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V E'~ o-! + \ ~ S N 5- -... n.~~i~i [.E1I-ESpt.rF l~~ %.~,(Aust fiC LINES O~~~~~~~~~~~~~~~~~~~~~~:Ir?;~~~~~~~~~~~~~~~~~< ~i.J~!l FI'i!-::'8 /~"~t~~ L~ }C..~.vea /r, r, ~~~~~~~~j7,, ~~~~.'~. ~_ ——,T - E:,,,,:'.~''~ ~. ~ FOE TIIE BI~~~~~~~~~~~~~~~~BR 1SaO 6~~~~~~~~~~~~~~~~ Q "3~~~~~~,, Sc~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~alvet o'0000 Ciilles~~~~~~~~~~~~~~~~~wie's Land Sulaeying. j i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.&l'' A TREATISE ON LAND-SURVEYING: COMPRISING THE THEORY DEVELOPED FROMI FIVE ELEIIIENTARY 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 NV. M. GILLESPIE, LL. D., Civ. ENG., PROFESSOR OF CIVIL ENGINEERING IN UNION COLLEGE. AUTHOR OF "A MANUAL OF ROAD-MAKING," ETC. EIGHTH EDITION. NEW YORK: D. APPLETON & COMPANY, 549 & 551 BROADWAY. LONDON: 16 LITTLE BRITAIN. 1875. Entered, according to the Act of Congress, in the year 1855, by WILLIAM MITCHELL GILLESPIE, [m t e U(lerk's Office of the District Court of the United States for the Southerv District of New-Yoi k. PREFACE4 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 mowing lot "to that of the Heavens), has been the earnest endeavor of the. present 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. 6v LLAND-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 founda tion 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 tnose 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 prin. eiples, is then explained in its turn, in the successive Parts, and eacb 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 founa 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, iz 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 mag netic 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. LAND-SURVEYING. XI. Thv' Tables comprise a Traverse Table, computed for this volume.nd giving increased accuracy in one-fifteenth of the usual space; a Table of Chords, appearing for the first time in English, and supplying he most accurate method of platting angles; and a Table of natural Sines and Tangents. The usual' Logarithmic Tables are also givei,, The tables are printed on tinted paper, on the eye-saving principle of Babbage. XII. The great number of engraved illustrations, most of them criginal, 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 pres,,'it volume will be followed by another on LEVELLING AND IIaIER S'URVEYINcG: 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; Malritime Surveying; and Geodesy, with its practical Astronownv GENERAL DIVISION OF THE SUBJECT. [A full Analytical Table of Contents is given at the end of the volutme.] Pasg PART I. GENERAL PRINCIPLES AND OPERATIONSo CHAPT. 1. DEFINITIONS AND METHODS......... o......o....... CHAPT. 2. MAKING THE MEASUREIENTS......................... 16 CHAPT. 3. DRAWING THE MAP.e oo.......o................... e 25 CHAPT. 4. CALCULATING THE CONTENT.......................... 38 PART If. C. AIN SURVEYINGl 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............ 82 CHAPT. 5. OBSTACLES TO MEASUREMEENT....................... 96 PART iI1, COMPISS SURVEYiSNG CHAPT. 1. ANGULAR SURVEYING IN GENERAL................... 122 CHAPT. 2. THE COMPASS.................................... 127 CHAPT. 3. THE FIELD-WORK........... o........ oo......... 138 CHAPT. 4. PLATTING THE SURVEY............................. 157 CHAPT. 5. LATITUDES AND DEPARTURES....................... 169 CHAPT. 6. CALCULATING THE CONTENT.......................... 180 CHAPT. 7, MAGNETIC VARIATION. 0............. ~......... 189 CHAPT. 8. CHANGES IN THE VARIATION........................ 203 PART IVo TRANSIT AND THEODOLITE SURVEYINY. CHAPT. 1. THE INSTRUMENTS................................... 211 CHAPT. 2. VERNIERS..................... 2.... 228 CIAPT. 3. ADJUSTMENTS...................................... 240 CHAPT. 4. THE FIELD-WORK.................................. 250 PART V. TRIANGULAR SURVEYING................... 260 PART VI. TRILINEAR SURVEYING.................... 27 PART VII. OBSTACLES IN ANGULA SURVEYING. CHAPT. 1. PERPENDICULARS AND PARALLELS................... 279 CHAPT. 2. OBSTACLES TO ALINEMENT........................ 282 CHAPT. 3. OBSTACLES TO MEASUREMENT...................... 287 CHAPT. 4. SUPPLYING OMISSIONS.............................. 297 PART VII. PLANE-TABLE SURVEYSIN............... 303 PART IX. SURVEYING WITHOUT NSTRUMENTS...... 1 PART. X, MAPPINs. CHAPT. 1. COPYING PLATS..................................... 316 CHAPT. 2. CONVENTIONAL SIGNS.............O.............. 322 CHAPT. 3. FINISHING THE MAP.........................2........ 32S PART XI. LAiNG OUT AND DIVIDIG ITP LAND. CHAPT. 1. LAYING OUT LAND................... 330 CHAPT. 2. PARTING OFF LAND................................. 34 CHAPT. 3. DIVIDING UP LAND.............................. 347 PART XIle UNITED STATES' PUBLIC LANDS........... 36 APPENDIX A-SYNOPSIS OF PLANE TRIGONOMETRY................... 379 APPENDIX B-DEMONSTRATIONS OF PROBLEMS, &C................ 387 APPENDIX C-INTRODUCTION TO LEVELLING........................ 409 ANALYTICAL TABLE OF CONTENTS.................... 415 TO TEACHERS AND STUDENTS. As it is desirable to obtain, at the earliest possible period, a sufficient knowledge of the generae principles of Surveying to commence its practice, the Student at his first reading may omit the vortions indicated below, and take them up subsequently in connection with his review of his studies. The same omissions may be made by Teachers whose classes have only a short time for this study. In PART I, omit only Articles (46), (47), (48), (51), (72), (84), (85). In PART I, omit, in Chapter IV, (127), (128), (129), (130); and in Chapter V, learn at first urnder each Problert, 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 Trigonometryp nd omit (423); (431) to (438); and all of Chapter IV, except (439) and (440), PART VIII may be passed over; and PARTS IX and X may be taken in fll. ilc PART XI, take all of Chapter I; and in Chapters II and III, take only the simpler ca cructions, not omitting, however, (517), (518) and (538). In PART XII, take (560), (561), (565), (566). Appendix C, on LEVELLING, may conclude this abridgea wurse. LAND-SURVEYING PART I. GENERAL PRINCIPLES AND FUNDAMENTAL OPERATIONS. CHAPTER Io. LDEFINITIONS A)ND METHlODS. (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 oi 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 PRICIPLES. [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 dimen. sions 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; all the points being supposed to be in the same plane. (5) First Method,. By measuring the distances from the re quired point to two given points. Thus, in Fig. 1, the point S is " deter- Fig. 1. mined,' 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 Geogralphy, 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.] Defintions and Method:s 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 Mlethod, By measuring the perpendicular distance from the required point to a given line, and the distance thJence along the line to a given point. Thus, in Fig. 2, if the perpendicular dis- Fig. 2. tance SO be half an inch, andl CA be one inch, the point S is "determined": for, its place could be again found by measuring Ione inch from A to C, and half an inch from C, A -.I 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 Geography, 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 lineperpendicular 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 a third of a right angle, and So 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 Ac — AB, and measurng on it the given distance. 12 GENERAL PRINCIPLES. [PART I In applying this principle in surveying, S, as before, may repre. sent 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 direoe tion of which is given by the magnetic needle of the compass. In Geography, this principle is employed to determine the rela. tive 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 Method, By measuring the angles mnade with a given line by two other lines starting from given points upon it, and passing through the required points 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 f. of a right angle with the line AB, which A. 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 determining the same point S, by each of the four preceding methods. In the First Method, we measure the distances AS and a 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. I. 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 mneasuring 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, A B made by the three lines SA, SB and - SC. Geographically, the position of Chi- cago would be determined by three —straight lines passing from it to Wash-; ington, Cincinnati, and Mobile, and mak- 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-... uated in the intersection of AB and CD. A.This method is sometimes employed to fix: the position of a Station on a Rail-Road "'C. B 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. 14 GENERAL PRINCIPLES. [PART I (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 suil terranean 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 GCeodesic. 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: 1 Measuring certain lines and angles, and recording them; 2o Drawing them on paper to some suitable scale; 30 Calculating the content of the surface surveyed. The three following chapters will treat of each of these topies in their turn. 16 FUNDAMENTAL OPERATIONS. [PART L CHAPTER II. MAKING THE MEASUREMENTS. (18) TiHE Measurements which are required in Surveymg, 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. l 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 to be given. Lines are usually measured with chains, tapes or rods, di. vided into yards, feet, links, or some other unit of measurement. (20) Gunters Chaal, 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 ring. Sometimes two or three rings are placed between the links. The chain is then less liable to * This length was chosen (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 Uaited States Coast Survey, the unit of measurement (which at some future time will be tue universal one) is the French Metre, equal to 3.281 feet, nearly. CHAP. II.1 Iaking thBe cleasr'eimnclts. 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.j 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 wvill 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 80. 3960. 5. 7.6 80. 121.2 0.70 46.20 70. 4620. 6. 9.1 1 35. 128.8 0.80 52.80 80. 5280. 7. 10.6 0. 136.4 0.90 59.40 90. 5940. 8. 12.1 95. 143.9 1.00 66.00 100. 6600. 9. 13.6 00. 151.5 * To pre-eent the very common mistake, of calling forty, sixty; or thirty, severity; it has been stggested 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 Thie- 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 8 rIng, o-" more, according to the way in which the chain is made. 2 [8 FFUFNDF AMENTAL OPERAITIONS [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 ie 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 Guntei'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. 19 (21) Pins.0 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 Fig. 9. Fig. 9. Fig. o1. 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 ases 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 ChaMin 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 "6 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 aover 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 are explained in Articles (23) and (24). 20 FUIPDAIrMENiAL 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. lie 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. lIe 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 downl 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. an]m the chain. men are afraid of wetting their feet, they can mueasure part of a chain, to the edge of the water, then stretch the chain across it, and then rneasnre aniother portion of a chain, so that with the former portion, it may namke up a' full clhain. CHAP 1i.] Making the Measurements. 21 end of the eleventh chain; another being substitute 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 1" 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, ncluding 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 completecl, each brass pin held by the follower counts ten chains, anld each iron pin one, as before. (25) Clhaining 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 F iBIDAMENTAL OPERATIONS. [PAn1T I dovinwards, (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 he 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 cown a steep hill, stretch the whole chain in line. Hold the Fg. i. upper end fast on the ground. liaise 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 cfown 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 "T 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 tie 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 horizcntal base, which is the distance desired. CIAP. II.] Making the IMeasurements 28 TABLE FOR CHAINING ON SLOPES. CORRECTION CORRECTION ANGLE. SLOPE. IN LINKS ANGLE. SLOPE. IN LINK 30 1 in 19 0.14 130 1 in 4 2.56 40 1 in 14 0.24 140 1 in 4 2.97 50 1 in 11- 0.38 150 1 in 4 3.41 60 1 in 91 0.55 160 1 in 3 8 3.87 70 in 8 0.75 170 1 in 3h 4.37 80 in 7 0.97 180 1in 3 4,89 90 1 in 61 1.23 190 1 in 3 5.45 103 1 in 6 1.53 200 1 in 2- 6.03 11 1 in 5 1.84 250 1 in 2 9.37 120 Iin 43 2.19 300 lin 1 13.40_ (27) Chaining is the fundamental operation in all kinds oi Surveying. It has for this reason been very minutely detailed. The "I 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 cons venient 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 FJUDAI, ENTAL OPElRAT'l. [PART L been recently manufactured, in whlich 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 "T 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. The Stac7lc is described in Art. (375.) (32) A Peramnbulator, or " LMeasuring lWheel," 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 311 inches, the circumference, and therefore each revolution, will be 8- feet, or half a rod. The roughnesses of the road and the slopes necessarily cause the registered distances to exceed the true measure. CHAP. iii.],aking the MIeasurements. 2 MEASURING ANGLES, (33) The angle made by any two lines, that is, the difference of their directions, is measured by various 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 consideration of this kind of measurement will be postponed to Part III. NOTING THE MEASUREMENTS. (31) 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. DRAWINP G THE UE Io (85) 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 projeetion. (36) Platting' A p1lat 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 OPERATIOiNS. [PART. all the measured distances being diminished in exactly the same ratio. PLATTING is repeating on paper, to a smaller scale, the mea suremnents 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 describing 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 Lineso 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) Arcso 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 hcle in which, at the desired dis tance, 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 lince is by the aid of a triangular piece of wood and a ruler. Let XB CHAP. I.] 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 the cross-piece, hold it firm, and slide the T square along, till its' edge pauses 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) Perpendtlclars. 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 Fi. 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 he long side be perpendicular to the 28 FUNDIIMENTAL OPERATIONS. [PART 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 rmost easily set out with an instrument called a Protractor, usually a semi-circle of brass. But the description of its use, and cf 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 fve 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 37- 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 630-7 inches, or nearly 64- inches; and on the latter scale 41-o. inches, coming midway between the 2d and 3d tenth of an inch. CHA. 11I.] 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) (41) Seales. 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 expressed fractionally; 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,-o, To Co I 09 2^ o o( &c. For convenience in printing, these may be written thus: 1: 500, 1:1000,: 2000, 1: 2500, &c. Plats of Fearm 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 = X = =-1:1584) is convenient 0B FUNDRAMENTAL OPERATIONS. rPART. 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 Commis, sioners 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:1000; about 1 chains to 1 inch. Fig. 16. OINE ACSIRE ON SCALE 01 1 CHAIN TO I INCIL' ____-. ___________ The choice of the most suitable scale for the plat of a farm sur vey, 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.] Drawin thIe Map. 31 (46) State Sarveys, On these surveys, smaller scales are necessarily employed. On the admirable United States Coast Survey, all the scales are expressed fractionally and decimally. 6 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. 10, 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 il 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 Gereat Britain is called, from the corps employed on it, the " Ordnance Survey." The "1 Ordnance Survey" of the. southern counties of Englanct 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' Comlnmuicated from the U. S. Coast Survey office. 32 FUTNDAMEiTAL OPEAT ONSE$, [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 fcr 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 11 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 X 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) Ra1l-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 M[aps 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. CHAP. III.] Drawing the alap..3 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; 10). 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 dcial or fu eather r 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 vary in length with changes in the moisture in the air. The student can, however, make for himself platting scales ol 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. i F 1i___ _- I~ i 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 1 inch. On it the distance 220 links, would extend a 84 FUNDtAMENTAL OPERATIONS. [PART X, 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 platting 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 2 oo 50 I; Fig. 19. Scale of 2 chains to 1 inch. Fig. 20. Scale of 3 chains to 1 inch. 0 1 2 3 4 5 6 7 8 9 K~ i-___ -_= — In the fourth figure he distances between the arrow heads are respectively 310, 270, and 540 links. Fig. 21. Scale of 4 cains to 1 ilch. O 1 2 3 4 5 6 7 8 10o 11 II 13 o=1 ==!FC- CHAP. 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 12; 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 chazns to 1 inch.'i a x i o 5 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. (5M) 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. 100io sO JO 290 _ 33 65544 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 di. vision 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 II. 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 386 FUNDAMENTAL OPERATIONS. LPART 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 Incminally 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 a 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- quiredo Take 1 inch in the dividers, and open the see- 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 cIIAP. III.] Drawing the lap. 37 scale, and the angle of its opening must not be again chaiged. 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 dis tance 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 Sc is the square of the assumed scale To the square of the true scale, I; FUNDAMENTAL OPERATIONS. LPAT I CHAPTER IV. CALCULATING TIlE CONTENT. (55) The CONTENT of a piece of ground is:ts 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. GCeometrically, it is plain that this horizontal measurement is absolutely necessary for the purpose of obtaining a correct plat. In Fig. 25, let ABCD, and 3BCEF Fig. 25. 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 Ba, DC, measured on ~ the slope, are longer than if measur- A _j 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 B 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 measuremnents 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 S'ocial aspect of this'question, the horizontal measurement is justified by the fact thatmo 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 Fi. 29. Figure.* Even if a side hill should pro- duce more of certain creeping plants, the G_|_ | | | \.. 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 measturea on the surface would give a larger content than it would after the hollow hat been filled up by the hill, while it would yet really be of greate o 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 lattew so named because it strips off or unfolds, as it were, the surface. t0 FUNDAMENTAL OPERATIONS. [PART 1 (57) anit of Content. The Ace e 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 16 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 4350 square feet; and consequently ten square chains makce one acre.* In different parts of England, the acre varies greatly. The statute acre, as in the United States, contains 160 square perches of 161 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 Wilt. shire is 120 perches of 16- feet, or 32,670 square feet. The acre i 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 lMetres, = 0.0247 acre, = one fortieth of an acre, nearly; and the flectare== 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. ILet the yoi:ng student beware of confounding 1C square chains with 10 cohaiis sqcare. The former make one acre; the latter space contains ten acres CHAP.:v.j Calculating the Conateni, 11 Multiply tle remainder by 40, and the outside figures will be Perches. The nearest round number is usually taken for the PeTches; 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 I. 20 P. " 71.1055 doA.-7. 0 R. 18 P. " 82.50 do. 8 A. I R. 0 P.'" 8.250 do. -0 A. 3 R. 12 P. "4 0.8250 do. =0 A. O 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. 1 Perces. 1" 2' 3 Perches..000.250.500.750 + 0.131.381.631.881 +21.006.256.506.756 + 1.137.387.637.887 +22.012.262.512.762 + 2.144.394.644.894 +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 + 5.162.412.662.912 +26.037.2871.537.787 + 6.169.419.669.919 +27.044.294.544.794 + 7 /.175.425.675.925 +28 C.050.300.550 800 + 8.181.431.681.931 +29!.056.306.556.806 + 9.187.437.687.937 +30 4.062.312.562.812 +10.194.444.694.944 +31.069.319.569.819 +11.200.450.700.950 +32.075.325.575.825 +12 1.206.456.706.956 +33.81.3 581831.58 31 3.212.462.712.962 +34.087 337.587.837 +14.219.469.719.969 +35.094.344.594.844 +15.225.475.725.975 +36.100.350.600 850 +16.231.481.731 981 +37.106.356.606.856 17.237.487.737.987 +38'.112.362.612.8621 +18.244.494.744.994 +39.119.369.6191.869 +19.250 500.750 1.000 +40 l125i.375.65.5.8 20 21______ (60) Chain Correctfion 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 On-tthird of one-tenth of itself. 42 FUNDABMENTAL OPERATI0NS. [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 true links; and 1002 (10199):: 100 square chains: 102.03 square chains. (61) Boandary Lanes, 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 dJtches 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) FITRST METHOD.-ARlTHMETICALLY. Prom diret 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 wlich 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, 6, c, and their sum by s, this rule will give its area& VE[. (US ~a) (s ~ b) (9-c ~ ] * When two sides of a triangle, and the included Fig. 30. angle are given, its content equals half the product B cf its sides into the sine of the included angle. Deaignating the an'les 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. "L _ C When one side of a triangle and the adjacent an- 1 gles 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. in. G these angles. Using the same symbols as before, the area =a2 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. sin. C 44 FUTDAMENT. L OPERATIONS. [PART 1 (66) Parallelogra ns; 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. ( 7) Trapezoids; or four-sided figures, t;o 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 b + d of the trapezoid will =- b (Q- a) (q-c) (q-b + d).]" (68) Quadrilaterals, or Trapezium'; 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 equivalent 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 byArt. (65). " 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=2Sd. sin. C. t When two opposzte sides, and all the angles are given, take one side and its ad. jacent 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. Vhen three sides and their two included angles are given, multiply together the si no 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 siles, 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.. sin. B I p. sin. (180~ -A -B)]. When the four sides and the sum of any two opposite angles are given, proceed thus: Take half the sum of the four given sides, and fiom it subtract each side in turn Multiply together the four remainders, and reserve the product. MIultiply together the four sides. Take half their product, and multiply it by the &osine of tare given sum of the angles increased by unity. Regard the sign of CHAP. IV.] Calculating the Comtent. 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 II, Chap. III. (70) SECO D METHOID,- E0MEIE ICALLY.r From meao surements of the necessary lines upon the plat. (71) Divislon Ito Triangles. 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. Subtract this product from the reserved prolduct, and take the square root of the remainder. It will be the area of the quadrilateral. When the four sides, and the angle of intersection of the diagonals qf dhe quadri&a teral 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. W'hel 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 O FJUDAllENTAL 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 ( so) 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 V/2:1; 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 mn the plat, of known length originally.'hen CHAP. IV.] Calculatin g 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 obtain ed by the present measurement To the true content. (72) Graph7eal tMultiplication. 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. — C _I 9. ____ ______ 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'0 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 fiom the scale. This ingenious operation is due to lM. Cousinzery. 48 FUPNDAENTAL OPERATIONS. [PART I. (73) )iviBtso into Tralpezoids 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, (ex-, | \ cepting a triangle at each end), and their content can be calculated by Art.' (6). 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.'. — This method is very advantageously applied to surveys by the compass; as will be explained in Part III, Chap. VI. (71) DBlvsion 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 Content, 49 same paper ill.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 1 inch, 12 inch; for a scale of 4 chains to 1 inch, I 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 triangle to an equivalent / _A rectangle, as at A in the figure. When an angle occurs between two parallels, as at B \ 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 be 10 10 10 tween the parallels inch, this width will represent X b = -— chains; and I 0n M the inch length represents chains, their productl, - X n = 10 square chains I acr e. 4 50 F BUNDAMENTAL OPERATION S. [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 -f 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." (7 ) 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 oms. When all have been thus measured, the total length, multiplied by the uniform width, will give the content. (77) THIRD METHOD-INTSTRUMENTALLY, By performn ing certain instrumental operations on the plat. (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. 41, be required to be reduced to a single equivalent triangle. / Produce one side of the figure, as 4 — 1. Draw a line from / \ Ahe first to the third angle of- a'. CHAP. iv.] Calculatin the Content 51 the figure. From the second angle draw a parallel to the line just drawn, cutting tile produced side in a point 1'o 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 trapezium.* 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'ig. 42. of the figure will be /4' named as numbered in the engraving, Produce 5 —1. Join 1 —. From./ 2 draw a parallel to 1-3, cutting the g' 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 — 3 -- 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 FFUNDAMIENTAL OPERATION&~ [vaRT 6 Fig. 43 \ ad.,..'/, 8 r ~ i~-~ (8l) 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 removeu by the following General Rule: 1o 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; frim the 4th angle. Its intersection with the 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 n —+3 angle. Draw a parallel to it through the t —+2 angle, and the intersection with the base will be the n+ 1 mark. CH.AI. Iv.] 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; / / \ triangle is here 1' — 3-2'. e \\ When the figure has many angles, / \ / \ \ they should not be numbered con- ~1 ~ —— s, 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 of the final triangle, the numbeFig. 45. of tile final triangle, the numbers should be continued from the 154 FUNDF AMENTAL OPERATIONSo [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 I in Fig. 46, an indefinite line, usually a perpendicular to a lino Fig. 46. 1 1 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. (81) $Speckl Instruimeuts, A variety of instruments have been invented for the purpose of determining areas rapidly and correctly. One of the simplest is the "G Computing Scale'" which is on the same principles as the Method of Art. (T ). It is repre~ sented in Fig. 47, given on the following page. It consists of a scale divided for its whole length from the zero point intc divisions, each representing 2, 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 partso 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.] CalIculatin the Content. 5i instrument is laid on this paper, with its edge on one of the U; i 47 parallel lines, and the slider is moved over one of the divi - sions of 21 chains, that one rood, or a quarter of an acre, has been measured between two of the parallel lines on the.i. o' paper (since 10 square chains make one acre); and that - one of the smaller divisions measures one perch between K 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 held, or farm, lay the I" 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. at A andB. Lay A the scale, with the 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 i 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. (86) A purely mechanical means of determining the area ot 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) FOURTHl MEETHOID.-TRIGOS NOMETRICALL1. By calcutating, 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 III, 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 chain, 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. SIURVEYING DIAGONILS; OR By the Pirst Method. (89) Surveying by bDiagonals 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 I 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. b8 CHAIN SURVEYI IG. [PART ii The lines which are measured in order to determine the cor. ners of the field are usually sides and diagonals of the irregular polygon which is to be surveyed. They therefore divide it up into triangles; whence this mode of surveying is sometimes call" ed " Chain Triangulation." A few examples will make the principle and practice perfectly clear. Each will be seen to require the three operations of rneasuring, plattling, and calculating. (90) A three-sided field; as Fig. 49. Fig. 49. ield-worle. Measure the three sides, AB, BC, and CA. 3Measure also, as a proof line, the distance from one of the cor- A i 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 i 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 (4D), 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 pIact, or miniature copy -as explained in Art. (35)of the fielc 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 settiri,ff from A the proper distance, and measure the length of the line CHAP. i.] arveying by Diagonals. 59 DC, by Art. (43). If its length on the plat corresponds to ita 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. Exam2ple 1. Figure 49, is the plat, on a scale of two chains t 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 CE 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. 10A. OR. 10P. (91) A four-sided field; Fig. 50. as Fig. 50. _ c Pield-wor/c. Measure the four sides. Measure also / a diagonal, as AC, thus di- viding the four-sided field / _ into two triangles. Mea- A" 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 re. sult must be arrived at by two different methods. The necessary length of this proof line can also easily be calculated by the principles of Trigonometryv 60 CHAIN SURVEYING. [PART 1 Calculation. Find the content of each triangle separately, as m 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 ii 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 tc the inch, AB = 588 links, BC = 210, CD - 430, DA = 274, the diagonal AC = 626, and the proof diagonal BD - 500. The total content will be 1Ao 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. 21. 38P. Xx. 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.:- > ^'/ angles formed outside of it,) but not for figures \ \ of a greater number of sides. CHAPTER III. SUIRVEtING BY PEERPENDICULARS: OR By the Second Method. (193) 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 " IDiagonals and Perpendiculars," or to measuring a crooked boundary by 6 Off-sets." Each will be considered in turn. CHAP. 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 pointed staff, on which it can turn freely, and which 6 should be precisely 8 links (63- 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, 2 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- Fg. 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 turnits A —.... 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 torre point, C as it did before to its left. 70 CHAIN SUJVE NiG LPART Is. This 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. P 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 Fi. 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 direc- tion of the line and at right angles to it. (10) Optical Square. The most convenient and accurate instrument 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 looling-glass, from the upper half of which the silvering is removed. This glass is placed *'lt French call Iile irrm'owv o)perng ailleton,;ild t-l, wide one croi.ee. CHAP. iii.] To set out Perpendiculars 7] so as to make precisely half a right Fig. G2. 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 l 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 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 perpendica lar, 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. Another form of the optical square contains two glasses, fixed ai an angle of 45~, and giving a right angle on the principle of the Sextant. 72 HltAIN SURVEYI;G, [PART 1. (108) Chain Perpendlcalars. 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 exe. cuted on paper in practical geometry, the chain being used, in the lace 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 L) 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 Perlendiculars, 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 c. 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. i \ D Then the content is equal to half the product \ of the side by the perpendicular. If obsta I - " 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.] Dlaonals and Perpendiculars, 73 (111) A fnim-sided field, g. 64. Measure the diagonal AC. Leave marks at the points on this diago- nal at which perpendiculars from B A' 2 8 c: and from D would meet it; find- ing these points by trial, as previously directed in Arts. (104) and (107). The best marks at tlese 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 measurenent. 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 ig. 64. Ans. OA. 2R. 29P. The field may be plattec 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 complicatec cases, by the column method, as below. A new symbol may be employed, this mark, k, or =H, to show the False Station, from which a perpendicular is to be measured. jxample 10. Calculation. ^aa I00 o 110 to B sq. ks. From 200 on 480 F.S. H ABC — 480 x 110 -26400 _ I 17S ItoD - ADCO- x 480 x 175-42000 I From 280 on 480 F. S. sq. chains 6.8400 _480 to C Acres 0.684 - 280 I3 It is still easier to take the two ha9,-j 200 triangles together; multiplying rom A 0 the diagonal by the sum of the perpendiculars and dividing by two. T4 [lCIN SURVEVINGo [PART II (112) A many-sided field. lFig. 65, and the accompanying field-notes represent the field which was surveyed by the First Method and platted in Fig. 51. Fig. 65. t13 ^~ - T-"-~ - -- - E 0~~~~5. 1.54to The content of the triangles may 1i'room 5.07 on 7.3 7 F. S. be expressed thus: Ia) 2.53 toD sq' Ilks /x aN 4" t / From 5.45 on 11.42 F.. CDE - 2x 775x253= 98037 From 4.95 on 11.42 F.S. mle. halc 58.8at746 Frorm 5.0 7 o 3. be expresse thusR. 22 FFrom 1 60 on 7.75. S. ABC = x1 142X267=1527 4. 1 to60 e ben taken tog= 1 ether, as Fro m C r the previous field. r 5 n 11.42. to C Content calculat fromx253 98037.... t.- AEF — x 737x154= 56749 ~ 4 9 perpendiculars will generally vaF rom 4.95 on 1 1.42i F.. sq. cobtains 58.8746 eas37 to A on cre 5.88746'-ll 5.07 / or, 5A. 3R. 22P. From B - to B The first two triangles might -~ i.c0 have been taken together, as in From C 0,F the previous field. 11.42 t'o C' Content calculasde from thi,5.45 /P- perpendiculars will generally va, H 4.95 Fr~l~om A ry slightly from that obtained by measuring on the plat. CHAP. ImI] 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 the plat less accurate. This procedure supplies a transition to the method of " Offsets," which is 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. 6. let ACDB be a crooked fence, D bounding one side of a field. Chain A._.-_ -. -.... -. - 11.B 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 Methlod, Art. (6). The Field-notes, corresponding to Fig. 66, are as in the margin. The measure-___ ments along the line are written in the 0 300 to B. column, as before, counting from the be- ginning of the line, and the offsets are Di 25 250 written beside it, on the right or left, oppo- C1 30 100 site the distance at which they are taken. A sketch of the crooked line is also usually Freom A 01 0 made in the Field-notes, though not absotitely 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 CHAII 8URVETNG. [PART Ii In taking the Field-Notes, the widths of the offsets should nor 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 offsets are exaggerated to four times their true proportional dimensions. Fig. 67. B B 1500 0 1250 20 0 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. 250 to 80 185 o 201 150 90 10ji 50 11 0 30 From A. 0 A (17) 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 form9 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 I. I \ - B B 260 30+60 260 30 to 90 240 10+70 240 10 to 80 0 220 50 220 50 20 200 30 20 200 30 40 180 10 40 180 10 45 160 0 45 160 0 50+ 0 40 50 to 0 140 55+ 5 120 60 to 5 120 50 20 100 70to 20 100 45 +15 80 60 to 1 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 (11S) These offsets may generally be taken with sufficient accura cy 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 CHIIN 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 offseistaff 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 modifled. 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 I -- 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 i1 L^- I/ ~.'~....~/f,/',....... 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 scale, 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) Calculatit g Contenti When the crooked line determined by offsets is the boundary of a field, the content, enclosed 80 CHAIN S$URVEYING [PART II Detween 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 takcen 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, Simpson'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) yEqualizing, 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 Fig. 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-hail 82 CHAIN SURVEYNG, [PART IL stretched straight by a light bow of whalebone, In practical hands, this method is sufficiently accurate in most cases. The stu< dent 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 q Equalized," and the fewer resulting zigzags reduced to one line by the method of Art. (78), &c, CHAPTER IV. SURVEYING BY THE PRECEDING METHODS COMlBINED. 125) All the methods which have been explained in the three preceding chapters- Surveying by Diagonals, 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 CHAP. IV.] Diagonals, Tie-lines and Offsets, 83 ber of the sides, diagonals, and proof-lines, to prove the work, should be platted before platting the offsets. Fig. 73. 1 3. 60. C D C ~B 0 360 $ 340 6 315 I D 10 275 = C 5 215 0 150 0 115 10 A 80 5 A 65 8 248 B O r 11 180 - 105 0 B 65 5 0 125 D Or u 11 o 90D__ a D 23 62 0 135 12 22. 15 110 0 A, ] ^ 13 90 _ ~~ 0O 50 0 Example 13. Required the con- 30 9 tent of the above field. Ans. C 0 r (127) Field-books, The difficulty and the importance of keep ing 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. [PART XT FIELD-BOOK No, 1. f a 10o0 I0 OT /. \ l..: so " 900 6Q 2 10 "IG G (6 00"- ~'9 \,,%/..X Q ^o ^> r00 p2 \'\ /...2 0 dBook No. (Fig. 74) shew he Sketch 3eho d, ---— xl' ed'in Art. (94). \ 0 sd i Art. (91). caP v.] S Diagonals, Tie-lines and Offsets, 85 FIELD-BOOK No. 2, -4570^ A 4A-0800 -.1! 0^ 1300 lie N,1,k~~ ~" >~ e~1.2 50 3480 1 020 200 \.3080 / L40 680 190 320( ) a"0 e e x 2300'1300 ^1390 1 0 12 C 0 ^^^I ^00 140 6Q20 260 0 ield-Book No. 2 (Fig. 2 75) shews bhe Column metocL exolau ed in Art. (95). @dir rt. % (9 86 CH1tIN SURVEYING. PaX x FIELD-BOOK No, 30 \, " 1230 /[':20 ~120 1120 0/ 130 f0/ i- / Jo )01COo800 ~ ( 0ig 0 580) / Tf 1220 50 ~7 II / 10 6, Field Book Mo. 3 (Fig. 76) is a convenient combination of the two preceding methods. The bottom of the Book is at the side of dis 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 tc 17 chains, or C, leaving a pole at 12.30 as a new starting point by and by. Then from C measure 1 9 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 Fielc-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. y \ "Capt. FROME, in his' Trigonometrical Survey," from which this examp'e nas been condensed, remarks, " It may, perhaps, be thought that too much stress:s laid on forms; but method is a most essential part of an undertaking of magnitu'le: and without excellent preliminary arrangements to ensure uniformity in all the most trifling details, the work never could go on creditably." 88 CIIHIN SJURVEYING. [PART II C__ _ 2564 80 D A F 2452 FX____ ~^~~ ~ 100 4050 Q 1700 0 62 3890 N 2324 150 84 42 3730 1420 40 0 3540 0 0 1340 0 3420 30 i 1264 0 Fro.