of the symmetrical triangles which are formed, proves this method. (174) ABP is a transversal to the triangle CDE. Then, by Theorem I. of "Transversals," CA X EB X DP = AE X BD X CP; whence we have CP: DP:: CA X EB: AE X BD. By "division," CP - DP: DP:: CA X EB- AE X BD: AE X BD. DC X AE X BD Hence, since CP - DP = CD, we obtain DP =A EB-AE X BD CA X EB —AE XBD' The other formulas are simplified by the common factors obtained by making AE = AC, or BE -= BD. (l175) By Theorem VII. "Harmonic Division," in the quadrilateral ABED, the line OF cuts DE in a point, L, which is the harmonic conjugate of the point at which AB and DE, produced, would meet. So too, in the quadrilateral DEHK, this same line, CG, produced, cuts DE in a point, L, which is the harmonic conjugate of the point at which DE and KH, produced, would meet. Consequently, AB, DE, and KH must meet in the same point. Otherwise; this problem may be regarded as the converse of Theorem X. of "Transversals," BCA being the angle, and P the point from which the radiating lines are drawn. (176) EGCFDH is the " Complete Quadrilateral." Its three diagonals are FE, DC, and HG; and their middle points A, B, and P lie in the same straight line, by our Theorem VIII. (182) This instrument depends on the optical principle of the equality of the angles of incidence and reflection. (184) The first method given, Fig. 120, is another application of the Theory of Transversals. The second method in the article is proved by supposing the figure to be constructed, in which case we should have a triangle QZR, whose base, QR, and a parallel to it, BD, would be cut proportionally by the required line PSZ; BD x QP so that QR: BD::QP: BS= (189) By "Transversals," Theorem I., we obtain, regarding CD as the transversal of the triangle ABE, CB X AF XED = AC X FE XDB; and since ED = DB, this becomes CB X AF = AC X FE; whence the proportion CB: AC::FE: AF. By "division," we have CB — AC: AC:: FE — AF: AF. Observing that AC CB -AC AB, we obtain AB =. (FE - AF). AF APP. B.] For Part li,9 Chapter V. 395 (190) Take CH = CB; and from B let fall a perpen- Fig. 24, bis. dicular, BK, to AC. Then, in the triangle CBH, we have A (Legendre IV. 12), E HK CE2- BH2 - BC2 BH2 H= 2 CRH 2[BC1] since CH -- BC. In the triangle ABH, we have (Leg. IV. 13) AB' = AH2 + BH2 + 2 AH. HK. Substituting for HK, its value from [1], we get AH AB2= AH2 + BH2 (1 + )' c But AH - AC - CI = AC — B.. AB2 = A2+BR2 (1+ AC- BC =AH2+BR 2.. A[2] In the above expression for AB, BH is unknown. To find it, proceed thus. Take CF = CD. Then DF is parallel to BH; and we have CD CB::DF: BH; whence CB2 BR2 = DF2. -. [3] In this equation DF2 is unknown; but by proceeding as at the beginning of this CE investigation, we get an equation analogous to [2], giving ED2 -EF2 + DF2. CD' CD whence DF2 = (DE'- EF2). E. Substituting this value of DF2 in [3], we have BH2 = (DE'- EF) CB2 EF2)CD XCE Substituting this value of BI-2 in [2], we have ACx~ BC ACXBC AB2=AH2+(DE2-EF2) ACXBC = (AC —BC)2+[DE2 —(CE-CD)2 ] X CD XE'CD X CE CD-CE(191) Since BCD is a right angle, AC is a mean proportional between AB and AD. (192) The proof follows from the similar triangles constructed. (193) The similar triangles give DE: AC:: DB: AB; whence, by "division," DE —AC: AC:: DB -AB: AB; whence, since DB - AB -= AD, we have ACXAD DE-AC' (194) From the similar triangles, we have DE: CA:: EB: AB; whence DE —CA: CA: EB —AB:AB; whence, since EB —AB=AE, we get AC X AE DE - AC' (195) The triangles DEF and BAF, similar because of the parallelogram which EDXAF ACXAF is constructed, give FE: ED:: AF: AB = AF = FE The triangles DEF and BCD give similarly FE: ED:: DC: CB AC= XDFE FE 396 DEMONSTRATIONS [APP. B. (196) The equality of the triangles formed proves this problem. (]197) The proof of this problem also depends on the equality of the triangles constructed. The details of the proof require attention. (198) EB is the transversal of the triangle ACD. Consequently, CB X AFXDE =- AB XFD X CE; or, since CB = AB+AC, (AB+AC) XAF XDEAB XFD X CE; AC XAFXDE whence AB = whence FD XCE-AFXDE' Taking E, in the middle of CD, CE = DE, and those lines are cancelled. Taking F in the middle of AD, AF = FD, and those lines are cancelled. (199) The line BE is harmonically divided at the points H and A, from Theorem IX., ECFBGD being a " Complete Quadrilateral." Consequently, AE: EH:: AB:HB. Hence, by "division," AE - EH: AE:: AB - HB: AB. We therefore have, since AB -- HB AH, AB = AE EH' (200) For the same reasons as in the last article, CF is harmonically divided at H and D; and we have CI: HF:: CD: DF; whence CH - HF: CI:: CD - DF: CD. CH x CF Hence, since CD - DF = CF, CD -= CR HF' The other two expressions come from writing CF as CR + HF, and HF as CF - CH. (201) The equality of the triangles formed proves the equality of the corresponding sides KD and DE, &c. (202) The similar triangles (made so by the measurement of CE) give AC x DE CD DE: CA: AB AC X D CD (203) The similar triangles (made so by the parallel) give CE: EA:: CD: AB CDxEA _ CDX(AC-CE) CE CE DFX CD (204) The similar triangles DFH and BCD give IF: FD:: DC: BC = FR GR The similar triangles FGH and ABC give FG: GH:: BC: AB = BC FC. DF X CD X GH Substituting for BC, its above value, we have AB = FH X FG' When CD = CE, DF = CD, whence the second formula. (205) The equality of the symmetrical triangles which are formed, proves the equality of A'B' to AB. (206) The proof of this is similar to the preceding. (207) Because the two triangles ABC and ADE have a common angle at A, we have ADE: ABC:: AD X AE: AB X AC; whence the expression for ABC. (208) From B let fall a perpendicular to ACs meeting it at a point B'. Call this perpendicular BB' = p. From D let fall a perpendicular to AC, meeting it at a point D'. Call this perpendicular DD' - q. [APP. B.] For Part V. 397 The quadrilateral ABCD = AC X ~ (p + q). 2. BCE The triangle BCE = CE X I p; whence P = CE The similar triangles EDD' and BEB' give p: q:: BE: DE, whence DE 2. BCE X DE q' BE CE X BE BCE BCE XDE_ BE+DE BD Then J (p q) - ) BCE-X -BCE X CE + CEXBE CEXBE CE X BE BD ACXBD Lastly, ABCD = — AC X BCE X BE =BCE X CXBE CEXBE CEXBE DEMONSTRATIONS FOR PART V. (382) Let B = the measured inclined length, b =this length reduced to a horizontal plane, and A = the angle which the measured base makes with the horizon. Then 6 = B. cos. A; and the excess of B over b, i e., B - b = B (1 - cos. A). Since 1 - cos. A = 2 (sin. i- A)2 [Trigonometry, Art. (9)], we have B — b= 2 B (sin. A A)2. Substituting for sin.. A, its approximate equivalent, -J A X sin. 1' [Trigonometry, Art. (5)], we obtain B-b = 2 B (~ A X sin. 1')2 -- (sin. 1')2.A2. B, = 0.00000004231 A2 B. By logarithms, log. (B - b) = 2.626422 + 2 log. A + log. B. The greater precision of this calculation than that of b = B.. cos. A, arises from the slowness with which the cosines of very small angles increase or decrease in length. (386) The exterior angle LER = LCR + CLD. Also, LER = LDR + CRD..*.LCR+CLD = LDR+CRD, and LCR = LDR+CRD-CLD. CD From the triangle CRD we get sin. CRD = sin. CDR X CL CD From the triangle CLD we get sin. CLD = sin. LGD XC. As the angles CRD and CLD are very small, these values of the sines may be called the values of the arcs which measure the angles, and we shall have CD CD LCR - LDR + sin. CDR X -- sin. LCD X -. The last two terms are expressed in parts of radius, and to have them in seconds, they must be divided by sin. 1" [Trigonometry, Art; (5), Note], which gives the formula in the text. Otherwise, the correction being in parts of radius, may be brought into seconds by multiplying it by the length of the radius in seconds; i. e., 1800 X 60 X 60 = 206264".80625 [Trigonometry, Art. (2)]. 3.14159, &c.'(391) The triangles AOB, BOC, COD, &-c., give the following proportions [Trigonometry, Art. (12), Theorem I.]; AO: OB:: sin. (2): sin. (1); OB: OC:: sin. (4): sin. (3); OC: OD:: sin. (6): sin. (5); and so on around the polygon. Multiplying together the corresponding terms of all the proportions, the sides will all be cancelled, and there will result 1 1:: sin. (2) X sin. (4) X sin. (6) X sin. (8) X sin. (10) X sin. (12) X sin. (14): sin. (1) X sin. (3) 5< sin. (5) X sin. (7) X sin. (9) X sin. (11) X sin. (13). Hence the equality of the last two terms of the proportion. 398 DEMONSTRATIONS [APP. B. DEMONSTRATION FOR PART VI. (399) In the triangle ABS, we have AB. sin. BAS c. sin. U sin. ASB: sin. BAS:: AB: SB sinAS sin [1] sin. ASB sin. S In the triangle CBS, we have BC. sin. BCS a. sin. V sin. BSC: sin. BCS: BC: SB = sin. BSC - sin. S' [2] c. sin. U a. sin. V Hence,. S = sin.; whence, c. sin. S'. sin. U-a. sin. S. sin. V=0. [3] sin. S sin. S' In the quadrilateral ABCS, we have BCS = 3600 - ASB — BSC - ABC - BAS; or V = 360~ -S - S'- B -U. Let T = 3600 - S - S'- B, and we have V = T - U. [4] Substituting this value of V, in equation [3], we get [Trig., Art. (8)], c. sin S' sin. U- a. sin. S (sin. T. cos. U- cos. T. sin. U) = 0. Dividing by sin. U, we get sin. U' c. sin. St-a. sin. S (sin. T. sin U cos. T -- 0. Whence we have cos. U. sin. S'+ -- a. sin. S. cos. T - cot. U= ——. sin. U a. sin. S. sin. T Separating this expression into two parts, and cancelling, we get cot. U-= c. sin. S' cos. T a. sin. S. sin. T sin. T' Separating the second member into factors, we get cos. T C. sin. S' cot. U sin. T a. sin. S. cos.T + 1);or cot. U = cot. T (a.'sicS sin S' ) Having found U, equation [4] gives V; and either [1] or [2] gives SB; and SA and SC are then given by the familiar " Sine proportion" [Trig, Art. (12)]. APP. B.] For Part VII,. 