-,-A a to D —~ A 722484 S 1240 52 A 1140 86 A40 2332 950 100 60 2206 772 60 0 2056 0 0 604 (0 A1805 40 34 502 M 3296 0 1550 50 450 _ v~^ 70 342 3275 54 X 1442 0 82 220 3120 62 From C A to A 2940 85 romD A to C 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 li, tinctive 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 calculation of one of the figures, 1, 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 CIhP. iV.] DBagonals, Tc-llines and Offsets, 89 along the sides PW and WX. From the content thus obtained must be subtracted the offsets on PQ, belonging to the figure ~, and those on WX belonging to the figure jK. When the offsets arc 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 1ST 2D D CONTENT DIVISION. OR IN SIDE. SIDE. SIDE. TRAPEZOID. CHAINS. BDPQ 168016981078 86.2650 52 250.6500 PQ { 52 30 80.3280 ( 30 216.3240 M~/~ i.~~ t1.8020 Additives. DWX 1370 1442 770 51.8339 PW 301 310.4650 56' 114.3192 WX 56 36 104.4784 ( 36 - 90.1620.9596 Total Additives, 140.8255 I_ _ _('~ 50 1741.4350) PQ \ 50 30 292 1.1680 ( 30 66.0990 1.7020 mJ ( ~ 52 142.3692) Subtbractives. WX 52 64 232 1.3456 ( 64 88.2816) 1.9964 Total Subtractives, 3.6984 Total Additives, 140.8255 i ere 171271 |________ { ~Differencee, _ 137.1271 DO CAIN SURVEYING. [PART 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. SQUARE CHAINS. SQUARE CHAINS. T( DN8 DWX Z < and offsets. NUV 140.4893 _____________( _______and offsets. DNO PPQ 100.1882 _ and offsets. and offsets. 0 5 ANO INRM 103.9778 and offsets. and offsets. 1HTN |N t R Offsets. 81.6307 and offsets. CNS HTN and offsets. and offsets. 109064 DPQ J XY 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 E, 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, szx fiun damental 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. HAP. Iv.1 Diagonals, Tie-lines and Offsets. 91 Fig. 78. A > T /\ \ D./' T'hus in the above figuretwo straight lines AB aDnd CD are mleasured through the entire length and breadth of the farm, or township, which is to be surveyed. The connecting lines AC, 3B, 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: C IMN SURVEYING. [PART t1 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'9 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 13 longer than it is wide, two ^ systems, of six lines each may be used with much " advantage, as in Fig. 79. ) Several such may be conm- /. I J bined when necessary. C Ilg, 80. In a case like that in Fig. o.; 80, five lines will be better \ than six, and will tie one an- other together, their points of intersection being carefully noted,. 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. -, Z_ -,s (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 somne elevated position, will be of much assistance in planning the general direction of the lines to be measured. (134) Inaccessilbe Areas. Fig. 82. A combination of offsets and /\ A. — -'- ~. tie-lines supplies an easy me- i — thod of surveying an inaccessible area, such as a pond, swamp, forest, block of houses, / ) &c., as appears from the fi-" gure; in which external bound- ing lines are taken at will and- "' -~_ * To find the exact point of intersection c~ these lines, which are only visnal 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 CHAIN 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 rectangle, and to measure offsets from the / sides of this to the angles of the field. (135) When one of the lines with which Fi. 8 an inaccessible field is surrounded, as in 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, i added to it. The triangle DEF is also D — Jto be added, and the triangle FGH deducted. To do this directly, it would be necessary to find the points of intersection P 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 differ- ence of the offsets AB and DE by half of BE; and the difference of DEF and FGH by multiplying the difference of DE and GiH by half of EG.* * For, making the triangle Dmn = ABC, then mnEO =- En X (mat CEM (DE - AB) X A EB; and so with the other pair of triangles. CHAP. Iv.] Diagonals, Tie-lines and Perpendiculars, 05 (136) Roads. A winding Road may also be surveyed thlas as is shown in Fig. 85; straight lines being measured in the road, Fig. 85. /. \ \ /x s/ \- / / s 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 ang les by tie lines and taking offsets from ther ln the blocks of hou ses. toJ the blocks of houses. 9(5 CHAIN SURVEYINt. [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 valleys 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 direction. 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 thLs 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 element in some of these demonstrations. All of them will constitute excellent exercises for students. CuAP. v.] Obstacles to Measaremento 97 PROBLEMS ON PERPENDICULA RS* P'oblem 1. To erect a pempendicular at any point of a ine. (140) Sirst Mflethod. Let A be the g. 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 _ - a portion of the chain not quite 1- times as I ng 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. ( 41) Seconcl Method. Measure off as be- Fig. 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~ 3 it out on 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 a-t 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 fbr tlio next problem may be applied to this. Many of these methods would seldom be required in practice, but cases somelimes occur, as every surveyor of much experience in Field-work has found to his serious inconvenience, in which some peculiarity of the local circum-stances forbids any of the usual methods being applied. In suci cases the collection here given will be found of great value. In all the figures, the given and measured lines are drawn witn fine full lines,.he visual lines, or lines 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 9S CHAIN SURVEYING. [PART 11 Problem 2. To erect a perpendicular to a line at a given points when the point is at or near the end of the line. (i43) First Mletzhod. Measure i.88 5U 40 links along the line. Let one asdistant 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 B / __ A. third take the 50 link mark, and 0 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. s9. 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 o; draw it tight; then will the 28 feet A ___ Is ___ mark be in the perpendicular desired. 0 1i (145) Second lMethod. HIold one end Fig. 90. of the chain at A and fix the other end at a 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 I I (the other end remaining at B) till it comes / in the line of CB at D. AD is the perpendicular required. CHAP. v.j Obstaeces to Measurement. 9 (146) Third l;iethod. Let A be the given Fig. 91. point. Choose any point B. Measure BA. 3 Set off, on the given line, AC = AB. On CB 2 AC2 produced set off from C, a distance C This -ill 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 any distance AB. From B, in any convenient direction, set off BC = AB. B ~ Then on the given line, set off AD = AC. On CB prolonged, set off E = 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 Method. 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. (1 49) 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. Probale 4. To let fall a perpendicular from a given point to a given line. (150) First Method. Let P be Fig.3. 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- -T around till the same distance meets t00 CHIN SURVEYING. [PART IL the line at some point Do 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 jelhod. 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 ~ B 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 Mlethod. Stretch a chain from the given point P, to some point, as A, of the given line. Fix to Fig. 95. the ground the middle point B of the chain AP, and swing around the end which was at P, or at A, till it meets the given line in a B point C, which will be the foot of the re- quired perpendicular. / (151) Second iethod. Takeany point, Fig. 96 as A, on the given line. Measure a dis- I tance AB. Let the end of this distance N on the chain be held at B, and swing around / / the end of the chain, till it comes in the -. 33 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 AC cG&P. v.] Obstacles to Measurement, 101 (155) Third Method. At any convenient Fig. 97. point, as A, of the given line, erect a perpen- dicular, 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 CP. c I D Then the distance from C to the foot of the CAxGP perpendicular, i. e. CD = C~l Problem 6. To let fall a perpendicular to a line, from an inaccessible point. (1S5) First Jilethord. Let P be the gien Fig. 98. point. At any point A, on the given line, set out a perpendicular AB of any convenient length. Prolong it on the other side of the line the H,, same distance. Mark on the given line a / point D in the line of PB; and a point E in -D A \ /''I the line of PC. Mark the point F at the in- \ tersection of DC and BE prolonged. The line c'. FP is the line required, being perpendicular F to the given line at the point G. (5.7) Second Method. 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 perpendi-\ cular, BD, to the visual line AP. Find the point at which these perpendiculars intersect, as at E (see Art. (133)), and the A ~ B line PE, prolonged to F, will give the perpendicular required. Problem 7, To let fall a perpendicular from a given point to v; inaccessible line. l02 CHAIN SURVEYING [PIAT (158) First Miethod. Let P be Fig. 100. the given point and AB the given A - ~ ~ lane. By the preceding problem, let fall perpendiculars from A to BP, at C; and from B to AP, at ID; the ^j/ line PE, passing from the given point P to the intersection of these perpendiculars, is the desired perpeni.di cular to the inaccessible line AB. This method will apply when only two points of the line are visible. (159) Second Method. Through the given point, set out, by the methods of Art. (16B5), &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. Problesm 1, To run a line, from a gzven 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 paral. lel required. (161) Second MZethod. LetABbe Fig. 101. the given line, and P the given point. A — DB 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 CP. Measure P DE. From P set off PF = CE; and from F, with a distance DE, and from P, with a distance = CD, describe arcs intersect. ing in G. PG will be the parallel required. If it is more con" venient, PC may be prolonged, and the equal triangle, CDE, be formed on the other side of the line AB. eVAr. v.] Obstacles to Measurement. 10O (162) Third Method. Measure.from Fg. 102. P to any point, as C, of the given line, and A-L. -- -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 Fi,. 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, F CD may be made one-half, or any other fraction, of CP,. and El be then made twice, &c., DE. (16l) Fifth Jfethod. From any. 1 point, as C, of the line, set off equal A —- ~ ~," distances along the line, to D and E. Take a point F, in the line of PD., p,,I Stake out the lines FC and FE, and I 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. To run a line from a given point parallel to an inaccessible line. (I16) First MIethod. Let AB Fr. o15. be the given line, and P the given A point. Set a stake at C, in the line h of PA, and another at any conven- ient point, D. Through P, set out, \ by the preceding problem, a parallel to DA, and set a stake at the point, E,3 where this parallel intersects DC prolonged. Through B 104 CHIIN SURVEYING, LPART 11 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 F. 1 T. at any point, C, in the line of AP, and another at any convenient place, as at D. Through P set out a parallel to AD, \ intersecting CD in E. Through E set \I\ out a parallel to DB, intersecting CB in F. The line PF will be the parallel re- C quired. (167) Altsnement and Ileasturment. We are now prepared, having secured a variety of methods for setting out Perpendi ulars 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 determined0 This operation is called BRanging the line; or Aliing 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,,;he 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. I.H Of Obstacles to Measurement; or how to find the length of a line which cannot be actually measured. S. 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. uv interpolate points in the line. * This word, like many others used in Engineering, is derived fiom 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 Alirsement. CP. v.1 Obstacles to IMeasiraieent. 105 A, TO PiIROLONG A LINE (169) By ranging with rods. When two points in 3i line are given, and it is desired to Fig. 107. prolong the line by ranging no 4 A 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 thei, 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, "4 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. (1t ) 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 clown 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 froir top to bottom of the rods will approximately attain the same end. When the lines to be ranged are long, and great accuracy is re quired, the rods may have attached to them plates of tin with oper 106 CHAIN SURVEYINGo [PART ti. ings cut out of them, and black horse-hairs stretched from Fig'. 10 top to bottom of the openings. A small telescope must then be used for ranging these hairs n 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 perpendlir lars Fig. I 09. The straight line, AB in the - -- B- ----- figure, is supposed to be stop- -D E F ped by a tree, a house, or other obstacle, and it is desired to prolong tho 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 EG and FHI, eaual 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 trangles. Fig. 110 The obstacles, noticed in the last arti- A B. G cle, may also be overcome by means of \ / I / 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 83d link at A; take hold of the 66th D link, and draw thei clhin 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 I). Make another CHAP. v ] Obstacles to Measuremeat. 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. (13S) By symmetrical triangles. Let AB be the line to be prolonged. Take any conv- Fig. 111. nient point, as C. Range AI v,. out the line AC, to a poinlt m' \.",,', such that CA'= CA. ":'Jn /' Range out CB, so that CB' /' - CLo Range backwards C/''' C A'B', to some point D, such -/ \ that DC prolonged will pass B A 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 manmere In this, as in all similar operations, very acute intersections should be avoided as far as possible. (174) By transversals Let AB be F g. 11. the given line. Take any two points C 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. V i Then the distance from D to P, a point in the required prolongation, will be P CDxBDxAE - BE x AC-xBD 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 ease the new distances CA and CD wi alone require to be measured. 108 CHUN. /NUVIG LPARS IX f AE be made equal to AC, then is DP -= B-CDBD BE-1) CDxAE 1f BE be made equal to BD, then is DP - C-AE - 1he minus sign in the lenorinators must be understood as only meaning that the difference of the two terms is to be taken, without regard to which is the greater. (1T5) By harmonic conjtRgaes i. 113. Let AB be the given line. Set a stake at any point C. Set stakes at'- points, D, on the line CA, and at^^','~ " E? on the line CB; these points,' /\a' D andc E, being so chosen that the /'\,/ ine DE will pass beyond the obsta-, e / cle. Set a fourth stake, F, at the intersection of the lines AE and K \. DB. Set a fifth stake G, any- C where in the line CF; a sixth stake, H, 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 KCI, and their intersection at P, will be in the line AB prolonged. (17t) ly tie coliploete quadrilaeraa. Let AB be the given line. Take any conven Fig. 11i lent point C; measure C from it to B, and onward, /- mn the same line prolonged, an equal distance to D. Take any other convenient point, E, such that CE and I 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 t their intersection in G. Range out FD and CE to Inter. sect in H. Measure GlI. Its middle point, P, is the required point in the line of AB prolonged. The unavoidable acute inter. sections in this constructicn are objectionable. CHAP v.] Obstacles to Measumement. 109 B. TO INTERPLAETL E POINTS IN A 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 fiolure. Fig. 115. ),ramt 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,:s vrel aim" pie 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 CIIAIN SURVEYING, [PART IX (T79) Across a valley. When a valley, or low spot, inter venes between the two ends Fig. 116. of the line, A and Z in the figure, a rod held in the low place, as at B3 would seldom be high enough to _ v; be seen, from A, to cover the distant rod at Z. In such a case, the surveyor at A should h:ld up a plumb-line over the point, at arm's length, and place nis 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. 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 Measuremenit, 11. 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 "6 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 pohits' on the same principle, by laying a straight stick on some support, going to each end of it in turn, and making it point successivelv to each end of the line. Th. " 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- Fi,. S18. ment for the same object, is represented a in the figure. AB and CD are two tubes, about 11 inches in diameter, con- A 1 x nected by a smaller tube EF. A piece lliii~-1!. of looking-glass, GH, is placed in the lli _ll lower part of the tube AB, and another, KL, in the tube CD. The planes of F i the two mirrors are at right angles to i! i l each other. The eye is placed at A, and i l the tube AB is directed to any distant object, as X, and any other object be- hind the observer, as Z, will be seen, ap- parently 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 wheE the two objects are near together. 112 CHAIN SIRVEYING [PART II. (183) Through a wood. When a wood intervenes between any two given Fig. 119. points, pre- D _ venting one from being. _ seen from the B' other, as in the figure, in which A and Z are the given points, prm ceed 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. (116), 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. Measure Z'Z. Then move each of the stakes, perpendicularly fiom 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. (18i) To an nvsible intersection. Let AB and C1) be twu lines, which, if prolong- Fig. 1s0. ed, would meet in a,- -z point Z, invisible from /..._ either of them; and let, \1 P be a point, from which / >a line is required to be,',/ set out, tending t1 this I / invisible intersection. C Set stakes at the five given points, A, B, C, D, P. Set a sitb stake at E, in the alinements of AD and CP; and a seventh stake BHAP. 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 lne. 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 II. 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 = ~Q^ il. 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. WVhen both ends of the line are accessible. B. When one end of it is inaccessible. C. lWhen both ends of it are inaccessible. A. W I IIBH $ E E S THE LIi[E ARE ACCESI BLEo (188) By plrpejnlkeulars. On F. reaching the obstacle, as at A in A, l, D the figure, set off a perpendicular, AB; turn a second right angle at, B -B 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. (17S). Fig. 1211. (187) By equilateral trianles. A l The method given in Art. (172) for / determining the direction of a line through an obstacle, will also give its C length; for in Fig. 121' (Fig. 110 repeated) the desired distance AGis equal to the measured distances AD, or DG. I} 114 CHAIN SURVEYING. [PART U (188) By symmetrical triangles. Fig 122 Let AB be tie distance required. MIeasure from A obliquely to some A point C, past the obstacle. Mea dP 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; -- ~ set another stake, D, so that C andc 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. Set a fourth stake, F, at the intersection of EA and CD. Measure C, AF and FE. Then is AB = A (FE-AF). Fig. 124. (190) In a Town, Cases may occur, \ in the streets of a compactly built townm 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, be- D tween points situated on two streets which meet at the point C, and between which rims a cross-street, DE. In this case measure AC, CE, CD, DE and CB. Then is the required distance CHAP. v.] Obstacles to Measurement, 115 AB = /(AC - BC)2 + [DE2 (CE - CD) ACxBC L CDxCE As this expression is somewhat complicated, an example will ba given: Let AC = 100, CE -= 40, CD = 30, DE = 219 and CB 80; then will AB = 51.7. B. WIEN ONE END OF THE LINE IS INACCESSIBLE. (191) By perpendiculars. This principle Fig. 125. may be applied in a variety of ways. In Fig. /I 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 perpen- dicular to CB, and continue it to a point, D, in the line of A and B. Measure DA. Then is AC2 D -AD (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 AE x CD and EC. Then is AB= _ - CD i 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 = A AB. Fig. 127. (193) Otlerwise: At A, in Fig. 127, mea- B sure a perpendicular, AC, to the line AB; and 4 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. / AC x AD Then is AB AC xE A ^-DE-AC -,Z16 I 116 CHiAN SURVEYING. [PART [T 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. ~-~ T c Then is AB AC x AE -DE - AC' / (I95) By a parallelogram, Set a stake, C, 129 in the line of A and B, and set another stake, D, 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 arc from D, intersecting the first arc in E. Or, Ai\ F take AC and CD, so that together they make I one chain; fix the ends of the chain at A and; D; 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 T^ rn AO A~xAF AOxCD EF. Then is AB A A; or, CB A xC EF; or, EF (196) By symmetrical triaig'les, Fig.130. Let AB be the required distance. From / A measure a line, in any convenient di- - rection, as AC, and measure onward, in the same direction, till CD = AC. Take /, Ax any point E in the line of A and B. D - Measure from E to C, and onward in the / i 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 Bo If more convenient, make CD -= AC, and CF =- CE, as shown by the finely dotted lines in the figure. Then will DG = 2 AB. CHAP, v.] Obstacles to Measurement. 117 (197) Otherwise: Prolong BA to Fig. 131. some point C. Range out any con- ___ _ venient 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- - For this object, take, on AC, some point D X \-\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 trausversals. Set a stake, C, Fig. 133. 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 AC x AF x DE D CE x DF - AF x DE' E If the point E be taken in the middle of CD, (as it is in the AC x AF figure) then AB D - A AG x DE If the point F be taken in the middle ofAD, then AB = CE - DEo The minus signs must be interpreted as in Art. (174). (l9~) By hlarmonc 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, ii 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 " -— Di of AB and FG. Measure AE and ElH. Thenis AB AE Al AE -- EII 118 CHAIN SURVEYI4NG [PART 11 6(20) To an inaccessible line The Fig. 134. shortest distance, CD, from a given point, A -,^B 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, line AC; set a second, F, at the inter- \ \ section of EB and CD; a third, G, at c the intersection of AF and CB; and a fourth, H, at the interseo. tion of EG and CD. Measure CH and HF. Then is CDCH-HF; or, CD = CH CH+HF or CD CHx CF CH-xF o - C-H-F 2CH-CF 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: \ I intersection, DE, for example,. ) may be found thus. From any -- \point B, on one line, measure. 1i BD, and continue it, till DF- DB. From any other point, G, of the former line, measure GD, and continue the line till DH = GD. Continue HF to meet DC in some point I. Measure KD. KD will be equal to the desired distance DE. BE can be found by measuring FK, which is equal to it. If DF and DlI, be made respectively equal to one-half, or once third, &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..] Obstacles to Measurement. 119 C. WIHEN BIOTH ENDS OF THE LINE ARE INACCESIBLE. (202) By similar triangles, Let AB Fi 136. be the inaccessible distance. Set a stake at any convenient point C, and find the distan-' 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 x CDA from C, on the line CB, to some point E. Measure CA. DE. Then is AB A- x D CD Fig. 1.37. If more convenient, measure CD in the A. B contrary direction from the river, as in Fig. 1379 instead of towards it, and in other respects proceed as before. (203) By parallels, Let AB be the in- i. 3 accessible distance. From any point, as C, - range out a parallel to AB, as in Art. (1065), &c. Find the distance CA, by Art. (191), -X / &c. Set a stake at the point E, the inter- section of CA and DB, and measure CE. D/ Then s AB = CD x (AC —E) Then'is AB ~~CE (204) By a parallelogram, Set Fig 139. a stake at any convenient point C. Set stakes D and E, anywhere in - the alinements CA and CB. With K I[iCL / D as a centre, and a length of the'/' 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 CHI SUlYVEYISG, [PARa II CD x DF X GH and l-G. The inaccessible distance AB CD x DF- x Gl CD2 x GH If CD -CE, then AB CD x FH. (205) By symmetrical triangles.'Take any convenient point, as C. Set stakes at two other Fig. 140. points, D and D', in the same A line, and at equal distances from C. Take a point E, in the line of AD; measure from / it to C, and onward till CE' - CE. Take a point F in -, the line of BD'; measure from, it to C, and onward till CF' /'. CF. Range out the lines AC /' and E'D', and set a stake at, /- " theirintersection, 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 clistances -- on each side of it, in the line of CA, to D and D'. Set \ D i off the same distances from \ I / C, in the line of CB, to E and - /x E'. Through C, set out a \\ y \ parallel to DE, or D'E', and./ ----'D set stakes at the points F i/ \\ and F' where this parallel /', intersects AE' and BID'.:B1 Range out the lines AD' and EF', and set a stake at their intere section A'. Range out the lines BE' and DF, and set a stake at their intersection BP. Measure A'B' and it will be equal to the desired distance AB. CHAP. v.] lestadles to MIeaisurement. 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, as AD, AE,B \. in the figure in the margin, and also DE, and / the area of this triangle found by any of the methods which have beengiven. Then is the desired area of the triangle ABC = area of ADE AB x AC AD x AE' (2408) Quadrilaterals. In the case Fig. 143. of a four-sided field, whose sides cannot -- / be measured, or prolonged, but whose diagonals can be measured, the area i((( may be obtained thus. Measure the diagonals AC and:D D; and also the portions AE, EC, into which one of 9i' them is divided by the other. Calcu- - late the area of the triangle BCER by the preceding method, or any of those heretofore given. Then the area of the quadrilateral ABCD = area of BCE x A BD BE x CE (209) Polygons. Methods for obtaining the areas of inac cessible fields of more than four sides, have been given in Arts. (101,) &e. PART 11T COMPASS SURVEYING; OR By the Third MIethod. CHAPTER I. ANGULAR $URJVEYIN IN GENERAL (210) Aingulcar Surveying determines the relative positions of points, and therefore of lines, on the THiIRD 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 Lorizontal, or level, when it is parallel to the surface oi standing water, or perpendicular to a plumb-line. A line is horizontal whent lies in a horizontal plane. [CaAP. i. Angalar Surveying in geaeral. 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 \ (by any of the methods of Geometri-, cal Drafting) into three hundred andl cl sixty equal parts, or "s 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 gradu(ations. 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 Goniometer, or Angle-measurer. (211) Now let it be required to measure Fg. iz. the angle between the lines AB and AC. Fix the staff in the grcund, so that its centre shall be exactly over the intersection of the two lines. Turn the alidade, so that it points, (as ~-' -C determined 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 COMIPASS SRlVE INCG. [PanT 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. (2i5) 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. (106 ); 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- F-i 146. ed across the other end of it, as in the figure; so that () ( 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 sup" ported 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 Enyineer'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 li eridian. Thus, in the figure, the- N line AB makes an angle of 30 degrees with the line' oO' AN, and the line AC makes an ingle of 75 de- a, grees with AN The difference of these angles, /.:..- C or 45 degrees, is the angle which AC makes. with AB, agreeing with the result obtained in At (214). S [CHAP. I. Angular Surveying in general. 1 25 (218) Surveying with the compass is, therefore, a less direct operation than surveying with the Transit or Theocolite. 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,,s 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 ccnsidered in Part IV), we will first take up CorIPASS SuRVEYING. (21 ) 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, Trianguatr, 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 Chemrnement. It has no special name in English, but may be called (from the American verb, To progress), the Metlod of Progression. The other system, the iethod 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 Trianglu lar Surveying is the AIethod of Intersections; since it determines points by the intersections of straight lines. 126 COMPASS SURlVEYINS fPART IrT. - - - - - - N-~- -- ~ -- -,,,',,, ^ ^ ^ ^,\ ^ "i 1i, C.it Xe~s I Bo V~/ #77~i i ik.. _ _ ci d CHAPTER II. TIE 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, sa that it may turn freely, and thus be enabled to continue pointing in the same direction (that of the " Magnetic Meridian," 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 wire 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. IKater 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 CMIPASS SURVEYING. [PART III The needle should not cone 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 di rection 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 liable 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 be applied to the colm pass, a graduated circle with vernier should be added, thus converting 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 on each side of the figure of the Compass, on page 126. It is itself drawn iln' Mili tary Perspective." CHAP. 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 0~ around to 360; 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 dif ference 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 Y, or West; and the left hand 90-degrees-pomit is marked E, or East. The reason of this will be seen when the method ofusing the compass comes to be explained in the following chapter. (225) The Ponts. In or- Fig. 150. dinary lanid surveying, only four points of the compass have /. 1 names, viz: North, South, East j/i and West; the direction of a /iJ^ / / \ line being described by the an- gle which it m-akes with a North I $ and South line, to its East or tog 1 0Eb a its West. But for nautical pur- S5/ poses, the circle of the compass / is divided into 82 points, the names of which are shown in 9 180 COMIPA$SS SRVEHIN [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, 2~ 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 neecle be ___ straight. Fix the pivot temporarily, so that the ends of the needle may cut oppo- site degrees i. edegrees differing by 1800. The condition of things at this ~S C) F]F~ig o15'~. 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- \ Date1 by a dotted line.* Now bend the needle, as in Fig. 153, till it cuts divi- \ sions midway between those cut by it in * This is another example of the fruitful pr;uciole of Reverston, first noticed in Art. (105). CHAP, n.] The Compass, 181 its present and in its former position Eig 153. This makes it certain that the needle is straight, or that its two ends and its cen- tre lie in the same straight line. Secondly, to put the pivot in the cen- I tre. 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, 45~ and 90~. 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. X'S In Fig. 154, the line of sight AB is represented as passing to one side of the centre, and the needle as pointing to 460. 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 COMIPASS SURVEYING. [PART III, siglht 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 200, the 0~ 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 sigdal 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 0~ 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, wtlich carries it and the sights, so tha t t 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 irt CHAP.i.] Tlhe Compass, 138 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 posiion, 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. (,308) Tripod1 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 F. Fi. 157. and stiffness are the qualities desired. The most usual form is shewn in the figures l v 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 adcvantage of being ve- \ ry 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 i13$ COMPISS SURVEYING. [lART III Fig. 158. Fig. 159. Fig. 160 % 0. j iUU lilii ill1llllililillliilt1111111 illM i V li fflu~iI~w'lii'lj''' i I llS 911111,> 4 termlinates 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'G 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) acob'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 sav ing 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 joint, 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 conven iently 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 direc. tion 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 peculia 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- r s 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- c ed. The circumference of the card is usually di- D vided to' or 1 of a legree. C is a prism, which,, the observer looks through. 0,t l The perpendicular thread A of the sight-vane, E, and the divisions on the card, appear tcgether 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 in slide on or off the sight-vane, E; and it may be reversed at plea sure, 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 of 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, looling 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 40~ 30', and the latter 100 15', both east, or both west, Fi. 162. from the no-th or south, the angle will be 300 16'. The divisions are generally numbered 50, 100, 150 / \\ &c. around the circle to 3600. // The figures on the compass card. _ are ieversed, or written upside\ \ down, as in the figure (in which only every fifteenth degree is mark-' a ed), because they are again reversed by the prism. CHAP. II.] The Compass. 137 (233) The prismatic compass is generally held in the hand, tho 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 arith. rnetical 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 4- 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 1 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 tle 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 frecisely, but not always correctly. The statement that two andt two make five, 2s 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 corn pass, 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. TEI FIELD WORK. (235) Taking Bearings. The " Bearing" of a line is the an. gle 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 " plmnb," 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 obliquelv. CHAP. II.] 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 1I distant from the observer), the A North end of the needle was at the / i point A, the bearing of the line sighted on, would be North 450~: 0 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. 600 W; if at H-, North. D (23 ) We can now understand why W is cn 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 recuired, and suppose the sights in the first place to be pointing in the direction of the \ / \ needle, S N, and the North sight t \ / o\ to be ahead. When the sights (and 1 ~- - the circle to which they are fasten- \ ed) have been turned so as to point' 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 bearing of the line to C be desired, half-way between North and 140 COMIPASS SURVEYIGT. [PAET 11. East, i. e. N. 45' 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 cn the left. These letters must then be reversed in the mind before the bearing is noted clown. (237) Readung 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. 31~ E. and N. 31}~ 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 VernierA Vernier arc 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 CAP. 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 tne 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 X - 177 the wrong number of the two between l which the needle points, such as reading 340 for 26~, in " case like that in the figure. 142 COMPASS SIURVEYIIi,. [PART Iti 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 frieC tion 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 formei position, and repeat this a number of times. CHAP. 111.] The Field Work, 143 (240) Back Sightse 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. 30~ E. would then be S. 300 W., and so on. (241) Local attractioln 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 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 COIPISS SURVEYING. [PART Ixl. figure for the case, as in Fig. 167. In Fig. 167. it, suppose the true bearing of BC, as given by a fore-sight from B to C, to be N. 400 E., but that there is local at- / traction at C, so that the needle is drawn aside 10~, 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- If 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. When 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. (242) Ainles of deflectin. 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 ST. 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 S.), and the last letters also alike, (i. e. both E. or both WV.), take the difference K' of the bearings. Example. IfAB bears vr —~-~~~ N. 30~ E. and BC bears N. 10~ E., the angle of deflection CBB' is 200 /o.Fig. 169. N /A Case 2. When the first letters are \R o alike and the last letters different; take / the sum of the bearings. Ex. If AB bears N. 40~ E. and BC bears N. 20~ -' —-- W.; the angle CBB' is 600.,40~ I0 Fig. 170. N i Case 3. When the first letters are different and the last letters alike, subtract the sum of the bearings from 180~e 1/ Ex. If AB bears N. 30~ E. and BC wo - - bears S. 40 E.; the angle CBB'is 110~/ / 10 S 10 146 COMPASS SURVEYIG [PART III Fig. 170, Case 4. When both the first and 300last letters are different subtract the' difference of the bearings from 1800. I/ o Ex. If AB bears S. 300 W. and BC - -. —- -Wbears N. 700 E.; the angle CBB' is t40O / \, / S! 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 180o; 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 t. 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 G(eneral 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 suen 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 M ork, 14i 900, this shews: hat 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. 32~ E.; N. 80 E.; S. 480 E.; S. 180 W.; N. 7T3-o W.; North. Suppose the first side to become du- North; the changed bearings will then be as follows: North; N. 480 E.; S. 800 E.;. 140 E.; S. 7410 W.; N. 320 W. To apply the rule to the 6" 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. 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. 148 COiPASS SURVEYING [PART IIL which can be most conveniently measured in the road, and lmea 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) Cheeks 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. ciP. IIT.] The Field Work. 149 (217) 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 clown on the lines their bearings and lengths. An improvement on this is to actually lay down the precise bear mgs 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. (219) A third method is to represe-t 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 descriptior 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 Ones black; re- and blue lines being used on the real map. 150 COMIPA$S SURVEYIN. [PARIT Im Fig. 174. 4 O F S 3E I 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 s8 that he may readily find them again. Then set the compass al 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 lhe 1st 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) Fleld-nites. The Field-notes of a Farm survey may be kept by any of the methods which have been described with iefer. ence to a Line survey. Below are given the Field-notes of the same field recorded by each of the methods. First Method. Fig. 175. N 83IT 2,.t 4> J 0 / 1F9~"a 7 ^^~,~ ^ /'^^ >.~~ "^Y " 152 COMPASS SiRVEYING. [PART m1 Second Third Fourth Method. Method. Method.* o (1) -()- j 3.23 STATIONS. BEARINGS. DISTANCES. or;C' 1 N.350 E. 2.70! co, ^.~ 2 N. 831o E. 1.29 t } 3 S. 570 E. 2.22 (oi (5 ) |4 S. 341 W. 3.55 ( ^' 5 N. 56^ W. 3.^3 > ~ (5)- ~3~ ~ o 1t 3.54 most0 esyerdy1- heTrMtoFig. 176. 2.77 0(4) -0.90 — 1 o _0 2.22 -0.25CO o;S' (2)1.34 a a Art, (2)b t te d eeo m toahi a di 7culties-2.70 -(1)When the Field-notes are recorded by the Fourth Method, the offsets may be kept in a separate Table; in which the 1st eo.umn 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 oblionely upward as directed in Art, (249) but arn not so printed here, from typographical diffi culties. CHAP.. IIi.j The Field Work, 153 column the lengths of the offsets, and the 4th column the side of the line, " Right," or "1 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 prlceding columns, separated into Right and Left, one being additive to the field, and the other subtractive. (250) Tests of accuracy. 1st. The check of intersections doscribed 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 6 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 sun vey; 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 SIRVtLEIG. [PARr III augles which they make with NS-or incorrectly, as being the angles which they make with NS'. This rule would therefore prove the work in either case. (258) Method of Radiation, A field may be surveyed fronm one station, either within it or withiout 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 Mfethod 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) MCethod 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 fiethod 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 dissat isfaction 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) Townl Srveying. 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 offiets as you go along, with the offsetstaff. Proceed in this manner from street to street, measuring the distances and offsets as you proceed. Fig. 177. -~-~ —L~ —~Z-/\~' —---- — ~l —------:B I *I__' " In the description of land conveyed, the rule is, that known and fixed mon< aments 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 land marks maliciously is an indictable offence."-Kent's Commentaries, IV, 515 156 COMPASS SIUREYING., [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, 6 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 TiE SURVEY. (263) The platting of a survey made with the lompass, 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 plant ting angles was referred to in Art. (lI), but its explanation has been reserved for this place. (264) With a Protractor. A Protractor is an istrument 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. iegees, a bered in both irect degrees, and numbered in both directions. It is, in fact, a mnminr ture of the instrument, (or of half of it), with which the angles have been measured. To lay off any angle at any point cf 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. k s \\" \ \\\\- 7//- \/ / / / -, 1 1.03) ik 1 3 10(98,6) fO0 _ 0 I0 t / 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 Magnetic 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 instru. ment 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 f 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 procured, nail down on a table of any shape a straight-edged ru \ ler, and slide along against it the outside of the stock of a T 1 - 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, arproximately 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 reprewent the first station, 1 in the figure. Its place must be so chosen Fig. 181. S.!i adl: I) a 160 COMPASS STURVEYING, LPART iR 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. (261). 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 "6 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. rV 1 Platting the Survey. 1l8 Fig. 182. N 2 3 /1 h 3 5 (267) The Compass itself may be used to plat bearings. For tlis 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 B' lines platted according to'- o the given bearings and cl I ~; distances, and suppose that O the last course comes to E, AC' / instead of ending at A, as it should. Suppose also.. / that there is no reason to E suspect any single great D error, and that no one of the lines was measured over very rough 11 1 2 GOMPASS SURVEtiINe IPARTf i 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 figure, 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 here after, is, however, the best for effecting this object. (269) Field Plattilng 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 "Keeping the fieldnotes." One method of doing this 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 / l of Bristol-board, or other stout drawing paper,) with lines ruled across it at equal distances ~ —- -i of some fraction of an inch. A - _ bearing having been taken and _: _r _ 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 mny convenient number of links. This was demonstrated by Dr. BOWDITCH, in No. 4, of " The Analyst.' CHAP. 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. The parallel lines on the paper be equi-distant, as in common ruled writing paper) will now K / represent meridians. Place the centre of the protractor on the meridian nearest to the 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 - __ c _ I _ B2 the two sides of the right-an- / gled triangle, of which the line AB in the figure is the hypoth- E enuse, is 1, that line makes an angle of 450 with the meridian. - C ~ If the ratio of the long to the short side of the right-angled / / triangle of which the line AC - hs the hypothenuse, is 4 to 1 l the line AC makes an angle A 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 ben prepared. TABLE FOR PLATTING BY SQUARES. Raioo o. 0P Ratio of f i Ratio of Ratio of 2=d[ o long side to 0 a "' long side to o' longside to o 0 short side a short side. 0 ehort side. a. o 10 57.3 to 1890 160 3.49 to 1 740 310 1.664 to 11590 20 28.6 to 880 170 3.27 to 1 730 320 1.600 to 1 580 30 19.1 to 1 870 180 3.08 to 1 720 330 1.540 to 1 570 40 14.3 to 11860 190 2.90 to 1710 340 1.483 to 1 560 50 11.4 to 1850 200 2.75 to 1 70 350 1.428 to 1 550 60 9.5 to 1 840 210 2.61 to 1 690 360 1.376 to 1 540 70 8.1 to 1 83 220 2.48 to 11680 370 1.327 to 1 530 80 7.1 to 1 82 230 2.36 to 1 670 380 1.280 to 1 520 90 6.3 to 181 240 2.25 to 1660 390 1.235 to 1 510 100 5.7 to 1 800 250 2.14 to 1 650 400 1.192 to 1 500 110 5.1 to 1 790 260 2.05 to 1 640 410 1.150 to 1 490 120 4.9 to 11780 270 1.96 to 1 630 420 1.111 to 1 48 130 4.3 to 1 770 2801.88 to 1 620 430 1.072 to 1 470 140 4.0 to 1 760 290 1.80 to 1 610 440 1.036 to 1 460 150 3.7 to 1750 30011.73 to 1 600 450 1.000 to 1 450 To use this table, find in it the ratio corresponding to the angie 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 num ber 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 = 5-1 squares, should be counted upward, to E, and AE would be the required course. (272) With a paper protractor. Engraved paper protractori may be obtained from the instrument-mnakers, and are very convec This and all the following ratios may be obtained directly from Trigonome'rical Tables; for the ratio of the long side to the short side, the latter heinq;aken as unity, is the natural cotangent of the anale. MHAP. IV.] Platting the Survey. 1 05 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 i ig. 187. use. Choose a convenient Npoint for the first station. ~-/ Suppose the first bearing to /, C be N. 30~ E. The line pass- \ ng through the centre of the 2 \ E circle and through the oppo- ^^ site points N. 30~ E. and S. / \ 30" W. has the bearing re- O 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. 60~ 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 plats. 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 SURVEYINSG [PRT' III not taken on the ground, are likely to be required, If tie mnep 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 cenire, and so that its 0~ and 90~ lines are parallel 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 0. 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 CH, 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. 18. example, to draw a line making at A an angle n of 40~ with AB, take, in the dividers, the dis- C. tances from 0 to 60 on the scale of chords; with B this for radius and A for centre, describe an in- @ o Jefinite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the arc 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. CHAP. IV.] Platting the Survey. 167 tance will be the chord of 40~ to that racius. 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. (275) MWth 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, Dne 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 arc 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 1rmt 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 Sp,2?7ypment, in the language of Trigonometry. 168 COMPASS SlRVEINGt C[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 sises. 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 lor, the chord of any angle is equal to twice the sine of half the angle. To find the cosine, pro. ceed 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 fiactional are or angle. Fig. 189. t The reason of this is apparent from the figure. DE is the sine of half the angle /A. BAC, to a radius of 10 equal parts, and t i0 / BC is the chord directed to be set off, to a, radius of 5 (:qual parts. BC is equal to DE; / for BC = 2.BF, by Trigonometry, and DE i " -- 2.BF, by similar triangles; hence BC = ~ DE. CHAPTER V. LATITUDES AND DEPAR1TURES (27S) Definitionso 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 Df the line; or its Northing 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 Departure. 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 oa the earth not being a plane. 170 OINPASS S UIVEYIN, [PART III that of Art. (9)), the point S is determin- Fig 190. ed by the angle SAC and by the distance AS. It is also determined by the distances AC and CS, measured at right jangles to each other; and then, supposing c 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 Departnres, 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 -- 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. IBAC. Hence, to find the Latituce of any course, multiply the natural cosine of the bearing by the length of the course; and to find the aDeparture 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 +, plu2s, or additive; if it be Southerly, the Latitude will be South, and will be marked with the algebraic sign -, ginus, or subtractive. If the course be Easterly, the Depart6ure will be East, and marked 4-, or additive; if the course be Westerly, the Departure will be West, and marked -, or subtractive. rtHA. v.j Latitudes and Departuies. 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. f/When the Bearing and Latitude are given; Latitdle e Distance =.td Latitude x sec. Bearing, cos. Bearilng Departure = Latitude x tang. Bearing. VWhen the Bearing and -Departure are given; Distance = Departre = DepartuIt x cosec. ]Bearing, sin. Bearing Latitude = Departure x cotang. Bearing. When the Distance and Latitude are given; Latitldle Cos. Bearing = -Listace 0 Distance^ Departure = Latitude X tang. Bearing. When the Distance and Departure e re given ^. *^. Depa rture Sin. Bearing = Dietallce Distance Latitude = Departure x cotang. Bearing. VWhen the Latitude and Departure are given; rn i. _- ~ ~ Departure Tang. of Bearing -epLatiture 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 = V/(Distance2 -Departure2) Departure = V(Distance2 - Latitude2) (281) Traverse Tables. The Latitude and Departure of any tlistance, for any bearing, could be found by the method given in Art. (279), with the aid of a table of Natural Sines. But to * Whenever sines, cosines, tangents, &c,, are here named, they mean the nait val sines &c., of an arc described with a radius equal to one, or to tle unit by which the sines, &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 Tabie 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,f are the Latitudes and Departures for that bearing, and for the respective distances-1, 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 mere 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 1/91, 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 from 1 to 10, The Table ir this volume was calculated for it, and then compared with the one just mentioned. t In using this or any similar Tabla, lay a ruler across the page, just above or below the line to be followed out. Tnis is a very valuable mechanica. assistance CHAP. v.] Latitudes and Departures, 173 Write clown 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 hav ing a distance of 873 links, and a bearing of 20~, be required. In the Table find 20~, 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 Iere 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. It is better in such cases not 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 SIRVEIHNG. [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 50~, &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 BAG be the bearing of the course AB, then is AC the Latih tude, 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 63{~, and the distance 3469 links. Proceeding as before, we have I)istances. 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 10 ~, take the half sum of the Latitudes and Departures for 10I~ and 10~0. 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 themn; to Platting them, and to Caleulating their content.@ * The Traverse Table admits of many other minor uses. Tlhs, 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 tu be made in chaining on slopes (see Art. CAAP. v.] Latitudes and Departures. 175 (282) Application to Testing a Survey. It is self-evident, that when the surveyor has gone completely arouni 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.1 I ________ ______ N. S. E. W. N. 35 E. 2.70 2.21 1.55 2 N.8380 E. 1.29.15 1.28 3 S. 57~ E. 2.22 1.21 1.86 4 S.8344jW. 3.55 2.93 2.00 5 N. 5610 W. 3.23 1.78 2.69 4.14 4.14 4.69 4.69 The entire work of the above example is given below. 350 1638 1147 3410 2480 1688 57340 40150 4133 2814 4133 2814 270. 221.140 154.850 355. 293.463 199.754 (26)); ftr, look in the column of bearings for the slope of the ground, i. e. the angle it makes witi the horizon, find the given distance, and the Latitude aorreaponding will be the desired horizontal measurement, and the difference between it and the Disqance will be the allowance to be made 176 COMPASS SIURVElYINI, [PART IIL 83~0 113 994 560 1656 2502 226 1987 1104 1668 1019 8942 1656 2502 129. 14.579 128.212 323. 178.296 269.382 57~ 1089 1677 570 1089 1677 The nearest link is taken 1089 1677 to be inserted in the Table, 1089 1677~ and the remaining Decimals 222. 120.879 186.147 are neglected. 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 inaccuraf. cies, 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 I 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 omis sions 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. CHAP v.] 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.,TIT. D. CORRECTED CORRECTED STA. BEARINGU, DIST. E.__ _ l': LATITUDES, DEPARTURES. N.+ S.- E.+ W.- N.+ S.- E.-+ W.1 N. 520 E. 10.63 6.54 8.38 6.58 8.34 2 S. 294~ E. 4.10 3.56 2.03 3.55 2.01 3 S. 314 ~W. 7.69 6.54 4.05 6.51 4.08 4 N. 61~ W. t.13.46 6.24 3.48 6.21 29.55 10.00 10.10 10.41 10.29 10.06 10.06 10.35 10.35 The corrections are made by the following proportions; the nearest whole numbers being taken: For the Latitudes. For the Departures. 29.55 10.63: 10: 4 29.55: 10.63: 12 4 29.55 4.10::10: 1 29.55: 4.10: 12 2 29.55.69:10: 3 29.55: 7.69 12 3 29.55: 7.13 10: 2 29.55:.13 12 3 10 12 * A demonstration of this principle was given by Dr. Bowditch, in No. 4 of u The Analyst." 12 178 COIPISS$ SIRVEYING. [PART 1i 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 De. partures. 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 subtractiveo Taking the example given on page 175, Art. (282), and beginning with Station 1, the following will be the results: TOTAL LATITUDES TOTAL DEPARTURES SA. FROMr STATION 1. FROM 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 imks.* This gives the point 2. Join 1....2. From 1 again, set This is most easily done with the aid of a right-angled triangle, sliding one of the sides adjacent to the right angle along the blade of the square, to which the other side will then be perpendicular. CIIAP, V.] Latitudes and Departures 179 off, upward, 236 Fig. 192. links, to B, and from B, to the right, per- __ 2__ pendicularly, set off 283 links, which will fixthepoint 3. Join 2.... 3; and so pro- ------ ceed, setting off North Latitudes along the Meridian l upwards, and South Latitudes along it downwards; East Departures perpen dicularly to the right, D a ~ --- and West Depar- a tures perpendicularly to the left. The advantages of this method are its rapidity, ease and accua racy; the impossibility of any error in platting any one course affecting the following points; and the certainty of the plat "co6rn ig together," if the Latitudes and Departures have been " Balanced." CHAPTER VI. CALCULATING TIE CONTENT. (2S6) Mefhods. WHEN a field has been platted, by what ever 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 Rectaro gular, or the Pennsylvania, or Rittenhouse's, method of calculationl. (287) Definitions, Imagine a Meridian line to pass througt 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 perpendiculardlistance 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.~ The difference of the Longitudes of the two ends of a line is called the Departure 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. Thomas Bu1rgh'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 " Differencea af Latitude,' usually but anomalously united with the word " Departure," bor. rowed from Navigation, seems to put beyond all question the propriety of the hinovation 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 ABODE. From Fig. 193. the middle and ends of each side N C draw perpendiculars to the Meridi- I an. These perpendiculars will be ----- M the Longitudes and Departures of the respective sides. The Longi- tude, FG, of the first course, AB, is evidently equal to half its Depar- E-~7' ture HB. The Longitude JK, of the second course, BC, is equal to A JL + LM + MK, or equal to the V.a T I Longitude of the preceding course, --- / \ plus half its Departure, plus half the Departure of the course itself. A - i The Longitude, YZ, of some other course, as EA, taken anywhere, is $ equal to WX — VX - UV, or equal to the Longitude of the proe ceding 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 DTuble 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 tlhai 2.)urse, plus the Departure of the course itself. 182 COMiPASS SUJRVEYING [PART Inl 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 itsef*. 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 Con. tent of a field, by means of the "6 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 Nk plain that its content is equal to the differ- A ence of the areas of the Trapezoid DBCE, D 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 n --- ----- -- the product of AD by FG; i. e. equal to... \ the product of the Latitude of the 1st course by its Longitude. ) The area of the Trapezoid DBCE is equal C to the product ofDE by half the sum of DB and CE, or by HJ; i. e. to the product of i 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.oirn.,ach other on the ground XHAP. VI.] Calculating the Content. 183 ducts. The Difference 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. 11. fore, it is seen, on inspection, that its area N would be obtained by taking the two Trian- B gles, ABEADG, from the figure EBCDGE, or from the sum of the two Trapezoids EBCF and FCDG.:l —--------- The area of the Triangle AEB will be A 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. c found to equal the Latitude of the 2d course D 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 will 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 ther&o fore 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 SIURVEYINO [PART IIn arising from multiplying together the Latitude and Double Longi. tude 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 Distances, 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 Double Longitudes, as in Art. (288), with reference to a Mleridian passing through the extreme East or WVest Station, and place them in the eighth column. 5. Multiply the Doouble 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 anl South, and East and West lines, and consldering 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) Exanmple 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, + 1.55 D. L 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, + 7.52 D. L. i+ 1.G86 are multiplied by the corresponding Latitudes, - 2.00 and the content of the field obtained as directed 4 i- 7.38 D. L. in the General Rule. 2.69 This example may serve as a pattern for the 5 - 2.69 )D. Lmost compact manner of arranging the work.'' p LATITUDES. DEP'TURES. o DOUBLE AREAS. ~rATION.BEARINGS.'TANCES. N.+ S.- E.+ W.-LONGITUDES. N. + p S. I N. 35~ E. 2.70 2.21 1:55 + 1.55 3.4255 2 N. 83o E. 1.29.15 1.28 + 4.38 0.6570 3 S. 57 E. 2.22 1.21 1.86 + 7.52 9.0992 5 N. 560 W. 3.23 1.78 2.69 -+ 2.69 4.7882 4.14 4.1414.69 1 4.69 1 18.8707 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. station, 4 in the figure. The Double Longitudes - 2.69 could then have been calculated as in the mar- 5 - 6. 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"'"'"'"""z -+ 1.55 the North products will be found to be 29.9625, + 1.28 and of the South products 8.1106, and their - 5.00 D L. 4 1.28 difference to be 21.8519, the same result as be- + 1.86 fore, i (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 cheek 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 COIPASS SURVEYINS. [PART 1n, All the examples (except the last) are from the author's actual surveys. Fig. 196. Example 2, given below, is also fully worked out, as anoth er pattern for the student, whc need have no difficult 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). IS- LATITUDES. DEP'TURES. DOUBLE DOUBLE AREAS. STATION. BEARINGS. TANCES. N. +-. E.+ — W.-=LONGITUDES. N. + S. - 1 N. 12 E. 2.81 2.75.60 + 6.56 18.0400 2 N. 76~ w. 3.20.77 3.11 + 4.05 3.1185 3 S. 24,~ W. 1.14 1.04.47 +.47.4888 4 S. 48~ E. 1.53 1.02 1.14 + 1.14 1.1628 5 S. 12~ E. 1.12 1.09.24 +. 2.52 2.7468 6 S. 77~ E. 1.64.37 1.60 + 4.36 1t.6132 3.52 3.5213.58 13.581 121.1585516.01a 6.0116 Content OA. 31. IP. 2)15.1469 Square Chains, 7.5734 Example 3. Example 4. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N.52~ E. 10.64 1 S. 21~ W. 12.41 2 S. 293~ E. 4.09 2 N. 83~0E. 5.86 3 S. 36 1 3 W. 7.68 3 N. 12 E. 8.25 N. 610 W. 7.24 4 N. 470 W. 4.2.1 Ans. 4A. 3R. 28P. Ans. 4A. 2R. 37P. Ekxample 5. _Example 6. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N.3410E. 2.73 N. 350 E. 6.49 2 N. 85 E. 1.28 2 S. 56-~ E. 14.15 3 S. 561 E. 2.20 3 S. 340 W. 5.10 4 S. 341W W. 3.53 4 N. 560 W. 5.84 5 N. 561~ W. 3.20 5 S. 2910 W. 2.52 Ans. 1A. OR. 14P. N. 48}e W. 8 3 CHAP. VI.] Calculating the Content. 187 Example 7. Example 8. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 S. 21; W. 17.62 I S. 652 E. 4.98 2 S. 34" W. 10.00 2 S. 58" E. 8.56 3 N. 560 W. 14.15 3 S. 14" W. 20.69 4 No 340 E. 9.76 4 S. 470 W. 0.60 5 N. 670 E. 2.30 5 S. 570 W. 8.98 6 N. 230 E. 7.03 6 N. 56~ VW. 12.90 7 N. 18 E. 4.43 7 N. 34~ E. 10.00 8 S. 760 E. 12.41 8 N. 21. E. 17.62 Example 9. Example 10. STA. BEARING. DISTANCE. S. A. BEARING. DISTANCE. I S. 570 E. 5.77 1 N. 63~ 51' V. 6.91 2 S. 368~ W. 2.25 2 N, 63 44' W. 7.26 3 S. 397 W. 1.00 3 N. 69 35' W. 3.34 4, S. 704 XW. 1.04 4 N. 770 50' W. 6.54 5 N. 683~ W. 1.23 5 N. 310 24' E. 14.38 6 N. 56~ W. 2.19 6 N. 31 18' E. 16.81 7 N. 330 E. 1.05 7 S. 68~ 55' E. 13.64 8N. 561~ W. 1,54 8 S. 68~ 42' E. 11.54 9 N.33o E. 3.18 9 S. 33~ 45' W. 31.55 Ans. 2A. OR. 32P. Ans. 74 Acres. Example 11. Example 12. STA. BEARING. DISTANCE. STA. BEARING. DISTANCE. 1 N. 183 E. 1.93 1 N. 72~0 E. 0.88 2 N. 9 W. 1.29 2 S. 200 E. 0.22 3 N. 140 W. 2.71 8 S. 630 E. 0.75 4 N. 740 E. 0.95 4 N. 51 E. 2.35 5 S. 48l E. 1.59 5 N. 440 E. 1.10 6 S 1410 E. 1.14 6 N. 2510 W. 1.96 7 S 19^1 E. 2.15 7 N. 81~ W. 1.05 8 S. 231~0 W. 1.22 8 S. 290 W. 1.63 9 S 50 W. 1.40 9 N. 71. 0.81 10 S. 300 W. 1.02 10 N. 13 W. 1.17 11 S. 810 X. 0.9 11 N. 63~ W. 1.28 12 N. 3210 W. 1.98 12 West. 1.68 _...........i 13 N. 490 W. 0.80 14 S. 19 E. 0.2. 0 188 COIMPASS SURVEYING. [PART II1 E xavmpe 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 twer- 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-East- erly corner of the homestead; thence nineteen chains and thirty-two links North-Easterly twenty-six degrees to a stone set in a 3 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) Iascherounis Theorem. The surface of any pooygon 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 makge with each other. Fig. 198. Thus, take any polygon, such as the -fiveBided one in the figure. Express the angle which the directions of any two sides, as AB, CD, make with each other, thus (AB ACD). Then will A the content of that polygon be, as below; E - I [AB. BC. sin (AB BC) + AB. CD. sin (AB A CD) + AB.DE sin (AB A DE) + B CD. sin (BC, CD) + B. DE. sin (BC A DE) + CD. D, Esin (CD A DE)] CHAP. VII.] Variation of the Magnetic Needle, 185 The demonstration consists merely in dividing the polygon into triangles by lines drawn from any angle, (as A); then expressing the area of each triangle by half the product of its base and the perpendicular let fall upon it from the above named angle; and finally separating the perpendicular into parts which can each be expressed by the product of some one side into the sine of the angle made by it with another side. The sum of these triangles equals the polygon. The expressions are simplified by dividing the proposed polygon into two parts by a diagonal, and computing the area of each part separately, making the diagonal the side omitted.* CHAPTER VII. TIHE VARIATION OF THE IlCGNETIt NEEDLE. (298) Definitions. The laygnetic Mleridian is the Fi. 199 direction indicated by the Magnetic Needle. The True I Meridian is a true North and South line, which, if produced, would pass through the poles of the earth. The Variation, or _Declination, of the needle is the angle which one of these lines makes with the other.t.. In the figure, if NS represent the direction of the True Meridian, and N'S' the direction of the Magnetic Meridian at any place, then is the angle NAN' the TVariation of the Needle at that place. (299) Direction of Needle. The directions of these two meridians do not generally coincide, but the needle in most places points to the East or to the West of the tiue North, more or less' The original Theorem is usually accredited to Lhuillier, of Geneva, who published it in 1789. But Mascheroni, the ingenious author of the " Geometry of the Compasses," had published it at Pavia, two years previously. The method is well developed in Prof. Whitlock's " Elements of Geometry." t " Declination" is the more correct term, and "' Variation" should be reserved for the change in the Declination which will be considered in the next chapter; but custom has established the use of Variation in the sense of Declination. 190 COMPASS SURVEYINGT. [PART III according to the locality Observations of the amount and the direction of this variation have been made in nearly all parts of the world. In the United States the Variation in the Eastern States is Westerly, and in the Western States is Easterly, as will be given in detail, after the methods for determining the True Meridian, and consequently the Variation, at any place, have been explained. TO DETERMIN E THE TRUE MERIDIAN. (300) By equal shadows of the Sun. On the South side of any level surface, erect an up- Fig. 2 right staff, shown, in horizontal projection, at S. Two or three hours before noon, mark the extremity, A, of its shadow. Describe an arc of a circle with S, the foot of the staff, for centre, and SA, the distance to the extremity of the shadow, for radius. About as many hours after noon as it had been before noon when the first mark was made, watch for the moment when the end of the shadow touches the arc at another point, B. Bisect the arc AB at N. Draw SN, and it will be the true meridian, or North and South line required. For greater accuracy, describe several arcs before hand, mnark the points in which each of them is touched by the-shadow, bisect each, and adopt the average of all. The shadow will be better defined, if a piece of tin with a hole through it be placed at the top of the staff, as a bright spot will thus be substituted for the less definite shadow. Nor need the staff be vertical, if from its summit a plumb-line be dropped to the ground, and the point which this strikes be adopted as the centre of the arcs. This method is a very good approximation, though perfectly correct only at the time of the solstices; about June 21st and December 22d. It was employed by the Romans in laying out cities. To get the Variation, set the compass at one end of the True Meridian line thus obtained, sight to the other end of it, and take CHAP. VIl.] Variation of the Magnetic Necdlee 191 the Bearing as of any ordinary line. The number of degrees im the reading will be the desired variation of the needle. (301) By the North Star, when in the Meridian. The North Star, or Pole Star, (called by astronomers Alpha Ursce joinoris, or Polaris), is not situated precisely at the North Pole of the heavens. If it were, the Meridian could be at once determinedc by sighting to it, or placing the eye at some distance behind a plumba line so that this line should hide the star. But the North Star is about 110 from the Pole. Twice in 24 hours, however, (more precisely 23h. 56m.), it is in the Meri- Fig 201. dian, being then exactly above or below A the Pole, as at A and C in the figure. To know when it is so, is rendered easy by the aid of another star, easily identified, which o D at these times is almost exactly above or below the North Star, i. e. situated in the same vertical plane. If then we watch for ^. the moment at which a suspended plumb- ~ line will cover both these stars, they will then be in the Meridian. The other star is in the well known constellation of the Great Bear, called also the Plough, or the Dipper, or Charle's Wain. Fig. 202. Fig. 203. POOJ a, s five bgh strs (he righ-h ones n Fig. 202) known a its the Poit stars from the rirh iand ones ig. 20to he) are known as the' Pointers," from their pointing near to the North 192 1COMPASS SURVEYING. [PART III. Star, thus assisting in finding it. The star in the tail or handle, nearest to the four which form a quadrilateral, is the star which comes to the Meridian at the same time with the North Star, twice in 24 hours, as in Fig. 202 or 203. It is known as Alioth, or Epsilon Ursce Majoris.' To determine the Meridian by this method, suspend a long plumb-line from some elevated point, such as a stick projecting from the highest window of a house suitably situated. The plumbbob may pass into a pail of water to lessen its vibrations. South of this set up the compass, at such a distance from the plumb-line that neither of the stars will be seen above its highest point, i. e. in Latitudes of 400 or 500not quite asfarfrom the plumb-line as it is long. Or, instead of a compass, place a board on two stakes, so as to form a sort of bench, running East and West, and on it place one of the compass-sights, or anything having a small hole in it to look through. As the time approaches for the North Star to be on the Meridian (as taken from the table given below) place the compass, or the sight, so that, looking through it, the plumb-line shall seem to cover or hide the North Star. As the star moves one way, move the eye and sight the other way, so as to constantly keep the star behind the plumb-line. At last Alioth, too, will be covered by the plumb-line. At that moment the eye and the plumb-line are (approximately) in the Meridian. Fasten down the sight on the board till morning, or with the compass take the bear ing at once, and the reading is the variation.f Instead of one plumb-line and a sight, two plumb-lines may be suspended at the end of a horizontal rod, turning on the top of a pole. The line thus obtained points to the East of the true line when the North Star is above Alioth, and vice versa. The North Star is exactly in the Meridian about 17 minutes after it has been n the same vertical plane with Alioth, andmay be sighted to after that interval of time, with perfect accuracy. " The North Pole is very nearly at the intersection of the line fiom Polaris to Alioth, and a perpendicular to this line from the small star seen t, the left of it in Fig. 202. t If a Transit or Theodolite be used, the cross-hairs must be illuminated by throwing the light of a lamp into the telescope by its reflection from white paper CHAP. viI.] Varation of the Mag'etic dlee e, 193 Another bright star, which is on the opposite side of the Pole, and is known to astronomers as G-amma Cassiopeice, also comes on the Meridian nearly at the same time as the North Star, and will thus assist in determining its direction. (302) The time at which the North Star passes the Meridian above the Pole, for every 10th day in the year, is given in the following Table, in common clock time.' The upper transit is the most convenient, since at the other transit Alioth is too high to be conveniently observed. e MONTH. 1st DAY. 11th DAY. 21st DAY..1 e Me. HX M.t HH. M. R January, 6 21 PM..5 41 Pe. 5 02 P. M. February, 4 18 P. M. 3 39 P. M. 3 00 P. M. March, 2 28 P. M. 1 49 P. M. 1 09 P. M. c April, 0 26 P. M.11 47 A. M.11 08 A. M. I May, 10 28 A. M. 9 49 A. M. 9 10 A. M. I June, 8 27 A. M. 7 48 A. M. 7 08 A. M. < July, 6 29 A. M. 5 50 A. M. 5 lI A. M. August, 4 28 A. M. 3 49 A. M. 3 09 A.M. b September, 2 26 A. M. 1 47 A. M. 1 07 A. M. October, 0 28 A M. M1 45 P..11 06 p. M. I November, 10 22 p. M. 9 43 p. M. 9 04 p, M. December, 8 24 P. M. 7 45 P. M. 7 06 p. iM. To calculate the time of the North Star passing the Meridian at its upper cni miniation Find in the " American Almanac," (Boston), or the " Astronomical Ephemeris," (Washington), or the " Nautical Almanac," (London), or by interpolation from the data at the end of this note, the right ascension of the star, and from it (increased by twenty-four hours if necessary to render the subtraction possible) subtract the Right ascension of the Sun at mean noon, or the sidereal time at mean noon, for the given day, as found in the " Ephemeris of the Sun," in the same Almanacs. From the remainder subtract the acceleration of sidereal on mean time corresponding to this remainder, (3m. 56s. for 24 hours), and the new remainder is the required mean solar time of the upper passage of the star acrosg the Meridian, in "Astronomical" reckoning, the astronomical day beginning at noon of the common civil day of the same date. The right ascension of the North Star for Jan. 1, 1850, is lh. 05m. 01.4s.; for 1860, lh. 08m. 02.8s.; for 1870, lh. 11m. 16.9s.; for 1880, lh. 14m. 45.1s. for 1890 lh. 18m. 29.2s.; for 1900, lh. 22m. 31s. 13 194 COMPASS SJURVEYIfIGo [PART XM. To find the time of the star's passage of the Meridian for other days than those given in the Table, take from it the time for the day most nearly preceding that desired, and subtract from this time 4 minutes for each day from the date of the day in the Table to that of the desired day; or, more accurately, interpolate, by saying: As the number of days between those given in the Table is to the number of days from the next preceding day in the Table to the desired day, so is the difference between the times given in the Table for the days next preceding and following the desired day to the time to be subtracted from that of the next preceding day. The first term of the preceding proportion is always ten, except at the end of months having more or less than 30 days. For example, let the time of the North Star's passing the Meridian on July 26th be required. From July 21st to August 1st being 11 days, we have this proportion: 11 days: 5 days:: 43 minutes 19 6 minutes. Taking this fiom 5h. 11m. A. M., we get 4h. 514m. A. M. for the time of passage required. The North Star passes the Meridian later every year. In 1860, it will pass the Meridian about two minutes later than in 1.854; in 1870, five minutes, in 1880, eight minutes, in 1890, twelve minutes, and in 1900, sixteen minutes, later than in 1854: the year for which the preceding table has been calculated. fhe times at which the North Star passes the Meridian below the Pole, in its lower Transit, can be found by adding 11h. 58m. to the time of the upper Transit, or by subtracting that interval crom it.* (303) By the North Star at its extreme elongation. When the North Star is at its greatest apparent angular distance East or West of the Pole, as at B or D in Fig. 201, it is said to be at its extreme Eastern, or extreme Western, Elongation. If it be observed at either of these times, the direction of the Meridian can be easily The North Star, which is now about lo 28' from the Pole, was 12~ distant from it when its place was first recorded. Its distance is now diminishing at the rate of about a third of a minute in a year, and will continue to do so till it ap. proaches to within half a degree, when it will again recede. The brightest star in the Northern hemisphere, Alpha Lyrae, will be the Pole Star in about 12,000 years, being then w.thin about 50 of the Pole, though now more than 51~ distant from it CHAP. vii.] Variation of the Magnetic Needle, 195 obtained from the observation. The great advantage of this method over the preceding is that then the star's motion apparently ceases for a short time. (304) The following Table gives the TIMES OF EXTREME ELONGATIONS OF THE NORTH STAR.i1 MONTH. IST DAY. 11TH DAY. 21ST DAY. EASTERN. WESTERN. EASTERN. WESTERN. EASTERN. WESTERN. H.. H. M. H. M. H. n. H. M. H. HIM. Jan'y, 0 27 P.M.0 19.M. 11 47A.M.11 35 P.M. 11 08 A.M. 10 56 p.'. Feb'y, 1024A.M.1013P.M. 945A.M. 933P.M. 906A.M. 854P.M. March, 834A.M. 8 22 P.M. 7 55A.M. 743 P.I. 715A.Mi. 04 P.A. April, 632A.M. 6 20 P.M. 553A.M. 5 41 P.M. 514A.i. 5 02 P.M. May, 4 34A.M. 4 22 P.M. 3 55 A.M. 3 43 P.M. 3 16 A.M. 3 04 P.r. une, 2 33 A.M. 2 21 P.M. 1 53 A.M. 142 P.M. 1 14A.. 102 P.. July, 0 35 A.M. 0 23 P.M. 11 52 P.M. 11 44A.M.11 13 P.M. 11 05A.M. August, 10 30 r.. 10 22 A.M. 9 51 P.M. 943A.M. 911 P.9. 903A.lM. Sept'r, 8 28 P.M. 820A.M. 749 P.M. 741 A.M. 7 09 P.M. 7 01 AM. Oct'r, 630 P.M. 622A.M. 551.. 543. 5 12P.M. 504AxM, Nov'r, 428paii. 421A.M. 349P.M. 341A.M. 310P.M. 302A.M. Dec'r, 1 2 30 p.M. 222A.M. 151 P.M. 143A.M. 112 P.M. 104A.M. The Eastern Elongations from October to March, and the Western Elongations from April to September, occurring in the day time, they will generally not be visible except with the aid of a powerful telescope. * To calculate the times of the greatest elongation of the North Star: Find in one of the Almanacs before referred to, or from the data below, its Polar dis. tance at the given time. Add the logarithm of its tangent to the logarithm of the tangent of the Latitude of the place, and the sum will be the logarithm of the cosine of the Hour angle before or after the culmination. Reduce the space to time correct for sidereal acceleration (3m. 56s. for 24 hours) and subtract the result from the time of the star's passing the meridian on that day. to get the time of the Eastern elongation, or add it to get the Western. The Polar distance of the North Star, for Tan. 1, 1850, is 1~ 29' 25"; for 1860, 1 26' 12".7; for 1870, 1~ 23~ 01"; for 1880, L[ 1^ 50".4; for 1890, 1" 16' 40" 7; for 1900, 1" 13' 32".2. 196 COMPASS SUR EYING. [tART IIm The preceding Table was calculated for Latitude 40,. Ths Time at which the Elongations occur vary slightly for other Latitudes. In Latitude 50~, the Eastern Elongations occur about 2 minutes later and the Western Elongations about 2 minutes earlier than the times in the Table. In Latitude 26~, precisely the reverse takes place. The Times of Elongation are continually, though slowly, becoming later. The preceding Table was calculated for July 1st, 1854. In 1860, the times will be nearly 2 minutes later; and in 1900, the Eastern Elongations will be about 15 minutes, and the Western Elongations 17 minutes later than in 1854. (305) Observations. Knowing from the preceding Table the hour and minute-of the extreme Elongation on any day, a little before that time suspend a plumb-line, precisely as in Art. (301), and place yourself south of it as there directed. As the North Star moves one way, move your eye the other, so that the plumbline shall continually seem to cover the star. At last the star will appear to stop moving for a time, and then begin to move backwards. Fix the sight on the board (or the compass, &c.) in the position in which it was when the star ceased moving; for the star was then at its extreme apparent Elongation, East or West, as the case may be. (S06) Azimuths, The angle which the line from the eye to the plumb-line, makes with the True Meridian (i. e. the angle between the meridian plane and the vertical plane passing through the eye and the star) is called the Azimuth of the Star, It is given in the following Taole for different Latitudes, and for a number of years to come, For the intermediate Latitudes, it can be obtained by a simple proportion, similar to that explained in detail in Art. (302)." * To calculate this Azimuthl From the logarithm of the sine of the Polar dis tance of the star, subtract the logarithm of the cosine of the Latitude of the place; the remainder will be the logarithm of the sine of the angle required The PF fr distance can be obtained as directed in the last note CHAP. VIi.] Taltia oi f hQe Mlagnetic Needle, 197 AZIMUTHES OF THE NORTH STAR. Latitudes. 1854 1855 -1856 1857 1858 1859 1860 1870 50 2o 16j 20 16 20 16 2 15215' 5' 23 14/O20 141'20 09k' 493 20 14' 20 1321'20138'2012'120 12 20 12/ 2011/20 061 480 20 111' 2 11' 20 101'20 101'20 09 220092009 2' 04 470 20 0' 20 08 20 08' 20 073' 2 071J20 06 20 0620011/ 46 20620 061'2 20 053'/20 0512'c 05' 2o 04i',20 041' 10 591' 450 23 04' 20 04' 20 03'20 03' 20 023' 20 021' 20 02' 10 57-1 444 2~ 00o23 02' 20 001'120 01 120 01/ 20 00' 20 00' 1o 551/ 430' 20 001/20 00' 10 59'10o 59' 10 58'1o0 58110 58lo 10 5342 10 58/ to10 58' 10 571'1 57 /1 1 563/1 561 10o 56' 10 513' 410 10 563/10 561/10 551/ 10 55110o 55' 1o 541' 1 54' Io 50 400 Io 55' 10 541'10 54 10 5310o 5314o 53 10o 52'-10 481 390 10 531/10 5211052'10 52/ 0 51-o 51'10 51' 1 463' 380 10 513'10 511'10 51' 10501'10 50' 10 49, 10491,0 451' 3700 501'1 491 4 910 4911 49' 10 481'1 481 1 48/'1 44 36 10o 48/ 10 48'10o 48' o1 471047 471o' 0o 461'17o 421 350 1047'/10 47' 10461'10 461/1046 10 45, 10 451, 10 4111 40o 10461, 10451 1o 451/10 45 1o 44/o 44110 44/ 1 40 o330 o 45' 10441'10 44110 433'10431/1043/ 10 421 10 39 320 10 44' 110 431' 1~ 43' 101 423110 421' 1042' 10 41-'110 38' 3o 10 42'110 42 /10 42 10 41:1/1041 o 040 /10401 10 37' 30 1 411'1 41 11o 41 10 40 11 40 1o 40 10 394 1 36' (307) Settin out a Meridian, When two points in the direction of the North Star at its extreme elongation have been Fig. 204 I? S' obtained, as in Art. (305), the True Meridian can be -* found thus. Let A and B be the two points. Multiply the natural tangent of the Azimuth given in the Table, by the distance AB. The product will be the length of a line which is to be set off from B, perpendicular to AB, to some point C. A and C will then be points in the True, Meridian. This operation may be postponed till morning. c If the directions of both the extreme Eastern and extreme Western elongations be set out, the line lying midway between them will be the True Meridian. A 198 COMPASS SURTVEYING. [PART ir. (3S) Determining the Variation. The variation would of course be given by taking the Bearing of the Meridian thus obtained, but!i can also be determined by taking the Bearing of the star at the time of the extreme elongation, arid applying the following rules. When the Azimuth of the star and its magnetic bearing are one East and the other West, the sum of the two is the Magnetic Varition, which is of the same name as the Azimuth; i. e. East, if that be East, and West, if it be West. When the Azimuth of the star and its Magnetic Bearing.are both East, or both West, their difference is the Variation, which will be of the same name as the Azimuth and BeanlLg, if the Azimuth be the greater of the two, or of the contrary name if the Azimuth be the smaller. Fig. 205. All these cases are presented together in the! r N' V figure, in which P is the North Pole; Z the place / of the observer; ZP the True Meridian; S the star at its greatest Eastern elongation; and ZN, ZN', ZN", various supposed directions ofthe needle. Call the Azimuth of the star, i. e. the angle'ZS, 2~ East. Suppose the needle to point to N, and the Beartmg of the star, i. e. SZN, to be 5~ West of Mag- netic North. The variation PZN will evidently be 7~ East of true North. z Suppose the needle to point to N', and the bearing of the star, 1. e. N'ZS, to be 1{~ East of Magnetic North. The Variation will be 1~ East of true North, and of the same name as the Azimuth, because that is greater than the bearing. Suppose the needle to point to N" and the bearing of the star, i.e. eN"ZS, to be 10~ East of Magnetic North. The Variation will be 84 West of true North, of the contrary name to the Azimuth, because that is the smaller of the two.* * Algebraically, always subtract the Bearing from the Azimuth, and give the re' mainder its proper resulting algebraic sign. It wil be the Variation; East if plus, and West, if minus. Thus in the first case above, the Variation = — 20 - (- 5~) = - +~ 7~ East. In the second case, the Variation + 2~ - (+ 13~) + 1o == i o East. In the third case, the Variation == + 2~ - (Q+ 10 -8~ = 8~ West. CHAP. vii.] Variation of the Magnetic Needle. 