399 DEMONSTRATIONS FOR PART VII. CP (403) If kAPC be a right angle, cos. CAB [Trigonometry, Art. (4)]. (405) AC = PC. tan. APC; and CB = PC. tan. BPC [Trigonometry, Art. (4)]. Hence AC: CB:: tan. APC: tan. BPC; and AC: AC + CB:: tan. APC: tan. APC + tan. BPC. Consequently, since AC + CB = AB, AC - AB ~ t tan. A PC tan. APC + tan. BPC' (4141) The equal and supplementary angles formed prove the operation. (421) In Fig. 285, CA:EG::AB:GB. By "division," CA-EG: EG:: AB - GB: GB. Hence, observing that AB —GB-=AG, we shall have A =GB (CA - EG) EG (423) Art. (12), Theorem III., [Trigonometry, Appendix A,] gives; Cos. C +; or c2 = a J- b62 - 2 ab. cos. C. This becomes [Trig., Art. (6)], K being the supplement of C, c2 = 2 + b2 + 2 ab. cos. K. The series [Trig., Art. (5)] for the length of a cosine, gives, taking only its first two terms, since K is very small, cos. K = -- K2. Hence, C=a'+6 + 2 + ab - ab K =(a + b)"-ab K2 = (a + b)2(1 (a + b); whence, c =(a + b) l/-( _- b)2 j) Developing the quantity under the radical sign by the binomial theorem, and neglecting the terms after the second, it becomes ab K(2 (a + 6)2 Substituting for K minutes, K. sin. 1' [Trig., Art. (5)], and performing the multiplication by a + b, we obtain ab K2. (sin. 11)2 c - a - b -- ab (a. 6) Now. (sin. 1')2= 0.000000423079; ab K2 whence the formula in the text, c = a + b - 0.000000042308 X a a+ b' (430) In the triangle ABC, designate the angles as A, B, C; and the sides opposite to them as a, b, c. Let CD = d. The triangle BC13D gives [Trig., Art. (12), sin. BDC sin. ADC Theorem I], a = d sin. BD. The triangle ACD similarly gives b = d. sin. CAD In the triangle ABC, we have [Trig., Art. (12), Theorem II.], tan. (A - B): cot. C:: a - b: a +- b; a -b whence tan. i (A -B) =' cot. ~ C. [1] b Let K be an auxiliary angle, such that b6 a. tan. K; whence tan. K = -. 400 DEMONSTRATIONS [APP. B, Dividing the second member of equation [1], above and below, by a, and substitu6 1-tan. K ting tan. K for -, we get tan. ~ (A - B) cot. J C. a 1-~ tan. Kc Since tan. 450 =1, we may substitute it for 1 in the preceding tqilation, and ban. 450 tan. K we get tan. (A- B) = tan. 45 tan. cot. A C. tan. 450 + tan. K From the expression for the tangent of the difference oi two arcs [Trig., Art. (8)], the preceding fraction reduces to tan. (450 - K); and the equation becomes tan. A (A - B) = tan. (450 - K). cot. A C. [2] In the equation tan. K = -_, substitute the values of b and a from the formulas at the beginning of this investigation. This gives sin. ADC sin. BDC sin. ADC. sin. CBD tan. K -= d.. -' d. = sin. CAD * sin. CBD sin. CAD. sin. BDC (A - B) is then obtained by equation [2]; (A + B) is the supplement of C; therefore the angle A is known. a. sin. C d. sin. BDC. sin. ACB sin. A sin. CBD. sin. CAB The use of the auxiliary angle K, avoids the calculation of the sides a and b. (434) In the figure on page 292, produce AD to some point F. The exterior angles, EB C= A + P; ECD = — A + Q; EDF = A + P. The triangle ABE BE sin. A CE sin. gives- = - The triangle ACE gives - = Dividing member a sin. P' a - x sin. Q' BE a. sin. Q by member, we get C = (a + x) sin. P In the same way the triangles BED and CED give =sin. (AR-+ ) vb 6+ sin. (R P) CE sin. (A + R) BWence as bef BE (b + x) sin. (R- Q) and = i ( Whence as befoe, = b ( b sin. (R - Q) CE. sin. (R - P) Equating these two values of the same ratio, we get a. sin. Q _ (b + x) sin. (R - Q) and (a + x) sin. P b. sin. (R - P) ab. sin. Q. sin. (R- P) b. sin. P. sin. (R -- ) = (a + x) (b + x) = ab + (a + b) x + x: sin. P. sin. (R - Q) To solve this equation of the 2d degree, with reference to x, make tan.2 K = 4 ab sin. Q (sin. R - P) (a — b) sin. P (sin. R — Q) Then the first member of the preceding equation = ~ (a - b)2 X tan.2 K; and we get x2 + (a + b) x = ~ (a- b)2. tan.2 I - ab, and x=- (a + b) V [4 (a- b)"tan.2 K-ab + (a+ b)2] - (a + b) q / [4 (a - b)2. tan.2 I + a (a - b)] =- (a + b) + (a - b) / (tan.2 K + 1). 1 a+-b a —b Or, since e (tan.2 K + 1) = secant K -Kwe have x - 2 cos. K2 -2. cos. K' APP. B.] For Part XI. 401 DEMONSTRATIONS FOR PART XI. (493) The content being given, and the length to be n times the breadth; Breadth X it times breadth = content; whence, Breadth (= Given the content = c, and the difference of the length and breadth = d; to find the length 1, and the breadth b. We have I X b= c; and - b = d. From these two equations we get I = i d -+ i V (d2 + 4 c). Given the content = c, and the sum of the length and breadth = s; to find I and b. We have I X b = c; and 1 + b = s; whence we get l = s - + /(s2- 4 c). (494) The first rule is a consequence of the area of a triangle being the product of its height by half its base. To get the second rule, call the height h; then the base = mh; and the area = h X mh; whence h = (X rea). For the equilateral triangle, calling its side e, the formula for the area of a triangle [ ( s - a) ( s - b) (i s -c)] reduces to 4 e2 V/3. Hence e=2 (/3area) 1.5197 / area. (495) By Art. (65), Note, J. AB X BC X sin. B = content of ABC; whence, B=2 X ABC AB. sin. B' (496) The area of a circle = radius 2 X y; whence radius= 7 X (497) The blocks, including half of the streets and avenues around them, are 900 X 260 = 234000 square feet. This area gives 64 lots; then an acre, or 43560 feet, would give not quite 12 lots. (502) The parallelogram ABDC being double'the triangle ABC, the proof for Art. (495), slightly modified, applies here. (504) Produce BC and AD to meet in E. Fig. 846, bis. By similar triangles, ABE: DCE:: AB2: DC2. ABE- DCE: ABE:: AB2- DC2: AB2.; — Now ABE - DCE = — ABCD; also, by -__ Art. (65), Note, ABE = AB. sin. A. sin. B 2.sin. (A+-B)' A The above proportion therefore becomes sin. A. sin. B ABCD: AB2.2. sin. (A B) AB- CD2: AB2 Multiplying extremes and means, cancelling, transposing, and extracting the square root, we get CD = 4V [AB2 s2. ABD. sin. B(A s -~,in AT.-sin B) 402 DEMONSTRATIONS [APP. B. When A + B > 180~, sin. (A + B) is negative, and therefore the fraction in which it occurs becomes positive. CF being drawn parallel to DA, we have sin. B sin. B sin. B AD=F0 — FB. =FB. - =(AB - CD) sin BCF FBsin. (180 - A - B) sin. (A-+ B) sin. A BQ= (AB - CD) ( ) sin. (A + B) (505) Since similar triangles are as the squares of their homologous sides, BDE: BFG:: BD2: BF2; whence BF = BD,/B-)E (506) BFG =. BF X FG -=. BF X BF. tan. B; whence, B F ta2. BG' (510) By Art. (65), Note, BFG = BF2. sin. B. sin. F 2. sin. (B + F) whence, BF= A/(2 sin. (B + F). BFG) sin. B. sin.E(511) The final formula results from the proportion FAE: CDE:: AE2: ED2. (512) Since triangles which have an angle in each equal, are as the products of the sides about the equal angles, we have ABE: CDE:: AE X BE: CE X DE. sin. A. sin. B sin. B ABE =. AB2 sin. (A + B) AE = AB sin.. sin, (A + )'' sin. E' BE -= AB sin. A E - DE. sin. DE sin. E' sin. DC' Substituting these values in the preceding proportion, cancelling the common factors, observing that sin. (A + B) = sin. E, multiplying extremes and means, and dividing, we get DE /(2. CDE. sin. DOE }' sin. E. sin. CDE / (515) The first formula is a consequence of the expression for the area of a triangle, given in the first paragraph of the Note to Art. (65). (517) The reasons for the operations in this article (which are of very frequent occurrence), are self-evident. (51S) The expression for DZ follows from Art. (65), Note. The proportion in the next paragraph exists because triangles having the same altitude are as their bases. (519) By construction, GPO = the required content. Now, GPC = GDC, since they have the same base and equal altitudes. We have now to prove that LMC = GDC. These two triangles have a common angle at C. Hence, they are to each other as the rectangles of the adjacent sides; i. e., GDC: LMC:: GC X CD:: LC X CM. Here CM is unknown, and must be eliminated. We obtain an expression for it by means of the similar triangles LCM and LEP, which give LE: LC:: EP -= CD: CM. APP. B.] For Part XI. 403 Hence, CM= CD X L. Substituting this value of CM in the first proportion, LE' and cancelling CD in the last two terms, we get LC2 GDC:LMC::GC:; orGDO: LMC:: GOXLE:LC2. LC2 = (LH + HC)2 = LH + 2 LH X HO + HC2. But, by construction, LH =- HK2 = HE2- EK2 = HE" — EC2 = (HE+EC) (HE-EC) = HC (HE-EC). Also, GC = 2 HC; and LE = LH + HE. Substituting these values in the last proportion, it becomes GDC: LMC:: 2. HC (LH -+ HE): HO (HE- EC) + 2 LH X HC + HIa.:: 2 LH+ 2 HE: HE- E + 2 LH + HC.: HE - EC +- 2 LII + HE + EC.: 2 HE - 2 LH. The last two terms of this proportion are thus proved to be equal. Therefore, the first two terms are also equal; i. e., LMC = GDC = the required content. Since HK =- V (HE2 EK2), it will have a negative as well as a positive value. It may therefore be set off in the contrary direction from L, i. e., to L'. The line drawn from L' through P, and meeting CB produced beyond B, will part off another triangle of the required content. (520) Suppose the line LM drawn. Then, by Art. (65), Note, the required content, c = ~. CL X CM. sin. LCM. This content will also equal the sum of the two triangles LCP and MOP; i. e., c= A ~ CL X p + i ~ CM X q. The first of 2~ these equations gives CM = CL. i LCM' Substituting this in the second equation, we have cq.CL X ~p +CL. sin. LCM' Whence, A p. CL2. sin. LCM + cq - c. CL. sin. LCM. Transposing and dividing by the coefficient of CL2, we get 2 c 2 cq CL 2 —.OL - p p. sin. CLM' CL = —:e: 2 (p p. sin. LOM If the given point is outside of the lines CL and CM, conceive the desired line to be drawn from it, and another line to join the given point to the corner of the field. Then, as above, get expressions for the two triangles thus formed, and put their sum equal to the expression for the triangle which comprehends them both, and thence deduce the desired distance, nearly as above. (522) The difference d, between the areas parted off by the guess line AB, and the required line CD, is equal to the difference between the triangles APO and BPD sin. A. sin. P By Art. (65), Note, the triangle APC AP2. A in sin. (A - P) sin. B. sin. P Similarly, the triangle BPD BP sin. B B i P: sin. (B * - P)' sin. A. sin. P sin. B. sin. P ~' d( - APa - B P. sin. (A -- P) sin. (B *t- P))' 404 DEMONSTRATIONS [APP. B. By the expression for sin. (a + b) [Trigonometry, Art. (8)], we have sin. A. sin. P sin. B. sin. P sin. A. cos. P + sin. P. cos. B sin. B. cos: P-+ sin. P. cos. B COS., a Dividing each fraction by its numerator, and remembering that.