199 If the star was on the other side of the Pole, the rules would apply likewise. (309) Other Methods. Many other methods of determining the true Meridian are employed; such as by equal altitudes and azimuths of the sun, or of a star; by one azimuth, knowing the time; by observations of circumpolar stars at equal times before and after their culmination, or before and after their greatest elongation, &c All these methods however require some degree of astronomical knowledge; and those which have been expi ined are abundantly sufficient for all the purposes of the ordinary Land-Surveyor. "Burt's Solar Compass" is an instrument by which, "when adjusted for the Sun's declination, and the Latitude of the place, the azimuth of any line from the true North and South can be read off, and the difference between it and the Bearing by the compass will then be the variation.' (310) Magnetic variation in the United States. The variation of the Magnetic needle in any part of the United States, can be approximately obtained by mere inspection of the map at the beginning of this volume.* Through all the places at which the needle in 1850,t pointed to the true North, a line is drawn on the map, and called the Line of no Variation. It will be seen to be nearly straight, and to pass in a N.N.W. direction from a little west of Cape Hatteras, N. C. through the middle of Virginia, about midway between Cleveland, (Ohio), and Erie, (Pa.), and through the middle of Lake Erie and Lake Huron. If followed South-Easterly it would be found to touch the most Easterly point of South America. It is now slowly moving Westward. At all places situated to the East of this line (including the New-England States, New-York,New-Jersey, Delaware, Maryland, nearly all of Pennsylvania, and the Eastern half of Virginia and North Carolina) the Variation is Westerly, i. e. the north end of the needle points to the west of the true North. At all places * Copied (by permission) from one prepared in 1856, by Prof. A. D. Bache, Supt. U. 8. Coast Survey, from the U. S. C. S. Observations. The dotted portions of the lines are interpolations due to the kindness of J. E. HIilgard, Assist. U. S. Coast Survey. + A gradual change in the Variation is going on from year to year, as will be ex. plained in the next Chapter. 200 CO(IIPAiS SURVEYI G. [PART II. Bituated to the West of this line (including the Western and South. ern States) the Variation is easterly, i. e. the North end of the needle points to the East of the true North. This variation increases in proportion to the distance of the place on either side of the line of no variation, reaching 21~ of Easterly Variation in Oregon, and 18~ of Westerly Variation in Maine. Lines of equal Variation are lines drawn through all the places which have the same variation. On the map they are drawn for each degree. All the places situated on the line marked 1~, East or West, have 1~ Variation; those on the 20 line. have 2~ Variation, &c. The variation at the intermediate places can be approximately estimated by the eye. These lines all refer to 1840. The lines of equal Variation, if continued Northward, would all meet in a certain point called the Magnetic Pole, and situated in the neighborhood of 96~ West Longitude from Greenwich, and 70~ of North Latitude. Towards this pole the needle tends to point. Another Magnetic pole is found in the Southern hemisphere; but the farther development of this subject belongs to a treatise on Natural Philosophy. The Variation on the Pacific slope of this country has been very imperfectly ascertained. A few leading points are as below. California; Point Conception, Sept. 1850, 130 49-' E. Point Penos, Monterey, Feb. 1851, 14~ 58' E. Presidio, San Francisco, Feb. 1852, 15~ 27' E. San Diego, Mar. 1851, 12~ 29' E. Oregon; Cape Disappointment, July, 1851, 200 45' E. Ewing Harbor, Nov. 1851, 18~ 29' E. Wash. Ter'y. Scarboro HIarbor, Aug. 1852, 21~ 30' E. (311) To correct Magnetic Bearings. The Variation at any place and time being known, the Magnetic Bearings taken there and then, may be reduced to their true Bearings, by these Rules. RULE 1. When the Variation is West, as it is in the NorthEastern States, the true Bearing will be the sum of the Variation and a Bearing which is North and West, or South and East; and the difference of the Variation and a Bearing which is North and East, or South and West. To apply this to the cardinal points, a CHAP. II.] Variation of the l agnetle Nedle 201 North Bearing must be called N. 0~ West, an East Bearing N. 90~ E., a South Bearing S. 0~ E., and a West Bearing S. 900 W.; counting around from N' to N in the figure, and so onward, " with the Sun." The reasons for these corrections Fig. 206. are apparent from the Figure, in which " the dotted lines and the accented letters represent the direction of the needle, and the full lines and the unac- cented letters represent the true North wr —- -.. and South and East and West lines. - \ When the sum of the Variation and the Bearing is directed to be taken, and comes to more than 90~, the sup- s plement of the sum is to be taken, ani the first letter changed. When the difference is directed to be taken, and the Variation is greater than the Bearing, the last letter must be changed. A diagram of the case will remove all doubts. Examples of all these cases are given below for a Variation of 8~ West. MAGNETIC TRUE MA GNETIC TRUE BEARING. BEARING. BEARING. BEARING. North. N. 8 W. South. S. 8& E. N. 1~E. N. 7~ W. S. 2~ W. S. 6~ E. N. 40~ E. N. 32~ E. S. 60~ W. S. 52~ W. East. N. 82~ E. West. S. 820 W. S. 5S. 50. 58" E. N. 700 W. N. 78" W. I S. 890 E. N. 83" E. N. 83" W. S. 89" W. RULE 2. When the Variation is Fig. 07. East, as in the Western and Southern States, the preceding directions must / be exactly reversed; i. e. the true /Bearing will be the difference of the — ^ / Variation and a Bearing which is' ~ -._. —s North and West or South and East; / and the sum of the Variation and a Bearing which is North and East, or South and West. A North Bearing' i 202 COMPASS SURVEYING. [PART III must be called N. 00 E., a West Bearing N. 90~ W., a South Bearing S. 0~ W., and an East Bearing S. 90~ E., counting fronm N' to N, and so onward, "against the sun." The reasons foi these rules are seen in the Figure. Examples are given below, foi a Variation of 5~ E. MAGNETIC TRUE MAGNETC TRUE BEARING. BEARING. BEARING. BEARING. North. N. 5~ E. South. S. 5~ W. N. 40~ E. N. 450 E. S. 60~ W. S. 65~- W. N. 890 E. S. 86~ E. S. 87~ W. N. 88~ W. East. S. 85~ E. West. N. 85~ W. S. 10 E 40.. 040. N~. 60~.6W. S. 50~ E. S. 45~ E. N. 2~ W. N. ~0 E. (312) To survey a line witl true Bearings. The compass may be set, or adjusted, by means of the Vernier, (noticed in Arts. (229) and (237), and shown in Fig. 148, page 126) according to the Variation in any place, so that the Bearings of any lines then taken with it will be their true Bearings. To effect this, turn aside the compass plate, by means of the Tangent Screw which moves the Vernier, a number of degrees equal to the Variation, moving the S. end of the Compass-box to the right, (the North end being supposed to go ahead) if the Variation be Westerly, and vice versa; for that moves the North end of the Compass-box in the contrary direction, and thus makes a line which before was N. by the nee, die, now read, as it should truly, North, so many degrees, West if the Variation was West; and similarly in the reverse case. CHAP. VIII, 203 CHAPTER VIII. CHANGES IN THIE VARIATION. (313) The Changes in the Variation are of more practical importance than its absolute amount. They are of four kinds: Irregular, Diurnal, Annual and Secular. (314) Irregular changes. The needle is subject to sudden and violent changes, which have no known law. They are sometimes coincident with a thunder storm, or an Aurora Borealis, (during which, changes of nearly 1~ in one minute, 2-1 in eight minutes, and 100 in one night, have been observed), but often nave no apparent cause, except an otherwise invisible "' Magnetic Storm." (315) The Diurnal change. On continuing observations of the direction of the needle throughout an entire day, it will be found, in the Northern Hemisphere, that the North end of the needle moves Westward from about 8 A. M. till about 2 P. M. over an arc of from 10' to 15', and then gradually returns to its former position.' In the Southern Hemisphere, the direction of this motion is reversed. The period of this change being a day, it is called the Diurnal Variation. Its effect on the permanent Variation is necessarily to cause it, in places where it is West, to attain its maximum at about 2 P. M., and its minimum at about 8 A. M.; and the reverse where the Variation is East. This Diurnal change adds a new element to the inaccuracies of the compass; since the Bearings of any line taken on the same day, at a few hours interval, might vary a quarter of a degree, which would cause a deviation of the end of the line, amounting to nearly half a link at the end of a chain, and to 35 links, or 23 feet, at the end of a mile. The hour of the day at which any important Bearing is taken should therefore be noted. ^ A similar but smaller movement takes place daring the night. 204 COMPASS SIUVEVINnG [PAIT in, (316) The Annual change. If the observations be continued throughout an entire year, it will be found that the Diurnal changes vary with the seasons, being about twice as great in Summer as in Winter. The period of this change being a year, it is called the Annual Variation. (317) The Secular change. When accurate observations on the Variation of the needle in the same place are continued for several years, it is found that there is a continual and tolerably regular increase or decrease of the Variation, continuing to proceed in the same direction for so long a period, that it may be called the Secular change of Variation.* The most ancient observations are those taken in Paris. In the year 1541 the needle pointed 7~ East of North; in 1580 the Variation had increased to 111~ East, being its maximum; the needle then began to move Westward, and in 1666, it had returned to the Meridian; the Variation then became West, and continued to increase till in 1814 it attained its maximum, being 22~ 34' West of North. It is now decreasing, and in 1853 was 200 17' W. In London, the Variation in 1576 was 11~ 15' E.; in 1662, 0~; in 1700, 9" 40' W.; in 1778, 22" 11' W.; in 1815, 24~ 27' W.; and in 1843, 23" 8' W. In this country the north end of the needle was moving East ward at the earliest recorded observations, and continued to do so till about the year 1810 (variously recorded as from 1793 to 1819), when it began to move Westward which it has ever since continued to do. Thus, in Boston, from 1708 to 1807 the Varia tion changed from 90 W. to 6~ 5' W., and from 1807 to 1840, it changed from 6~ 5' W. to 90 18' W. Valuable Tables of the Secular changes of the Variation in van ous parts of the United States have been published by Prof. Loomis mn Silliman's "American Journal of Science," Vol. 34, July, 1838, p. 301; Vol. 39, Oct. 1840, p. 42; and Vol. 43, Oct. 1842, p. 107. An abstract of the most reliable of them is here given. froy and Schenectady are from other sources. * If the term " Declination of the Needle" could be restored to its proper nse. bhis " Change of Variation' worid be properly called the' Variation of the De clination." CIAP. VIII.] Changes in the Variationo 205 PLACE, LATITUDE. LONGITUDE. DATES. ANNUAL MIOTION. iBuriington, Vt. 440 27' 73~ 10' 1811...1834 4'.4 Chesterfield, N..I 420 53' 72~ 20' 1820...1836 6'.4 IDeerfield, Mass. 42~ 34' 720 29' 1811.. 1837 5'.7 lCambridge, Mass. 420 22' 71 7 7' 1810...1840 3'.4 New-Haven, Conn. 410 18' 720 58' 1819...1840 4'.6 Keeseville, N. Y. 440 28' 730 32' 1825...1838 5'.4 Albany N. Y. 42~ 39' 73~ 45' 1818...1842 36. "re ";; " 121842...1854 4'.9 Troy, N. Y. 42~ 441 73~ 40' 1821...1837 6'.2 Schenectady, N. Y. 420 49' 730 55' 1829...1841 7'.2 66^ "C 4,".1841...1854 6'.0 New-York City. 40~ 43' 740 01' 1824...1837 38.7 Philadelphia. 390 57' 750 11' 1813...1837 3'.6 Milledgeville, Ga. 330 7' 830 20' 1805...1835 1'.7 Mobile, Ala. 300 40' 880 11' 1809...1835 2'.2 Cleveland, 0. 410 30' 810 46' 1825...1838 4.5 Marietta, 0. 390 25' 810 26' 1810...1838 2'.4 Cincinnati, 0. 390 6' 840 27' 1825...1840 2'.0 Detroit, Mich. 420 24' 820 58 1822...1840 4'.3 Alton, Ill. 380 52' 900 12' 1835...1840 3'.0 From these and other observations it appears that at present the lines of equal variation are moving Westward, producing an annual change of variation (increasing the Westerly and lessening the Easterly) which is different in different parts of the country, and is about five or six minutes in the North-Eastern States, three or four minutes in the Middle States, and two minutes in the Southern States. (1I8) Determnnation of the chanige, y Interpolation, To determine the change at any place and for any interval not found in the recorded observations, an approximation, sufficient for most purposes of the surveyor, may be obtained by interpolation (by a simple proportion) between the places given in the Tables, assuming the movements to have been uniform between the given dates; and also assuming the change at any place not found in the Tables, to have been intermediate between those of the lines of equal varia. tion, which pass through the places of recorded observations on each side of it, and to have been in the ratio of its respective dis. 206 COMP8SS SURVEYING. [PART Ill. tances from those two lines; for example, taking their arithmetical mean, if the required place is midway between them; if it be twice as near one as the other, dividing the sum of twice the change of the nearest line, and once the change of the other, by three; and so in other cases; i. e. giving the change at each place, a "s weight" inversely as its distance from the place at which the change is to be found. (319) Determination of the change, by old lines, When the former Bearing of any old line, such as a farm-fence, &c. is recorded, the change in the Variation from the date of the original observation to the present time can be at once found by setting the compass at one end of the line and sighting to the other. The difference of the two Bearings is the required change. If one end of the old line cannot be seen from the other, as is often the case when the line is fixed only by a " corner" at each end of it, proceed thus. Run a line from one corner with the old Bearing and with its distance. Measure the distance from the end of this line to the other corner, to which it will be opposite. Multiply this distance by 57.3, and divide by the length of the line. The quotient will be the change of variation in degrees.* For example, a line 63 chains long, in 1827 had a Bearing of North 10 East. In 1847 a trial line was run from one end of the former line with the same Bearing and distance, and its other end was found to be 125 links to the West of the true corner. The 1.25 x 57.3 change of Variation was therefore 63 - 1.137 - 1 8' Westerly. L et AB be the original line; AC the trial line, Fig. 208. and BC the distance between their extremities. AB and AC may be regarded as radii of a circle " --— _ and BC as a chord of the arc which subtends their angle. Assuming the chord and arc to coincide (which they will, nearly, for small angles) we have this proportion; Whole circumference arc BO:: 600: BAC: or, 2 X AC X 3.1416: B. 360 o: BAG, whence BAC = X 57.3; or more precisely 57.29578. CaAP. VIII.] Changes in the Variationo 207 (320) Effects of the Secular change. These are exceedingly important in the re-survey of farms Fig. 209. by the Bearings recorded in old deeds. Let SN denote the direc- K / tion of the needle at the time of the original Survey, and S'N' its direction at the time of the re-sur- X / "..' vey, a number of years later, Suppose the change to have been' V 0~ the needle pointing so much / \\, farther to the west of North. The / \ line SN, which before was due North and South by the needle will now bear N. 3~ E. and S. 3~ W; the line AB, which before was N. 40" E. will now bear N. 43~ E; the line DF which before was N. 40~ W. will now bear N. 37~ W; and the line WE, which before was due East and West, will now bear S. 87~ E. and N. 87~ W. Any line is similarly changed. The proof of this is apparent on inspecting the figure. Suppose now that a surveyor, ignorant or neglectful of this change, should attempt to run out a Fig. 210. farm by the old Bearings of the deed, none of the old fences or corners i-emaining. The full lines in the figure represent the original bounds of the farm, and the dotted\ lines those of the new piece of land \ which, starting from A, he would muwittingly run out. It would be of the same size and the same shape as \ the true one, but it would be in the \ wrong place. None of its lines would agree with the true ones, and in some places it would encroach on one neighbor, and in other places would leave a gore which belongs to it, between itself and another neighbor. Yet this is often done, and is the source of a great part of the litigation among farmers respecting their " lines." 208 COMPASS SURVEYIIN LPART IIL (321) To run out old lines, To succeed in retracing old lines, proper allowance must be made for the change in the variah tion since the date of the original survey. That date must first be accurately ascertained; for the survey may be much older than the deed, into which its bearings may have been copied from an older one. The amount and direction of the change is then to be ascertained by the methods of Arts. (318) or (319). The bearings may then be corrected by the following RULES. Wehen the North end of the needle has been moving Westerly, (as it has for about forty years), the present Bearings will be the sums of the change and the old Bearings which were North-Easterly or South-Westerly, and the differences of the change and the old Bearings which were North-Westerly or South-Easterly. If the change have been Easterly, reverse the preceding rules, subtracting where it is directed to add, and aclcing where it is directed to subtract. Run out the lines with the Bearings thus corrected. It will be noticed that the process is precisely the reverse of that in Art. (311), The rules there given in more detail, may therefore be used; RULE 1, when the Variation is West," being employed when the clenge has been a movement of the N. end of the needle to the East; and RULE 2, " when the Variation is East," being employed when the N. end of the needle has been moving to the West. If the compass has a Vernier, it can be set for the change, once for all, precisely as directed in Art. (312), and then the courses can be run out as given in the deed, the correction being made by the instrument. (322) Example. The following is a remarkable case which recently came before the Supreme Court of New-York. The North line of a large Estate was fixed by a royal grant, dated in 1704, as a due East and West line. It was run out in 1715, by a surveyor, whom we will call Mr. A. It was again surveyed in 1765, by Mr. B. who raa a course N. 87~ 30' E. It was run out for a third time in 1789, by Mr. C. who adopted the course N. 86~ 18' E. In 1845 it was surveyed for the fourth time by CHAP. vIII.] Cha nges if the Variationi 209 Mr. D. with a course of N. 88~ 30' E. He found old " corners," and " blazes" of a former survey, on his line. They are also found on another line, South of his. Which of the preceding courses were correct, and where does the true line lie? The question was investigated as follows. There were no old records of variation at the precise locality, but it lies between the lines of equal variation which pass through New-York and Boston, its distance from the Boston line being about twice its distance from the New-York line. The records of those two cities (referred to in Art. (317)) could therefore be used in the manner explained in Art. (318). For the later dates, observations at New-Haven could serve as a check. Combining all these, the author inferred the variation at the desired place to have been as follows: In 1715, Variation 80 02' West. In 1765, " 5~ 32' " Decrease since 1715, 2S 30'. In 1789, " 5~ 05' " Decrease since 1765, 0~ 27'. In 1845, " 70 23' "4 Increase since 1789, 20 18'. We are now prepared to examine the correctness of the allowances made by the old surveyors. The course run by Mr. B. in 1765, N. 870 30' E. made an allowance of 20 30' as the decrease of variation, agreeing precisely with our calculation. The course of Mrl. C. in 1789, N. 860 18' E., allowed a change of 1~ 12', which was wrong by our calculation, which gives only about 27', and was deduced from three different records. Mr. D. in 1845, ran a course of N. 880 30' E, calling the increase of variation since 1789, 20 12'. Our estimate was 20 18', the difference being comparatively small. Our conclusion then is this: the second surveyor retraced correctly the line of the first: the third surveyor ran out a new and incorrect line: and the fourth surveyor correctly retraced the line of the third, and found his marks, but this line was wrong originally and therefore wrong now. All the surveyors ran their lines on the supposition that the original "' due East and West line" meant East and West as the needle pointed at the time of the original survey. The preponderance of the testimony as to old land marls agreed with the results of the above reasoning, and the decision of the court was in accordance therewith. 14 210 COMPASS SURVEYING. [PART m Fig. 211 In the above figure the horizontal and vertical lines represent true East and North lines; and the two upper lines running from left to right represent the two lines set out by the surveyors and in the years, there named. (323) Remedy for the evils of the Secular change. The only complete remedy for the disputes, and the uncertainty of bounds, resulting from the continued change in the variation, is this. Let a Meridian, i. e. a true North and South line, be estab. lished in every town or county, by the authority of the State; monuments, such as stones set deep in the ground, being placed at each end of it. Let every surveyor be obliged by law to test his compass by this line, at least once in each year. This he could do as easily as in taking the Bearing of a fence, by setting his instrument on one monument, and sighting to a staff held on the other. Let the variation thus ascertained be inserted in the notes of the survey and recorded in the deed. Another surveyor, years or centuries afterwards, could test his compass by taking the Bearing of the same monuments, and the difference between this and the former Bearing would be the change of variation. He could thus determine with entire certainty the proper allowance to be made (as in Art. (321)) in order to retrace the original line, no matter how much, or how irregularly, the variation may have changed, or how badly adjusted was the compass of the original survey. Any permanent line employed in the same manner as the meridian line, would answer the same purpose, though less conveniently, and every surveyor should have such a line at least, for his own use.' * this remedy seems to have been first suggested by Rittenhouse. It has since been recommended by T. Sopwith, in 1822; by E. F. Johnson, in 1831, and by W. Roberts, of Troy, in 1839. The errors of re-surveys, in which the change is neglected, were noticed in the " Philosophical Transactions," as long ago as 1679 PART IV. TRANSIT AND THEODOLITE SURVEYING: By the Third Method. CHAPTER I. THE INSTRUIMENTS. (321) THE TRANSIT and THE THEODOLITE (figures of which are given on the next two pages) are Goniometers, or Angle-MIeasurers. 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 circu-.ar 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 ots 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 hanl 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. TASHT AND TIE@D9OLITE ~ S$rTEEIf6. [PART ~ THE TRANSIT, Fig o1~ LI, S WTX cHAP. I.J The Instruments. 213 THE THEODOLITE. Fig. 213 ~l. 214 TRANSIT AND THEODOLITE SURVEYING. [PART Il (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 corn pletely over, so as to look both forward and backward, while ii; the Theodolite it cannot do so. Hence the name of the Transit.' Thllis 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 Vernier Scale, which enables minute portions of any arc 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. In this country it has almost entirely supplanted the Theodolite. CHAP. I.] The Instrumeats, 21 The inaccurate division of the limb of the instrument is alsc 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 wh: vh 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 "I 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 arc 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 P' 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 thQroughly 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 fog some valuable information respecting the details of the instrument. The Theodo lits is of the favorite English form. 216 TRtNSIT AND THEODOLITE SURVEIrIT. [PART Iv as determined by the bubbles in the small spirit levels, 13, B, of which there are two at right angles to each other. In the figure of the Theodolite, the large level 6, and the semicircle NN are for the purposes of Levelling, and of measuring Vertical angles. They will therefore not be described in this placec (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 Fin. 2J4. inverted 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 MSAP. 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. (399) 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 the 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 reizlered visible by every point of it sending forth rays of light in every direction. In this figure,. the highest and lowest 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, 0, 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 X C and D, which act like one magnifying glass, so that l the rays, after being refracted by them, enter the eye at such angles as t) form there a magnified and inverted image of the object. This combination of the two plano-convex lenses, C and D, is known as "6 Ramsden's Ey-piece." 21a TRANSIT AI T THEODOLITE SURVEEYING. [PAIrA IV This Telescope, inverting objects, shows them upside Ei. 21 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 les- sen the distinctness of vision, the former arrangement is preferable for the glasses of a Transit or a Theodolite. A little practice makes it ecually 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 3, 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-hlaiHr Since a considerable field of view is seen in looking 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 cAAP. I.] The Instruments. 219 ot the eye-piece. These lines are called the cross-hairs, 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 work manship 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 of collimation of the Telescope.* The cross-hairs are attached to a ring, or short thick tube of brass, placec within the Tele- Fig. 217. scope tube, through holes in which pass loosely four screws, V =.f. (their heads being seen at a, a, a, in Figs. 212 and 213), - whose threads enter and take 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 Fg. 18. 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 preci- sion. A horizontal hair is sometimes added. The cross-hairs are best made of platinum wire, drawn out very fire by being previously enclosed From the Latin word Collimo, or Collineo, meaning to direct one thing to wards another in a straight line, or to aim at. The line of aim would express th meaning. 220 TRNSIT AND THEODOLITE SURVEYING. [ART 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 cut 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 m the end of the ring, and are fastened by gum, or varnish, or bees wax. 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 cho- \/\/\ sen, 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 J.] The Instruments. 221 (332) The milled-headed screw seen at M, passing into the tele scope has a pinion at its other end entering a Fig. 220. toothed rack, and is used to move the object glass, kA v,e W 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." (334) 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 uppel oue, and we will therefore always speak of it as such. 222 TRANSIT AND THEODOLITE SURVEYING LP[ART 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 any error of eccentricity, arising from the centre of the axis which carries 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. 21. be the true centre of the graduated cir- /: - cle, but C' the centre on which the plate carrying the indexes turns. Let AC'B represent the direction of a sight taken { C 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 ID'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 19~, 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 inequali ties of graduation on different parts of the limb, and thus diminish ing their effect on the resulting angle. Fig. 22., 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, 120~ from A and from each other. The half-sum or arithmetical mean, of A and B was taken, then the mean of A, C, and D, and then the mean of these two means. But this was wrong, for. cHAe. 1.] flhe 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, = ( B + + + +- = D (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'o 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. (383) Iovements, When the line of sight of the telescope is directed to a distant well-defined point, the unaided hand of the observer cannot move it with sufficient delicacy and precision to make the intersection of the cross hairs 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 clanp. This consists of two pieces of brass, which, by turning the clamp-screw e, which passes through them on the outside, can be made to take * In some instruments there is another concentric circle on which the de gles are also numbered from 0~ to 900 as on the compass circle. 224 TRATN$IT AND TIEODOELITE SURVEYING. LPART 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 " 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 clamp screw, 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 piece 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 lhorizontal 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 highest part 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 plato sHAP. 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. i is in te fige in? one in and the other out, as slhown 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. 1.5 2^0 TRANSIT AND THEODOLITE SURVTEYING. [PART IT 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 hle its bubble brought to its centre by the third screw. If there is only one level on the instrument, it is first brought parallel tc 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 Lhe 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, (Lunette 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." (311) The Surveyor's Transit, In this instrument (so named by its introducers, Messrs. Gurley, and shown in Fig. 224), the Vernier-plate, which carries the standards and telescope, is under the plate which carries the graduated circle, and the compass is attached to the latter. By this arrangement, when the Vernier is set at any angle, the line of sight of the telescope will make that angle with the N. and S. lines of the compass. Consequently, this instrument can be used precisely like the Vernier compass CHAP. I.] The Instruments, 27 to allow for magnet- Fig. 24. ic variation and thus - v to run out a line with true bearings, as in Art. (312), or to run out old lines, allowing for the secular variation, as in Art. (321 ) The instrument may also be used like the co mmonr E n gin e er's Transit~ The compass, however, will then not give the bearings of the l lines surveyed, but they can easily be deduced from that of any one line. (312) Gonlasmometre. A very compact in- ig.22' strument to which the above name has been i given in France, where it is much used, is shown in the figure. The upper half of the cylinder is movable on its lower half. The observations may be taken through the slits, as in the Survey- l or's Cross, or a Telescope may be added to it. Readlngs may be taken both from the compass, and from the divided edge of the lower half of 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.Jlean the glasses with kid or buck skin. Wash them, if dirtied, with alcohol. 228 [PART I CHAPTER II. VERNIERSo (313) 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 di. vided. (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. c,! 5 tCs IO t e I m V 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 N" Nonins," 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-hun drelths 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 one. hundredth 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. R co, V- I - 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 qwhat 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 * The student will do well to draw such a scale and Vernier on two slips of Ihick paper, and move one beside the other till he can read them in any possible position; and so with the following Verniers. 230 TRANSIT'ASD THEODOLITE SURVEYING. [PART L.V. 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 Ver nier, (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 cases 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 ite 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 ms = (m -- 1) v. v -- s. s -v = - 7;m 8 S ~s- +7 ~~S If ms = (m ) v, then v - s ~.,1 is lm v- 1' m- 1 m rn * e. Algebraically, v = i. e. When v = ------ m+ -1 n- 1 CHAP. Jii Verniers, 231 (347) Illustration, In this figure, the scale, as before, repre. sents (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. -1 I.I t t t I in II I I-{ 1 ILI 0 1i 1_ i t t i parts, each of which is therefore one-tenth of eleven-tenths of az 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 30.16o 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.'IiH ~ This figure, and others in this chapter, are from Bree's " Present Practice." 232 T IRANSIT AND THEODOLITE $UREYISTG [PART IV longer lines, and the shorter lines again divide these into half tenths, 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 = 2 ~ =.048 inch, and is shorter than a division 25 cf 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 leading in the figure is ~0.686, (30.65 by the scale and.036 by the Vernier), the dotted line marked D showing where the coincidence takes place. (349) Circle 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. 0. 15 o The above figure shows such an arrangement. The index, or zero, of the Vernier is at a point beyond 358~, 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 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 00, 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 TIEODOLITE SURVEYING. [PARr IV. Fig. 231. by the Vernier to be 10'. Fig. 232. 2I 0 -- 1 210 C 19 $1 ~: i~~-)ll i ^ ii 1 ~~~~~~~~~~~~~~~~~~~~~~~~~i~l ^v~ ab ~ ~~~~ \ " CIAP. Ii.1 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 = t30 = 31', and 30 will be longer than a spac o e on the circle by to which it will therefore read, as in the last case, but the Vernier will be 4 retrograde." This is the Vernier of the compass, Fig. 148e 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 = 5 - 32', and the Vernier will therefore read to 2'. (351) Circle divided to 201. If 20 spaces on the Vernier are equal to 19 on the circle, each space of the latter will be 19 x 20' 920'= 19', and the Vernier will read to 20'- 19' -1'. 20 If 40 spaces on the Vernier are equal to 41 on the circle, each 41 x 20' space on the Vernier will be = 40 = 20-'; and the Ver" nier will therefore read to 20' -- 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 13~ 40' on the limb, i. e. to 41 spaces, each of 20'. Fig.233. 36[0' }'0_ 5_ - 10 15 2 If 60 spaces on the Vernier are equal to 59 on the circle, each 59 20 1' of the former will be -== ~ === 19' 40"' and the Vernier o0' 236 TRANSIT AND THIEODOLITE SURVEYINIG [PART IV will therefore read to 20'- 19' 40" 20". The following figure shows such an arrangement The reading in that position would be 40" 46' 20". Fig. 234. 50 iO 40 I 9 8 16 5 4 32 1 \i^,i (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 59 x 15' ~ti -14' 45", and the Vernier will read to 15". In the following figure the reading is 10~ 20' 45", the index pointing to 10~ 15', and something more, which the Vernier shows to be 5' 45' Fig. 235. I - I T i 10k'~ 9 8 5 A- 3 2? * 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 7~ 25' 40", the reading on the circle being 7~ 20', and the Vernier showing the remaining space to be 5' 40". Fig. 236. 1J-^II1i t __' 1-1 ~lt~ t u?! >k I III J (354) Reading tackwards. 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 smafl angles of deflection for railroad curves. 238 TRiNSIT AND THEODOLITE $BURVEY1NG. [PART Iv. (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 Fiz. 23" 1035 cide, is 3' 20", (or it is 10' - 6' 40" = 3' 20"), and the entire reading is therefore 4 23' 20".5 (3376) DoulIe Verniers, To avoid the inconveniences of read\ I zero of the Vernier is 4~ 20', and something more, and the reading of the Vernier from 10 towards to the right, wvhere the lines coincide, is 3' 20"' (or it is 10'-d' 40" = 3' 20"), an8 the entire esacling is therefore 4~ 23' 20". (35g) Doubl~ Ves user se To avoicd 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. __ 1_ 10 I I! \/\\\ I ~~~o~~~~~-h~~~~~~~i 45I o- 8~~~~~~~~~~~~~~~~~~~~ cHAP. 11.] 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 3600 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. 12 0 i 30 215 O 5 0 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 rhore, read the ipper figures on the right hand Vernier: and vice versa. 240 [LART 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 lectifications.* (359) In observations of horizontal angles with the Transit or the Theocolite,f 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. * 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 TheTheodolite acjustments which relate only to levelling or to measuring vertical angles, will not be here discussed. MHAP. III. Adjustments. 241 We shall see that all these adjustments are finally resolvable into these; lst. Making the vertical axis of the instrument perpendicular to the plane of the levels; 2d. Making the line of collimation 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 se, that is, to "' verify" the adjustments; and, secondly, if we find that they are not so, to zmake 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. BErin 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. Rectification.-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. IC i I D D plane, seen edgeways, and CD the centre line of the vertical anx, this applies equally to the Transit and the Theodolite. 16 242 TRANSIT AND THEODOLITE SUIRVEYING, [PART IV which is here drawn as making an acute angle with this plane gn 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 haf 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 Be. Direct the telescope I his adjustment is not the same in the Transit and in the Theodolite. That For the Transit will be first given, and that for the Theodolite in the next article OAP II.] Adjustments, 248 Fig. 242 B E A - 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 tohe righ t he ht 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 Fig. 243.B~ 3-3~^ ~ Fig. 244. ofT Fig. 245. B —--------------- 0k. Fig. 246. 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 B. 2-44 TRmSIT AND THEODOLITE SURIEYINC G [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 I. 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 leftband screw and ti.ghtening 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' capstan-headed." One of the other pair of screws may need to be loosened to avoid straining 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 adjusteCd; 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. 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. 240 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 O', and the desired perFig. 248 B- --— a —- -'~~Fig. 28. —----- 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 graduar tion 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 TiEBDOLITE SURlVETI[, [PART IV, the defect. This error of eccentricity is similar to that explamied with respect to the compass, in the latter part of Art. (226). (363) Third adjustment. To cause the line of collimation to revolve 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 firm 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 nillion 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 "6 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 / l has moved in a vertical plane. If not, the apparent P p I 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 steepl~ and pin, which l',plies 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. /' ( i ",, (364) Centring eye-piece. In some in-' I struments, 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 B^r^,,pr 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 248 TRNSTIT AND THEODOLITE SURTVEYIN. [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, tils line of Fig. 251. 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 un screwing 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 acljusted 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 In Theodolites, the Telescope is revolved in the Ys. It 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. 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 adjnstment of' Centring the (,lject-,Jlass is the invention of Messra Gurley, of Troy. 250 fPART Ii CHAPTER IVo 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 in- tersection 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 190~, 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, 360~ + 90~ = 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 mears 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.' 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 360O. He will probably find quite a difference fiom that. ~o2 TRAN$ST AND THEODOLIT'E SURVEIMNG [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~ and 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 1800, will be on the left of the first line, and those to which the readings were more than 1800, will be on its right, looking in the direction in which the survey is proceeding, from A to B, and so on.* (369) Probable error. When a number of separate observa tions 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 "I Probable error" of this determination. This probable error is equal to the square root of the sum 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 6 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. (S66), an 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 tanr *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 are 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).Anles of deflection The angle of deflection of one line from another, is the Fig. 253. angle which one line /'c makes with the other line produced. Thus, in A 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 IX 1800 and more than 180~ for objects on the left; conversely tc 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, and 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 obfained by using three tripods. One would be set at the first star tion 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 360~9 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 P J ) / \EAS 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 j200"i The survey represented in the figure, B 20~ C 50~ is recorded in the first of the accompa- C 50~ D 180~ nying Tables, as observed with the The- i 0~ F 230o odolite; and in the second Table, as 300 F 210 F 300 G 2500 observed with the Transit. G 250o 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.' (374) 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 X If there are two verniers; take care always to read the degrees from the same vernier. Mark it A. 256 TRANSIT AND THEODOLITE SRVEYLNG. [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 rods On the cross-hair ring, described in Art. (330), stretch two more horin zontal 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 stactia. 