- -= cot. a, we have d AP' i BP3 d — cot. P + cot. A cot. P + cot. B' For convenience, let p = cot. P; a= cot. A; and b = cot. B. The above equation will then read, multiplying both sides by 2, AP BP2 p+ a -p+ b Clearing of fractions, we have 2 dj2 + 2 dap +-2 dbp.+ 2 dab =p. AP2 + b. AP' -p. BP2- a. BP2. Transposing, dividing through by 2 d, and separating into factors, we get i~L.ol AP2 — BP b. AP2 a BP2 P2 + (a+ b P- b 2d 2d Ap2-BP_ AP2a.BP2 -a+(+ AP2-BP) 2] If A - 900, cot. A =- a 0; and the expression reduces to the simpler form given in the article. (523) Conceive a perpendicular, BF, to be let fall from B to the required line DE.; Let B represent the angle DBE, and 1f the unknown angle DBF. The angle BDF=-=900 —; and the angle BEF = 90 - (B - P) = 90 -B --. By Art. sin. BDE. sin. BED (65), Note, the area of the triangle DBE= - DE2BED) sin. (BDE + BED) DE. sin. (900 -13) sin. (900~- B +- 1) sin. B 2 X DBE X sin. B 2 X DBE X sin. B Hence, DE = sin. (90~ - 3). sin. (900 - B +-13) - cos. 3. cos. (B -)' Now in order that DE may be the least possible, the denominator of the last fraction must be the greatest possible. It may be transformed, by the formula, cos. a. cos. b = 3 cos. (a + b) + a. cos. (a - b) [Trigonometry, Art. (8) ], into j cos. B + J. cos. (B - 2 13). Since B is constant, the value of this expression depends on its second term, and that will be the greatest possible when B - 2 3= 0, in which case 3 = B. It hence appears that the required line DE is perpendicular to the line, BF, which bisects the given angle B. This gives the direclion in which DE is to be run. Its starting point, D or E, is found thus. The area of the triangle DBE = i BD. BE. sin. B. Since the triangle is isosceles, this becomes /12 DBE\ DBE =- BD'. sin. B; whence BD- sin. Bt)' DE is obtained from the expression for IDE2, which becomes, making f- j B, 2 X DBE X sin. B (2.DBE.sin.B) DE-.. -. B whence D E (2.DBE.sin.B) ao8; ~3 -B~ ccs. ~ B.'B APP. B.] For Part XI. 405 (524) Let a= value per acre of one portion of the land, and b that of the other portion. Let x =the width required, BC or AD. Then the value of xXBE xX AE BCFE =a X, and the value of ADFE -= b X 10 10 Putting the sum of these equal to the value required to be parted off, we obtain value required X 10 a X BE + b X AE (525) All the constructions of this article depend on the equivalency of triangles which have equal bases, and lie between parallels. The length of AD is derived from the area of a triangle being equal to its base by half its altitude. (527) Since similar triangles are to each other as the squares of their homologous sides, ABC': DBE:: AB': BD2; whence BD-= AB DB - AB m The construction of Fig. 363 is founded on the proportion BF: BG:: BG: BA; when BD = BG = (BA X BF) =BA- (528) By hypothesis, AEF: EFBC:: m: n; whence AEF: ABC: m: m + n; and AEF = ABC m -.ACDB m Also, AEF = A ~ AE X EF m n - 2 m n- +r DB x AE The similar triangles AEF and ABD give AD: DB:: AE: EF = ADB- The DB x AE second expression for AEF then becomes AEF=- AE. D Equating AD this with the other value of AEF, we have ACXDB m AE2 X DB M 2s I. m 6n Ae. XrA De wBhence AEeX,+AC X AD X (530) In Fig. 366, the triangles ABD, DBC, having the same altitude, are to each other as their bases. In the next paragraph, we have ABD: DBC:: AD: DC:: m: n; whence AD:AC:: m: m-n; and AC: DC:: m-+ n:n; whence the expressions for AD and DC. In Fig. 367, the expression for AD is given by the proportion AD: AC:: m: n + n. Similarly for DE, and EC. (531) In Fig. 368, conceive the line EB to be drawn. The triangle AEB = i ABC, having the same altitude and half the base; and AFD = AEB, because of the equivalency of the triangles EFD and EFB, which, with AEF, make up AFD and AEB. The point F is fixed by the similar triangles ADB and AEF, The expression for AF, in the last paragraph, is given by the proportion, ABC: ADF:: AB X AC: AD X AF; AB X AC ADF ABXAC X m whence, AF AD ABC - AD m + (532) The areas of triangles being equal to the product of their altitudes by half their bases, the constructions in Fig. 369 and Fig. 370 follow therefrom. 406 DEMONSTRATIONS LAPP. B. (533) In Fig. 371, conceive the line BL to be drawn. The triangle ABL will be a third of ABC, having the same altitude and one-third the base; and AED is equivalent to ABL, because ELB = ELD, and AEL is common to both. A similar proof applies to DCG. (534) In Fig. 372, the four smaller triangles are mutually equivalent, because of their equal bases and altitudes, two pairs of them lying between parallels. (535) In Fig. 373, conceive AE to be drawn. The triahgle AEC = -. ABC, having the same altitude and half the base; and EDFC =-AEC, because of the common part FEC and the equivalency of FED and FEA. (536) In Fig. 374, in addition to the lines used in the problem, draw BF and DG. The triangle BFC = - ABC, having the same altitude and half the base. Also, the triangle DFG = DFB, because of the parallels DF and BG. Adding DFC to each of these triangles, we have DCG = BFC = i ABC. We have then to prove LMC = DCG. This is done precisely as in the demonstration of Art. (519), page 402. (537) Let AE = x, ED = y, AH = x', HF= y, AK = a, KB = b. The quadrilateral AFDE, equivalent to ~ ABC, but which we will represent, generally, by m2, is made up of the triangle AFH and the trapezoid FHED. AFH = i. x'y'. FHED = i (x-') (y + y')..: AFDE=m2=. x'y' + (x -x') (y - y') = x (y + y') - x'y. The similar triangles, AHF and AKB, give bx' a: b:: x': y'=-. Substituting this value of y' in the expression for m2, we have hence = a(2 n xy/) AK (2 ABC - AE X ED) bx -ay KB X AE - AK X ED' The formula is general, whatever may be the ratio of the area m2 to that of the triangle ABC. (53S) In Fig. 376, FD is a line of division, because BF =-the triangle BDF divided by half its altitude, which gives its base. So for the other triangles. (539) In Fig. 377, DG is a second line of division, because, drawing BL, the triangle BLC = I ABC; and BDGC is equivalent to BLC, because of the common part BCLD, and the equivalency of the triangles DLG and DLB. To prove that DF is a third line of division, join MD and MA. Then BMA- = BGA. From BMA take ]MFA and add its equivalent MFD, and we have MDFB = j BGA = ~ (ABDG - BDG) = ~ (2 ABC - BDG) = - ABC - - BDG. To MDFB add MDB, and add its equivalent, ~ BDG, to the other side of the equation, and we have MDFB + MDB = ABC - BDG - + BDG; or,; BDF - ~ ABC. (540) In Fig. 378, the triangle AFC = ~ ABC, having the same base and onethird the altitude. The triangles AFB and BFC are equivalent to each other, each being composed of two triangles of equal bases and altitudes; and each is therefore one-third of ABC. APP. B.] For Part XI. 407 In Fig. 319, AFC: ABC:: AD: AB; since these two triangles have the common base AC, and their altitudes are in the above ratio. So too, BFC: ABC::BE: BA. Hence, the remaining triangle AFB: ABC:: DE: AB. (541) By Art. (65), Note, ABC = i AC X CB X sin. ACB. But the angle ACB =-ACD+DCB =- (180~-ADC)+i (180~-CDB) = 180~ — (ADC+CDB). Hence, ABC -= AC X CB X sin. J (ADO + CDB) = J AC X CB X sin. ~ ADB. Let r = DA = DB = DC. Since AB is the chord of ADB to the radius r, and therefore equal to twice the sine of half that angle, we have AB AB AB X BO X CA sin.. ADB =; whence, ABC = AC X CB X 2; and r = 4. ABC Also, since the area of each of the three small triangles equals half the product of one of the equal sides (= r), by the sine of the included angle at D, these triangles will be to each other as the sines of those angles. These angles are found thus: AB BC AC sin. I ADB= 2; sin. J BDC =; sin. ADC =-. 2r 2r 2r (542) The formulas in this article are obtained by substituting, in those of Art. (523), for the triangle DBE, its equivalent + X AB X BC X sin. B. BD thus becomes = ( m ABXBCXsin. B ( m ABBC d / (,-~- XABXBCX sin.2B) sin. B and DE = -XABXBcos. B cos.-B m n (543) The rule and example prove themselves. (544) In Fig. 383, conceive the sides AB and DC, produced, to meet in some point P. Then, by reason of the similar triangles, ADP: BCP:: AD2: BC2; whence, by "division," ADP - BCP = ABCD: BCP:: AD2- BC2: BC2. In like manner, comparing EFP and BCP, we get EBCF: BCP:: EF2-BC2: BCG. Combining these two proportions, we have ABCD: EBCF:: AD2 - BC2: EF2 - BC2; or, m n: mn:: AD2- BC2: EF2_- BC2. Whence, (m + n) EF2 - m. BC2 -n BC2 = m. AD2 - m. BC2;..EF_= V(m X AD + n X BC) Also, from the similar triangles formed by drawing BL parallel to CD, we have AL: EK:: BA BE- -BA X EK AB (EF - BC) AL AD-BC (545) Let BEFC=- ABCD = a; let BC = b; BH =- h; and rn + AD - BO = c. Also let BG = x; and EF = y. Draw BL parallel to CD. By sim. ilar triangles, AL:EK:: BA: BE:: BH: BG; or, AD-BC:EF-BC:: BH:BG; i.e., c: y-b:: h: x; whence x = y-b) Also, the area BE a.BG (EF BC) (y b); whence y = —. x 408 DEMONSTRATIONS. [APP. B. Substituting this value of y in the expression for x, and reducing, we obtain 2bh 2ah b6 k i2ah b2h/2 X2+2x — -; whence we have x —b = +iC + - 2 h The second proportion above gives yb whence y - b - x. A A Replacing the symbols by their lines, we get the formulas in the text. (546) ABEF - ABCD. But ABRP = ABEF, because of the common part ABRF, and the triangles FRP and FRE, which make up the two figures, and which are equivalent because of the parallels FR and PE. So for the other parts. (547) The truth of the foot-note is evident, since the first line bisects the trapezoid, and any other line drawn through its middle, and meeting the parallel sides, adds one triangle to each half, and takes away an equal triangle; and thus does not disturb the equivalency. (548) In Fig. 885, since EF is parallel to AD, we have ADG: EGF:: GH2: GK2. EGF is made up of the triangle BCG =a', and the quadrilateral BEFC m m + ABOD - + * (a —a'). Hence the above proportion becomes a: a'+ m (a a'):: GH12: GK2; or, (m+- n) a n: ma + na':: G: G2; whence GK = GH V /(-m- +-)). GK GE is given by the proportion GH: GK:: GA: GE = GA ~ G. In Fig. 386, the division into p parts is founded on the same principle. The triangle EFG-GBC + EFCB = a' + Q-. Now ADG: EFG:: Ao: EG2; p ~or, ~': AG2: EG~; whence GE-AG / ( ( P GL is obtained by taking the triangle LMG = a' -+; and so for the rest. (552) In Fig. 390, join FO and GO. Because of the parallels CA and BF, the triangle FCD will be equivalent to the quadrilateral ABCD, of which GCD will therefore be one half; and because of the parallels GE and CH, EHDC will be equivalent to GCD. (553) In Fig. 391, by drawing certain lines, the quadrilateral can be divided into three equivalent parts, each composed of an equivalent trapezoid and an equivalent triangle. These three equivalent parts can then be transformed, by means of the parallels, into the three equivalent quadrilaterals shown in the figure. The full development of the proof is left as an exercise for the student. In Fig. 392, draw CG. Then CBG ABCD. But CKQ = CGQ. Therefore CKQB = ~ ABCD. So for the other division line. (556) The division of the base of the equivalent triangle, divides the polygon similarly. The point Q results from the equivalency of the triangles ZBP and ZBQ, PQ being parallel to BZ. APPENDIX C. INTRODUCTION TO LEVELLING. (1) The Principles. LEVELLING is the art of finding how much one point is higher or lower than another; i. e., how much one of the points is above or below a level line or surface which passes through the other point. A level or horizontal line is one which is perpendicular to the direction of gravity, as indicated by a plumb-line or similar means. It is therefore parallel to the surface of standing water. A level or horizontal surface is defined in the same way. It will be determined by two level lines which intersect each other.* Levelling may be named VERTICAL SURVEYING, or Up-and-down Surveying; the subject of the preceding pages being Horizontal Surveying, or Right-and-left and Fore-and-aft Surveying. All the methods of Horizontal Surveying may be used in Vertical Surveying. The one which will be briefly sketched here corresponds precisely to the method of " Surveying by offsets," founded on the Second Method, Art. (6), " Rectangular Co-ordinates," and fully explained in Arts. (114), &c. The operations of levelling by this method consist, firstly, in obtaining a level line or plane; and, secondly, in measuring how far below it or above it (usually the former) are the two points whose relative heights are required. (2) The Instruments. A level Fig. 415. line may be obtained by the following simple instrument, called a " Plumb-line level." Fasten together two pieces of wood at right angles to each other, so as to make a T, and draw a line on the upright one so as to be exactly perpendicular to the top edge of the other. Suspend a plumb-line as in the figure. Fix the T against a staff stuck in the ground, by a screw through the middle of the crosspiece. Turn the T till the plumb-line exactly covers the line which was drawn. Then will the upper edge of the cross-piece be a level line, and the eye can sight across it, and note how far above or below any other point this level line, prolonged, would strike. It will be easier to look across sights fixed on each end of the cross-piece, making them of horsehair stretched across a piece of wire, bent into three sides of a square, and stuck into each end of the cross-piece; taking care that the hairs are at exactly equal heights above the upper edge of the cross-piece. * Certain small corrections, to be hereafter explained, will be ignored for the present, and we will consider level lines as straight lines, and level surfaces as planes. 410 LEVELLING. [APP. C. A modification of this is to fasten a common Fig. 416. carpenter's square in a slit in the top of a staff, by means of a screw, and then tie a plumb-line at the angle so that it may hang beside one arm. When it has been brought to do so, by turning the square, then the other arm will be level. Another simple instrument depends upon the principle that "water always finds its level," corresponding to the second part of our definition of a level line. If a tube be bent up at each end, and nearly filled with water, the surface of the water in one end will always be at the same height as that in the other, however the position of the tube may vary. On this truth depends the " Water-level." It may be easily constructed with a tube of tin, lead, copper, &c., by bending up, at right angles, an inch or two of each end, and supporting the tube, if too Fig. 417. flexible, on a wooden bar. In these A. ends cement (with putty, twine dipped in white-lead, &c.), thin phials, with their bottoms broken off, so as to leave a free communication between them. Fill the tube and the phials, nearly to their top, with colored water. Blue vitriol, or cochineal, may be used for coloring it. Cork their mouths, and fit the instrument, by a steady but flexible joint, to a tripod. Figures of joints are given on page 134, and of tripods on page 133. To use it, set it in the desired spot, place the tube by eye nearly level, remove the corks, and the surfaces of the water in the two phials will come to the same leveL Stand about a yard behind the nearest phial, and let one eye, the other being closed, glance along the right-hand side of one phial and the left-hand side of the other. Raise or lower the head till the two surfaces seem to coincide, and this line of sight, prolonged, will give the level line desired. Sights of equal height. floating on the water, and rising above the tops of the phials, would give a better-defined line. The "Spirit-level" consists essentially Fig. 418. of a curved glass tube nearly filled with _ alcohol, but with a bubble of air left' within, which always seeks the highest spot in the tube, and will therefore by its movements indicate any change in the position of the tube. Whenever the bubble, by raising or lowering one end, has been brought to stand between two marks on the tube, or, in case of expansion or contraction, to extend an equal distance on either side of them, the bottom of the block (if the tube be in one), or sights at each end of the tube, previously properly adjusted, will be on the same level line. It may be placed on a board fixed to the top of a staff or tripod. When, instead of the sights, a telescope is made parallel to the level, and various contrivances to increase its delicacy and accuracy are added, the instrument becomes the Engineer's spirit-level. APP. c.] The Practice, 411 (3) The Practice. By whichever of these various means a level line has been obtained, the subsequent operations in making use of it are identical. Since the " water-level" is easily made and tolerably accurate, we will suppose it to be employed. Let A and B, Fig. 419, represent the two points, the Fig. 419. difference of the heights of which is i required. Set the instrument on any spot from which both the points can be seen, and at such a height that the level line will pass above the highest one. At A let an assist- G ant hold a rod graduated into feet, tenths, &c. Turn the instrument towards the staff, sight along the level line, and note what division on the staff it strikes. Then send the staff to B, direct the instrument to it, and note the height observed at that point. If the level line, prolonged by the eye, passes 2 feet above A and 6 feet above B, the difference of their heights is 4 feet. The absolute height of the level line itself is a matter of indifference. The rod may carry a target or plate of iron, clasped to it so as to slide up and down, and be fixed, at will. This target may be variously painted, most simply with its upper half red and its lower half white. The horizontal line dividing the colors is the line sighted to, the target being moved up or down till the line of sight strikes it. A hole in the middle of the target shows what division on the rod coincides with the horizontal line, when it has been brought to the right height. If the height of another point, C,jFig. 420, not visible from the first station, be required, set the instrument so as to see B and C, and proceed exactly as with A Fig. 420. 