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 6 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 FIeld-work. 25/7 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 i the tops of two stakes set in the ground about in the line. It 17 258 TRANSIT AND THEODOLITE SURVEYINGS [PART IT 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. " Mason & 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, &e, 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. 256. keep the field on the left C hand, as in following the or- _,' der of the letters in this |. figure, and turn the telescope |?200~ around "with the sun," and ~ 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~ = 180~ 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, and call them both plus. When the angle is between 90~ and 180~, call the difference between the angle and 180~ the Bearing, and call its Latitude minus and its Departure plus. When the angle is between 180~ and 270~, call its difference from 180~ the Bearing, and call its Latitude minus and its Departure minus. When the angle is more than 270~, call its difference from 360~ the Bearing, and call its Latitude plus and its Departu-e 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 "s angles" for "' Bearings."' Traversing" is the best method in all these cases. Inacccessible areas would be surveyed nearly as in Art. (131), except that the angles of the lines enclosing the space would be measured with the instrument, instead of with the chain. (378) Plattinge 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 beer 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 Mlethod. (379) TRIANGULAR SURVEYING is founded on the Fourth A2etswa of determining the position of a point, by the intersection of two known lines, as given in Art. (8). By an extension of the princi. pie, 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 6 Geodesic Surveying," but improperly, 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 calculated 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.] TRIANGM LAR SURVEYINGG 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 ccnsidered 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) Measaring a Base. Extreme accuracy in this is necessary, because any error in it will be rmultiplied 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, Suipports, 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 TRIAiNGULAR SURVEYING. [PART V mistakes in recording. They must be very carefully compared with a standard measure. QSupports 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 then, 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; these legs bearing pieces which can slide up and down them and on which the rods themselves rest. 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, horizonltal 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.] TRUANGULR SURVEYING'. 263 of the next rood The rods thus never tonch. The distance between the two sets of cross-hairs is of course to be added. A Base could be measured over very uneven ground, or eves 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) Correctioas 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 )ther; 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 cirTected for temperature. Thus, if an iron bar expands -Tunuv of its length for 1~ Fahrenheit, and had been tested at 32~, and a Base had been measured at 720 with such a bar 10 feet long, and found to contain 3000 of them, its apparent length would be 80,000 feet, but it? real length would be 8.4 feet more. An iron and a brass ba can be so combined that the difference of their expansion, causes two points attached to their ends to remain at the samta distance at all temperatures. Such a combination is used lon the U. S. Coast Survey. (383) Choice of Stations, The stations, or "' Trigonometrical points," which are to form the vertices of the triangles, and to be observed tc 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 equi. lateral as possible. This is seldom attainable, but no angle should be admitted less than 30~9 or more than 1204.t * More precisely, A being this angle, and not more than 2~ or 3", the differ snce 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 secondari triargulation, the unaobserved angle ought to be nearly a right angle. ;64 TRISNE ULAi S TRVEYI NG [CPAT V. To extened 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. 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 diagonals determined, take the place of simple triangles. The engraving is on a scale of 1:1200,000. S AC 14 US TS EW4AMPSH4IRE~ ^ RHODE ISLANDQ MAsACHOS^TTS^ ^ ^^ b,^~ 9^ TLA TIC 0 0M A0 Psi Vne T 1 L A N T I C A N C 266 TRINGULAIR SURVEYfING LPART v (384) 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. 2f2 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. 2G3 tecting tent. The heights of these signals varied between 15 and 80 feet. Another good signal consist3 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 can be turned down and the Theodolite set exactly under the former summit of the signal, i. e. in its vertinal axis. a Signals should have a height equal to at least ^-^ of their d' PART V.] TRIIhGULIR SURVEYING. 267 tance, sc as to subtend an angle of half a minute, which expel rience 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 refect 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, Drum. mond'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. (S85) Observations of the Aiig'les, These should be repeated as often as possibleo 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, the state of the atmosFig. 264, To determine ac a station A, z C whether its signal can be seen from B, projected against the sky or not, measure the vertical" angles BAZ and ZAG. If their A bum equals or exceeds 180,o A B -- will be thus seen from B. If e//,,././ not, the signal at A must be rais. ed till this sum ecuals 180. 268 TRIANGILAR 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. MEAN RIGHT OR LEFT OF I 2 REMARKS. COBSERVED TO VERNIER A. VERNIER B. READING. PRECED G OBJ T. A 700 19' 0" 18' 40" 70" 18' 50" B 1030 32' 20" 32' 40" 103~ 32' 30" R. C 115~ 14' 20" 14' 50" 1150 14' 35" R. When the angles are "6repeated," 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 - one. Set the instrument at D, 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. Calculato the distances CR and CL with the angle RDL instead of RCL, since they are sufficiently nearly equal. Then PART v.] TRIANULAR SURVEYING. 269 RL CDL + D. sin. (RDL + LDC) CD. sin. LDC, CRCL + R. sin. I. si. 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 fiaction, 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.GCL. Much calculation may be saved by taking the station D so that all the signals to be observed canr 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 F, i 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. 2 Take any point G on DE, and on DF set off DH II -DG-E Then is GI- parallel to EF, (since E DG: DIH: DE: DF) and the telescope directed G\C to its middle, K, will point to the middle of the PE diagonal EF. We shall also have DC = DK D-. A.ny such case may be solved by similar methods. For the investigation, see Appendix B. 270 TRIASNGLAR SURVEYINo. [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 correctedo 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; f + q - + +: 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 1800. It is still more accurate but laborious, to apportion the total error, or difference from 1800, among the angles inversely as the "' Veights,' 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) Caleulation and platting. The lengths of the sides of the triangles should be calculated with extreme accuracy, in t-wo 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 ono If the trianles were very large, they would have to be regarded as spherical, and the SuIP of their angles would be more than 180~; but this " spherical ex cess" would be only 1" for a triangle containing 76 square miles, 1 for 4500 aqua,:e miles, &c.; and may therefore be neglected in all ordinary surveying ope. ratious, PART v.] TRIAN4GULJR SUIRVEYING, 271 twentieth of a foot in a distance of 11- 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 "; Rectangular 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 Terication, 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 filliag up, 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 com pass 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. FPAUT 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 "6 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~, as explained in Art. (377). (391) Radiating Triangulatiion This name may be given t a method shown in the figure. Choose Fig. 268. a conspicuous point, 0, nearly in the 0. centre of the field or farm to be sur- veyedo Find other points, A, B, C, D, &c. such that the signal at O can be G( o'" / seen from all of them, and that the tri- \14', angles ABO, BCO, &c, shall be as A / nearly equilateral as possible. Mea- sure one side, AB for example. At A c 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 molre Foi the demonstration, see Appendix B. PART v.] TRIANIGLAR 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 / \o. \ each corner of the field, if the Theodo- F,' c lite or Transit be used, or take the \ / bearing of each, if the Compass be used. Consider first the triangles which have i XY for a base, and the corners of the field, A,, 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 h soehat 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 be calculated as in the last article. The /,/,>6\ \l / figure will then shew in what manner the --',,/ content of the field is the difference between ~ Y the contents of the triangles, having X (or Y) for a vertex, which 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 fiekl, 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 TRIANGULAR SURVEYING. [PART v (394) Inversion of the Fourth lethode In all the opera. tions 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 oppoo site bearings, for it will be at the intersection of the lines thus determined. (395) Defects of the Method of Intersectiosn 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 VT. TRILINEAR SURVEYING; By the Fifth Method. (396) TRILINEAR SURVEYING iS founded on the Fifth Method of Oetermining 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. 12, 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 ETU1VEYINGS [PARI Vi to 90~-ASB. The intersection of these lines at O 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 90~ —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 50 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 prob. lem. 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, 0 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 =- 180> -2 (90o ASB) = 2 ASB. Therefore, any angle inscribed in the cilcumfer ence 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, cr 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 = c; BC = a; ABC = B; ASB= S; BSC= 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. Si s. + 1' (a. sin. S. cos. T V=T-U c. sin. U a. sin. V SB-;or,= sin. S or Sin. S S A sin. ABS a. sin. CBS 8A =A sin. AS SO = S. sin. 8 sin. S' 278 TRILINEAR SURVEYINGE [PART Vi Attention must be given to the algebraic signs of the trigonometrical functions. Example. ASB = 33 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 Trio linear Method is to Maritime or Ilydrographical 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 thel 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 pro. cess will determine the depths and the places of each point of the bottom. ?ART VII OBSTACLES IN ANGULAR SURVEYING. (401) THE obstacles, such ad 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 1. PERPENDICULARS AND PARALLELSo (402) Erecting Perpendicalars, To erect a perpendicular to a line at a given point, set the instrument at the given point, and, if it be a Compass, 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 readin of the vernier and then turn it till the new reading is 900 more oi less than the former one. l'he Demonstrations of the Problems which require them, and fiom which they can conveniently be separated, will be found in App.c B. -a280 OBSTACLES IN ANTGLAR $URTV INGo [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 A 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- pL \C tion thus obtained a distance CP = CA. cos. CAB. PA will be the required perpendicular. (404) Letting fall perpendiculars, To let fall a perpendis cular 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, 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 complements of the angles measured. Then is A ACAB __tan. iAPO_ tan. APC + tan. BPC] CHAP. i.] Perpendiculars and Parallels, 281 (406) To let fall a pependicular to an inaccessible line from a given point. Let C be the point and Fig. 275. AB the line. Calculate the angle CAB _ 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 point of the given line, as A, direct Fig. 276. it on this line, and note the read- A', - 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 either the same as it was when the telescope was directed on the given line, or is 180~ different. It will then be directed (forward or backward) on PQ, a parallel to AB, since equal angles have been measured at A and P. The manner of readinl cllem is similar to the method of "Traversing," Art. (373). (408) To trace a line through a given point parallel to alt. ilaccessible 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 tNGBUIA $SIRVEYIN, [PART m CHAPTER II. OBSTACLES TO ALIINEMENT, 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 II off there a perpendicular, to C; set off - another at C to D, a third at D to E, c D making DE = B3C, 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 _ 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 All prolonged. At D, turn 600 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. CUAP. II.] Obstacles to Alinement. 28. (412) By triangulation, Let Fig. 280. AB be the line to be prolonged. A G _o E Choose some station C, whence 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). (41114) 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 an- gles of 45" in the directions C A - iL P and D. Set at some point, C, on one of these lines and deflect from CB 450~ and mark the point D where this direction intersects the direction BD. Also, at 0, deflect 900 from CB. Then, at D, deflect 900 from 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. m84 OBSTACLES IN ANGULAR SURVEYINGt. [PART vI, 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-piate 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)> (i16) By a random line. When a wood, hill, or other obsta. cle, prevents one end of the line, Z, Fig. 282. from being seen from the other, A, run, x 57 3 0~.2865 = 17'. The cor2000 - rect Bearing is therefore S. 80~ 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. X 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 ~B_ AB In this example, Bz 10.005 = tan. 17'.j 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 ____ z with the Compass. Run any number of zig-zag I courses, AB, BC, CD, DZ, in any convenient i, direction, so as at last to arrive at the desired point. Calculate the Latitude and Departure of each of' ic these courses and take their algebraic sums. The sum of the Latitudes will be equal to AX, and that,Bj XZ7...1B of the Departures to XZ. Then is Tan. ZAX =X A; 1. e. the algebraic sum of the Departures divided by the algebraic sum of the Latitudes is equal 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 referied; 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 repreThe 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. 283 OBSTACLES IN ANGULAR 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 perpen- dicular to it. As before, a right-angled triangle \ will be formed, and the angle ZAY will be the angle at A between the first line AB and the desired line AZ. /B This method of operation has many usefull 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. C E Take a point E, on the line CB, beyond the obstacle, and fromtit A.a set off a parallel to CD, to some point, F, in the line DBo Measure EF, CD, and CA. Then this proporcion 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 Fig. 286. obstacles prevent the preceding me- A -, B thods being 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. Mea& sure it, and a point, D, in the desired line, will be obtained. CHAP. x11.] 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 prolong ing 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(AB + BZ2); in Fig. 283, AZ = V(AX2 + XZ2); in Fig. 284, AZ V (AY2 + YZ2); in Fig.285,AG GB (CA- EG) t YZ2); in Fig. 285, AG' - -^ = l in Fig. 286, the, triangle ACD will give the distance AD. The method of Latitudes and Departures, Arts. (417) and (418), is very generally appliaEble. So is the following. (422) By triangulation, Let AB Fig. 287. be the inaccessible distance. From A,. __/\ 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 ineluded angle are known to find the c side AB. If all the angles can be measured, they-may be cor. rected, as in Art. (387).* (423) A broken Bases 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 tlis figure, and the following ones,the angular point enclosed in a circle Indicates the place at which the instrument is set 288 OBSTACLES IN ANGULAR SURVEYING. [PART vn. 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. C shall have abK2 AB c = a + b - 0.00000042308 x a-. as o 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. (424) 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 _( JL. I{ AC, perpendicular to AB, and as )1 nearly equal to it, by estimation, as the ground will permit. At C, mea- \~ sure 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 pro-,dwd. Then is AD = AB. CHAP. III.] Obstacles to Measuremesnt 289 (427) By triangulation, Measure a distance Fg. 290. AC, about equal to AB. Measure the angles at A J ] A and C. Then in the triangle ABC, two angles and the included side are known, to find another I AC sin.AOB ACBr side, AB == A~ B i (( sin. ABC' When the compass is used, the angles between the lines will be deduced from their respective Bearings, by the principles of Art. (213). If the angle at A is 90~, AB = AC. tang. ACB. If A 90~, and C = 45~, then AC = AB; but this position could not easily be obtained, except by the use of the Sext nt, 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 C i beseen, and from D, A and B. Measure / AC, AD, and the angles C and D. Then, | 1: in the triangle BCD, are known two angles and the included side, to find CB. Then, in the triangle ABC, are known Cal two sides and the included angle, to find the third side, AB. (429) To and 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 OB]STACLES IN XANGULAR SURVEYING, [PAlT vnI C. WHEN BOTH ENDS OF THE LINE ARE INACCESSIBLE. (430) General Methode Let Fig. 292. AB be the inaccessible line. 0 / ~ Aeaslas any convenient distance -, / \ CD, and the angles ACD, BCD, o, / \ ADC,BDC. o / 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. CB. Then, in the triangle AB C, two sides and the included angle care 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. ADO. sin. BD sin. CAD, sin. BDO Then find the difference of the unknown angles in the triangle CAB from the formula Tang. g (CAB-ABC) tang. (45~-KI). cot. ACB. Then is CAB = ( (CAB -ABC) + ( (CAB + ABC). Finall AB= CD ~sin. BDC. sin. ACB Finally, AB=CI ssin. CBD. sin. CAB Exanmple. Let CD = 7106.25 feet; ACD = 95 17' 20"; BCD = 61 41' 50"; ADC = 39~ 38' 40"; BDC = 78~ 35' 10"; required AB. The figure is constructed with these data on a scale of 5000 feet to I inch =1: 60000. By the above formulas, K is found to be 30~ 26' 5'; CAB 1130 55' 37"; and lastly AB = 6598.32. Both the methods may be used as mutual checks in any im portant case. CHAP. III. Obstacles to leasurement, 291 Fig. 293. If the lines AB and CD crossed c each other, as in Fig. 293, instead of \' x..tl being situated as in the preceding S; figure, the same method of calcula- \ \ tion would apply. (431) Problem, To measure an inaccessible distance, AB,'hen 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\ A 3 the angles CDA and ADB. Measure also the line. DC and the angle C. Then in the triangle ACD two angles, and the \, ~/ includel side are given to find AD. In the triangle DAB, the 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 ABo (432) Problem, To measure an inaccessible distance, AB, when only one point, C, can be found from wlich both ends of the line can be seen. Consider CA Fig. 295. and CB as distances to be deter- A —---------- B mined, having one end accessible. A Determine them, as in Art. (427),? by choosing a point D, from which /I7 C and A are visble, and a point E /', from which C and B are visible. "' At C observe the anglesDCAACB 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 uagle are kqown, to find AB. 292 OBSTACLES IN ANGULAR SURVEYING, [PART VIX (433) Problemsa To measiure 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 - CD. Find the distances CA and,, I DB, as in the preceding problem;,'/' i. e. choose a point E, from which A - CJ' F and C are visible, and another E 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. Pour inaccessible objects, A, B, C, D, being in a right Fig. 297. line, and visiblefrom only onepoint, A B \ i f. C D E, it is required to determine the dis- ~ -,/ tance between the middle points, B' _ and C, the exterior distances, AB. and CD, being known.'\ / LetAB = a, CD = b, BC = x; AEB = P, AEC = Q, AED == R. Calculate an auxiliary angle, K, such that 4ab sin. Q. sin. (R -P) tang. 2 K -- - (a -b)y sin.. sin. (R- Q) ~. ra+b a-b Then is x a- a-b 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 a Base line passes over water, &c. CHAP. III.] Obstacles to Measurement, 293 (435) Problem. Given the angles observed, at the ends of a lnze which cannot be measured, 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 CD AB sin. CBD. sin. CAB sin. B1DC. 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 CD can also be obtained Fig. 298. graphically. Take a line of any length,' as C'D', and from C' and D' lay off angles \\ I equal to those observed at C and D, and thus fix points A, B'. Produce AB' till it'I - equals the given distance AB, on any de- - i 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 ^ciqired distance to the same scale as AB.* (43B) Probleml Three points, A, B, C, being gwen 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. * See Article (458) for a solution of this problem by the Plane-Table. 294 OBSTACLES IN ANGULAR SURVEYINIS [PART vII CONSTRUCTION. Begin, Fit. 299. as in Art. (397), by describ-,' ^ ing a circle passing through' A and B, and having the cen- 1,/ I\1 tral angle subtended by AB, B i, _/ equal to twice the given an- p_ --- --— X~ —-- u 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 zircumference. 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 = 1QP, 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 == BCY == 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 known 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 mn position, by their mutualc 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 measzrerd, it is required to determine the position of P and Q CHAP. III.] Obstacles to measurenment 295 Fig. 300. C,, /.... i',',. ^J: //.-',; \ -", \,............. ——:'..G^~~~~~ ------—.. —- -"' 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; wherce to find EB. In the same way, the triangle CFD gives FCO. In the triangle EBC are known EB and BC, and the angle BBC ABC -ABE; whence EC and the angle ECB are found. In the triangle ECF are known EC, FC, and the angle ECF B3CD 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 i is measured by the same arc as the angle ABE, equal to that supplement by construction, So too with the angle DQP. 296 OBSTACLES:N AWt JLR SURVEYI T G [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, 1BP 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, BC, 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 vith one of tho given points, the problem is indeterminate. (413) Problem of the eight points PFour points, A, B, C, D, are inaccessible, but visible Fig. 301. from four other points, E, 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 sys- tem, of the angles under i which are seen the points E uv the first system. 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. 387E i.AP. IV. To Supply Omissions 297 CHAPTER IV. TO SUPPLY OlMISSIONS, (439) Any two omissions in a closed survey, whether of the irection 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. (853); 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 C is given in the margin on the scale of 40 chains to I inch (1:31,680), and 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 GBSTACLES IN ANGULIR SURVE INTG [PART VIL DIST. LATITUDES. DEPARTURES. STA. BEARING. IN LI N;-. 8.'E IN LINKS. N. E. ____ _ _ W. A North. 1284 1284.00000 0 0 B N. 32~ E. 1782 1511.22171 944.31619 C N. 80~ E. 2400 416.75568 2363.53872 D S. 48" E. 2700 1806.652Ct 2006.49096 E S. 18~ W. 2860 2720.02159 883.78862 F N. 730 28' 21" W. 4621. 1314.69682 4430.55725 4.526.67421 4526.67421 5314.34587 5314.34587 CASE 1. eWen the length and the Bearing of any one side are wanting. (4,0) 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. (4117). 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 73~ 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 thle length of one side and the Bearing of another are wanting, (441) When the defieent sides adjoin each othero 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 Omisslons, 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 | 5314.34587 East Departure. The course, EA, corresponding to this de- E ficiency we find, byproceeding 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. 73~ 28' 21" W.; and the side EF = 2860. (442) When the defcient 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 the courses to change places without changing Bearings or lengths, so as to / ( bring the deficient lines next to each B 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, 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 Bear. ng of FA. (443) Otherwise, by changing the 3Ieridian. 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 Southb 800o OBSTCLES IN.ANGJLARI SURVEIING. [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)o 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 * t''''''''a'-'' 2of CD and the Bearing of FA. STA. OLD BEARING. NEW BEARING. f CD and the Bearing of F -~ ~ Make CD the Meridian. The changA North. N. 80 W. B N. 32~ E. N. 480~. ed Bearings will then be found by C N. 80~ E. Norli. Art. (244) to be as in the margin. D S. 48~ E. N. 52~ E. To aid the imagination, turn the E S. 18 W. S. 6- 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 thl e lengtls of two sides are wantiny. (441) When the defcient 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 Prox 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. TV.] ~ Supply Omissions. 301 Thus, in Fig. 303, page 299, let EF and FA be the sides whose.engths are unknown. EA is then to be calculated, and its length will be found, as in Art. (41), to be 5497.026, and its bearing S. 75" 11' 15" W., whence the angle AEF 75~ 11' 15" —18" - 570 11' 15"; AFE =18" + 730 28' 21" 91~ 28' 21"; and EAF = 31~ 20' 24" whence can be obtained EF= 2860 and FA = 4621.5, (445) When the deficient sides are separated from ealc 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. B1.. Join BG. Then in the figure A / ABGEFA, are known the Latitudes and Departures of all the courses ex- cept BG, whence its length and Bearing r 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 = 1782, and CG = DE = 2700. (446) Otherwise, by changing the Mferidian. As in Art. (443), imagine the field to turn around, till one of the sides whose length is wanting, becomes a Mleridian 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 S 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-. LDBE ~IG.- -ceding example. Make BC STA.OLD BEARING. NEW BEARING. ST.. NEW BE.RING. a Meridian. The other Bears A North. N. 320 I. ings are then changed as in B N. 320 E. North. C N. 80 E. N. 480. the margin Calculate new D s 480 E. S. 80 EB. Latitudes and Departures. E S. "S~ W. S. 14~ E. The difference of the Depart F N. 738' 21" W. S. 740 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 obtained. This Latitude being put in the table, and the columns of Latitude then added un. their difference will be the length of BC. CASE 4. When the Bearings of two sides are wantting. (447) When the defieent sides adjoin each other, 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 defcient 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 m \ I E of the line joining the extremities of 1 / the deficient sides, CG in the figure. A / This line and the deficient sides form a triangle in which the three sides are 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 Geometrie pour les Arpenteurs," and Lhii illier's ".olygonometie." The method of Arts. (442), (445), and (448) is new. PART VI. PLANE TABLE SURVEYING. (149) 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 Method, " 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 Tab lo 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. * The Plane Table is not a Gonfometer, or Angle measure, like the Compass, Transit, & but a Gonigraph4 or Angle.drawer. 304 PLANE TABLE SURVEYING. [PART IIm. A pair of sights, like those of the compass, are sometimes placed under the board, serving, like a 6 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 thei r 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 Gonio meter, like that of Art. (213). (451) The Mltdade 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 ruler 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 Al.dade. 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 supoorted above the ruler by a standard, and capable of pointing upward or downward. It admits of adjustments similar in principle PART vIII.] PLANE TABLE SURVE ING. 305 to the 2d and 3d adjustments of the Transit, Part I.V 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 would 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) Iethod of Radiatiol. This is the simplest, though not the best, method of surveying with the Plane-table. It is especi. ally applicable to survey- Fig. 307. ing a field, as in the figure. In it and the following fi- A gures, the size of the Table // C is much exaggerated. Set the instrument at any conve- / -- nient point, as 0; level it, / 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. Mleasure 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 (3B1). 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 PLbNE TABLE SURVEING. [PART VIII (453) Method of Progression. Let ABCD, &c., be the line to be surveyed. Fig. 308. Fix a needle at a/ convenient point \ of the Plane-table, a \ near a corner so as to leave roonm for the plat, and \7 _ t 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 6, shall be exactly over that station. Sight to A, pressing the fiducial edge of the ruler against the needle, and draw a lhie by it. Measure BA, and set off its length, to the desired scale, on the line just drawn, from 6 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 cb 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, ~d on the plat, as be was fixed. Set up at D, make do 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 6 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, ca, on the plat. This method is substantially the same as the method cf surveying a line with the Transit, explained in Art. (372). It requires all the points to be accessible. It is especially suited to the sur vey of a road, a brook, a winding path through woods, &c. The offsets required may often be sketched in by eye with suflicient precision. PART VIii.] PLANE TABLE SURVEYINGe 307 When the paper is filled, put on a new sheet, and begin by fixing on it two points, such as C and D, 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. (45f) 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 (832). Set up the instrument at any convenient point, as X in the figure, and sight to all Fig. 309. B c......., 3 x Y the desired points A, 3, 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 XY, 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 tho 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. 30~ 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. 808 PLANE TABLE SRTVEYINGe [PART VILl (45a) Miethod of Resections This method (called by the French Recoupement) is a modification of the preceding method of Inter Fig. 310. 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 6a 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 Ae 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. i 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,he leading one, well deserves naturalization in our language. PART VIII.] PLANE ABLE 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 ta- -- ble represent a line AB on the ground. Set up at A, make ab. -. —... coincide with AB, and draw a. _line from a directed towards a steeple, or other conspicuous ob-'Lt: ject, as S. Do the same at B. Draw a line cd, parallel to ab, and intercepted between aS, and 6S. Divide ab and cd 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. (i4 ) To And one's pace oA tie grout 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. 31. O in the figure, whose place on the plat already on the table is desired, and orient it, by one of \ the means described in the last \ 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 10 PLANE TABLE SURVEYIN e [PART VII 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. (4i8) Iaaccessible distanees Many 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. (435) 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 &'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'. 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, abcd, similar to ab'6cd', and the fine ed thus determined will represent CD on the same scale as AB IPAIRT IX. bURVEYING WITHOUT INSTRUMENTS. (459) TIlE 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) DiBtances Sy 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 812 8URtEYING 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 main'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. thk 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 corn mon gallop 1010. For longer times, the difference in horses is more apparent fART IX.] jRVEtYIN WITHOUT INSTRTUIENTo 8313 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) Dstances 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 2t 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 corre spending scale for himself. (463) Distances by sounda Sound passes through the air Mwth 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, 32~ Fahrenheit. For higher or lower temperatures add or subtract 1- 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 8SURVEYIN WITHOUT INSThUDENT$S [PAIT 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 91 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 /,ilength of strings the time of its vibration, in seconds, = / lt 39t - (41b) Algl.es 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 distan ces, thus forming chain angles, explained in Art. (10o). (465) illethods of operation, The' First Method" of deter mining 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 de termined. The "1 Second Method," Art. (6), is also much employed, the right angles being obtained by eye, or by the easy methods given in Part II, ChapterV, 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. (464). The " Fourth Method,' Art. (8), may also be employed, the angles being similarly determined. The' Fifth Method," Art. (SO), 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 blrisk 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 hurricang l20 feet; and a violent hurricane, that tears up trees, &c., 150 feet per second PART Ix.] SURVEIIRNG WITIOUT INSTRIMENT$ 315 The method referred to in Art. (11) may also be employed. When a sketch has made some progress, new points may bl 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. t G 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 Methodl, chain angles being measured at B and HII, a point in AB prolonged. The pond K 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, HIIM is fixed by a chain angle at HI 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 con. trivances. 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. MAPPING CHAPTER 1. COPYING PLITS. (466) THE flat of a survey necessarily has many lines of construe tion drawn upon it, which are not needed in the finished map These lines, and the marks of instruments, so disfigure the papei 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. (4167) Stretdcing 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, passmg 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 mode rately 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 tie 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 trating. 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 Jf 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 interseaio5ns 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 arc 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 oairs of dividers may be used as explained in Art. (90). " Tri CHAP. i.] Copying Plats. 319 angular compasses," having three legs, are used by fxing 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 Reduction and Enlargement of plats in any desired ratio; under which head Copyingby the Pantagraph and Camera Lucida will be noticed. (474) ediuci g by squares. Begin, as in the preceding article, by drawing 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. 820 AMAPPING.O [PAT x. (475) Reducing by proportional seales. 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 con structed thus. Draw any line \c AB. With it for radius and A Fi.35 for centre, describe an indefinite \\ arc. With B for centre and a \\\\\ radius equal to one-half, one-third, \ \\\ \ \ &c., of AB according to the de-, \\\\\\\ \.. \ \\ \\ sired reduction describe another A D B arc intersecting the former arc 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)o 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 "1 Angle of reduction" may be simplified Fig. 316. thus. Draw a line, AB, parallel to one side B- C 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 re- ~ quired to be reduced. Apply a T square so / as to pass thorough 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 i, to,<'\= be half the original scale. Divide the outermost \,/ -= arc into three equal parts, and draw a line from \', — A to one of the points of division, as D. Then ieach 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 ares on. CHAP. i.] Copying Platse 321 the right hand side of the figure it corresponds to. The other part of that arc 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. (478) Redulcing by a pantagraph. This instrument consists of two long and two short rulers, connected so as to form a parallels gram, 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 lucidao 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) Eilarging 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 3 ". 2L MAPPING, IPART X CHAPTER II CONVENTIONAL SIN8, (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 pro" ducts, 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 Topographieal Miappiny. (480) The relief of grounde 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 "6 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 bo 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 equib distant horizontal planes, and their intersections with it to be drawn as they would be seen from above, or horizontally projected on the CIAP II.] Conventional Signs. 323 map. These are'" Contour lines." They are 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. The marginal figure shews an elongated W half-egg, or oval hill, -" ____ thus represented. s o t pf y The spaces between m'-/ the rows of vertical' 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. B24 MAPPING, [PAtT 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. Rocks are drawn in their proper places in irregular anulalar forms, imitating their true appearance as seen from above. The nature of the rocks, or the G-eology 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 scol. loped circles, irregularly disposed, Fig. 32. imitating trees seen "in plan," and: d 1 closer or farther apart according to \4.,: 1.1 M. the thickness of the forest. It is C1 usual to shade their lower and right' hand sides and to represent their 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 Ve figure, the lines being arranged in odd -n V v - ZnWc f \V Vv iumbers, and so that the top of each group is "" -.' v convex and its bottom horizontal or parallel,,,,"ili - ""'IL...;. to the base of the drawing. Mleadows are " - ^ sometimes represented by pairs of diverging lines, (as on the ri ght 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.''l'i jil l' Crops are so temporary that signs for them are,' j i unnecessary, though often used. They are usu- Il 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 Sec-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.,..... AIcarshes and cSwamps are represented by an -~ ~:,,_ irregular intermingling of the preceding _ _ _.. _..,-_,_ sign with that for grass and bushes, as in the =_ - "' figure. -.g h ha (481) Colorled Topograplhy. 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. G-amboge (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. Tc Those in Part IT, Chapter V, have the lines too close together in the middle, 326 lMAPPING. [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. MJarbling, 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 clouded appearance. Scratched parts of the paper may be painted over by first applying strong alum water to the place. The conventions for colored Topo0graphy, adopted by the French Military Engineers, are as follows. WOODS, yellow; using gamboge and a very little indigo. GRASS-LAND, green; made of gainboge and indigo. CULTIVATED LAND, brown; lake, gamboge, and a little India ink. "L 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. HEATH, FURZE, &c., marbled green and pink. VINEYARDS, purple; lake and indigo. SANDS, a light brown; gamboge and lake.; Yellow ochre" will Io. LAKES and RIVERS, light blue, with a darker tint on their upper and left hand sides. SEAS, dark blue, with a little yellow added. MIARSHES, 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, yreenish 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 uppei and left hand side) and of indigo on the opposite side. CHAP. ii.' Conventional Signs, 327 (185) 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, 3?26. Wind qmill 334. Post office, 37. Steam mill, 33. Taver'n, 5 328. Furnace, 8 336 Blacksmith's sholw, 329. NWoollen factory,. 337. G-uide board, t 330. Cotton factory,: 338. Quarry, 331. Glass works, 339 Grist mill, 0 332. Church, ~ 340. Saw mill, 333. Grave yard, 4341. 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 imitation of the realities; and, if desired, colored appropriately. Mines 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. A. large black circle, 0, 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.