2 I- — 1 c2 — 1 C and B. If 0 be 1 foot below B, as in the figure, it will be 5 feet below A. If it were found to be 7 feet above B, it would be 3 feet above A. The comparative height of a series of any- number of points, can thus be found in reference to any one of them. The beginner in the practice of levelling may advantageously make in his notebook a sketch of the heights noted, and of the distances, putting down each as it is observed, and imitating, as nearly as his accuracy of eye will permit, their pro 412 LEVELLING. [APP. C. portional dimensions.* But when the observations are numerous, they should be kept in a tabular form, such as that which is given below. The names of the points, or "Stations," whose heights are demanded, are placed in the first column; and their heights, as finally ascertained, in reference to the first point, in the last column. The heights above the starting point are marked +, and those below it are marked -. The back-sight to any statlon is placed on the line below the point to which it refers. When a back-sight exceeds a fore-sight, their difference is placed in the column of "Rise;" when it is less, their difference is a "Fall." The following table represents the same observations as the last figure, and their careful comparison will explain any obscurities in either. Stations. Distances. Back-sights. Fore-sights. Rise. Fall. Total Heights. A 0.00 B 100 2.00 6.00 - 4.00 - 4.00 C 60 3.00 4.00 -1.00 - 5.00 D 40 2.00 1.00 + 1.00 - 4.00 E 70 6.00 1.00 + 6.00 + 1.00 F 60 2.00 6.00 - 4.00 - 3.00 15.00 18.00 - 3.00 The above table shows that B is 4 feet below A; that C is 5 feet below A; that E is 1 foot above A; and so on. To test the calculations, add up the back-sights and fore-sights. The difference of the sums should equal the last " total height." Anotherform of the levelling field-book is presented below. It refers to the same stations and levels, noted in the previous form, and shown in Fig. 420. Stations. Distances. Back-sights. Ht. Inst. above Datum. Fore-sights. Total Heights A 0.00 B 100 2.00 + 2.00 6.00 - 4.00 C 60 3.00 - 1.00 4.00 - 5.00 D 40 2.00 - 3.00 1.00 - 4.00 E'70 6.00 + 2.00 1.00 + 1.00 F 50 2.00 + 3.00 6.00 - 3.00 15.00 18.00 - 3.00 In the above form it will be seen that a new column is introduced, containing the Height of the Instrument (i. e., of its line of sight), not above the ground where it stands, but above the lDatum, or starting-point, of the levels. The former columns of "Rise" and "Fall" are omitted. The above notes are taken thus: The height of the starting-point or " Datum," at A, is 0.00. The instrument being set up and levelled, the rod is held at A. The back-sight upon it is 2.00; therefore the height of the instrument is also 2.00. The rod is next held at B. The fore-sight to it is 6.00. That point is therefore 6.00 below the instrument, or 2.00 - 6.00 =- 4.00 below the datum. The instrument is now moved, and again set up, and the back-sight to B, being 3.00, the Ht. Inst. is -4.00 + 3.00 -- 1.00; * In the figure, the limits of the page have made it necessary to contract the horizontal distances to one-tenth of their proper proportional size. APP. C.] The Practice. 413 and so on: the Ht. Inst. being always obtained by adding the back-sight to the height of the peg on which the rod is held, and the height of the next peg being obtained by subtracting the fore-sight to the rod held on that peg, from the Ht. Inst. The level lines given by these instruments are all lines of apparent level, and not of true level, which should curve with the surface of the earth. These level lines strike too high; but the difference is very small in sights of ordinary length, being only one-eighth of an inch for a sight of one-eighth of a mile, and diminishing as the square of the distance; and it may be completely compensated by setting the instrument midway between the points whose difference of level is desired; a precaution which should always be taken, when possible. It may be required to show on paper the ups and downs of the line which has been levelled; and to represent, to any desired scale, the heights and distances of the various points of a line, its ascents and descents, as seen in a side-view. This is called a " Prof le." It is made thus. Any point on the paper being assumed for the first station, a horizontal line is drawn through it; the distance to the next station is measured along it, to the required scale; at the termination of this distance a vertical line is drawn; and the given height of the second station above or below the first is set off on this vertical line. The point thus fixed determines the second station, and a line joining it to the first station represents the slope of the ground between the two. The process is repeated for the next station, &c. But the rises and falls of a line are always very small in proportion to the distances passed over; even mountains being merely as the roughnesses of the rind of an orange. If the distances and the heights were represented on a profile to the same scale, the latter would be hardly visible. To make them more apparent it is usual to "exaggerate the vertical scale" ten-fold, or more; i.e., to make the representation of a foot of height ten times as great as that of a foot of length, as in Fig. 420, in which one inch represents one hundred feet for the distances, andten feet for the heights. The preceding Introduction to Levelling has been made as brief as possible; but by any of the simple instruments described in it, and either of its tabular forms, any person can determine with sufficient precision whether a distant spring is higher or lower than his house, and how much; as well as how deep it would be necessary to cut into any intervening hill to bring the water. He may in like manner ascertain whether a swamp can be drained into a neighboring brook; and can cut the necessary ditches at any given slope of so many inches to the rod, &c., having thus found a level line; or he can obtain any other desired information which depends on the relative heights of two points. To explain the peculiarities of the more elaborate levelling instruments, the precautions necessary in their use, the prevention and correction of errors, the overcoming of difficulties, and the various complicated details of their applications, would require a great number of pages. This will therefore be reserved for another volume, as announced in the Preface. ANALYTICAL TABLE OF CONTENTS. PART I. GENERAL PRINCIPLES AND FUNDAMENTAL METHODS. CHAPTER I. Definitions and Methods. ARTIOLE PAGE ARTICLE PAGE (1) Surveying defined........... 9.Division of the subject. (2) When a point is determined.. 9 (12) By the methods employed.. 14 (3) Determining lines and surfaces 10 (13) By the instruments........ 14 To determine points. (14) By the objects............. 14 (5) First Method........... 10 (15) By the extent............. 15 (6) Second do............ 11 (16) Arrangement f f this book... 15 (7) Th7ird do............... 11 (17) The three operations common (8) Fourth do.............. 12 to all surveying......... 15 (10) Fifth do............... 13 CHAPTER II, Making the Measurements, Measuring straight lines. (25) Chaining on slopes......... 21 (19) Actual and Visual lines.... 16 (28) Tape..................... 23 (20) Gunter's Chain............ 16 (29) Rope, &c.................. 24 (21) Pins..................... 19 (30) Rods..................... 24 (22) Staves................... 19 (32) Measuring-wheel.......... 24 (23) How to chain............. 19 I (33) Measu ring Angles......... 26 (24) Tallies................... 21 (34) Noting the Measurements... 26 CHAPTER III. Drawing the Map. (35) A Map defined............ 26 (45) Scales for farm surveys 29 (36) Platting............ 25 (46) Scales for state surveys..... 31 (37) Straight lines.......... 26 (4X7) Scales for railroad surveys 32 (35) Arcs..................... 26 (49) How to make scales........ 33 (39) Parallels............ 26 (50) The Vernier scale......... 35 (40) Perpendiculars........ 27 (51) A reduced scale............ 36 (41) Angles................... 28 (52) Sectoral scales... 36 (42) Drawing to scale........... 28 (53) Drawing scale on map...... 37 (44) Scales.................... 29 (54) Scale omitted............ 37 CHAPTER IV. Calculating -the Content. (55) Content defined........... 38 (6S) Quadrilaterals............ 44 (56) Horizontal measurement.... 38 (69) Curved boundaries....... 45 (57) Unit of content............ 40 (70) Second Method, Geometrically 45 (5 5) Reductions............. 40 (71) Division into triangles..... 45 (59) Table of Decimals of an acre. 41 (72) Graphical multiplication... 47 (60) Chain correction........... 41 (73) Division into trapezoids.... 48 (61) Boundary lines............ 42 (74) Do. into squares...... 48 (75) Do. into parallelograms 49 METHODS OF CALCULATION. (76) Addition of widths........ 50 (63) First Method, Arithmetically. 43 (77) Third Method, Instrumentally 50 (64) Rectangles.............. 43 (7S) Reduction to one triangle.. 50 (65) Triangles................ 43 (S4) Special instruments....... 54 (66) Parallelograms........... 44 (87) Fourth Method, Trigonometri(67) Trapezoids........... 44 cally.................... 56 416 CONTENTS. PART II. CHAIN SURVEYING. CHAPTER I. Surveying by Diagonals. ARTIOLN PAGE ARTICLE PAGE (90) A three-sided field......... 58 Keeping thefield-notes...... 62 (91) A four-sided field.......... 59 (94) By sketch.............. 62 (92) A many-sided field......... 60 (95) In columns............. 62 (93) How to divide a field...... 61 (96-97) Field-books........... 64 CHAPTER II. Surveying by Tie-lines. (98) Surveying by tie-lines.... 66 (101) Inaccessible areas..... 67 (100) Chain angles........... 67 (102) Without platting...... 67 CHAPTER III. Surveying by Perpendiculars. To set out Perpendiculars. Qffsets. (104) By Surveyor's Cross..... 69 (114) Taking offsets........... 75 (107) By Optical Square...... 70 (117) Double offsets.......... 76 (108) By the Chain......... 2 (118S) Field work.............77 Diagonals and Perpendiculars. (119) Platting................ 79 (110) A three-sided field...... 72 (120) Calculating content.... 80 (Il1) A four-sided field....... 73 (121) When equidistant..... 80 (112) A many-sided field...... 74 (122) Erroneous rules....... 81' (113) By one diagonal....... 75 (123) Reducing to one triangle 81 (124) Equalizing............ 81 CHAPTER IV. Surveying by the methods combined. (125) Combination of the three (132) Exceptional cases........ 92 preceding methods...... 82 (134) Inaccessible areas......... 93 (127) Field-books.............. 83 (136) Roads.................. 95 (130) Calculations............. 88 (137) Towns.................. 95 (131) The six-line system....... 90 CHAPTER V. Obstacles to Measurement in Chain Surveying. (13S) The obstacles to Alinement and Measurement................... 96 (139) LAND GEOMETRY............................................... 96 Problems on Perpendiculars. (140) PROBLEM 1. To erect a perpendicular at any point of a line......... 97 (143) 2. I" " when the point is at or near the end of the line......... 98 (14S) 3. " when the line is inaccessible... 99 (150) 4. To let fall a perpendicular from a given point to a given line 99 (1153) 5. " when the point is nearly opposite to the end of the line... 100 (156) 6. " when the point is inaccessible.. 101 (158) 7. ", when the line is inaccessible... 101 Problems on Parallels. (160) PROBLEM 1. To run a line from a given point parallel to a given line. 102 (165) 2. Do. when the line is inaccessible............. 103 CONTENTS. 417 OBSTACLES TO ALINEMIENT. ARTICLE PAGE A. To prolong a line........................................... 105 (169) By ranging with rods..... 105 (:174) By transversals........... 107 (171) By perpendiculars........ 106 (175) By harmonic conjugates.... 108 (172) By equilateral triangles... 106 (176) By the complete quadri(173) By symmetrical triangles.. 107 lateral................. 108 B. To interpolate points in a line............................ 109 (177) Signals................. 109 (181) With a single person..... 111 (178) Ranging................ 109 (182) On water..............111 (179) Across a valley.......... 110 (183) Through a wood......... 112 (18O) Over a hill.............. 110 (184) To an invisible intersection. 112 OBSTACLES TO MEASUREMENT. A. W1Vhena both ends of the line are accessible..................... 113 (186) By perpendiculars........ 113 (8S9) By transversals.......... 114 (1S7) By equilateral triangles... 113 (1190) In a town............... 114 (188) By symmetrical triangles.. 114 B. IWhen one end of the line is inaccessible....................... 115 (191) By perpendiculars........ 115 (119) By transversals........... 117 (194) By parallels............. 116 (199) By harmonic division..... 117 (195) By a parallelogram....... 116 (200) To an inaccessible line.... 118 (196) By symmetrical triangles.. 116 (201) To an inacc. intersection.. 118 C. ITZhen both ends of the line are inaccessible............... 11 9 (202) By similar triangles....... 119 (04) By a parallelogram....... 119 (203) By parallels............. 119 (205) By symmetrical triangles.. 120 INACCESSIBLE AREAS.......................... 121 (207) Triangles............... 121 1(209) Polygons................ 121 (208) Quadrilaterals........... 121 PART III. COMPASS SURVEYING, CHAPTER 1. Angular Surveying in general. (210) Principle................ 122 (217) The Compass............ 124 (2311) Definitions.............. 122 (219) Methods of Angular Sur(2113) Goniometer............. 123 veying................ 125 (214) How to use it........... 123 (220) Subdivisions of Polar Sur(215) Improvements........... 124 veying................ 125 CHAPTER.I, The Compass. (22:1) The Needle.............. 127 (228) Tangent Scale........... 132 (222) The Sights.............. 128 (229) The Vernier.......... 132 (223) The Telescope........... 128 (230) Tripods............... 133 (224) The divided Circle........ 128 (231) Jacob's Staff............. 134 (225) The Points.............. 129 (232) The Prismatic Compass... 135 (226) Eccentricity............. 130 (234) The defects of the Compass. 137 (227) Levels................. 132 27 418 CONTENTS. CHAPTER III. The Field-work. AlTICLE PAGE ARTICLE PAGL (235) Taking Bearings......... 138 (242) Angles of deflection.... 144 (236) Why E. and W. are re- (243) Angles between courses.. 145 versed............. 139 (244) To change Bearings...... 146 (237) Reading with Vernier.. 140 (245) Line Surveying......... 147 (238) Practical Hints....... 140 (246) Checks by intersecting bearings........... 148 Mark stations. Set beside (247) Keeping the Field-notes 149 fence. Level crossways. Do (2 ) N ew York Canal Maps. 149 not level by needle. Keep 1) ew York Canal Maps 149 same end ahead. Read from (252) Farm Surveying......... 150 same end. Caution in read- (254) Field-notes........... 151 ing. Cheek vibrations. Tap (256) Tests of accuracy...... 153 compass. Keep iron away. Electricity. To carry com- (258) Method of Radiation. 154 pass. Extra pin and needle. (259) Method of Intersection. 154 (260) Running out old lines.. 154 (239) To magnetize a Needle. 142 (261) Town Surveying......... 155 (240) Back-sights........... 143 (262) Obstacles in Compass Sur(241) Local Attraction...... 143 veying................ 156 CHAPTER IV. Platting the Survey. (263) Platting in general....... 157 (273) Drawing-board protractor. 166 (264) With a protractor........ 157 (274) With a scale of chords.... 166 (265) Platting bearings......... 158 (275) With a table of chords.... 167 (268) To make the plat close.... 161 (276) With a table of natural sines 168 (269) Field platting......... 162 (277) By Latitudes and Depart(272) With a paper protractor 164 ures................... 168 CHAPTER V. Latitudes and Departures. 27S) Definitions........... 169 Applications. 279) Calculation of Latitudes (282) Testing survey......... 175 and Departures......... l10 (283) Supplying omissions...... 176 (280) Formulas................ 171 (284) Balancing............... 177 (281) Traverse Table........... 171 (285) Platting................ 178 CHAPTER VI. Calculating the Content. (286) Methods................ 180 (292) General rule............ 184 (287) Definitions............. 180 (293) To find east or west station 184 (288) Longitudes.............. 181 (294) Example 1........... 184 (289) Areas.................. 182 (296) Examples 2 to 13........ 186 (290) A three-sided field...... 182 (297) Mascheroni'sTheorem.... 188 (291) A four-sided field........ 183 CHAPTER VI11 The Variation of the Magnetic Needle. (298) Definitions.............. 189 (306) Table of Azimuths....... 196 299) Direction of the needle.... 189 (307) Setting out the meridian.. 197 To determine the true meridians. To determine the variation. (300) By equal shadows of the (308) By the bearing of the star. 198 sun.................. 190 (309) Other methods........... 199 (301) By the North Star when in (310) Magnetic variation in the the meridian.......... 191 United States........ 199 (302) Times of crossing the me- Line of no variation. 199 ridian.............. 193 Lines of equal variation.. 200 (303) By the North Star when at Magnetic Pole....... 200 its extreme elongation.. 194 (311) To correct magn. bearings. 200 (304) Table of times......... 195 (312) To survey a line with true (305) Observations........... 196 bearings.............. 202 CONTENTS. 419 CHAPTER VIII, Changes in the Variation., ARTCILE PAGE ARTICLE PAGE (3]14) Irregular changes........ 203 (31 S) By interpolation...... 205 (3][5) The Diurnal change...... 203 (319) By old lines.......... 206 (316) The Annual Change...... 204 (320) Effects of this change.... 207 (317) The Secular change....... 204 (321) To run out old lines..... 208 Tables for United States. 205 (322) Example............ 208 To determine the secular (323) Remedy for the evils of change.............. 205 the secular change..... 210 PART IV. TRANSIT AND THEODOLITE SURVEYING. (BY THE 3d METHOD.) CHAPTER I. The Instruments, (324) General description of the (333) Supports................ 221 Transit and Theodolite.. 211 (334) The Indexes. Eccentricity. 221 The Transit.......... 212 (335) The graduated circle...... 223 The Theodolite....... 213 (336) Movements. Clamp and (325) Distinction between them. 214 Tangent screw......... 223 (326) Sources of their accuracy.. 