* V m t ti t e o t d d+++++++++++++++++b i t * Very minute directions for the execution of the details described in this chap ter, are given in Lieut. R..S. Smith's "Topographical Drawil,' " 828 MAPPING. [PART X CHAPTER IIT FINISHING TIE 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 roints. 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 Italic, and GOTHI O R E G Y P T 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 may 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. Cx&P [II.] Finishing the Mapo 32S (489) Joining papers If the map is larger than the sheets of paper at hand, they should be joined with a feather-edge, by pro. ceeding 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. (490) 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 XLo LAYING OUT, PARTING OFF, AND DIVIDING UP LAND. CHAPTER I. LXYN OPUT LANTDo (491) Its nature. This operation is precisely the reverse ot 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 (101), 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 am 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. HAP. I.] Laying out Land, 331 chain; or 208y1o- feet; or 69150 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. h71e 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 times 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 147- feet by 295 feet; or 49D yards by 982 yards. 50 paces by 100 is often used as an approximation, easy to be remembered. The content being given, and the diference between the length and breadth. Let c represent this content, and d this difference. Then the longer side = d + V'(d2 +4 c) Examp)le. 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. 832 L NIL @iOUT O NID DI BVDING UP LAUND. [PART X1 The content being given, and the sum of the length and breadth Let c represent this content, and s this sum. Then the longei side = + V(s -- c)..Exanmple. 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. (491) To lay out triangiles 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 bp the length of the side required. This rule makes the sides of an equilateral triangle containing one acre to be 480, 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, tbhe length of the side BlC = AB AiB. sin. B Exa?2mle. Eighty acres are to be laid out in the form of a triangle, on a base, AB, of sixty chains, bearing N. 80~ We. A the bearing of the side BC being N. 70~ E.e ere the angle B is found from the Bearings (by Art, (243), reversing one of them) to be 30~o 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 aay out circles, Multiply the given content by 7: divide the product by 22, ana take the square root of the quotient CHAP. I.] Laying out Land. 33b 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 178I links. A circle containing a quarter of an acre will have a radius of 89 links. (49T) Towan lotse 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 31 acres, and 171 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 inew 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 Fi. 343. on which taxes are unpaid is 3 -~__ _ 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. - 1 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 384 LALING OUT ANDI DIVIDING UP LAND, [PART xs 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. (4199) ew 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) wvith 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 constructiou by broken lines, and the lines of the result by heavy full lines. Ca.. IX.] 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 oif 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-, ~, ogram, and its perpendicular width, / CE, will be obtained as above. The length of one of the parallel. B B CE ABDC sides, as AC = A 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 9~~ y~: base AB. The quotient will be an approximate breadth, CE, or DF; too small if the sides con- verge, as in the figure, and vice / versa. Measure CD. Calculate E I' the content of ABDC. Divide the difference of it and the required 836 LYALIN OUT AND DIVIDIX4 UP LUNDe [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, G-H, 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' and B. The required sides will then be given by c these formulas: CD =/(AB2 2 x ABCD. sin. (A + B)) sin. B AD, -AB - CD) i. B sin. (A ~ B) BC- (AB CD) sin. A -) sin. (A A+ B) 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 - 80~ B - 60~o 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 D, of the given side AB, set out a "guess line," DE, perpendicular to AB, and calculate the content of B )D DEB. Then the required distance BF, from the angular point to the foot of the desired perpendicular, =- DD /( ). CHAP. i.] Parting off Land, 337 ExampZle. Let IBD = 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. Ind the angle B from the Bearings; then is c G BE u2 x B F G7 tang._ B B]l' Example. Let BA bear S. 75~ E., and BC F 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 quadrllateral Produce the converging sides to meet at B. Calculate the Fig. 369. content of the triangle HKB, whe- s C ther on the ground or plat, or froma s Bearings. Add it to the content of the quadrilateral required to be B H F parted off, and it will give that of the triangle FGB, and the me. thod of the preceding case can then be applied. (50S8) 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. (055), setting out a line in the required direction, calculating the triangle thus formed~ and obtaining BF by the same formula as in that Article. r?^ 388 LiYING 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 G BF /(2 x BFG sin (B + F) B T i 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- 300. 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 quadrila -.. lI teral from a field in which -c are given the side AB, and\, 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 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 /FAE' will fix the point D, since AD = AE - ED. (51S) When the field and the dividing line are given by Bear ings, 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 CHAP. nI.] Parting off Land, 339 desired content from this to obtain CDE. Calculate the side AE -sin. B. 12The D2 x CDE. sin. DCE * sin, hen A - sin. E. sin. CDE / Example. Let DA bear S. 201: W.; AB, N. 51 ~ W., 8.19; 1C) NO. 73-i~ E.; and let it be required to part off two acres by a fence, DC, running N. 45~ 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. (514) 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 D / line starting from a given point i on one of the sides, A in the figure, the base, AB, of the desired tri- 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 perpendieular distance from A to BD, and the quotient will be BD. * This original formula is very convenient for logarithmic computation. 40 LAYING OUT AND DIVIDING UP LANTDT [PART xi (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 Bearings, as in Art. (243). Then is BD =- 2x AB 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, since ED ED tang. BAD. -E AB - BE;ExamplZe Eighty acres are to be set off from a corner of a field, the course AB being N. 803 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, NO 17~ 23' W. The scale of Fig. 352 is 40 chains to I inch = 1: 31680. 2 ARD If the field were right angled at B, of course BD = AB o AB' (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 "6 guess lines" from the given point to various angles, and roughly measur. ing the content thus parted off. Fig. 353. If, as in the figure, A being the D given point, the guess line AD c - parts off less than the required coi / I tent, and AE parts off more, then / the desired division line AZ will B, 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. i1.] Parting off Lands 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 m-ake this proportion; ADE: AIDZ:: 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), D 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.* JExample. 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 divi sion 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. 842 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. 80 E., 3.63. The content of ABCD will be 3.19 square chains. Sub. tracting this from one acre, the remainder, 6.81 sq. chs., is the content of ADZ. AP 3.63 x sin. 65 = 3.29. Dividing ADZ by half of this, we obtain DZ = 4.14 chains. By the Second 1Method, 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 A 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 /' \ ground or on a plat, the division// can be made by construction,/ thus. From P draw PE, paral / / /.-P -lel to the side BC. Divide the //, /\ given area by half of the perpen- / /' \ dicular distance from P to AC,, / /\ \ and set off the quotient from C,/ / // I \ to G. Bisect GC in H. On <__ G EL' -', C HIE describe a semi-circle. On \ / it set off EK EC. Join. 1-1; Set off HL = HK. The line LM, drawn from L through P, will he the division line required.* If HK be set off in the contrary flirection, it will fix another line L'PM', 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 fmnce 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 Ileoded ior the Demolstration, the stud;ict should draw it himself to a large scale. CHAP. ii.] Parting off Lanld 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. i given by Bearings or angles, proceed p-\ thus. Find the perpendicular distances, PQ and P1R, from the given / \ point to the sides, by the formulas PQ -PC e sin. PCQ; and PR= / PC sin. PCR. Let PQ= q, PR =p, and the required content =y c. R I c Then CL - / ( - B r' V V p2 sin. LCMI Example. Let the angle LCM = 82~. Let it be required to part off the same area as in the preceding example. Let PC = 19.75, PCQ-170 30 1' PCR= 640 291'o 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 quadrilateral. Conceive the two sides of the field which the division line will intersect, Fig. 356. DA and CB, produced till they meet at a C point G, not shown in the figure. Calculate 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 _ giveinm Gummere's Surveying. The figure A E represents it on a scale of 20 chains to 1 inch =- 1 15840. 344 LI YNG OUT AND DIVIDING UP LANDO [PART XI Example. A field is bounded thus: N. 140 W., 15,20; N. 700 E., 20.43; S. 60 E., 22.79; N. 86 2 W.9 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 aiy figureo Let it be required to part off from a field a certain area by Fig. 357. 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. A + cot. B P — B ) L /rAP2.cot. B-BP.cot. oA - V4 IL~ --- cot. A o. B + i(/ot. Ao..' - Ap2 - BPd 2 2 ~ (cot. A + cot B ~ - j 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- A2- / AP2 Cot, -co AP — BP2 2 2d + cot.B- 2d' 2 d ~~A2 dP22 CIAP. ii.] Parting off Lando 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 81~. 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. 200 45'; or — 86~ 25'. The heavy broken line, C'D', shows the latter. The scale of the figure is 10 chains to I 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 /(. B). The line DE thus obtained will be / perpendicular to the line, BF, which bisects the angle B. The length of DE = v(2 DBE. sin. B) cos. ~ B 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. (5'2) 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 B46 LiAYlN OUT IND DIVIDIN) G P LANDB [PART xi. by the above sum, and the quotient will be the desired breadth, BC or AD, in chains. Example. Let the land on one side of EF be worth $200 pet acre, and on the other side $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 I 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. i. STRAIGHTENING CROOKED FENCES. (525) It is often required to substitute a straight fence for a crooked one, s~ 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......5 is rt t e 15.. 10.,,2s... 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, Ixx.] Dividing up Land. 847 Fig. 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. DIVIDIN5T UP LANBD (526) 1MOST 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, accord. ing 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 BD D - AB v /. So, too, E is fixed by BE BC V/. 348 LAYING OUT AND DIVIDING IP LANDe [PART XL Generally, to divide the triangle into two parts, BDE and ACED which shall have to each other a ratio = m: n, we have BD AB in/- c This may be constructed thus. Describe a Fig. 363. semicircle on AB as a diameter. From B set /" ~ —'\ off BF. BA. At F erect a perpendi- (\ \ \ cular meeting the semicircle at G. Set off BG ^ / from B to D. D is the starting point of the divi- A c 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 V-5; BD" -AB VA; BD'"t-AB V E The same method will divide the trian- D, gle into any desired number of parts hav- " "' ing any ratios to each other. A C (528) By ines perpendicular to a side, Suppose that ABO is to be divided into two parts having Fi. 3C5. 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 A E c AC. Then is AE = (AC x AD X — ) In this figure, AFE: EFBC: mgn: 1: 2. If the triangle had to be divided into two equivalent parts, the above expression would become AE = V (- 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 = m: n, by a line making a given angle with a side. Part off, as in A.rt (509) or (51O), Fig. 350, an area BFG m ABC. m + n CHAP. IIi.] Dividing up Land, S49 (530) By lines starting from an angle. Divide the side oppo site 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 = AC ~~ and DC =AC G. mz + n If the triangle had to be divided into three Fig. 367. parts which should be to each other: nm: n:p1) we should have AD = AC DE m +n + p/ / \ =AC, and EC = AC - _ D E m + n +-+ 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 B]D, and from E draw a / parallel to it, meeting AB in F. DF will be A -~ D c the dividing line required. The point F will be most easily obtained on the ground by the proportion AD: A:: AE = AC: AF. The altitude of AFD of course equals 2 ABC e- Al). 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. A x m before. Let this ratio = m: n, then AF =. AD In+ n1 o50 LHLNG OUT AND DIVIDING UP LAND, [PART xs (532) Next suppose that the trian- ig.369. gle ABC is to be divided into three equivalent parts, meeting at D. The / altitudes, EF and GH, of the parts kDE and DCG, will be obtained by A/ X~ v H — lividing ~ ABC, by half of the respective bases AD and DCo 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 GiH = G l ABC * A AD./ In practice it is more convenient to de- termine the points F and G, by these A \ proportions; z K B BK: AK:: EF: AF; and BK: 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 Fi 371, parts, graphically. Divide AC into three equal parts at L and M. Join E. \ BD, and from L and M draw paral- \ lels to it, meeting AB and BC in E ^ i B 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, D the middle of AC, draw FD and FE. The problem is now solved. f ~, CHAP. 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 n the triangle into two equivalent parts. / Join AD. Take E in the middle of BC, a,< -' and from it draw a parallel to DA, meeting AC in F. EDF is the fence required. (536) If it be required to di- B vide a triangle into two equiva- Fig. 374. lent parts by a straight line pass- // ing through a point within it, pro- //'I/ 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. JoinFB. From // / ~ B draw BG parallel to DF. Then // bisect GC in H. On HE de-,/\ \ scribe a semicircle. On it set off / EI = EC. Join KH. Set off, // \ 1L = HK. The line LM drawn 4_- _ IJ C A L k 17n C from L, through P, will be the'\ " / division line required.. 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 A the rectangular co-ordinates AE and /I 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; AH and HF being its rectangular co-ordinates. From B let fall the perpendicular BK on AC. 352 LAItNG OUT AND DIVIDIN UP LAND. [PAR XI. Then is AHll = AK ( ABC -~AE x ED) The position of the AE x BK-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 A ABC by half of the perpendicular DH, let fall from D to AB, and the quotient will be the distance BF. To find G, if, as in this figure, the trian- A- ~ c gle BDC (= BC x 1 DI) 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 = 5.17k. The content of the triangle ABC will be 666.6 square chains. Each of the small triangles must therefore contain 222.2 sq. chs., BID being one division line. We shall therefore have BF = 222.2 - DH =- 22.2 chains, BDC =45 x 2 x 5ol.7 = 116.4 sq. chs., not enough for a'second portion, 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. ^77. be also solved graphically, thus. Take CL -= AC. Jon DL, and from lB draw BG parallel to DLo Join DG. It will be a second line of division. Then take a point, / M, in the middle of BG, and from A it 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. mEAP. IlI.] iviling up Land, 353 (540) Let it be recuired to divide Fig. 378. the triangle ABC into three equiva- L lent triangles, by lines drawn from the three angular points to some unknown point within the triangle. This point is now to be found. On any -A e side, as AB. take AD =- AD. 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:: n: n, 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 intersec- tion of these parallels, in F, is the aC. point required. In the figure the parts ACF, ABF, BCF, are as 2: 3: 4. (H11) Let it be required to find Fig. 3G. 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 c 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. By calculation, the distance DA = D DC -AB X BC x CAB 4 X areanABC' The three slnall 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 e the equal distances. 25 P354 LYvxUII j OUT AD DIVIDING UP LAND, [PART xi (512) By the shortest possible lime. 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:: m: n; or DBE:ABC::m /: + n. - I _.From the smallest angle, B, of the triangle, / measure along the sides, BA and BC, a distance BD - BE - X AB x BC). DE is the - line required. It is perpendicular to the line BF which bisects sin. B U M the angle ABC; and it is = X X AB xBC) DIVISION OF RECTANGLES. (53) 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. Example. A rectangular field Fig. 383 ABCD, measuring 15.00 chains B 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 F D 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'9 and from F to F', will divide the land as required. The scale of the figure is 10 chains to I inch = I: 7920. The other modes of dividing up rectangles will be given undey the head of " Quadrilaterals," Art. (548), &c. CHAP. III.] Dividing up Land, 355 DIVISION OF TRAPEZ IDS. (544) By lnes 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 BGFE: B EFDA: m: n. The length of the desired dividing line, EF i/(m X AD~ + n x BC E V/^ m + n. Et;'-; AB(EF-B- BC) The distance BE B I i ( AD- BC zExamle. Let AD = 30 chains; BC = 20 chs. and A = 54 chs.; and the parts to be as 1 to 2; required EF and BE. Ans, EF=23.80; andBE 20.65. The figure is on a scale of 30 chains to 1 A 1 inch = 1: 23760. (515) Given the bases of a trapezoid, and the perpendicular distance, 3BHi between them; it is required to divide it as before, and to find EF, and the altitude, BG, of one of the parts. Let BC X BI BCFE: EFDA: m: n. Then BG = -B-C + AD-BG x/r 2 x ABGD x BH C-x B 12 1F- X + FT' V Lr + n AD~BC AD BG JAD - BC EF = BC BG x B 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 ABGD = 1350 square chains. Substituting these numbers in the above formula, we obtain BG = 20.96, and EF = 23.88. (546) By ines starting from peints 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 i. If it be also required that the division lines shall start frow yiven points on a side, proceedig. 3 thus. Let it be required to / I ~ divide the trapezoid AIBCD /,\ into three equivalent parts by / fences starting from P and Q Divide the trapezoid, as above / directed, into three equivalent A ~.~ P i - trapezoids by the lines EF and GiH. These three trapezoids mu 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 cf dividing trapczolds, 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- lel to AD, into two parts,.'" BEFC and EFDA, which shall be to each other as / n'n. Prolong AB and CD to intersect in G. Let a be.' the area of the triangle \ ADG, obtained by any me- K thod, graphical or trigonometrical, and a' = the area B H G[ i of the triangle BCG, obtained by subtracting the area of the given quadrilateral from that of the triangle ADG. Then GK = G-H i(7 ma_+~~ ~). Having measured this length of GK from G oin (v + n) al' GIl, 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 trape Mcid, any line drawn through the middle of the first line, and meeting the paral lei bases, will divide the trapezoid into two equivalent parts. CHAP. III.] Dividing lp Land. 357 Otherwise, take GE = GA / -- -; and from E run' V(m +) n a' a parallel to AD. If the two parts of the quadrilateral were to be equivalent, gm =n, and we have GKI GIH -t- a ); and consequently GE to GA in the same ratio. Example. Let a quadrilateral, ABCD, be required to be thus livided, and let its angles, B and C, be given by rectangular co-ordinates, viz: AB' = 6.00; B'B = 9.00; DC' = 8.00; CC = 13.00; B'' = 24.00. Here Gi 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 IH = 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 =: 15840. If the quadrilateral be given by Bearings, part off the desired area = -- ABCD, by the formulas of Art. (501). mn + n Suppose now that a quad- Fig. 386. rilateral, ABCD, is to be divided into p equivalent parts, ^' by lines parallel to AD. - Measure, or calculate by Trigonometry, AG. Let Q be EF the quadrilateral ABCD, and, N as before, a' = BCG. Then A GE-AG j+ Q; GL=-AG/ p 2; at' + Q a' + Q 1 2 - A D, then p; so in any similar ce If the quadrilateral be given by ]Bearings, part off, by Art. (504), - ABCD, then part off 2 ABCD; &c.; so in any similar case, pP i,58 OTUIN& OUT AND DIVIDIN UP L ND [PART xI (59) 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. parts having a ratio = n. By hypo-, thesis, ABEF m. ABCD. \ Taking away the triangle ABG, the A G P i B remainder, G-BEF, 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 Hines runaing in any given directfoni To divide a quadrilateral ABCD into two parts: m' n, part off from it an m area -. 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. (55l) By Iines startfin from asn angle, ABCD is to be divided, by the line CE, into two Fig. 388. parts having the ratio mn n. n Since the area of the triangle ~ CDE =-. ABCD, DE will 2n __ n be obtained by dividing this area A, by half of the altitude CF. (552) By limes starting from points in a sidee 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=. ABCD) as m+n/ \ n also ABE; AB, BE, and EA be- A E —~ ing given on the ground. BEF will then be known = ABFE - - BEF ABE. Then GF =,BE and the point F is obtained by running a parallel to BE, at a perpendicular distance from it = GF. CHAP. iir.] Dvidling up Land, 359 To divide a quadrilateral, ABCD, Fig. 390. graphically, into two equivalent parts by a line from a point, E, on a BP / side, proceed thus. Draw the diago- nal CA, and from B draw a parallel \ / \ to it, meeting DA prolonged in F. \ \ Mark the middle point, G, of FD. ~ A, G Join G E. From C draw a parallel to EG, meeting DA in I. EHl 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 into three equal parts, at / F, F', and G, G'. Draw BF, BF'. From G draw GiH, parallel to FB, and _ A2 from G' draw G'H', pa- F' rallel to F'B. FH and F'tI are the required lines of division. Let it be required to make Fig. 392. the above division by lines C starting from two given points, P and Q. Reduce D \ the quadrilateral to an equi- // \ valent triangle C1E, as in / \ Art. (87). Divide EB into // three equal parts at F and'__ G. Join CQ, and, from G, 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 lanes passing through a point within the tigm'e. Proceed to part off the desired area as in Arts. (519), (520), or (521), according to the circumstances of the case. B60 LAYING OUT AND D ID iG UP LAND. [PART XI DIVISION OF POLYGONS. (555) By lines running in any directio Let ABCDEFG be a given polygon, and BH the di- Fig. 393. rection parallel to which is to be drawn a line PQ, dividing the polygon into two parts in any de- C sired ratio= n: n. The area \ rn PCDEQ i. ABCDEFG. P\ -~ ~\~ m ~B~ Bj Taking it from the area BCDEH, B~ —-- the remainder will be the area BPQIH The quadrilateral A ^ F BCEH, CQE being supposed to be drawn, can then be divided by the method of Art. (458), into two parts, BPQH and 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 froman aangReo Produce one side, AB Fig. 394. /\N\\ \ A W B' l\\ / i \ \'X B ~/ /^ \/ \~'. \ \ of the given polygon, both ways, and reduce the polygon to a single / \, \- N \ I, \ \,. "\k / t \, \ / \\ evalente triangle, Y, bys the aethod of Artp (82) aThen duivalent triangle XYZ by the me thored ratio, as at and dra 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. coU. II.] Dividing tip Land. 36. 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. 350 E., 23.00; N. A 7510 E., 30.50; S. 381 E., 46.49; N. 66~0 W., 49.64. It is to be divided into four equivalent parts by two straight lines, i one of which is to run parallel to the third A side; required the distance of the parallel division line from the first corner, mear sured on the fourth side; also the Bearing A D 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. 880 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 Trigoomemetry 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. 862 IT S. ITBLIC LANDS PART XIHI Fig. 396. ^^^ ^ R )A0 E" I N(, 1_____2L1 1, nh//?. o -— i _T_ L o ~7lleL K( ~L - G A * _ Wou1E 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,ither, 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 lion. J. Butterfield, " Commissioner of the General Land Office," and communicated to the author by Hon. John Wilson, the present Commissioner.o The aim of the " Instructions" is stated to be 6' simplicity, uniformity and permalnency." 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 IU 8. 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 8 910 1112 acres. The sections in each Township are 17 16 15 14 13 numbered, as in the margin, from 1 to 36, 19 20 21 22 2324 beginning at the North-east angle of the 29 28 27 26 25 Township, and going West from 1 to 6, 31 32 3333536 then East from 7 to 12, and so on alternately 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) l fficulty. " 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 wvestern 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~ of North Latitude. The length of a degree of Longitude in Lat. 460 N. is taken as 48.0705 statute miles, and in Lat. 470 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 (111- per degree of Latitude) equals the above divided by 11g, i. e. 0076 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 Double 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 Ue S. PUBLIC LANDS, [PART XI1 (5'6) Running Township linese " 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 prineipal 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 coreers 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., I. 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 sholld prepare a diagram of the townships, with the lumbers here referred to. in their proper places, as here indicated PART xii.] RuMning Townsshi Lines,.67 ships north of the base line); throwing the excess over, or dteiciency under, four hundred and eighty chains, on the last half-mile, according to law, and at the intersection establishing the " CLOSING CORNERt," 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. 1 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. Townuships situated SOUTI 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, &co 368 ITO $. PUBLIC LANDS, [PART XII1 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 6 97 7 53 5 17 1 6 5 4 3 2 1 99 98 96 72 70 54:52 36f34 1816 100 94 9568 9 50o 51 32 33 14 r15 12 7 8 9 10 1 12 7 92 93 91 67 49 31 13 89 9065 66 47 482 9 30 11 12 13 18 17 16 15 11 13 18 87 86 64 46 28 10 88 84 85 62 634 4526 27 8 9 2 1 9 20 21 22 23 2 19 82 81 61 43 25 7 83 79 80 59 60 41 4223 245 ( 25 30 2 28 27 26 25 130 77 76 58 40 22 4 78174 75 56 57 38 39 20 212 3 36 31 32 33 34 35 36 31 73.._5 37 19 1 6 5 4 3 2 1 In the above Diagram, the squares and large figures repre. sent 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 XTI.] Rnmliag Section Litest 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 random'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 trua 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 random 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 south]wardly 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 through 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 post 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, bat 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 meat surement 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, 3,n 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 NORTH 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 publie 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 6 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 the PART XIi.] Exceptional Mlethods, 871 uplands for quantity and timber, the principles of the act of 5May 24th, 1824, may be adopted, and you are authorized to enlarge the quantity, so as to embrace four acres front by forty in depth, form ing 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 front and in rear, at the distances requisite to secure the quantity of 100 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 MIethod." 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 from 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 i'eander 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 "6 meander corners;" and in meandering you are to commence at one of those corners on the.ownship line, coursing the banks, and measuring the distance of mach course from your commencing corner to the next' meandec 372 U., S, PUBLIC LA t )5 [PART XII 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 IEANDER CORNERSn 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 lae.s 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 fiactional 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) Il1arkiig 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 6 sight 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 lile passes from the blazed trees. Due care must ever be taken to have the lines so well marked as to be readily followed." (566) lMarkiing 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 nAt, a post is to be planted thereat and the position of the corner post is to be PART XII.] Marking Corners. 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 monument 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. MiEANDER 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. (564). When posts are used, their lengtn 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 diagonally, thus: Wp->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 itfaces. Thus, if the post be a common boundary to four townships, say one and tzwo 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: ( R. 1 W. The position of the post which Friom. tc E.. S. is here taken as an example, is S. 31 shewn in the following diagram. ( 2X2W. fromn N. to W. 1 S.. 2 W. R. 1 WV ( 36 ) T. 1S. T. I S. ( 1 7;)I AV. 31 from E. to S. _ 2 S _ $ 1.2W. 12 W. II_ froti W. to S. S 2 S. ( R. 2 W R. IV. ( 1 T. 2S. T. 2 S. 8374 U. S, PUBLIC LANDS. [i'AR' xlI 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 thefour 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' 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 distant 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 aotched, 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 blaze 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 of the blaze. The kind of tree and the diameter of each are facts tc 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-section 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, or 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 mivzay 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 Stan lard Parallels or Correction lines, whereon are to appear both the Car 376 iU So PUBLIC, LANDST [PART X1 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 Bookso There should be several distinct and separate field-books; viz.: " 1. Field-notes of the MERIDIAN and BASE lines, showing the establishment of the township, 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. Fielcl-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 TOWNSHIPS, 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, by 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 fronm the, Instructions " which we have followed throughout, will shew what is required. The ascents and descents are recorded in the right-hand columns, r.ArT Xir.] Field-Notes' 37T FIELD NOTES OF THE EXTERIOR LINES OF AN ISOLATED TOWNSHIP. resil no/es of the S'urvey of township 25 north7, of range 2 west, of the i[TilameZntt mnerldian, in thte T7erritory of OREGON, by Robert Acres, deputy surveyor, under hi conltract No. 1, bearing dtae the 2d day of January, 1851. 2'GChs lks. Fet. _c |TOWNSHIP LINES commenced January 20, 1851. Southern boundary variation 180 41' E. E East. On a randomA line on the south boundaries of sections 31, 32, 33, 34, 35, and 36. Set temporary mile and half-mile posts, and intersected the eastern boundary 2 chains 20 links nortll;- of the true corner 5 miles 74 chains 53 links. Therefore the correction will be 5 chains 47 links WV. 37.1 E links S. per mile. TIUE SOUTHERN BOUNDARY variation 18~ 41' E. West On the southern boundary of sec. 36, Jan. 24, 1851. 40.00 Set qr. sec. post from which a 10 a beech 24 in. dia. bears N. 11 E. 38 Iks. dist. a do 9 do do S. 9 E. 17 do c 62.50 a brook 8 1. wide, course NW....................... d 10 80.00 Set post cor. of secs. 35 & 36, 1 & 2, from which............. a 5 a beech 9 in. dia. bears S. 46 E. 8 1. dist. a do 8 do do S. 62 W. 7 do a W. oak 10 do do N. 19 W. 14 do a B. oak 14 do do N. 29 E. 16 do Land level, part wet and swampy; timber beech, oak, ash. hickory, &c. West. On the S. boundary of sec. 3540.00 Set qr. sec. post, with trench, from whhich a 10 g a beech 6 in. dia. bears N. 80 E. 8 1. dist. C planted SW. a yellow locust seed.' 65.00 To beginning of hill a 5 80 00 Set post, with trench, cor. of sees. 34 & 35, 2 & 3, from whicl a 20 a beech 10 in. dia. bears S. 51 E. 13 1. dist. IE ~ do 10 do do N.56 W. 9 do planted SW. a white oak acorn, NE. a beech nut. Land level, rich, and good for farmiing timher same. a West, On the S. boundary of sec. 3440.0C Set qr. sec. post, with trench, from which a 5 a B. oak 10 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 mound with trench as per instructions, and Planted NE. a W, oak ac'n; NW. a yel. locust seed. feaX ~ SE. a butternut; SW. a beech nut Land level, rich and good for farming, some scattering oal andd walnut. &c., &c., &c 878 U. S. PUBLIC LANDSe PART XII. FIELD NOTES OF THE SUBDIVISIONAL OR SECTIONAL LINES, AND MEANDERS. T7ownship 25 NV., Range 2 W'T., Willaametie 31er. Chs. lks. Feet. SUBDIVISIONS. Commenced February 1, 1851. Nortl. Between sees. 35 and 369.19 A beech 30 in. dia....................................... d 10 29.97 A beech 30 in. di....................... 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 51.90 A beech 18 in. dia..oa.........oo....o...........ooo...... d 5 76.73 A sugar 30 in. dia........................................... d 8 80.00 Set a post cor. of sees. 25, 26, 35, 36, from which d 2 a beech 24 in. dia. bears N. 62 W. 17 1. dist. a poplar 36 d o 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 lnd'gr. spice, &c. East. On random line between sees. 25 a.cd 369.00 A brook 30 1. wide, course N.... o.................... d 10 5 15.00 To foot of hillo.....o..................... d 10 c 40.00 Set temporary qr. sec. post........................... a (60 55.00 To opposite foot of hill.........d 940, 72.00 A brook 15 1. wide, course N................... d 20 80.00 Intersect E. boundary at post...................... a 10 Land level, second late; timbler, 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 bankl dowl n stream. Comenelced Feberuary 11, 1851. Dist. Courses.. REARKS. Cbs. lI... S. 76 W. 18.46 In section 4 bearing to corner sec. 4 on rihlt bank N. 700 W S. 61 W. 10.00 Bearing to cor. sec. 4 and 5, right baik N. 52~ VW. S. 61 W. 8.18 To post in line between sections 4 and 5, lbreadth of river by triangulation 9 chains 51 Ilnks. 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 Smn th's (caim, coulrse E. S. 44 W. 21.96 S. 36 W. 27 53 To post in line between sections 5 and 8, breadth of riNer e trtiant1lation S chains 78 li ks. &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 trian. gles. 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 ~' ". Thus 28 degrees, 17 minutes, and 49 seconds, are written 28~ 17' 49". Fractions of a second are best expressed decimally. An arc, including a quarter of a circumference and measuring a right angle, is therefore 90~. A semicircumference comprises 180~. It is often represented by 7, 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"= 0.00000405. The length of the radius of a circle in degrees, or 360ths of the circumference - 57~.29578 = 57~ 17' 24".8 = 3437'.47 = 206264".8.t An arc may be regarded as generated by a point, M, Fig. 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 -- 90~ -=; ABC = 180~ -=; = ABCD 2700 ABCDA =3600 2 7r. The point may still continue its motion, and generate C 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, D OM, indefinitely produced, generates corresponding angles. * For merely solving triangles, only Articles (1), (2), (3), (5), (6), (10), (11), and (12), are needed. r The number of seconds in any arc which is given in parts of radius, radius being unity, equals lbe length of the arc so given divided by the length of the arc of one second; or multiplied by thb tumber of seconds in radius. 380 TIGIRONiOMETRY. [APP. A. If the point, M, shculd move from the origin, A, in the contrary direction cu 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 oo in one direction, and fiom 0 to - o in the contrary direction. The Complement of an arc is the arc which would remain after subtracting the,rc from a quarter of the circumference, or from 90~. If the arc be more than 90~, its complement is necessarily negative. The Supplement of an arc is what would remain after subtracting it from half the circumference, or from 180~. If the arc be more than 180~, its supplement is necessarily negative. (8) rE'iigolBa.ometseaal ]Lines. The relations of the sides of a triangle to its angles are what is required; buitit 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 Trigonometrical Lines; or r Circular Functions, because their length is a function of that of the circular arcs. The principal Trigonometrical lines are Sines, YTangents, and Secants. 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. 898. the diameter which passes through the other ex- B _ tremity. The TANGENT of an arc, AM, is the distance, 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 extremrity. The SECANT of an are, AM, is the part, OT, of the prolonged radius, comprised between the ] 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, MQ is the cosine of AM, BS its cotangent, and OS its cosecant. The cosine MQ is equal to OP, the part of the radius comprised between the centre and the foot of the sine. The chord of an arc 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 be. tween the radius and the sine. The Trigonometrical lines are usually written in an abbreviated form. Calling the are AM = a, we write, iMP = sin. a. AT = tan. a. OR = sec. a. MQ = cos. a. BS cot. a. OS = cosec. a. The period after sin., tan., &c., indicating abbrevration, is frequently omitted, The arcs whose sines, tangents, &a., are equal to a line =a, are written, sin.-' a, or arc (sin. = a); tan.-1 a, or arc (tan.= a); &c. * For the vreat value of this Indirect mode of comparing the sides and angles of triangles, ta Oomte's " Philosophy o Mathematics," (Harpers', 1551,) page 225. pp. A.] TRIGONOMETRY. 381 (i) The lines as ratios. The ratios Fig. 899. 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 ra- dius or arc. Consequently, radius being unity, \ these lines may be expressed as simple ratios. Thus, in the right-angled triangle AB C, we _ would have A C in. A BC opposite side AC_ adjacent side AB hypothenuse' AB hypothenuse' BC opposite side AC adjacent side tan.A - - cot. A= = AC adjacent side BC opposite side' ec A AB hypothenuse AB hypothenuse a A = - ajcensieccosc. 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 "homogeneous." 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 t. (5) Their variationsB iBIL lenIgtIae 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 arc has Fig. 400. become 90~, the sine has become equal to $' B S radius, or unity, and the tangent and se- cant have become infinite. The complee -___ ____ mentary lines have decreased; the cosine being equal to radius or unity at starting and becoming zero, and the co- \ A tangent and cosecant passing from infin-' A ity to zero. When the point M has passed the first quadrant at B and is proceeding towards C, the sines, tan- N' N T 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 882 indicates these variations. The sines and tangents of very small arcs may be regarded as sensibly proportional to the arcs themselves; so that for sin, a", we may write a. sin. I"; 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 379 may read thus: The number of seconds in any very small arc given in parts of radius, radius be',ig unity, is equal to the length of the are so given divided by sin. 1 382 TRIGONOMETRY. IAPP. 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: a9 a5 aa sin. a =a - - - 2.3v 2etc. 2.3 2.3.4.5 2.3.... as" a a6 cos. a=l~ — -- +~ - etc. 2 2 3.4 2....6 Let it be required to find cos. 30~, by the above series. 30 30~ =- T10 = X 3.1416 =.5236. Substituting this number for a, the series becomes, taking 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. 30~ 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. 30~ =. sin. 450 =- /2. sin. 60~ -- v/3 tan. 30~ = ~/3. tan. 45~ = 1. tan. 60~ = /3. sec. 30~0 = /3. sec. 45~ -'/2. sec. 60~ - 2. (6) Theiri 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 follow. ing 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 Triigonometrical Linesfor Arcs from 0~ to 3600 Arcs. 00 Between 0O and 900, 900 Between 900 and 1800. 1800 Sine... 0 +, and increasing, +1 +, and decreasing, 0 Tangent.. 0 +, and increasing, o -, and decreasing, 0 Secant..+ 1 ~ -, and increasing, ~oo -, and decreasing, -1 Cosine..+1 -, and decreasing, 0 -, and increasing, -1 Cotangent c. oo +, and decreasing, 0 -, and increasing, Fco Cosecant.. co -, and decreasing, +1 +, and increasing, cI Arcs. 1800 Between 1800 and 2T70. 270 o Between 2700 and 3600. 8600 Sine. 0 -, and increasing, 1 ~, and decreasing, 0 Tangent 0 -, and increasing, c -", and decreasing, 0 Secant. - -, and increasing, ic +, and decreasing, + 1 Cosine.. — 1 -, and decreasing, 0', and increasing, + 1 Cotangent. 