214 (337) Levels............ 224 (327) Explanation of the figures. 215 (33S) Parallel plates.225 (32S) Sectional view............ 216 (339) Watch Telescope......... 226 (329) Telescopes.............. 217 (340) The Compass........... 226 (330) Cross hairs.............. 218 (341) Theodolites............. 226 (331) Instrumental parallax..... 220 (342) Goniasmometre.......... 227 (332) Eye-glass and object-glass.. 221 CHAPTER II. Verniers. (343) Definition............... 228 (351) Circle divided to 20'........ 235 (344) Illustration.............. 228 (352) Circle divided to 15'....... 236 (345) General rules............ 229 (353) Circle divided to 10'....... 237 (346) Retrograde Verniers...... 230 (354) Reading backwards....... 237 (347) Illustration.............. 231 (355) Arc of excess.......... 238 (34S) Mountain Barometer...... 231 (356) Double Verniers....... 238 (349) Circle divided into degrees. 232 (357) Compass Verniers....... 239 (350) Circle divided to 30'........ 233 CHAPTER III. Adjustments, (35S) Their object and necessity. 240 1 Rectification'....... 243 (359) The three requirements in (362) In the Theodolite...... 245 the Transit............ 240 (363) Third Adjustment. To cause (360) First Adjustmnent. To cause the line of collimation to the circle to be horizontal revolve in a vertical plane 246 in every position........ 241 Verification(plumb-line; Verification.......... 241 star; steeple and stake) 246 Rectification......... 241 Rectification......... 246 (361) Second Adjustment. To (364) Centring eye-piece........ 247 cause the line of collima- (365) Centring object-glass. 24 tion to revolve in a plane 242 Adjusting line of colliVerification....... 242 mation........... 248 420 CONTENTS. CHAPTER IV. The Field-work. ARTICLE PAGE ARTIOLE PAGE (366) To measure a horizontal (372) Line-surveying........... 254 angle................. 250 (373) Traversing, or surveying (367) Reduction of high and- by the back angle... 254 low objects......... 251 (374) Use of the Compass.... 255 (.368) Notation of angles..... 252 (375) Measuring distances with (369) Probable error........ 252 a telescope and rod.. 256 (370) To repeat an angle.... 252 (376) Ranging out lines......... 257 (371) Angles of deflection.... 253 (377) Farm-surveying.......... 258 (378) Platting......i.......... 259 PART V. TRIANGULAR SURVEYING. (BY THE 4th METHOD.) (379) Principle................ 260 (385) Observations of the angles.. 267 (380) Outline of operations...... 260 (3S6) Reduction to the centre... 268 (381) Measuring a base......... 261 (3S7) Correction of the angles... 270 Materials.............. 261 (388) Calculation and platting. 270 Supports............... 262 (3S9) Base of Verification....... 271 Alinement............. 262 (390) Interior filling up...... 271 Levelling.............. 262 (391) Radiating Triangulation. 272 Contacts............... 262 (392) Farm Triangulation..... 272 (382) Corrections of Base....... 263 (393) Inaccessible Areas.. 273 (383) Choice of stations......... 263 (394) Inversion of the Fourth U. S. Coast Survey Ex- method............... 273 ample............... 265 (395) Defects of the Method of In(3S4) Signals................. 266 tersections............. 274 PART VI. TRILINEAR SURVEYING. (BY THE 5th METHOD.) (396) The Problem of the three (398) Instrumental Solution. 277 points................. 275 (399) Analytical "... 277 (397) Geometrical Solution 275 (400) Maritime Surveying...... 278 PART VII. OBSTACLES IN ANGULAR SURVEYING. CHAPTER I. Perpendiculars and Parallels. (402) To erect a perpendicular to a line at a given point................. 279 (403) To erect a perpendicular to an inaccessible line, at a given point of it 280 (404) To let fall a perpendicular to a line, from a given point........ 280 (405) To let fall a perpendicular to a line, from an inaccessible point.... 280 (406) To let fall a perpendicular to an inaccessible line from a given point.. 281 (407) To trace a line through a given point parallel to a given line........ 281 (408) To trace a line through a given point parallel to an inaccessible line.. 281 CONTENTS. 421 CHAPTER IIo Obstacles to Alinement. ARTIOLE PAGE A. To prolong a line......................... 282 (409) General method......... 282 (413) When the line to be pro(410) By perpendiculars....... 282 longed is inaccessible... 283 (411) By an equilateral triangle. 282 (414) To prolong a line with only (412) By triangulation......... 283 an angular instrument... 283 B. To interpolate points in a line.............. 284 (415) General method......... 284 (418) By Latitudes and Depart(416) By a random line........ 284 ures, with transit....... 285 (417) By Latitudes and Depart- (4L19) By similar triangles...... 286 ures, with compass..... 285 (420) By triangulation......... 286 CHIAPTER III. Obstacles to Measurement. A. When both ends of the line are accessible........... 287 (421) Previous means.......... 287 1 (423) A broken base...........28 (422) By triangulation......... 287 (424) By angles to known points. 288 B. When one end of the line is inaccessible....... 2... 288 (425) By perpendiculars........ 288 (429) To find the distance from a (426) By equal angles........... 288 given point to an inacces(427) By triangulation.......... 289 sible line.... 289 (42S) When one point cannot be seen from the other.... 289 C.' TWhen both ends of the line are inaccessible.......... 290 (430) General method.......... 290 (433) When no point can be found (431) To measure an inaccessible from which both ends can distance, when a point in be seen............... 292 its line can be obtained.. 291 (434) To interpolate a base..... 292 (432) When only one point can be (435) From angles to two points.. 293 found from which both (436) From angles to three points 293 ends of the line can be (437) From angles to four points. 294 seen................. 291 (435) Problem of the eight points 296 CHAPTER IV. To Supply Omissions. (439) General statement.................................... 297 (440) CASE 1. WYhen the length and bearing of any one side are wanting.... 298 CASE 2. When the length of one side and the bearing of another are Woanting............................... 298 (441) When the deficient sides adjoin each other............... 298 (442) When the deficient sides are separated from each other..... 299 (443) Otherwise: by changing the meridian................. 299 CASE 3. When the lengths of two sides are swanting................ 300 (444) When the deficient sides adjoin each other............. 300 (445) When the deficient sides are separated from each other.... 301 (446) Otherwise: by changing the meridian............. 301 CASE 4. When the bearings of two sides are wanting................ 302 (447) When the deficient sides adjoin each other............... 302 (44S) When the deficient sides are separated from each other..... 302 422 CONTENTS. PART VIII. PLANE TABLE SURVEYING. ARTIOLU PAGE (449) General description....... 303 (455) Method of Resection...... 308 (450) The Table............... 303 (456) To Orient the Table...... 308 (451) The Alidade............ 304 (457) To find one's place on the (452) Method of Radiation...... 305 ground................ 309 (453) Method of Progression.... 306 (45S) Inaccessible distances..... 310 (454) Method of Intersection.... 307 PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) General principles........ 311 (463) Distances by sound....... 313 (460) Distances by pacing....... 311 (464) Angles............... 314 (461) Distances by visual angles. 312 (465) Methods of operation...... 314 (462) Distances by visibility.... 313 PART X. MAPPING. CHAPTER I. Copying Plats. (466) Necessity............... 316 (474) Reducing by squares...... 319 (467) Stretching the paper...... 316 (475) " by proportional (4,68) Copyingaby tracing...... 317 scales... 320 (469) " on tracing-paper.. 317 (476) " by a pantagraph 321 (470) " by transfer-paper. 317 (477) "by a camera luci(4$71) by punctures.... 318 da........... 321 (472) " by intersections.. 318 (478) Enlarging plats.......... 321 (473) by squares...... 319 CHAPTER II. Conventional Signs. (479) Object................. 322 (483) Signs for water.......... 325 (480) The relief of ground...... 322 (484) Colored topography..... 325 (481) Signs for natural surface... 324 j (485) Signs for detached objects. 327 (4S~2) Signs for vegetation...... 324 CHAPTER III. Finishing the Map. (486) Orientation............. 328 (489) Joining paper.329 (487) Lettering............... 328 (490) Mounting maps.......... 329 (4X8) Borders................. 328 CONTENTS. 423 PART XI. LAYING OUT, PARTING OFF, AND DIVIDING UP LAND. CHAPTER I. Laying out Land. ARTICLE PAGE ARTICLE PAGE (491) Its object.............. 330 (496) To lay out circles........ 332 (492) To lay out squares........ 330 (497) Town lots.............. 333 (493) To lay out rectangles..... 331 (498) Land sold for taxes....... 333 (494) To lay out triangles...... 332 (499) New countries........... 334 CHAPTER II. Parting off Land. (500) Its object.................................................... 334 A. By a line parallel to a side. (501) To part off a rectangle........................................ 335 (502)."' a parallelogram..................................... 335 (503) " " a trapezoid.................................. 335 B. By a line perpendicular to a side. (505) To part off a triangle......................................... 336 (507) " a quadrilateral...................3................. 37 (i50S) " " any figure........................................ 337 C. By a line running in any given direction. (509) To part off a triangle......................................... 337 (511) " " a quadrilateral.............................. 3. 338 (5113) i. any figure...................339....................... 39 D. By a line starting from a given point in a side. (514) To part off a triangle.................................... 339 (516) "' a quadrilateral.............. 340 (51B 7).. any figure...................................... 340 E. By a line passing through a given point within thefield. (519) To part off a triangle.......................................... 342 (520) a quadrilateral..................................... 343 (522) " " any figure....................................... 344 F. By the shortest possible line. (523) To part off a triangle..................................... 345 (524) G. Land of variable value.......................... 345 (525) H. Straightening crooked fences..................... 346 CHAPTER III. Dividing up Land. (626) Arrangement.........347 Division of Triangles. (527 ) By lines parallel to a side..................................... 347 (528) By lines perpendicular to a side................................ 348 (529) By lines running in any given direction................... 348 (530) By lines starting from an angle................................. 349 (531) By lines starting from a point in a side...................3........ 49 (535) By lines passing through a point within the triangle............... 351 (540D) Do. the point being to be found............................ 353 (541) Do. the point to be equidistant from the angles............... 353 (542) By the shortest possible line.................................... 354 Division of Rectangles. (543) By lines parallel to a side................................ 354 424 CONTENTS. ARTICLB,PAGS Division of Trapezoids. (544) By lines parallel to the bases........... 355 (546) By lines starting from points in a side............................ 355 (54'7) Other cases........................................ 356 Division of Quadrilaterals. (54S) By lines parallel to a side.,........... 356 (549) By lines perpendicular to a side........................... 358 (550) By lines running in any given direction.......................... 358 (551) By lines starting fiom an angle............................ 358 (552) By lines starting from points in a side........................... 358 (554L) By lines passing thi'ugh a point within the figure................. 359 Division of Polygons. (555) By lines running in any given direction...........................360 (556) By lines starting from an angle.................................. 360 (557) By lines starting from a point on a side........................... 361 (555) By lines passing through a point within the figure.................. 361. (559) Other Problems......................................... 361 PART XSI. UNITED STATES' PUBLIC LANDS. (560) General system.......... 363 Meandering............ 371 (561) Difficulty...........3.... 364 (56,5) Marking lines............ 372 (562) Running township lines.... 366 (566) Marking corners.......... 372 (563) Running section lines..... 368 (567) Field-books........ i...... 376 (564) Exceptional methods...... 370 Township line s......... 377 Water fronts........... 370 Section lines....... 378 Geodetic method....... 371 Meandering.......... 378 APPENDIX. APPENDIX A. Synopsis of Plane Trigonometry. (1) Definition................ 379 (7) Their mutual relations...... 83 (2) Angles and Arcs............. 379 (S) Two arcs......... 383 (3) Trigonometrical lines........ 380 (9) Double and half arcs....... 384 (4) The lines as ratios........... 381 (10) The Tables............... 384 5) Their variations in length 38.. 381 ( 11) Right-angled triangles...... 85 (6) Their changes of sign....... 382 (12) Oblique-angled triangles. 385 APPENDIX B. Demonstrations of Problems, &c. Theory of Transversals........... 387 Proofs of Problems in Part V.... 397 Harmonic division.............. 390." " in Part VI..... 398 The Complete Quadrilateral...... 391 " " in Part VII.... 399 Proofs of Problems in Part II., in Part XI..... 401 Chapter V..... 393 APPENDIX C. Introduction to Levelling. (1) The Principles............. 409 (3) The Pratice............... 411 (2) The Instruments.......... 409 VroAi1 olun;. WVEST SIDE. Double Double Factors. Log Double Factors. Log. Area. I~og. tArea. 97 o.98677 97 0.98677 JI rSI? 9.41300 I 15 9.41300,99,. 0.99564 4 1 18 IO.2 I.00860 I.3954I 1 24.854.40837 25.607 r 9- 9 0.-99564 i 1I0.2 I.oo86o 22 2I 9.55433 2 1I7 9.46594 IO.5 I.02IIg Io.7 I.02938 I.57II6 37-252. 50392 3I.90I r 10.5 I.0211g 10.7 I.02938 3 I7~ 9.46594 3 I 941300 ( 9-4 o.97772 ( I(.3 I.OI284 I.46485 29. I6I I1.45522 28.524 9-5 0O.97772 10.3 1.01284 4t I7; 9.46594 4 130 9.35209 8. I o.90849 9. 0.95424 I.352I5 22.498 1I.3I9I7 20.852 5 I3 I 9.35200 5 19 9.51264 7. I o0.85126 7.5 0.87506. III84 I2.937 | 1. 34I94 21.976 7.I 0.85I26 r 7.5 0.87506 20~ 9.53405 6I 260 9.64I84 ( ~7.0 ~ o o. 845 IO0 2 6.7 o. 82608 I. 2304I I6.998 1.34298 22.028 240 9.6093I 7 I8~ 9.48998 77 86.3 I 6.3 0.79934 6.7 o.82608 I. 25375 I7.937 I. I4204 I3.868 6.3 0.79934 6.7 o.82608 8 2I 0 9 55433 8| 220 9.57358 5.4 0.73239 5.7 0.75588 I.o86o6 I2. IgI I I5554 I4.306 5-4 0-73239 5.7 0.75588 300 0 9.69897 9 200 9.53405 4.9 0.69020 5. 0.69897 I.I2156 13.230.98890 9.747 _- 5.0 o 69897 I87.058 Io o I5~ 9.4I300 I95. I49 4.9 o.69020 2)382. 207 o0.80217 6.340 Total Area, 19I. 103 I95-I49 As we have only one grade of goods, and one price, and never -lo-;jA+e from piter.it is not abso!l-telv necessarv for narties to aDnlv to Topography by Means of Contours. UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, NOV. 20, I875. Messrs. Heller & Brightly: GENTLEMEN: In compliance with your request, I have prepared a reduced plot to show the method of representing topography by means of contours. As such information can only be intended for amateurs, I will explain the method in detail and show some of its applications to location. If a series of horizontal planes be assumed at equal distances apart, they will intersect an irregular or inclined surface in lines; or if a water level be supposed to be raised or lowered by equal stages, the successive water-lines or edges thus formed will represent the contour lines of the surface. These contours or horizontal sections are shown by projecting them upon a horizontal plane. In the case in question, the lowest point of the field was found, and the horizontal plane passing through it assumed as the zero or datum plane, upon which all the other contours were projected. The interval between the planes was assumed to be three feet, and the points on the surface at these various elevations, having been first determined by the level and rod, were located by the plane table. This method of representing topography is vastly superior to any other, as it exhibits exactly the slope at any portion of the ground; gives the elevation of the base of any object within the tract; enables one to make vertical sections in any direction with accuracy from the plot, and to locate roads, paths or other features upon a given grade or at any desired elevation, and furnishes the means of calculating the contents of irregular solids with great precision. To locate a road upon a given grade, say-3-Ioo: take with the dividers from the scale one hundred feet, and, starting from a given point-say P-describe an,arc from it as a centre, cutting the next lower contour (27) from this intersection as a centre; with the same radius describe an arc cutting (24) and so on. The line joining these centres will be a surface line on the given grade, and will be the cheapest road that can be built upon the given conditions. Crossing the stream at (T T'), upon a level 6' above the water, to reach the top of the hill at V the grade must be increased to 5-Ioo or I-20; and the centre line is located in the same manner, but with a radius of 60~, as that will be the base for a rise of three feet. Descending from V to W, the grade is the same; but from W to X it is reduced to 5-200 or I-40, to prevent cutting at X. From Z back to P two routes are shown, which are the shortest for the given grade 3-Ioo, although a third might be located following the stream to'a point M, where it must be diverted to N, and thence zigzag up the hill, but it would be longer than the others. The two distances Z x y z S R and Z x' y' z' R are equal; but the first gives a more graceful curve and will require less construction, the part S R being supposed to have been already built for the descending grade. These are a few of the applications of this system of projections. It is also of great use in dividing up land, computing areas, locating a system of drainage, etc. Yours very truly, LEWIS M. HAUPT, Prof. of Civil Enayineerino 2~~~~~~~~~~~~~~~~~~~~, I; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Vr~ ~~~~~~~'.f'-' i-'- - -.. \'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' I >~~~~~~_ / 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i'ii Ki~~. ~i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'~-4 —-- C13,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'"'~ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i!',',, lii~l, Xx ~'7' I' I I i:~~~~~~~~~~~