1o —, and decreasing, 0 -, and increasing, F1r Cosecant..:c —, and decreasing, 1 -, and increasing, F oo PDP. A.] TRIGONOI ETRY. 38 From this table, and Fig. 400, we see that an arc and its suppllement 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. (a~+ 180~) = cot. a'. sec. (0a+ 180~) — sec. a~. cosec. (a~+ 180~) — cosec. a" sin. (-a ~) =-sin. a~. cos. (-a) 0 cos. a~. tan. (-a~)= —tan. a~. cot. (-a~) = —cot. a. sec. (-a0)= sec. a~. cosec. (-ao) =-cosec. a~. An infinite number of arcs have the same trigonometrical lines; for, an arc a, the same arc plus a circumference, the same arc 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, iproceed thus: Let 1029~ be an arc the sine of which is desired. Take from it as many times 360~ as possible. The remainder will be 309~o Then we shall have sin.309~=sin.(1880~- 3090)=sin.- 129~=- sin.129~= —sin.(l 80~ -' 9)= — sin.51" (7) Taheir mutual relatiosas. Radius being unity, sin. a~ cos. a~ tan. a =. cot. a~ = -- cos. a0 sin. a 1 1 sec. a - cosec. a cos. a sin. a~ tan. a~ Xcot. a~ = 1. (sin. a0)2 + (cos. a0)2 1* I + (tan. a~) = (sec. a0)2. 1 + (cot. a~0) = (cosec. a~)2. Hence, any one of the trigonometrical lines being given, the rest can be found from some of theses equations. (8) TWO arcs. 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. 5. 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 + b) tan a tan. b 1 - tan. a. tan. b tan a -tan. b tan. (a - b) tan. tan. b I +t tan. a. tan. b cot. a. cot. b l I cot. (a + 6) = cot. ot. cot. b + cot. a cot. a. cot. b - 1 cot. (a - ) = cot. b - cot. a * The square, &c., of the sine, &c., of an arc, 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 are,bus, sin.2 a~, tan.2 aO, &c. But the notation given above, places the index as used by Gaunss belambre, Arbogast, &c., though the first two omit the parentheses. 384 TRIGONOMETR., APP. A. sin. a. sin. b -. cos. (a -- ) cos. (a +- ). cos.. acos. b =. cos. (a + b) + A cos. (a b). sin. a. cos. b =. sin. (a + b) + sin. (a - b). cos. a. sin. b -. sin. (a - b) -- sin. (a - b). sin. a - sin. b =2 sin. i (a -- ) cos. ~ (a -b). cos. a + cos. 6a = cos. ~ (a + b) cos. - (a - b). sin. a - sin. b = 2 sin. ( ( — b) cos. (a + b). cos. b cos. a = 2 sin. (a - b) sin. - (a + b). sin. (a + b) cos. a. cos. b tan. a - tan. b =si ( -- cos. a. cos. b sin. (a - b) tasin. a - ta n. b cos. a. cos. b sin. (a - b) cot. b - cot. a =.. sin. a. sin. b (9) Doubale ianld Baalf ares Letting a represent any arc, as befcat we have the following formulas: sin, 2 a = 2 sin. a. cos. a. cos. 2 a = (cos. a)' - (sin. a) = 2 (cos. a)2 - 1 1 - 2 (sin. a,. 2 tan. a 2 cot. a 2 tan. 2 a= 1 — (tan. a)2 (cot. a)2-1 cot a - tan. a cot. 2 a (= cot. a = (cot. a - tan. a). 2 cot. a sin. 3 a= / [ (1-cos a)]. cos. a a = [ 1 (i + cos. ) ]. sin. aos. a c. a cos. a tan. A a - os. a 1 -+.os. a. a n a + s. 1+ cos. a_ sin. a _ + cos. a\ cot. a_ a - n -s sin. a - cos. a 1 -cos. a (10) Trigononsetricsal Tablles. Tn 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 45~ to 90~ at the bottom; the latter being complements of the former. Consequently, the columns which have Sine and Tangent at top have Cosine and Cotangent at bottom, since the cosine or cotangent of any arc is the same thing as the sine or tangent of its complement. The minutes to be 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 proportion. 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 cotangenta from 90~ to 45~. Beyond those points they are integers and decimals. The calculations, like all others involving large numbers, are shortened by the ise 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 necesary part of trigonometry. APP. A] TRIGONOMETRY. 38& 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 any two sides being given, the other parts can be obtained by one of the following equations: A Given. Required. Formulas. a a. a, b c, A, B c =( 2(a6 + b); tan. A =; cot. B=. a a a, c b, A, B 6 =/(2 _ a2); sin. A -; cos. B -. C C a, A b, c, B b =a.cot. A; c= B=90o-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) Obliqufe-agle~d Triai- Fig. 402. gles. Let ABC be any oblique-angled 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 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.-In every plane triangle, the sum of two sides is to their diference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference. THEOREM III. —I every plane triangle, the cosine of any angle is equal to a fra. tion 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 twoice 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. sin. B sin. C Calling the given side a, we have b - a. sin'. and c a sin sin. A sin. A 25 2S6 TSRIGONOMETRY. [rAPP. CASE 2.-Given two sides and an angle opposite one of them. The angle oppo. site the other given side is found by Theorem I. The 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 8.-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 180~), 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=180~0C. Then a -b tan. (A - B) tan. j (A + B). +. sin, C (A+B) + (A- =A. B)=A. (A+ )-(A-+ B)3=B. c=a. sin. A' In the first equation cot. i C may be used in the place of tan. A (A + B). CASE 4.-Given the three sides. Let s represent half the sum of the three sides i (a + b + c). Then any angle, as A, may be obtained from either of the following formulas, founded on Theorem III.: Bin. A -= / [(s-B) ( osA. A= L be ]'ttan. A A / = (8 b) (s-c) sin. A = 2/ [s (s-a) (s-b) (s-c)] be 62 + c2 - al cos. A= - 2 2 be The first formula should be used when A < 90~, and the second when A > 90~. 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. DElff0STRATIONS OF PROBLEMS, ETC. MANY of the problems, &c., contained in the preceding pages, require Demonstra tions. 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, BEX AD X CF=EAXD XBF. 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 BFG and CFD, we have CD: CF:: BG F: BF. Multiplying these proportions together, we have F BGXCD: BEXCF:: ADXBG: AEXBF. Multi- / 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 / 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 _ three non-consecutive parts equals the produet of the other three parts; then will these three points lie in the same straight lineg 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 -trzanyle, lines Fig. 405. be drawn, to a point situated either within or without the A triangle, and prolonged to rmeet 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 = =\ = That is, in Fig. 405, or Fig. 406, AE X BF X CD=EB X FC X DA. Fig. 406. A For, the triangle ABF being cut by / 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 / = —: \ s transversal DB, gives / - -- DC X FB X PA= AD X CB X FP. / -- D Multiplying these equations together, and suppressing the common factors PA, CB, and FP, we have AE X BF X CD EB X FC X DA. TH:EOREM IV. —Conversely: If three points are situated on the three sides of a tri. angle, 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-con secutive 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. CoR. 1.-The MEDIANS of a triangle (i. e., the lines drawn from its summits to the middles of the opposite sides) meet in the same point. For, supposing, in Fig. 405, the points D, E, and F to be the middles of the sides, the products of the non-consecutive parts will be equal, i. e., AE X BFX CD= DAXEB XFC; since AE - EB, BF = FC, CD - DA. Then Theorem IV. applies. COR. 2.-The BISSECTRICES of a triangle (i. e., the lines bisecting its angles) mzeet in the same point. For, in Fig. 405, supposing the lines AF, BD, CE to be Bissectrices, we have ILegendre IV. 17): BF: FC::'AB AC, BFXAC FCXAB, CD: DA:: BC: BA,. whence CDX BA= DA X BC, AE: EB:: CA: CB, AE X CB =EB X CA. Multiplying these equations together, and omitting the common factors, we hav4 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 linus crawn fiom 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 eEB XFC XDA; 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 BDC. For, since BF = FC, the equation of Theorem III. reduces to AE X CD-EB XDA; vlience AE: EB:: AD: DC' consequently ED is parallel to BC. Con. 5.-Conversely: If EE 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. CoR. 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. 7. —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.-If 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 ine on which the sides meet, the axis of homology. 390 ARMIONIC DIVISION, [AprP B. HARMONIC DIVISION. DEF'INITrIOS.-A straight line, AB, is said to Fig. 408. be harmonically divided at the points C and D, Il —-- ------- -- when these points determine two additive seg- A C B 1 ments, AC, BC, and two subtractive segments, AD, BD, proportional to one an. other; 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 proportional 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 thi 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 GH, also parallel 0 to OA. We have, GB: OA::BD: AD; and BH: OA:: BC: AC. But, by hypothesis, AC: BC:: AD: BD. Hence, the first two proportions reduce to ____/ _ GB = BH; and consequently, EK = KF. \ 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 har. monic 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, 3, 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; o AC: CB:: AD: BD. The series of fractions,, -, 1 I, I, &c., is called an harmo i r progres. Bion, became any consecutive three of its terms form an harmonic nroportion. srp. B.] THE COMPLETE QUADRILATERAL. 391 THEOREM VI.-If any transversal to an harmonicpencil be drawn, it eill 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 VII.-If, from the summits of any Fig. 411. triangle, ABC, through any point, P, there be drawn the transversals AD, BE, CF, and the trans- versal ED be drawn to meet AB prolonged, in F', / the points F and F' will divide the base AB bar-.-' monically. _-'I _ X A F B F' This may be otherwise expressed, thus: The line, OF, which joins the intersection of the diagonals of anS quadrilateral, ABDE, with the point of meeting, C, of two opposite sides prolonged, cuts the side AB in apoint 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 giye six different points of intersection. 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, composed of two opposite triangles. The complete quadrilateral, AEBCDF, has three diagonals; viz., two interior, AC, BD; and one exterior, EF. E/ THEOREM VIII.-In every COMPLETE QUADRILATERAL the middle points of its three liagonals lie in the same straight line. AEBCDF is the quadrilateral, and LMN the middle points of its three diago. nals. From A and D draw parallels to BC, and from B and 0 draw parallels to 392 THE COMPLETE QTAfDRlLATERAL, LAPP. B. AD. The triangle EDC being cut oy 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 CB = E'B', EA = CA', EB = DB', DA = E'A'. Hence, the aoove 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 ECA'A bisect each other at N, and those of the parallelogram EBB'D at M, 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 quadrilateral each of the three diagonals is divided harmonically by the two others. CEBADF 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 K/ AEF through the same point C; and there- fore, by Theorem VII., DBG and ACH di- /' vide EF harmonically. - - - ~- ~ G E H F So too, in the triangle ABD, CB, 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. I HEOREM X.-If from apoint, A, any num- Fi. 41. her 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. / < l-\ By the preceding Theorem, the diagonal "'i'" _ BC' of the complete quadrilateral, BAB'C'CO, 0 C C C" Q is divided harmonically at D and E. Hence, OA, OP, OD, and OQ, form an harmionic 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 iole with respect to tLe formel. D E M 0 N S ~ RAT I O N. PART II.; CHAPTER V. (1 40), (141) The equality of the triangles formed in these methods proves their correctness. (143)9 (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), AC2 + BC2 — AB2 AB2 = ACa + BC 2 AC. CE; whence CE 2= A- C-, 2 AC BC2 When AC = AB, this becomes CE - AC The similar triangles, BCE and DCA, give EC CB:: AC: CD; whence CD CB X AC 2 2 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 - A AGD; and GAD = - FGA. Therefore FAG + GAD = FAD = (AGD + FGA) = (- (180~) = 90~. (149) See Part VII., Art. (403). (150) 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. AC2 As in the proof of Art. (146), changing the letters suitably, we have AE- A= 2 AB' rhe similar triangles AEC and ADP give AP AP AC2 AP X AC AC: AE:: AP: AD - X AE =- - X --- = 2 AC AC 2 AB 2 AB (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. (15~) 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 iraw them on the figures, as directed. 394 DEMONSTRl TIONS [4PP, 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. (1.64) 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: OE:: 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 ED Hence, since CP - DP = CD, we obtain DP = - CA X EB —AE XBD The other formulas are simplified by the common factors obtained by making AE = AC, or BE = BD. (175) By Theorem VII. "Harmonic Division," in the quadrilateral ABED, the line CF 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. (1~2) This instrument depends on the optical principle of the equality of the angles of incidence and reflection. (1~4) 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, woumd be cut proportionally by the required line PSZ; BD X QP so that QR: BD: QP: BS =. QI{ (1~9) By "Transversals," Theorem I., we obtain, regarding CD as the transversal of the triangle ABE, CB X AF X ED - AC X FE XI)B; 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 thai AC = AAC, we obtain AB = OB -AC = AB, we obtain AB= ~-.(FE - AF). APP. B.] For Part 1.,, Chapter V. 395 (190) Take CH - CB; and from B let fall a perpen- ig.124, bis. dicular, BK, to AC. Then, in the triangle CBH, we have A B ~Legendre IV. 12),. CHr+BH2-BC2 BH 2 2 CH - 2 BC' [1] since CH =BC. In the triangle ABH, we have (Leg. IV. 13) D AB - AH1 + BHt + 2 AH, HK. Substituting for HK, its value from [1], we get AB2=AH2+BH2 (1+ ). But AH AC - CH=A-BC. AB2=AH2+BH2 + C -- CBC AHR2+ BH,. B [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 B2 BH DF. [3] In this equation DFP is unknown; but by proceeding as at the beginning of this CE investigation, we get an equation analogous to [2], giving EDa =EF2 + DF2. CD GD whence DF2 = (DE2 — EF 2).. Substituting this value of DF' in [3], we have CB' BH2 = (DE - EF)D X E Substituting this value of BHR in [2], we have ACXBC GCXBC AB2=AH1'+(DE2- EF2). AXE- =(AC-BC)+ [DE2- (CE- CD)2 ] X CDX C (191) Since BCD is a right angle, AC is a mean proportional between AB and AD. (192j fhe proof follows from the similar triangles constructed. (193) The similar triangles give DE: AC:: DB: AB; whence, by "division," DEAC: 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 EDLXAF ACGXAF A constructed, give FE: ED: A: AF:AB = EDFX = F- -. A GxDC The triangles DEF and BCD give similarly FE: EDI:: DC: CB = F~. FE 396G DEMO)lNSTRATIONS LAPP. B, (196) The equality of the triangles formed proves this problem. (I19T) The proof of this problem also depends on the equality of the triangles constructed. The details of the proof require attention. (19~) EB is the transversal of the triangle ACD. Consequently, CB X AFXDE ABXFDXCE; or, since CB =AB+AC, (AB+AC) XAFXDE=ABXFD X CE: AO X AF X DE whence AB ACAFXDE FD X CE AFxDE' Taking E, in the middle of CD, CE = DE, and those lines are cancelled. Taling 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 The rem IX., ECFBGD being a " Complete Quadrilateral." Consequently, AE: EH:: AB: HB. Hence, by "division," AE -EH: AE: AB — HB: AB. We therefore have, AE X AH since AB -HB = AH, AB = A --' AE -E HiU (200) For the same reasons as in the last article, CF is harmonically divided at H and D; and we have CH: HF:: CD: DF; whence OH -HF: CH::CD - DF: CD Hence, since CD - DF = CF, CD = CH- HF' The other two expressions come from writing CF as CH + 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 ACXDE CD: DE: GCA: AB = C (203) The similar triangles (made so by the parallel) give CE: EA:: CD: AB3 CDXEA CDX(AC-CE) CE CE DFx CD (204) The similar triangles DFH and BCD give HF: FD:: DC: BC = FH GThe similar triangles FGH and ABC give FG: GH:: BC: AB = BC FG. DF X CD X GHSubstituting for BC, its above value, we have AB = F X FG 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. (20~) From B let fall a perpendicular to AC, 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 YV.9 The quadrilateral ABCD = AC X i (p + q). 2. BCE The triangle BCE = CE X X p; whence p - CE The similar triangles EDD' and BEB' give p::::BE DE, whence DE 2. BCE X DE PBE CEX BE' Then BCE BCE XDE X E+BE BE D Then 3 (p +q) - - BCE X BCE X CE + CEXBE CEXBE CE X BE' BD AC XBD Lastly, ABCD = AC X BCE X CB BCE C CEXBE CEXBE' DEMONSTRATIONS FOR PART V. (3~2) 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 b = B. ces. A and the excess of B over b, i e., B - b = B (1 - cos. A). Since 1 - cos. A = 2 (sin. ~ A)" [Trigonometry, Art. (9)], we have B -b =2 B (sin. j A)2. Substituting for sin.. A, its approximate equivalent, i A X sin. 1' [Trigonometry, Art. (5)], we obtain 3 -b 2 B (I A X sin. 1')2 =- (sin. 1')2.A2. B, =0.00000004231 A2 B By logarithms, log. (B- b) = 2.626422 4- 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. (3~6) 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. CRLD = sin. CDR X. CD From the triangle CLD we get sin. CLD = sin. LDC X CD CL' 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. LDC X C The last two terms are expressed in parts of radius, and to have them in seconds, they must be divided by sin. I"' Trigonumetry, 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., 180~ X 60 x 60 180X0 X 6 (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) X sin. (5) X sin. (7) X sin. (9) X sin. (11) X sin. (18). tence the equality of the last two terms of the proportion. 908 DEMONSTRATIONS APPr., DEMONSTRATION FOR PART VI. (399) In the triangle ABS, we have AB. sin. BAS c. sin. IU sin. ASB: sin. BAS:: AB: SB A sin. BAS sin. sin. ASB sin. S In the triangle CBS, we have BC sin. BCS a. sin. V sin. BS: sin CS BC: SB= sin s. S' [2] sin. BS C sin. S 2 Hence, U a 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 360~- ASB — BSC-ABC-BAS; or V=360 —S-S'- — U. Let T = 360- S -S' - B, and we have V T -- U. 41 Substituting this value of V, in equation [3], we get [Trig., Art. (8)],. sin S' sin. U - a, sin. S (sin. T. cos. U - cos. T. sin. U) = 0 Dividing by sin. U, we get c. sin. S'- a. sin. S (sin. T Uos. U cos. T = 0 sin. U - Whence we have cos. I c. 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. U sin. S' cos. T a. sin. S. sin. T sin. T Separating the second member into factors, we get cot. U=os. T c.sin. S' +; sin.-T a. sin. S. cos. T + 1; or cot. U = cot. T. sin S. s + \a. sin. S. cos. T Having found U, equation [4] gives V; and either [1] or [2] gives SB; an:t SA and SC are then given by the familiar "Sine proportion" [Trig, Art. (12)1 PP., B.] For Part VlI. 399 DEMONSTRATIONS FOR PART VII. CP (403) If APC be a right angle, CA =cos. CAB [Trigonometry, Art. (4)]. CA. (405) AC = PC. tan. APC; and CB = PC. tan. BPC [Trigonometry, Art. (4)]. Hence AC: CB:: tan. APC: tan. BPC; and AC: A + CB:: tan. APO: tan. APC + tan. BPO. tan. APC Consequently, since AC + CB AB, AC = AB. tan. tAP tanP A+C - tan. *PC' (4114) 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 AG GB (CA - EG) EG (423) Art. (12), Theorem III., [Trigonometry, Appendix A,] gives cos. C== +-; or c2= a + 6 —2 ab. cos. C. This becomes [Trig., Art (6)], K being the supplement of C, c = a2 - 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 = 1- - K2. Hence, c = a+ b2+ 2 ab-ab K =(+ b)2- ab 1) = (a )+ b) ( a + b); whence, c =(a + b) (- (a - b2) Developing the quantity under the radical sign by the binomial theorem, and neg. lecting the terms after the second, it becomes ab K2 (a+ b) Substituting for K minutes, K. sin. 1' [Trig., Art. (5)], and performing the multiplication by a - b, we obtain. abK2.(sin. 11)2 (sin. 11)2 c a+ b - K (s. Now ) =0.0000000423079; 2 (a +) 2 ab K2 whence the formula in the text, c = a + b - 0.000000042308 X a+ b' (430) In the triangle ABC, designate the angles as A, B, C; and the sides op. posite to them as a, b, c. Let CD = d. The triangle BCD gives [Trig., Art. (12), sin. BDC sin. ADC Theorem I], a = d in. The triangle ACD similarly gives b = d si CD' sin. CBD' sin. CAD' In the triangle ABC, we have [Trig., Art. (12), Theorem II.], tan. ~ (A-B): cot. i C:: a-b: a +b; whence tan. ~ (A -B) = b cot. C [1] Le be an aubiliary angl ha Let K be an auxiliary angle, such that 6 = a. tan. K; whence tan. K = 400 DEMONSTRAT 5IONS [APP. B. Dividing the second member of equation [1], above and below, by a, and substituting tan. K for, we get tan. j (A -B ) = 1 - cot. C. a 1I - tan. K. Since tan. 45~0-, we may substitute it for 1 in the preceding edaation, and tan. 45 — tan. K we get tan. (A- B) = t 45 t. cot. C. tan. 450 + tan. K From the expression for the tangent of the difference ot two arcs [Trig., Art. (8)], the preceding fiaction reduces to tan. (45~ - K); and the equation becomes tan. ~ (A- B)- =tan. (45 - K). cot. G C. [2] In the equation tan. K =, substitute the values of b and a from the formulas a at the beginning of this investigation. This gives sin. ADO sin. BDC sin. AD. sin. CBD tan. K=d. I- 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 Then c - AB = - - ~ 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, EBC= A +P; ECD= A+Q; EDF =A+R. The triangle ABE BE sin. A C. E sin. A gives -=. The triangle ACE gives + =. Dividing member a sin. P a-i x sin. Qm BE a. sin. Q by member, we get ( sin. P CE (a+ x) sin.P BE sin. (A' It) In the same way the triangles BED and CED give b si. (+ ); b 6+x sin. (R-P)' CE sin. (A h+ R) BE (b + x) sin. (R - Q) and - hence as before, = a sin. (R - )Whece as b. sin. ( -- P) Equating these two values of the same ratio, we get a. sin. Q _ (b + ) sin. (R - Q) (a + x) sin. P b. sin. (R P); and the ab. sin. Q. sin. (R- P) x ( b x sin. P. sin.(R Q) + -a +(a To solve this equation of the 2d degree, with reference to x, make tan.2 - 4 ab sin. Q (sin. R - P) (a- b)2 sin. P (sin. -Q)' Then the first member of the preceding equation = 1 ~ (a- )2 X tan.2 K' and we get x2 - (a - b) x = 4 (a -)2. tan.2 K- ab, and.z =- (a+ b) ~ (a- b)' tan.2- - IK -ab + 4 (a+ b)] =- (a + 6) t / [4 (a - )2tan.2 K + 2 (a - b)2] — 4 ( b+ ( ) V (L-a.) / +(t-n 1). 1 a + b a- c Or, since / (tan.2 K- +1) -= secant K = w have x= - cos. K' 2 2. cos. K. APP. B.1 For Part XlI 401 )ESMONSTRATIONS FOR PART XI, (493) The content being given, and the length to be n times the breadth Breadth X n times breadth = content; whence, Breadth =/ (cnt ). Given the content = c, and the difference of the length and breadth = d; to find the length I, and the breadth b. We have I X b = c; and l - b d. From these two equations we get I -= d + / (d2 - 4 c). Given the content = c, and the sum of the length and breadth = s; to find I and b, We have 1 X b = c; and 1+& b - s; whence we get 1 -= s +4 V (s2/ - 4 c). (494) The first rule is a consequence of the area of a triangle teing 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 / 2 Xaea). For the equilateral triangle, calling its side e, the formula for the area of a triangle (4" y ul I e -- 2 //area\ / [( s) ( s -a) (iss-b) ( s-c)] reduces to e2 V3. Hence e = 2/( 3 1.5197 / area. (495) By Art. (65), Note, ~. AB X BC X sin. B = content of ABC; whence, 2 X ABC BC —-— = sAB sin, B (496) The area of a circle = radius 2 X2; whence radius -/( are 7 a y\ 22 (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, b1s. By similar triangles, 3 ABE: DCE: AB2: DC2. ABE - DCE: ABE: iAB2 ~DC'2: AB" - Now ABE - DCE = ABCD; also, by Art. (65), Note, ABE = AB2 sin. A. sin. B 2.sin. (A+B) A The above proportion therefore becomes sin. A. sin. B ABCD~: AB. 2. sin. (A + B) AB CD: AB Multiplying extremes and means, cancelling, transposing, and extracting the equbre /r 2.ABCD.sin. (A +B)S root, we get CD = / JAB" 2. sA.. A sin. (A 26 t02 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 - FC =- FB. -FB. -(AB - CD) sin. BCF -B in. (180 — A B) s(A~ in. (A + s ) BC = (AB-CD)sin. (A sin. (A + B) (505) Since similar triangles are as the squares of their homologous side, BD1: BFG: BD: BF2; whence BF =BD,/ B-FG (506) BFG=. BF X FG --. BF X BF. tan. B; //2.BFG\ whence, BF /(nB ] tan. B/' (510) By Art. (65), Note, BFG = BF2 in B;s 2. sin. (B + F)' whence, BF = /(2 sin. (B + F). BFG) sin.B.sin. F (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 productse the sides about the equal angles, we have ABE: CDE:: AE X BE: CE X DE. sin. A. sin. B sin. B ABE= =. AB. AE -- AB.. sin. (A B)' sin. E_ B sin A. sin. CDE sin. E' sin. DC Substituting these values in the preceding proportion, cancelling the common fac tors, observing that sin. (A + B) = sin. E, multiplying extremes and means, and V 2. CDE. sin. DCE dividing, we get DE = /2 CDEi. sin. CDE r x sin. E.s~. C-D-E/' (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. (51 ~) 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, GPC = the required content. Now, GPC = GDC, since they have the same base and equal altitudes. We have now to prove that LIMC = 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 = D CD M. apP. B.] For Part XI. 403 CD x La Hence, CM = ~-L Substituting this value of C. in the first proportion, LE and cancelling CD in the last two terms, we get LC2 GDC: LMC: G: L-; or GDC: LM:: GC X LE: LC2. -LE LC2 = (LH + HC)2 = LH2 + 2 LH X HC + HC2. But, by construction, LH = HK2= HE2 — EKE= IHE2- 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): HC (HE- EC) + 2 LH X HC + HCa.:: 2 LH + 2 HE: HE- EC + 2 LH + HC.:HE -EC + 2 LH+ 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 -- / (HEa - 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 c=. CL X CM.sin. LCM. This content will also equal the sum of the two triangles LCP and MCP; i. e., c = ~ CL X p + -' CM X q. The first of these equations gives CM CL sin LC Substituting this in the second equation, we have L Xp + c =.CL XP p 4CL. sin. LCM, Whence, ~ p. CL. sin. LCM + cq = c. CL. sin. LCM. Transposing and dividing by the coefficient of CL2, we get L 2 c L 2 cq CL2 —CL=- 2 p p. sin. CLM CL 2V- 4M p 4 2 Po, sin. LCM/ 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 APC and BPD sin. A., sin. P By Art. (65), Note, the triangle APC =.AP2.' sin. (A +' P)' sin. B. sin, P Bimilarly the triangle BPD -. BP2 sin n P sin. (-B - P)' sin. A. sin. P sin. B sin. P'. d-. (APa A P) - sin. (B sin. (A + P) sin. (B + P) 404 DEMONSTRATIONS [APP. B By the expression for sin. (a + b) [Trigonometry, Art. (8)], we have d=1 AP2. sin. A. sin. P _ B 2 sin. B.sin. P sin. A. cos. P + sin. P. cos. A sin. B. cos. P + sin. P cos. B cos. a Dividing each fiaction by its numerator, and remembering that. = cot. a, w sin. a have d AP2 BP' 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, Ap2 Bp2' 2 d -- - p+a-a p+b' Clearing of fractions, we have 2 dp2 + 2 dap + 2 dbp + 2 dab =p. AP + b. AP -p. BP2 —a. BP Transposing, dividing through by 2 d, and separating into factors, we get p / (a+ AP2 -BP\ ) b.AP - a. BP2 p2 + va 4- b- ~ ~ 2d- - ~- ~ -2 b. /: A P2-~BP1^rb AP2, B. AP -a. AP2 - BP A B If A - 90~, 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 f the unknown angle DBF. The angle BDF = 90~ -; and the angle BEF 90~ -(B ) = 90~ B +. By Art. sin. BDE. sin, BED (65), Note, the area of the triangle DBE Eo i (BD EBED sin. (BDE -+- BED). E2' sin. (900 -) sin. (90~ B + B ) sin. B Hence DE2 = -2 X DBE X sin. B 2 X DBE X sin. B Hence, E~ ~ -~-_ __ sin. (90~ - ). sin. (90~ - B + c) cos...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. -= - cos. (a -+ b) + J. cos. (a - b) [Trigonometry, Art. (8) ], into + cos. B + -. cos. (B - 2 ). Since B is constant, the value of this expression de. pends on its second term, and that will be the greatest possible when B - 2 / - 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 direction n which DE is to be run. Its starting point, D or E, is found thus. The area of the triangle DBE = -- BD. BE. sin. B. Since the triangle is isosceles, this becomes DBE B= BD2. sin. B; whence BD s= / ( B- ) DE is obtained from the expression for DE2, which becomes, making, = sB, = 2 X DBE X sin. B (2. DBE. sin. B) DE ns. B s.; whence, DE - D cos. B. cos. I B cos. i B APP. B.] For Part Xi. 405 (524:) Let a =value per acre of one portion of the land, and' that of the other portion. Let x =the width required, BC or AD. Then the value of x X BE x x AE BCFE =a X X, and the value of ADFE= b X 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 (5S2) 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. (52/7) Since similar triangles are to each other as the squares of their homolo. gous sides, ABC: DBE: AB': BD2; whence BD ==AB DE AB/ AB / ABC V m -f-n The construction of Fig. 363 is founded on the proportion in BF: BG:: BG: BA; when BD = BG - - (BA X BF) - BA f/ -n~' (528) By hypothesis, AEF: EFBC:: m: n; whence AEF: ABC:: m: + n rn AC X DB ne and AEF = ABGC - -- Also, AEF = A AE X EF. m2 + n 2 -f- n DB x AL The similar triangles AEF and ABD give AD: DB:: AE: EF -- -D- The 0AD DB x AL second expression for AEF then becomes AEF- = AE AD Equating AD this with the other value of AEF, we have AC X DB m AE2 X DB wenc /(AX AD 2 m n- n 2. AD 1\ m + n (530) In Fig. 366, the triangles ABD, DBC, having the same altitude, are tc each other as their bases. In the next paragraph, we have ABD: DB: AD: DC:: m: n; whence AD: AC:: mn: m-n; and AC: DC::n t +n: n; whence the expressions for AD and DC. In Fig. 367, the expression for AD is given by the proportion AD: AC:: m: m + r. Similarly for DE, and EC. (S31) In Fig. 368, conceive the line EB to be drawn. The triangle &EB = ABC, having the same altitude and half the base; and AFD = AEB, necause 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 XC A ADF AB X AC m whence, AF --- - - - AD-' AD ABC AD min + n' (532) The areas of triangles being equal to the product of their altitudes by tfalf their bases, the contrtnsuctions in Fig. 369 and Fig. 370 follow therefrom. 106 DEMONSTRTAIONS LAPP. B. (583) 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 DOG. (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 triangle 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 = lABC, 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 = ~ ABC. We have then to prove LMC = DCG. This is done precisely as in the demonstration of Art. (519), page 402. (637) Let AE = x, ED =y, AH = x, F', HF 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 -. xy'. FHED x-') (y + y').'. AFDE -- 2 -= y. x'y' + (yx x y) (y + y' ) y + X'y. The similar triangles, AHF and AKB, give bx' a: b:: x': y a Substituting this value of y' in the expression for 2, we have n2 =4 x (y+-) - 6 x'y; w, ar (2 mn2 - xy) AK (2 ABC-AE X ED) whence, x -' bx - ay KB X AE-AK X ED The formula is general, whatever may be the ratio of the area mi to that of the triangle ABC. (38N) 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 - 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 = BGA = (ABD (BA -- BD) = -1 ( AC BDG) = ABG -- BDG, To MDFB add MDB, and add its equivalent, ~ BDG, to the other side of the equation, and we have MDFB + MDB = l ABC-G BDG + i ]BDG; or, BDF - ABC. (540) In Fig. 378, the triangle AFC = ~ ABC, having the same base and one. third 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. 379, AFC: ABC:: AD: AB; since these two triangles have the commos 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 =- AC X CB X sin. ACB. But the angle ACB=-ACD+DCB = (l80~-ADC)+- (180~-CDB) = 180~ — (ADC+CDB), Hence, AB = AC X CB X sin. J (ADC + CDB) =- 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 A B X BC X CA sin... ADB - -; whence, ABC -- AC X CB X -; and r = B —. 2r 2r' 4. ABC Also, since the area of each of the three small triangles equals half the product of the two equal sides into 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, BDC =; sin. ADO = 2 (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. -n i/ ABxBGXsin. /B m ^ ^^\ BD thus becomes =/( ABXB Xsi B) = /( XABXBC) \m + n sin. B' \ n.(f,-,XABXBCXsin.2 sin.B m and DE = -- -. X AB X BC). cos. C 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 - BCa: BC2. In like manner, comparing EFP and BCP, we get EBCF:BCP:: EF2-BC2: BCa Combining these two proportions, we have ABCD: EBCF:: AD2 - BC2: EF2 BC2; or, m + n: m:: AD2 - BC2: EF2 — BC2. Whence, (m + n) EF2 - m. BC2 - n BC2 -= m. AD2 - m. BC2 /(m X AD+ - nX BC2) Also, from the similar triangles formed by drawing BL parallel to CD, we have ALT E E BA X EK AB (EF-BC) AL:EK:: BA: BE= I- - AL AD-BC (545) Let BEFC = — ABCD = a; let BC = 6; BH = —; and gn -+ n AD - BC = 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; h (y -b) L e, c: y -b::: x; whencex = 2a Also, the area BEFC = a -. BG (EF+ BC) x(y+); whencey -- X BU8B DEMONSTRATIONS. [APP. n Substituting this value of y in the expression for x, and reducing, we obtain 2 bh 2 ah b i //2 ah b2h2 \ + x =-; whence we have = - + V - + ex e The second proportion above gives y- b =-; Twhence y = 6 + - ~ x. 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 tra. pezoid, 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. (54) 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 m ~n ABCD = ~- (a a'). Hence the above proportion becomes m~n mi n + n a: r + m (a a'):: G2: GK2; or,, e' + 5+ (m n) a: ma- a':: GHl: GK2; whence GK = GH /V(4 + -) GI~ 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' + -. Now ADG: EFG:: AG2: EG2; Q /a'~ o a, a' Q:'+-:: AG2:EG2; whence GE =AG / $,) p a + Q'/ GL is obtained by taking the triangle LMG- =a'+ 2-; and so for the rest. (55a) In Fig. 890, join FC and GC. 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, EHDO 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 OKQ = 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 1i 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 m 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) TSae instr'uments. A level Fig. 415 line may be obtained by the following T - 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 crossniece. 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, pro. longed, 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. LAPP. I A modification of this is to fasten a common Fig. 416. carpenter's square in a slit in the top of a staff, i-. -~ - 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 )rinciple 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 " later-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 - -- ----- 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 L spot in the tube, and will therefore by t. J-.1 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 expan. sion 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 cr tripod. When, instead of the sights, a telescope is made parallel to the level, and varl ous contrivances to increase its delicacy and accuracy are added, the instrumen becomes the Engineer's spirit-leveL APPI. c.] The Practice. 411 (3) The Practice. By whichever of these various means a level line nas 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 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- 2 — - - - ant hold a rod graduated into feet, tenths, &c. Turn the instrument to- wards the staff, sight along the level line, and note what division on the 3 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 1B, 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, Fig. 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, -~ 6 and... If abe.1.fot... Ffu C and B. If C be 1 foot below B, as in the figure, it will be 5 feet below A. It 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 noteoook 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 112 LEVELLING. [API'. c. portional dimensions.* But vhen 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 station 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 eareful comparison will explain any obscurities in either. Stations. Distances. Back-sights. Fore-sights. Rise. Fall. Total Heights. A 1 0.00 B 100 2.00 6.00 4.00 -4.00 0 60 3.00 4.00 -1.00 5.00 D 40 2.00 1.00 + 1. -~ 4.00 E 10 6.00 1.00 + 5.00 + 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." Another fornm 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. lt. Inst. above Datum. Fore-sights. Total Heights. A 1 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 _Datum, 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 distanoes to one-tenth of their proper proportional size. &pp. c.l 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, fiom 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 "Profile." 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 dis tances 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, and ten 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 an. other volume, as announced in the Preface. 414 APPENDIX D. MAGNETIC VARIATIONS IN THE UNITED STATES. [From a Report by C. A. SCHOTT, Assistant U. S Coast Survey]. See Silliman's Journal, May, 1860, p. 335; and U. S. Coast Survey Report for 1859, App. 24, p. 296. TW. and E. indicate lVest and East Declinations. They are given below in Degrees and tenths. U K aP.-~ ~ ~. ~ D R" -J e aO a O n Q _o.w. __ W. W. W. W. W. W. W W. W. W.. W.W. 1680 8.8 4.8 W. 1690 8.7 4.8 W. 1700 9.7 8.5 8.8 1710 9.0 10.4 8.0 8.4 1720 8.3 9.5 7.6 7.9 1730 7.8 8.9 7.0 7.1 1740 7.4 8.3 6.4 6.3 1750 7. 7. 7 5 8 5.3 1760 8.1 7.0 6.9 6.1 5.2 4.4 1770 8.1 6.8 6.3 5.5 4.7 3.5 1.2 wT. 1780 8.3 6.8 6.1 5.2 5.0 4.4 2.8 0.7 W. 1790 8.5 6.8 7.8 6.3 6.3 5.0 4.8 4.2 3.0 2.2 0.2W. 1800 8.9 7.0 7.5 6.2 6.4 5.0 4.6 4.2 3.0 2.0 0.4 0.2 E. 1810 9.4 7.3 7.3 6.3 6.5 5.2 4.7 5.4 4.3 3.1 1.9 0.5 0.4E. 1820 10.0 7.8 7.6 6.7 6.8 5.6 5.0 5.8 4.7 3.4 2.2 0.8 0.4 E. 1830 10.6 8.4 8.3 7.3 7.5 6.1 5.4 6.3 5.2 3.8 2.7 1.1 0.2 E. 1840 11.2 9.1 9.1 8.1 8.4 6.7 6.0 7.0 5.7 4.4 3.4 1.5 0.1 W. 1850 11.8 9.9 9.7 8.9 91 7.4 6.7 7.7 6.4 5.2 4.3 2.0 0.6W. 1860 12.3 10.6 10.3 9.9 9.7 8.1 7.5 8.3 7.0 6.0 5.2 2.6 1.2W. Minimum 1765 1782 11813 1800 1779 1794 1801 1787 1795 1799 1805 1798 1815 0 " "~0-; _. _ 6 I I Il O cfm w a 0 ) 0 E. E. E. E. E. E. E. E. 1770 3.7 1780 4.0 1790 4.1 11.1 11.4 13.6 15.1 18.9 1800 4.1 4.1 7.1 11.4 12.3 14.1 15.4 19.1 1810 4.0 4.2 7.2 11.7 13.0 14.5 15.7 19.3 1820 3.6 4.2 7.3 12.0 13.6 14.8 16.0 19.5 1830 3.2 4.1 7.2 12.2 14.2 15.1 16.3 19.7 1840 2.8 4.0 7.1 12.3 14.6 15.4 16.6 19.8 1850 2.2 3.7 7.0 12.5 15.0 15.6 16.9 20.0 1860 1.7 3.5 6.8 12.6 15.3 15.8 17.2 20.2 1870 1.2 3.2 6.6 12.6 15.4 15.9 17.2 20.4 Maximumn. 1794 1817 1820 _ I Not yet attaine. ANALYTICAL TABLE OF CONTENTS. PART I. GENERAL PRINCIPLES AID FUNDAMENTAL iETiODS. CAPTER 1. Definitions and Methods. LBCOLBE PAGE ARTICLE PArI (1) Surveying defined.......... 9.Division of the subject. (2) When a point is determined.. 9 (X2) 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 (1) By the extent............ 15 (6) Second do................ 11 (16) Arrangement of this book., 15 (7) Third do............... 11 (17) The three operations common (S) Fourth do............... 12 to all surveying........ 16 (10) Fifth do................ 13 CfHAPTER II, IMaking the Measurements. MIeasuring straight lines. (25) Chaining on slopes......... 21 (19) Actual and Visual lines.... 16 (28) Tape..................... 25 (20) Gunter's Chain.......... 16 (29) Rope, &c.................. 24 (21) Pins................... 19 (3s) Rods..................... 24 (22) Staves............... 19 (32) Measuring-wheel.......... 24 (23) How to chain............ 19 (33) Measuring Angles........ 25 (24) Tallies................ 21 (34) Noting the Mlieasurements... 25 CHAPTER III. Drawing tlhe Iap. (35) A Map defined............ 25 (45) Scales for farm surveys..... 29 (36) Platting................. 25 (46) Scales for state surveys.... 31 (37) Straight lines............. 26 (47) Scales for railroad surveys.. 32 (38) 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 (68) Quadrilaterals........... 44 (56) Horizontal measurement.... 38 ( )9), Curved boundaries...... 45 (57) Unit of content............ 40 (70) Second Method, Geometrically 45 (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 (7~) Reduction to one triangle. 50 (65) Triangles............... 43 (~4) Special instruments....... 54 (66) Parallelograms........... 44 (87) Fourth Method, Triyonometri. (67) Trapezoids............... 44 cally.........5......... 56 416 CONTENTSe PART II. GCHAIN SUJRVEYING. CHAPTER 1. Surveying by Diagonals, ATIaOL PAGE I ARTICLE AGB (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 HI, Surveying by Tie-lines. (98) Surveying by tie-lines.... 66 (101) Inaccessible areas... 67 (100) Chain angles........... 67 (102) Without platting..... 67 CHAPTER HI1. Surveying by Perpendiculars. To set out Perpendiculars. Offsets. (104) By Surveyor's Cross,,,o. 69 (114) Taking offsets........... 75 (107) By Optical Square...... 0 (117) Double offsets....... 76 (1S0) BytheChain...... 72 (l81 ) Field work............. 77 Diagonals and Perpendiculars. (19 ) Platting............. 79 (110) A three-sided field...... 72 (120) Calculating content..... 80 (11 ) 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........ 9 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 Prooblens on Perpendiculars. (140) PROBLEM 1. To erect a perpendicular at any point of a line......... 97 (143) 2. " *" 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 firom a given point to a given line 99 (153) 5. " " when the point is nearly opposite to the end of the line... 100 (156) 6. " " when the point is inaccessible.. 101 (X15) 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.....o......, 103 CONTENTS, 417 OBSTACLES TO ALINEMENT. A^TIOXL PAGE A. To prolong a line..... 105 (169) By ranging with rods.... 105 (174) By transversals.......... 107 (A 71) 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 (~1) With a single person..... 111 (17~) Ranging........ 109 (1~2) On water............... 111 (179) Across a valley.......... 110 (1S3) Through a wood......... 112 (1~0) Over a hill.............. 110 (1~4) To an invisible intersection. 112 OBSTACLES TO MEASUREMENT. A. When both ends of the line are accessible..................... 113 (1 6) By perpendiculars...... 113 ( ~9) By transversals.......... 114 (1~7) By equilateral triangles... 113 (190) In a town............... 114 (1 L~) By symmetrical triangles.. 114 B. WVthen one end of the line is inaccessible....................... 115 (191) By perpendiculars........ 115 (19~) 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 (211) To an inacc. intersection.. 118 C. IVhen both ends of the line are inaccessible.................... 119 (2OS) By similar triangles....... 119 ( 4) By a parallelogram....... 119 (203) By parallels........... 119 (205) By symmetrical triangles. 120 INACCESSIBLE AREAS.......................... 121 (207) Triangles............... 121 (209) Polygon............. 121 (20@) Quadrilaterals........... 121 PART III. C01MPSS SURTVEYIG. CHAPTER I. Angular Surveying In general, (210) Principle................ 122 (217) The Compass............ 124 (211) Definitions.............. 122 (219) Methods of Angular Suro (21 3) Goniometer............ 123 veying................ 125 (2B14) How to use it........... 123 (220) Subdivisions of Polar Sur-'215) Improvements........... 124 veying................ 125 CHAPTE R I. The Compass. (2,21) The Needle.............. 127 (22~) 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 Comyass. 13* (227) Levels.................. 132 27 m4 8 @CONTENTS, CHAPTER 1II. The Field-work. AtR TYci, PAGE ARTICLE P A S (235) Taking Bearings......... 138 (242) Angles of deflection.. 144 (236) Why E. and W. are re- (24S3) Angles between courses.. 145 versed........ 139 (244) To change Bearings...... 146 (23T) Reading with Vernier.. 140 (245) Line Surveying.......... li7 (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!) Ne Yok Canal p. 14 not level by needle. Keep ( ) ew York Canal Maps 14 same end ahead. Read from (252) Farm Surveying.......... 150 same end. Caution in read- (i5s4) Field-notes........... 151 ing. Chetk vibrations. Tap (2 5) Tests of accuracy...... 153 compass. Keep iron away. Electricity. To carry corn- (2 ~) 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 CHIAPTER IV. Platting the Survey. 2'63) Platting in general....... 157 (273) Drawing-board protractor. 166 (2 4) With a protractor........ 157 (274) With a scale of chords.... 166 (265) Platting bearings......... 158 (27) With a table of chords.... 167 (26 ) 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. (27~) Definitions.............. 169 Applications. (279) Calculation of Latitudes (2~2) Testing survey.......... 175 and Departures......... 170 (283) Supplying omissions..... 176 (2~O) Formulas............... 171 (2~4) Balancing............ 177 (2@1) Traverse Table.......... 171 (2~8 ) Platting................. 178 CHAPTER VI. Calculating the Content. (S~ ) Methods................ 180 (292) General rule............ 184 (297) Definitions............, 180 (293) To find east or west station 184 (26~) Longitudes.............. 181 (294) Example 1............. 184 (269) Areas................. 182 (296) Examples 2 to 13........ 186 (290) A three-sided field...... 182 (297) Mascheroni's-Theorem..... 188 (9 1) A four-sided field....... 183 TCHiPTER VII. The Variateon of the Iaguetic Needlee (296) Definitions.............. 189 (306) Table of Azimuths....... 196 (299) Direction of the needle... 189 (307) Setting out the meridian. o 197 To determine the true meridian. To determine the variation. (3OO) By equal shadows of the (30O) By the bearing of the star. 198 sun.................. 190 (309) Other methods.......... 199 (3Ol1) By the North Star when in (310) Magnetic variation in the the meridian.......... 191 United States....... 199 (3O2) 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. 20C (304) Table of times......... 195 (312) To survey a line with true (3905) Observations......... 196 bearings........... 20I CONTENTS. 419 CHAPTER VIII. Changes in the Variation, tRTICLE PAGE ARTICLE PAGe (314) Irregular changes........ 203 (31 8) By interpolation...... 206 (315) The Diurnal change...... 203 (319) By old lines.......... 206 (316) The Annual Change...... 204 (320) Effects of this change.... 207 l317) The Secular chang e....... 204 (321) To run out old lines..... 20S 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. irANSIT AND THEODOLITE SURVEYING, (BY THE 3d METHOD.) CAliPTER 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 (338) Parallel plates........... 225 (32~) 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 IH. 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 (384) Mountain Barometer..... 231 (356) Double Verniers......... 238 (349) Circle divided into degrees. 232 (357) Compass Verniers........ 239 (350) Circle divided to 30'....... 288 CHAPTER I1I. Adjustments. (3,5) Their object and necessity. 240 Rectification......... 243 (359) The three requirements in (362) In the Theodolite...... 245 the Transit............. 240 (363) Third Adjustment. To cause (360) First Adjustment. 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..,e..... 241 star; steeple and stake) 246 Rectification...O..... 241 Rectification......... 246 (361) Second Adjustment. To (364) Centring eye-piece........ 247 cause the line of collima- (365) Centring object-glass...... 247 tion to revolve in a plane 242 Adjusting line of colli. Verification........, 242 mation............. 24g 4 20 CONTENT~, ClAPTER IV. The Flekl-worl. &FITICLE PAGE ARTICLE PAGO'366) To measure a horizontal (372) Line-surveying........... 54 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 (37%) Measuringdistances with (369) Probable error...... 252 a telescope and rod.. 256 (370) To repeat an angle... 252 (376) Ranging out lines....... 257 1371l) Angles of deflection.... 253 (377) Farm-surveying......... 258 (37~) Platting................. 259 PART V. TRIANGULAR SURVEYING, (BY THE 4th METHOD.) (379) Principle............... 260 (3~5) Observations of the angles.. 237 (3~0) Outline of operations..... 260 (3~6) Reduction to the centre... 268 (3~1) Measuring a base........ 261 (3~7) Correction of the angles... 270 Materials.............. 261 (3~~) Calculation and platting... 270 Supports.............. 262 (3~9) Base of Verification...... 271 Alinement............. 262 (390) Interior filling up........ 271 Levelling.............. 262 (39g) Radiating Triangulation... 272 Contacts.............. 262 (392) Farm Triangulation....... 272 (3~2) Corrections of Base....... 263 (393) Inaccessible Areas........ 273 (3S3) 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(3~4) Signals................. 266 tersections............. 274 PART VI. TRILINEAR SURVYEING (BY THE 5th METHOD.) (396) The Problem of the three (39~) Instrumental Solution..... 277 points................ 275 (399) Analytical " o..... 277 (397) Geometrical Solution..... 275 (400) Maritime Surveying...... 278 PART VII. OBSTICLES IN INGULAR SURTVEYING CAIPTER. 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 e.......... 280 (4@0) 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 9407) To trace a line through a given point parallel to a given line........ 281 (40~) To trace a line through a given point parallel to an inaccessible line. 281 CONTENTSo 421 (:iAPTER II. Obstacles to Alinement, KTMICLE PAGY A. To prolong a line..........8......... 282 (409) General method......... 282 1 (4a1) When the line to be pro(4-10) By perpendiculars....... 282 longed is inaccessible... 283 4II11) By an equilateral triangle. 282 (41i4) To prolong a line with only (412) By triangulation......... 288 an angular instrument... 283 B. To interpolate points in a line.............. 284 (415) General method.......... 284 (41~) By Latitudes and Depart(416) By a random line........ 284 ures, with transit...... 285 (417) By Latitudes and Depart- (419) By similar triangles...... 286 ures, with compass..... 285 (42@) By triangulation......... 288 CAPFTER III, Obstacles to Measurement. A. Bihen both ends of the line are accessible.......... 287 (421) Previous means.......... 28 (43) A broken base......... 287 (42) By triangulation......... 287 (424) By angles to known points. 288 B. WThen one end of the line is inaccessible............ 288 (425) By perpendiculars........ 288 (429) To find the distance fiom 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. T/hen both ends of the line are inaccessible......... 200 (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............... "92 its line can be obtained.. 291 (434) To interpolate a base..... I92 (432) When only one point can be (4 35) From angles to two points.. 293 found from which both (436) From angles to three points 293 ends of the line can be (4137) From angles to four points. 294 seen................ 291 (43S) Problem of the eight points 296 CHAPTER IV. To Supply OmisSions, (439) General statement......................................... 297 (440) CASE 1. When 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 wanting........................................... 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. MThen the lengths of two sides are wanting................. 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................ 02.47t) When the deficient sides adjoin each other.............. 3802 (4,s) When the deficient sides are separated from each other..... 802 422 CONTENTS. PART VIII. PLANE TABLE SURIVEING' ARTIOLE PAn8 1449) General description....... 303 (455) Method of Resection...... 308 (450) The Table............... 303 (456) To Orient the Table......08 (451) The Alidade............. 804 (4e7) To find one's place on the (452) Method of Radiation...... 305 ground................ 309 (453) Method of Progression.... 306 (45~) Inaccessible distances..... 310.154) Method of Intersection.. 307 PART IX. SURVEYING WITHOUT INSTRUMENTS. (459) General principles......... 311 (463) Distances by sound....... 31l (460) Distances by pacing...... 311 (464) Angles.................. 314 (46 1) Distances by visual angles. 312 (465) Methods of operation... 314 (462) Distances by visibility.... 313 PART X. MAPPING. CHAPTER I. Copying Platse (466) Necessity............... 316 (474) Reducing by squares...... 819 (467) Stretching the paper...... 316 (47T) " by proportional (46S) Copying by tracing....... 317 scales........ 320 (469)' on tracing-paper. 317 (476) " by a pantagraph 321 (470) " by transfer-paper. 317 (47) " by a camera luci(471) " by punctures.... 318 da........... 21 (47) " by intersections.. 318 (47~) Enlarging plats.......... 321 (473) " by squares...... 319 CHAPTER SH. Conventional Signs. (479) Object.............. 322 (413) Signs for water.......... 325 (4~0) The relief of ground...... 322 (4~4) Colored topography...... 325 (4~1) Signs for natural surface... 324 (4~5) Signs for detached objects. 327 (4S2) Signs for vegetation..... 324 CHAPTER III. Finising the Map, (486) Orientation............. 328 (4~9) Joining paper.,....... 329 (4~7) Lettering.............. 328 (490) Mounting maps.,.,.o.e 329 (4S~) Borders....,,.......... 328 CONTENTS. 08 PART XI. LAYING OUT, PARTITG O0FF, AND IDIVIINM UP LIND. CIAPTER I. Laying out Land. ARTICLE PAGE 1 ARTICLE PAG) (491) Its object............... 330 (496) To lay out circles........ 332 (492) To lay out squares....... 330 (497) Town lots............. 333 (4 93) To lay out rectangles..... 331 (498) Land sold for taxes....... 33 (494) To lay out triangles...... 332 (499) New countries......... 334 CIHAPTER I, Parting off Land. (500) Its object.................................................... 83-1 A. By a line paarallel to a side. (501) To part off a rectangle....................................... 335 (502) " " a parallelogram................ a.............. 335 (503) " " a trapezoid..................................... 335 B. By a line perpendicular to a side. (505) To part off a triangle........................................ 336 (507) " " a quadrilateral.................................... 37 (50~) " " any figure....................................... 837 C. By a line running in any given direction. (509) To part off a triangle........................................ 337 (511) " " a quadrilateral............................... 338 (51 3)' " any figure................................ 339 D. By a line starting from a given point in a side. (51.4) To part off a triangle............................. 339 (516) ". a quadrilateral................................... 340 (517) " " any figure................ 340 E. By a line passing through a given point within the field. (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. StraigMhening crookedfences..................... 346 CiAPTER HIl Dividing up Land. (526) Arrangement................................................ 34 Division of Triangles. (527) By lines parallel to a side.............................. 347 "52 ) By lines perpendicular to a side.............................. 348 (529) By lines running in any given direction....... o......... o 348 (530) By lines starting from an angle................................ 349 (531) By lines starting from a point in a side...................... 349 (535) By lines passing through a point within the triangle............... 351 (5L9) Do. the point being to be found....................... 353 (541) Do. the point to be equidistant from the angles............. 352 (5' 2) By the shortest possible line........................... 354 Division of Rectangles. (543) By lines parallel to a side..................................... 354 (24 CO@ITENTS. ARTIOLB PAGB Division of Trapezoids. (544) By lines parallel to tne bases................................. 35 (546) By lines starting fiom points in a side............................ 355 (z4L7) Other cases.................................,............. 356 Division of Qaldrilaterals. (648) 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 from an angle............................. 358 (552) By lines starting from points in a side.......................... 358 (554) By lines passing through a point within the figure................. 359 Division of Polygons. (555) By lines running in any given direction......................... 360 (5>,6) By lines starting from an angle.................................. 360 ('57) By lines starting from a point on a side......................... 361 (55~) By lines passing through a point within the figure................. 361 (59) Other Problems.............................................. 381 PART XII. UNITED STATES' PUBLIC LANDS. (560) General system.......... 363 Meandering............ 371 (661) Difficulty............... 364 (e506) Marking lines............ 372 (562) Running township lines.... 366 (566) Marking corners......... 372 (563) Running section lines..... 368 (567) Field-books.............. 6 (36 4) Exceptional methods...... 370 Township lines......... 377 Water fronts.......... 370 Section lines........... 878 Geodetic method...... 371 -Meandering.......... 378 APPENDIX. APPELDI X A. Synopsis of Plane Trigonometry. (I) Definition.................. 879 (7) Their mutual relations..,,. S..38 (2) Angles and Arcs............ 379 (~) Two arcs................. 383 (3) Trigonometrical lines........ 380 (9) Double and half arcs....... 384. (4) The lines as ratios........... 381 (1O) The Tables.............. 384 (5) Their variations in length.... 381 (I1) Right-angled triangles...... 385 (e) Their changes of sign........ 382 (12) Oblique-angled triangles.... 385 APPTEiDI B, Demonstrations of Pr ibems, &, 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 }'roofs of Problems in Part II.,.. in Part XI..... 401 Chapter V................. 393 APPETDIX C. itroduiction to Levelling0 (1) The Principles.............. 409 (3) The Practice............... 411 (2) The Instruments.......... 409 1 OR, LATITUDES AND DEPARTURES OF COURSES, CALCULATED TO THREE DECIMAL PLACES FOR EACH QUARTER DEGREE OF BEABING, A LATITUDES AND DEPARTURES, o^ Lat, Dep. Lat. Dep. Lat Dep LatLat. 0 i1o000 0'000 2.000 0.000 3.000oo o. ooo 4.ooo 000 5.ooo 900 o io0ooo o.oo4 2.000.oo09 3.ooo o.oi3 4.oo00 oo07 5.000 891 oA Iooo o0 009 2.00oo 0.017 3.ooo 0.026 4.000ooo oo35 5.ooo 89[ o4 i.ooo oooi3 2.000 0.026 3.ooo 0.039 4.ooo 0oo52 500ooo 89 1 -'.000 o-OI7 2.000 0oo35 3.ooo 0.052 3.999 0.070 4-999 SWo It 1.ooo 0.022 2.000 0o.o044 2.999 o.o65 3999 00'87 4'999 884 1.00ooo 0.026 1.999 0-052 2.999 0.079 3.999 ooio5 4.998 8'8 i o000o o.o003I 999 o.o6i 2.999 0.092 3.998 o0i22 4.998 884 2 o0~999 o0.35 1999 0.070 2.998 oio5 3.998 o9 i4o 4'997 SW 2a o~999 0.039'.998 0.079 2.998 0.118 3.997 0.o57 4.996 871 2 o0.999 o0044 i'998 0.087 2.997 0.131 3.996 0.174 4.995 874 2 o0.999 0.048.998 o0.0o6 2.997 o0. 4 3.)995 0.192 4 994 87/ 30 o0999 9 0052 I.997 o.io5 2.996 0.157 3995 0.209 4.993 S7'0 34 0.998 0.057 I-997 o.113 2.995 0.170 3.994 0.227 4.992 861 34 0.998 o.o61i.996 0.122 2a994 0.183 3.993 0.244 4-99I 864 3J 0.998 o.o65 1.996 o'.31 2. a994 0.196 3.99 0.o262 4.989 864 4~0 0-998 0.070.(995 o.04o 2.993 0.209 3.990 0-279 4'988 S6~ 4i 0.997 0.074 -.995 o.i48 2.992 0.22,2 3.9891 0.296 4.986 854 4 o0997 0-078'99~4 0.57 2.991 o.235.988 0-: o 3i4 4.985 854 4j o-9- o-o83,.993 o.i66 2990'o 0.248, 9.86.0.33i 4.983 854 0 0. 996 o0087 1.992 0.174 2..989...-..261 — 3.~985 0.349 4.98a 5 1'5 0 o.996 0.092 1992 o083 2-87 0oI275'' "3.83 o.366 4-979 844 5 o0.995 0.096 1.991 0.12 2.986 o0288' 3-982 0.383 4 977 844 5: 0o995 0oio) x99o o 9 0.200 2,985 o'03o01 3:98o o.4oi 4~975 84'G 0.995 o-i0o 1.989 o20o9 2.984 o.3i4 3093 8 o.4i8 4"973 8A4 6i 0.994 0.109 i.988 ~'2'18' 2".982' 0o.3'7''3 "976 o.435 4.970 83tj 6 0-~994 oi3 -i987 0.226- 2.981 0.34o 3.974 10.453 4.968 834 6: 0.993 o.1i8 1.986 0 o.235 2.979 0o353 3-972 0.470 4~965 834' 0o-993 0.122 1985 0.244 2.978 o0.3.66 3:970 0.487 4.963 /30 74 0-992 0.126 1.984 0,252, 2.976. 0.379 3-968 0.5o5 4.960 824 74 0.991 0oi31 1.983 o.261 2-974 0. 392.3966 0.522 4-957 827 7i 0991 0-135 13.982 0.270 2 973 o.4o5. 3-963 0o539 4.954 824 8O 0-99go o0.139 981 0.278 2-971 o.418 3-.961 0o557 l4-951 /20 8 0o.990 0.143 1-979 0-287 2:969'o; - 43'6 3959'0-574 4-948 821 8 o0-989 o.i48 1.978 0-296 2.967 o'.443 13-.'56' 0.591 4-945 814 84 0-988 0.152 1.977 o-364 2.965 o.456 3-953 o.6o8 4-94/2 8it 9~ 0-988 o.156 1.975 o.3i3 2.963 0-469 11'3951 0626 4-938 810 94 0.987 /o.i6i I974 0o321 2.961.0.482 3-948 0.643 4.935 80o 94 o.986 o.i65 1-973 0o330 2.959 o0495 3-945 0o66o 4-93i 8o0 9 - 0986 0-169 971 0-339 2.957 0-508.-3942 0-677 4-928 80o 10~ 0-985 0.174 -1970 0.347 2-954 0o.521 3.939 0-695 4-924 / 0 io~ 0-984 0-178 i.968 o0356 2.952 0.534 3.936 0.712 4-920o 793 ]oA 0-983 0-182 1.967 0.364 2.950 0.547 3.933 0.729 4-916 794 o10 0.982 o0.187 i.965 0.373 2.947 6.56o 3-930 0.746 4-912 794 1 o 0o982 0-[191.963 0.382 2.945 0.572 3.927 0 o-763 4-908 9o01 ii 0~-981 01i95 1.962 0.390 2.942 0o585 3.923 0 780 4-904 784 11i 0o980 0og99.96o0 0.399 2.940 0.598 3.920 0.797 4-900 7.8 ill 0-979 0.204 i.958 0.407 2.937 o61i 3.916 1o-85 4-895 784 120 0.978 0.20o8.956 o.4i6 2.934 0o624 3.-93 0o832 4-891 /' ~ 124 0.977 02a12 1.954 0.424 2.932 0o637 3go09 0-849 4.886 /77i 124 0-976 0.216 i.953 0.433 2.929 0.649 3-905 0o866 4-881 774 2jI 0.975 0-22I i.95I o-44i 2926 0-662 3901go o-883 4-877 774i 130 0-974 0a225 1.949 0-450 2.923 0.675 3.897 0-900 4-872 77 0 13i 0.973 0.229 I.947 o.458 2.920 o0688 3.894 0.917 4.867 7641 i34/ 0-972 0.233 1.945 0.467 2-917 0.700 3.889 0.934 4.862 764 1x3 0-971 0-238 i.943 0-475 2.914 0-713 3-885 0o951 4.857 764 14 4 0-970 0.242.94x o0484 29I1r 0o726 3.88i 0.968 4-851i 7,0 144 0-969 0.246 x.938 0-492 2-o908 0-738 3.877 0-985 4.846 754 x44 0-968 0.250.x936 o05oi 2.904 o0751 3.873 1.oo002 4-841 754'44 o0967 0.255 1.934 0.509 2901 0.764 3-868 i.oi8 4-835 75 1' 0-966 0.259'i932 o058 2.898 0.776 3.864 1-035 4-830 70 Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. ^ 1re LATITUDES AND DEPARTURES. Dep. Lat. Dep. Lat. Dep. Lat ) Dep. Lat, Dep. 00 oooo o 6 0 ooooo.ooo000 o.oo 8.00ooo o-ooo 9.00 ooooo 9 00 o4 0o022 6.ooo 0.026 7.000 o.o3i 8.ooo oo035 900ooo o0.0o39 89 o4 o0o44 6I ooo 3.052 7.oo00 o.o6i 8.ooo 0.070 9000 0.079 89 o3 o{o65 5.9999 o079o5 69990.092 7999 8 10 0.087 5.999 ooio5 6.999 O.I22 7.999 0o.4o 8.999 0.157 o90 i oio109 5.999 0-131 6.998 0.153 7.998 0.175 8.998 ci96 884 i o.i3i 5.998 o0.157 6.998 0.183 7.997 0.20o9 8.997 0,236 88i 0.i53 5'997 o.i83 6.997 0.214 7.996 0.244 8.996 0.275 884 8/ 0.174 5:996 0.209 6.996 0.244 7~995 0.279 8.995 o.314 S0~ 2j o0.96 5.995 0.236 6.995 0.275 7-994 o.314 8.993 0.353 874 24 o02i8 5.94 0.262 6.993 o.3o5 7.992 0.349 8.991 0.393 87 2 J| 0-240 5c993 0.288 6.992 0.336 7.991 0.384 8.99o 0.432 874 8 0.262 5-.92 o0.3i4 6.99o 0.366 7.989 o.4i9 8.988 0.471 ~70 34 0.283 5:99 0.o'34o0: 6.989 0.397 7.987 o~454 8.986 o.5io 864 34 0o3o5 5:989 [0o366.6.987 0.427 7.985 o0488 8.983 0.549 864 3' 0.o327'5 987 o392 6.985 o0.458 7.983 0.523 8.981 0.589 861 40 03 149 5' 985.0.419 6.983 0.488 7-981 o.558 8-978 0.628 ~6~ 4 4 0o 371: 5' 984 o0445 6.981 0.519 7-978 o0593 8'975 o0.667 854 14^ o~392: 5.982'0.471 6-978 0.549 7-975 0o623 8-972 0.706 85 44 0.44'5-979 0o-497 6-976 o-580 7.973 0-662 8-969 0.745 854 50 o.436 5-977 0.523 6-973 0.610 7-970 0.697 8.966 0.784 SW) 5 o.458 5.975 0.549 ]6971 o.641 7-966 0.732 8.962 0.824 844 54 0 479 ]'59a 9: 0.575 6.968 0-671 7.963 0767 8.959 0o863 844 5. o 5o 5,g970d ois6oi 6.965 0.701 7-960 0.802 8.955 0.902 844'~ Q 0o523- 59.967.o627 6-962 0.732 7.956 0836 8.951 0.941 SJ0 61 o0 544 5.964..653 6.958 0.762 7.952 0o871 8.947 0.980 83 64 o0566, 5.'96. o.;-679 6.955 0.792 7-949 0-906 8.942 1-019 834 6: 0.588:5:.958 o 705 6-951 0.823 7-945 0-940 8-938 i~o58 834 0o.609 5-955 0.731 6.948 0.853 7-940 0.975 8-933, 1.097 3~c 74 o.631 5.952 0.757 6.944 o.883 7.936 1-oio 8-928 I1i36 824 74 o.653 5-949 0.783 6.940o 0914 7-932 1-o44 8-923 1.175 824 74 o.674 5.945 0-809 6.936 0-944 7'927. 1-079 8-918 -21,4 824 o~ 0.696 5.942 0o835 6.932 0.974 7-922 I.-i3 8-912 1253 S2o 84 0.717 5.938 o.861 6.928 1.oo4 7'917'1- 48 8'907 42991 8, 8 0.739 5-934 0.887 6.923 1.o35 7-912 1-182 890o1.330 813 84 0.76.:5.930 j0.913 6.919.o65 J",907 1-027 8-895 1.369 8i~ 90~ 0.782 ]5[976 0o939 6.914 1-095 7-902 1-251 8-889,'408 ~1~ 91 0o8o4 55.922 0-964 6-909 1-125 7-896 1-286 8-883 -~447 80o 9 o0.825 5.918 0-990 6-904 1-155 7-890 1-320 8-877 1-485 804 91 0o847 5.913.o0i6 6.899 i1-85 7.884 1.355 8-87o 1.524 801 10.868 590o9 l-042 6.894 1-216 7-878 i.389 8.863 i.563 SO0 o 0.890 5.904 i'o68 6.888 1.246 7.872 1.424 8.856 i.6oi 794 I10 oo.-911 /5-900.o093 6.883 1.276 7.866 1.458 8.849 i.64o 794 ]o0 0-933 5-895 1.119 6.877 1-306 7-860 1.492 8.842 x1679 794. V o0-954 5-890 1.145 6.871 I1336 7.853 1.526 8.835.7177 790 11i 0-975 5.885 1.171 6.866 i-366 7.846 1.561 8.827 -.756 784 11i 0-997 5,88o 1-196 6.859 1.396 7-839 x'595 8.819."794 784 Ii 1.o08 5.874 1-222/ 6.853 1-425 7.832 0,629 8.8ii i.833 784 a[2 i.o4o 5.869 1.247 6-847 i.455 7-825 i.663 8-803 0.871 7S0 121 ~ x.o6i 5.863 1.273 6.84i 1.485 7.818'.697 8-795 I.910 774 124 1/.082 5-858 1.299 6.834 1.5:5 7.810 1/732 8.787 1-948 774 122 1103 5.852 ~.324 6,827 1.545 7-803 1766 8-778 1.986 774 S.g~ I.:125 5-846 I.35o 6821 I1.575 7795 xo8oo 8-769 2-025 770 i3. 1-1 oi67 5-834 1 401 6.807 i-634 7"-779.868 8-75I 2-101 764 13 1-1:88 5-828 1.426 6.799 i.664 1777I -902 8.742 2.139/ 7-6 14~ 1-210 5.822 1.452 6.792 1.693 7.762 1.935 8'733 2.177 760 14- 1. 231 581,5 1 477 6.785 1.723 7'754 1969 8'723 2.215 75S I14 1-252 58o09 i.502 6.777 1.753 7-745 2.003 8-7I3 2253 2 75 141 I -.273. 5.8o2 1.528 6.769 J1782 7-736 2.037 8-703 2.291 754 15 1 ~"294 5-796.-553 6.761 1-812 7-727 2.071 8-693 2.329 750 Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. ~lp^~~~~~~~~~~ 0 0 ^ Q O c~~~~~~: "i~~~~~~~~~~~~~t ~ ~ ~..... LATITUDES AND DEPARTURES. c Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat,. 0.9 66 o0259 1.932 o0-518 2.898 0.776 3.864 i o35 4-830 o o 15~ o0965 0o263 1.930 o.526 2.894 0 789 3.859 io052 4.824 i74 1i5 0.964 0o267 1-.97 o.534 2.891 0.802 3.855 1.069 4.8i8 74 ] 51i 0.962 0.271 I.925 0o543 2.887 o0814 3.85o0.o86 481i2 74t 160 o0./9 0.oo276 1.923 o0.551 2.884 0.827 3.845 /.io3 4.8o6'O 1i6 0.960 0o28o I.920 0o56o 2.880 D.839 3.84o,911 I 4.800oo 731 164 0.959 0~284 1.918 0o568 2.876 0.852 3.835 io36 4.794 734 164 0.958 o0283 1.915 0.576 2.873 0.865 3.83o0 ~I53 4.788 734 11 0.956 0.292 1.913 o.585 2.869 0.877 3.825 z.169 I 4782'S0 174 0.955 0.297 1 91o 0-593 2.865 0.890 3.820 1.186 4.775 721 17 o-954 o.3oi 1.907 o.6o01 2.861 0.902 3.815 ~.2o3 4.769 724 17- 0-952 o.3o5 1.905 0o.610io 2.857 0.915 3.810 1.220 4.762 724 1~0 0.951 0.309 1-902 o.6i8 2.853 0.927 3.804'.236 4.755 C2~ x18 0.950 o.3i3 1.899 0.626 2.849 0.939 3.799 ~.253 4-748 71' 184 0.948 0.317 1.897 0.635 2.845 o0.952 3.793 "2969 4.742 714 184 0.947 0.321 1-894 o0643 2.841 0.964 3.788 ".286 4.735 714 19j 0.946 0.326 1.891 o.65i 2.837 0~977 3.782 1302 4.728 10 194 0.944 o.33o i/888 0o659 2.832 0o989 3.776'319 4-720 70o 19 0-943 o0334 i[885 o.668 2.828 I/ooi 3.771 i 335 4-713 70o 19g 0-941 o:338 1.882 o0676 2.824 1-014 3.765 1.352 4-706 70g 200 0.940 0.342 1-879 0o.684 2.819 1.-026 3.759 1-368 4.698 70 20 0.938 o0.346 1.876 0.692 2.815 i/o38 3/.753 3-384 4.691 694 20o 0-937 0.350 I.873 0.700 2'.810 1o051 3:747.4o0T 4.683 69 204 0.935 o-354 1.870 0.709 2.805 I.o63 3.741 1-417 4.676 69I 21~ 0.934 o.358 1-867 0-717 280oI 1.075' 3.734 1.4.33 4.668 6;o 214 0o932 0.362 i/.864 0.725 2.796 1.087 3.728 I.450 4.660 684214 o0930 0.367 1.86i 0.733 2-791 iiboo 3.722.i466 4.6-52 684 211 0.929 o0371 i.858 0.741 2-786 1I112 3.715 1.482 4.644 684 o! o0.927 0.375 1.854 0.749 2.782 1.124 -3.709 1495 4.636 4~~o 221 o0926 0o379 i.85s 0.757 i2777 1-36 307o2 -5I5 4.628 67i 224 0-924 o-383 1.848 0.765 2.772 i.i48 3.696.53i 4.619 674 22 0-922 o0387 i.844 0o773 2.767. i16o,:,3;689 1547/ 4.611 67 1 2o/ 0.92I 0.39I 3.9/ 841 o0.78I 2762 1172: 3:682 1. 563 4 o-63 @70 234 0.919 0395 1.838 0.789 2.756 ii8'4' 13.675'579 4-594 664 234 0-917 o0399 li834 0.797 2.75f1 1-96 j3.668 -595 4.585 664 234 o-915 o40o3 1.831 o.8o5 2.746 120o8 3.661,6ii 4-577 664 0.~ 0914 o0407 1.827 0.813 2-74 I 1220 3-654 i-627 4.568 6@lumn, at the top of the page; and for the minutes on the left. But if the aligic is between 450 and 9go, look for the degrees and the title of the column, at the bottomn; and for the mlinutes on the righlt. The Secants caad Cosecansts, which are not inserted in this table, may be easily suppliel. If I be divided by the cosine of an arc, the quotient will be the secanlt of that arco A nd if x be divided by the sine, the quotient will be the cosecant. The values of the Sines and Cosines are less than a unit, and are given in decimals, although the decimal point is not printed. So also, the tangents of arcs lees than 450, and cotangents of arcs greater than 450, are less thar a unit s:nl ar exIressed in decimals with tLe decirsrl pcint omitted. 64. NiATURAL SINES AND COSINES. TABLE III. _0 10_ ______1_20 30 40 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o ooooo Unit. 01745 99985 03490 99939 05234 99863 06976 99756 60 1 00029 Ul1it. 01774 99984 o3519 99938 05263 9986I 07005 99754 59 2 00058 Unit. oi8o3 99984 o354 99937 05292 99860 07o34 99752 58 3 00087 Unit. 01832 99983 03577 99936 05321 99858 07063 99750 57 4 ooii6 Unit. o01862 99983 36o6 99935 o5350 99857 07092 99748 56 5 oo00145 Unit. 01891 99982 03635 99934 05379 99855 07121 99746 55 6 0oo75 Unit. 01920 99982 03664 99933 05408 99854 07150 99744 54 7 00204 Unit. 01949 99981 03693 99932 05437 99852 07179 99742 53 8 00233 Unit. 01978 99980 03723 99931 05466 9985I 07208 99740 52 9 00262 Unit. 02007 99980 03752 99930 05495 99849 07237 99738 51 io 00291 Unit. 02036 99979 o3781 99929 05524 99847 07266 99736 50 ii 00320 99999 02065 99979 o38io0 99927 05553 99846 07295 99734 49 12 00349 99999 02094 99978 03839 99926 05582 99844 07324 9973i 48 i3 00378 99999 02123 99977 o3868 99925 o56ii 99842 07353 99729 47 14 00407 99999 02152 99977 03897 99924 05640 99841 07382 99727 46 i5 00436 99999 02181 99976 03926 99923 05669 99839 07411 99725 45 16 00465 99999 022 11 99976 03955 99922 05698 99838 07440 99723 44 7 00495 99999 02240 99975 03984 999921 05727 99836 07469 99731 43 8 o00524 99999 02269 99974 04013 9999 05756 99834 07498 99719 42 9 00oo553 99998 02298 99974 04042 99918 05785 99833 07527 99716 41 20 00582 99998 02327 99973 04071 99917 o58i4 99831 07556 99714 40 21 oo6II 99998 02356 99972 04100 99916 05844 99829 07585 99712 39 22 0oo64o 99998 02385 99972 04129 9991i5 05873 99827 07614 99710 38 23 00oo669 99998 02414 99971 o 04159 99913 05902 99826 07643 99708 37 24 00698 99998 02443 99970 04i88 99912 05931 99824 07672 99705 30 25 00727 99997 02472 99969 04217 99911 05960 99822 07701 99703 35 26 00756 99997 02501 99969 04246 99910 05989 99821 07730 99701 34 2 7 00785 99997 0253o 99968 04275 99909 o6o8 99819 07759 99699 33 28 oo8i4 99997 025601 99967 04304 99907 06047 99817 07788 99696 32 29 oo844 99996 02589 99966 o4333 999o6 06076 998i5 07817 99694 31 30 00873 99996 02618 99966 04362 99905 o61o5 99813 07846 99692 30 31 oo0902 99996 02647 99965 04391 999904 o6i34 99812 07875 99689 29 32 oo931 99996 02676 99964 04420 99902 o6i63 99810 07904 99687 28 33 00960 99995 02705 99963 04449 99901 06192 99808 07933 99685 27 34 00989 99995 02734 99963 04478 99900 06221 99806 07962 99683 26 35 o1oi8 99995 02763 99962 04507 99898 06250 99804 07991 99680 25 36 oIo01047 99995 02792 9996I o4536 99897 06279 99803 08020 99678 24 37 01076 999941 02821 99960 04565 99896 o63o8 9980- 08049 99676 23 /38 oi~o5 99994 o2850o 9999 04594 99894 06337 99799 08078 99673 22 39 o0i34 99994 02879 99959 04623 99893 o6366 99797 08107 99671 21 40 0o164 99993 02908 99958 o4653 99892 06395 99795 0o836 99668 20 41 01i1093 993 9 02938 99957 04682 9989o 06424 99793 0o865 99666 i9 42 0o222 99993 02967 99956 047 11 99889 o6453 99792 o08194 99664 o 43 or01251 99992 02996 995 04740 99888 o648 2 99790 08223 99661 17 44 01280 99992 o3o25 99954 04769 99886 o65i1i 9978 o8252 99659 16 45 o030o 9999 030o54 99953 04798 99885 o6540 99786 08281 99657 15 46 oi338 99991 o3o83 99952 04827 99883 0656 99784 o831o 99654 4 47 o01367 99991 03112 99952 04856 99882 06598 99782 08339 99652 13 48 o01396 99990 o3i4 99951 o4885 99881 06627 99780 08368 996491 2 49 oI425 99990 o3170 99950 04914 99879 o6656 99778 08397 996471 I 50 o01454 99989 03199 99949 04943 99878 o6685 99776 o08426 99644 o 0 5 01 o483 99989 03228 99948 04972 99876 o6714 99774 08455 99642 9 52 oi513 99989 o3257 99947 o5oo/ 99875 06743 99772 08484 99639 8 53 01542 99988 03286 99946 05030 99873 06773 99770 o8513 99637 7 54 oI571 99988 o33i6 99945 05059 99872 06802 99768 08542 99635 6 55 o016oo 99987 03345 99944 05088 99870 o683 99766 0857i 99632 5 56 o0629 99987 o33 74 99943 o5117 99869 06860 99764 o86oo 99630 4 57 o01658 99986 03403 99942 05146 99867 06889 99762 08629 99627 3 58 o687 99986 03432 99941 o0575 99866 06918 99760 o8658 99625 2 59 o01716 99985 3461 99940 o52o05 99864 06947 99758 08687 99622 1 60 o01745 99985 03490 99939 05234 99863 06976 99756 o8716 99619 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 89~ 880 870 860 850 TABLE III. NATURAL SINES AND COSINES. 65 50 60 _ _ _ 9 0 / |O!0 9 Sinle Cosine. Sine. Cosine. Sine. Cosine. Sine. Cos ine. Cne. o o8716 99619 1o453 99452 12187 99255 13917 99027 5643 98769 60 i 08745 99617 10482 99449 12216 99251 13946 99023 15672 98764 5 2 08774 996i4 10511 99446 12245 99248 13975 9909 1570 98760 5 3 o8803 99612 io54o 99443 224 0 994 244 14004 99015 15730 98755 57 4 o883 996o09 10569 99440 12302 99240 I4033 99011 15758 98751 56 5 o8860 99607 10597 99437 12331 99237 I406I 99006 15787 98746 55 6 08889 99604 I0626 99434 12360 99233 14090 99002 15816 98741 54 7 0898g 99602 I0655 9943I I2389 99230 1411I 98998 I5845 98737 53 8 08947 99599 10684 99428 12418 99226 I4148 98994 I5873 98732 52 9 o8976 99596 10713 99424 12447 99222 14177 98990 15902 98728 51 1o 09005 99594 10742 99421 12476 99219 14205 98986 15931 98723 50 II 09034 99591 1077I 99418 12504 99215 I4234 98982 15959 98718 4 12 09063 99588 o800o 99415 12533 99211 14263 98978 5988 98714 48 3 09092 99586 0o829 9941 2 2562 99208 14292 98973 I60o7 98709 47 14 09121 99583 10858 99409 12591 99204 14320 98969 I6046 98704 46 15 09150 99580 10887 99406 12620 99200 14349 98965 16074 98700 45 i6 09170 99578 Iog1 6 99402 12649 99197 14378 98961 161o3 98695 44 17 09208 99575 10945 99399 12678 991q3 14407 98957 I6132 98690 43 I8 09237 99572 10973 99396 12706 99189 14436 98953 i6160 98686- 42 19 09266 99570 11002 99393 12735 99186 14464 98948 16189 98681 41 20 09295 99567 11031 99390 12764 99182 14493 98944 I6218 98676 40 21 o09324 99564 iio6o 9936 12793 99178 14522 98940 16246 98671 39 22 09353 99562 I1089 99383 12822 99175 i455I 98936 i6275 98667 38 23 09382 99559 IIii8 99380 12851 99171 14580 98931 163o4 98662 37 24 o94 1 99556 11147 99377 12880 99167 I4608 98927 i6333 98657 36 25 09440 99553 II176 99374 12908 99163 14637 98923 16361 98652 35 26 o9469 99551 11205 99370 12937 99160 14666 98919 16390 98648 34 27 09498 99548 11234 99367 I2966 99156 14695 98914 16419 98643 33 28 09527 99545 11263 99364 12995 99152 14723 98910 16447 98638 32 29 09556 99542 11291 99360 13024 99148 14752 98906 16476 98633 31 30 09585 99540 11320 99357 i3053 99144 14781 98902 i65o5 98629 3 3I o9614 99537 11349 99354 13081 g9941 I4810 98897 i6533 98624 29 32 09642 99534 11378 99351 1311o 99137 I4838 98893 16562 98619 28 33 09671 99531 11407 99347 13139 g9933 14867 98889 16591 98614 27 34 09700 99528 11436 99344 i31i68 99129 14896 98884 16620 98609 26 35 09729 99526 11465 99341 13197 9912 1I4925 98880- 16648 98604 25 36 09758 99523 I 494 99337 13226 99122 I4954 98876 16677 98600 24 37 09787 99520 11523 99334 13254 99118 14982 98871 16706 98595 23 38 o9816 99517 1552 99331 13283 99114 15o11 98867 16734 98590 22 39 o9845 99514 |II580 99327 13312 99110 I5040 98863 i6763 98585 21 40 09874 99511 11609 69324 I334 99ggo6 15069 98858 16792 98580 20 4' 0o903 99508 ii638 99320 13370 99102 15097 98854 16820 98575 I9 42 o9932 99506 I1667 99317 13399 99o98 15126 98849 16849 98570 8 43 09961 99503 11696 99314 13427 99094 15155 98845 16878 98565 17 44 09990 99500 11725 9931o 13456 99091 I5184 98841 16906 98561 16 45 10019 99497 11754 99307 13485 99087 I5212 98836 16935 98556 15 46 10048 99494 1 1783 99303 13514 99083 I5241 98832 16964 9855I 14 47 10077 99491 1812 993oo 73543 97 15270 98827 16992 98546 13 48 ioio6 99488 11840 99297 13572 99075 15299 98823 17021 98541 12 49 1I013 99485 II86g 99293 13600 99071 15327 98818 17050 98536 ii 50 IOI64 99482 11898 99290 13629 99067 i5356 98814 17078 98531 o0 51 I0192 99479 11927 99286 i3658 99063 15385 98809 17107 98526 9 52 10221 99476 ii956 99283 13687 99059 5414 98805 17136 9852 8 53 10250 99473 ii985 99279 13716 99o55 15442 98800 17164 985 6 7 54 10279 99470 12014 99276 13744 99051 15471 98796 17193 9851i 6 55 10308 99467 12043 99272 13773 99047 15500 98791 17222 98506 5 56 0337 99464 12071 99269 13802 99043 15529 98787 17250 985o0 4 57 io366 99461 12I10 99265 1383 99039 15557 98782 17279 98496 3 58 10395 99458 2129 99262 1386o 99035 15586 98778 17308 98491 2 59 10424 99455 12158 99258 13889 9903I 15615 98773 17336 98486 60 io453 99452 12187 99255 3917 99027 15643 98769 17365 98481 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sinc. Cosine. Sine. I ________ __.__ 11 _______ 810 800 840 88 82 810 8~ r J0 66 NATURAL SINES AND COSINES. TABLE III. 100 lio 12~ 130 140 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine Cosine. Sine, Cosine. 0 17365 98481 1go081 98163 20791 97815 22495 97437 24192 97030 60 I 17393 98476 I9100 98157 20820 07809 22523 97430 24220 97023 59 2 17422 98471 19138 98152 20848 97803 22552 97424 24249 97015 58 3 17451 98466 19167 98146 20877 97797 22580 97417 24277 97008 57 4 17479 98461 19195 98140 20900 97791 22608 97411 24305 97001 56 5 17508 98455 I9224 98135 20933 97784 22637 97404 24333 96994 55 6 17537 98450 19252 98129 20962 97778 22665 97398 24362 96987 54 7 7565 98445 19281 98324 20990 97772 22603 97391 2439o 96980 53 8 17594 988840 1930 98II8 21019 97766 22722 97384 24418 96973 52 9 17623 98435 19338 98112 21047 97760 22750 97378 24446 96966 51 I o 7651 98430 19366 98107 21076 97754 22778 97371 24474 969059 5o i1 1768o 98425 19395 98101 21104 97748 22807 97365 24503 96952 49 12 17708 98420 19423 98096 21132 97742 22835 97358 24531 96945 48 13 17737 98414 19452 98090 21161 97735 22863 97351 24559 96937 47 14 17766 98409 19481 98084 21189 97729 22892 97345 24587 96930 46 15 17794 98404 19509 98079 21218 97723 22020 97338 24615 96923 45 16 17823 98399 19538 98073 21246 97717 22948 97331 24644 96916 44 17 17852 98394 19566 98067 21275 977311 22977 97325 24672 960o9 43 18 17880 98389 I1595 98061 21303 97705 23005 97318 24700 96902 42 19 17909 98383 19623 98056 21331 97698 23033 97311 24728 96894 41 20 17937 98378 19652 98050 21360 97692 23062 97304 24756 096887 4 21 17966 98373 19680 98044 21388 97686 23090 97298 24784 96880 39 22 17995 98368 19709 98039 21417 97680 23118 97291 24813 96873 38 23 18023 98362 19737 98033 21445 97673 23146 97284 24841 96866 37 24 18052 98357 19766 98027 21474 97667 23175 97278 24869 96858 36 25 1808i 98352 19794 98021 21502 97661 23203 97271 24897 9685i 35 26 8109 98347 19823 98o016 2530 97655 23231 97264 24925 96844 34 27 18138 98341 19851 98010oo 2559 97648 23260 97 2 57 249541 96837 33 28 18166 98336 19880 98004 21587 97642 23288 97251 24982 96829 32 29 i8195 98331 19908 97998 21616 97636 233i6 97244 25010 96822 3I 30 18224 98325 19937 97992 21644 97630 23345 97237 25038 96815 30 31 18252 9832o 19965 97987 21672 97623 23373 97230 25066 968o7 22 3o 18281 98315 109994 97981 21701 97617 23401 97223 25094 96800 23 33 1830 098310 200220 97975 21729 97611 23429 97217 25122 96793 27 34 18338 o83o4 20051 97969 21758 97604 23458 97210 25151 96786 26 35 18367 98299 20079 97963 21786 97598 23486 97203 25179 96778 25 2 36 18395 982094 20108 97958 21814 097592 23514 971.6 25207 96771 24 37 18424 98288 20136 97952 21843 975 5 23542 971 9 25235 96764 23 38 184.52 98283 20165 97946 21871 97579 23571 97182 25263 96756 22 39 18481 98277 20193 97940 21899 97573 2359.9 97176 25291 96749 21 40 185o0 98272 20222j 97934 21928 97566 23627 97169 25320 96742 20 4 xi8538 98267 202501 97928 21956 97560 23656 97162 25348 96734 i19 42 18567 98261 20279 97922 21985 97553 23684 97i55 25376 96727 1S 43 1855 982556 20307 97916 22013 975/47 23712 97148 25404 96719 1 7 44 i8624 98250 20336 97910 22041 97541 23740 97141 25432 96712 16 45 18652 098 240 20364 97905 22070 97534 23769 97134i 25460 967'05:i5 46 I8681 98240 20393 97899 22098 97528 23797 97127 25488 96697 141 47 8710o 98234 20421I 97893 22126 97521 23825 97120 25516 9669o 13 48 18738 98229 20450 97887 22155 97515 23853 97113 25545 96682 1 49 18767 98223 20478 97881 22183 97508 2.3882 97106 25573 96675 Ii 5o 18795 98218 20507 97875 22212 97502 23910 97100 25601 96667 1o 5 1i8824 98212 20535 97869 222240 97496 23938 97093 25629 96660 0 52 i8852 98207 2o563 97863 22268 97489 23966.97086 25657 96653 53 1888 98201 10502 97857 22297 97483 23995 97079 25685 96645 7 54 189'0 98196 20620 97851 22325 97476 24023' 97072 25713 96638 6 55 18938 98190 20649 97845 22353 97470 24051 97065 25741 96630 5 56 18967 98185 20677 97839 22382 97463 24079 97058 25769 96623 4 57 18995 98179 20706 97833 22410 97457 24108 97051 25798 96615 3 58 19024 98174 20734 97827 22438 97450 24136 97044 25826 96608 a 59 19052 98168 20763 97821 22467 97444 24164 97037 25854 96600 i 60 19081 98163 20791 978i5 22495 97437 24192 97030 25882 96593 o Cosine. Sine. Cosine Sine Cosine. Sine. osine. Sine. Cosine. Sine. 9o 8 0 60 50 TABLE III. NATURAL SINES AND COSINES. 6_ __150 _16~ 1 8~ 19~ Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 25882 96593 27564 96126 29237 95630 30902 95106 32557 94552 60 i 25910 96585 27592 96118 29265 95622 30929 95097 32584 94542 59 2 25938 96578 27620 96110 29293 95613 30957 95088 32612 94533 58 3 25966 96570 27648 96o102 29321 95605 30985 95079 32639 94523 57 4 25994 96562 27676 96094 29348 95596 3o1012. 95o070 32667 94514 56 5 26022 96555 27704 96086 29376 95588 3o40o 9506i 32694 945o4 55 6 26050 96547 27731 96078 29404 95579 3io68 95052 32722 94495 54 7 26079 96540 27759 96070 29432 95571 3o1095 95043 32749 94485 53 8 26107 96532 27787 96062 -29460 95562 31123 95033 32777 94476 52 9 26135 96524 27815 96054 29487 95554. 31151 95024 32804 94466 51 io 26163 96517 27843 96046 29515 95545 31178 95015 32832 94457 50 I1 26191 965609 27871 96037 29543 95536 31206 95006 32859 94447 49 12 26219 96502 27899 96029 29571 95528 31233 94997 32887 94438 48 13 26247 96494/ 27927 96021 29599 95519 31261 94988 32914 94428 47 14 26275 96486 27955 96013 29626 95511 31289 94979 32942 94418 46 15 26303 96479 27983 96005 29654 95502 313i6 94970 32969 94409 45 16 26331 96471I 28011 95997 29682 95493 31344 94961i 32997 94399 44 1 7 26359 96463 28039 95989 i29710 95485 31372 94952 33024 94390 43 18 26387 96456 280o67 95981 29737 95476 31399 94943 33o5 94380o 42 19 26415 96448 28095 95972 29765 95467 31427 94933 33079 94370o 41 20 26443 96440 28123 95964 29793 95459 3,454 94924 33o106 94361 40 21 26471 96433 28150 95956 29821 95450 31482 94915 33i34 9435i 39 22 26500 96425 28178 95948 29849 95441 31510o 94906 33i6x 94342 38 23 26528 96417 28206 95940 29876 95433 31537 94897 33189 94332 37 24 26556 96410 28234 95931 29904 95424 3i565 94888 33216 94322 36 25 26584 96402 28262 95923 29932 95415 313593 94878 33244 94313 35 26 26612 96394 28290 95915 29960 95407 31620 94869 33271 94303 34 27 26640 96386 28318 95907 29987 95398 31648 94860 33298 94293 33 28 26668 96379 28346 95898 3oo005 95389 3675/94851 33326 94284 32 29 26696 96371 28374 95890 3oo0043 95380 317o3 94842 33353 94274/ 31 30 26724 96363 28402 95882 30071 95372 31730 94832 3338i 94264 30 31 26752 96355 28429 95874 30098 95363 31758 94823 33408 94254 29 32 26780 96347 28457 95865 30126 95354 31786 94814 33436 94245 28 33 268o8 96340 28485 95857 30o54 95345 3i813 94805 33463 94235 27 34 26836 96332 28513 95849 30182 95337 31841 94795 33490 94225 26 35 26864 96324 2854, 95841 30209 95328 3i868 94786 335i8 94215 25 36 26892 96316 28569 95832 30237 95319 31896 94777 33545 94206 24 37 26920 96308 28597 95824 30265 95310 31923 94768 33573 94196 23 38 26948 96301 28625 95816 30292 95301 31951 94758 336oo00 94186 22 39 26976 96293 28652 95807 30320 95293 31979 94749 33627 94176 21 40 27004 96285 28680 95799 30348 95284/ 32006 9474o 33655 94167. 20 41 27032 96277 28708 95791 30376 95275 32034 94730 33682 94157 1 42 27060 96269 28736 95782 3o403 95266 32061 94721 33710 94147 18 43 27088 96261 28764 95774 30431 95257 32089 94712 33737 94137 17 44 27116 96253 28792 95766 30459 95248 32116 94702 33764 94127 16 45 27144 96246 28820 95757 30486 95240 32144 94693 33792 94118 15 46 27172 96238 28847 95749 30514 95231 32171 94684 33819 94108 14 47 27200 96230 28875 95740 30542 95222 32199 94674 33846 94098 13 48 27228 96222 28903 95732 30570 95213 32227 94665 33874 94088 12 49 27256 96214 28931 95724 30597 952o04 32254 94656 33901o 94078 II SO 27284 96206 28959 95715 3o625 95195 32282 94646 33929 940681 o 51 27312 96198 28987 957o07 30653 95186 32309 94637 33956 94058 9 52 27340 96190 29015 95698 30680 95177 32337 94627 33983 94049 8 53 27368 96182 29042 9569o 30708 95168 32364 94618 34o01 94039 7 54 27396 96174 29070 9568 30736 95159 32392 94609 34038. 94029 6 55 27424 96166 29098 95673 30763 95150 32419 94599 34065 94019 5 56 27452 96158 29126 95664 30701 95142 32447 94590 34093 94009 4 57 27480 96,50 2 91r54 95656 3o8i9 95133 32474 9458o 34120 93999 3 58 27508 96142 29182 95647 30846 95124 32502 94571 34147 93989 2 59 27536 96134 29209 95639 30874 95115 32529 94561 34,75 93979 i 6o 27564 96126 29237 9563o 30902 95106 32557 94552 34202 93969 o Cosine. Sine. Cosine. Se. ine.osineosine. sine... Sine. //'e'e Co s in \4 o'73~ ~2~! ^ 1o 70o 68 NATURAL SINES AND COSINES. TABLE III. 200 210 22~ 23~ 24~ Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Coeine. 0 34202 93969 35837 93358 37461 92718 39073 92050 40674 91355 6o I 34229 93959 35864 93348 37488 92707 39100 9203 4-0700 91343 50 2 34257 93949 35891 93337 37515 92697 39127 9202 40727 91331 58 3 34284 93939 35918 93327 37542 92686 39153 92016 40753 91319 57 4 34311 93929 35945 93316 37569 92675 39180 92005 407801 91307 56 5 34339 93919 35973 93306 37595 92664 39207 91994 4o8o6 91295 55 6 34366 93909o 36ooo 93295 37622 92653 39234 91982 40833 91283 54 7 34393 93899 36027 93285 37649 92642 39260 91972 4o86o 91272 53 8 34421 93889 36o54 93274 37676 92631 39287 91959 4o886 91260 52 9 34448 93879 36o81 93264 37703 92620 39314 91948 40913 91248 51 io 34475 93869 36o08 93253 37730 92609 3934I 91936 40939 91236 50 Ii 3/,503 93859 36i35 93243 37757 92598 39367 91925 40966 91224 49 12 3453o 93849 36162 93232 37784 92587 39394 91914 40992 91212 48 13 34557 93839 36190 93222 37811 92576 39421 91902 41019 91200 47 14 34584 93829 36217 93211 37838 92565 39448 91891 41045 91188 46 15 34612 93819 36244 93201 37865 92554 39474 91879 41072 91176 45 i6 34639 93809 36271 9390o 37892 92543 39501 91868 410o98 91164 44 17 34666 93799 36298 93180 37919 92532 39528 9856 41II25 91152 43 18 34694 93789 36325 93169 37946 92521 39555 91845 41151 91140 42 19 34721 93779 36352 93159 37973 92510 39581 91833 41178 91128 41 20 34748 93769 36379 93148 37999 92499 39608 91822 41204 91116 40 21 34775 93759 364o6 93137 38026 92488 39635 91810 41231 91104 39 22 34803 93748 36434 93127 38o53 92477 39661 91799 41257 91092 38 23 34830 93738 36461 93116 38o80 92466 39688 91787 41284 91080 37 24 34857 93728 36488 93106 38107 92455 39715 91775 4i3io 91068 36 25 34884 93718 36515 93095 38134 92444 39741 91764 41337 91056 35 26 34912 93708 36542 93084 3816i 92432 39768 91752 41363 91044 34 27 34939 93698 36569 93074 38188 92421 39795 91741 41390 91032 33 28 34966 93688 36596 93063 38215 92410 39822 9172 41416 91020o 32 29 34993 93677 36623 93052 38241 92399 39848 91718 41443 9100oo8 31 30 35021 93667 36650 93C42 38268 92388 398751 91706 4i469 90996 3 3i 35048 93657 36677 93031 38295 92377 39902 91694/ 4496 90984 29 32 35075 93647 36704 93020 38322 92366 39928 91683 41522 90972 28 33 35102 93637'36731 93010o 38349 92355 39955 91671 41549 90960 27 34 35130 93626 36758 92999 38376 92343 39982 91660 4157 90948 26 35 35157 93616 36785 92988 38403 92332 4000ooo8 91648 41602 90936 25 36 35184 93606 36812 92978 38430 92321 40035 91636 41628 90924 24 37 35211 93596 36839 92967 38456 92310 40062 91625 4i655 90911 23 38 35239 93585 36867 92956 38483 92299 4oo0088 9613 468I 90899 22 39 35266 93575 36894 92945 385o10 92287 40115 91601 41707 90887 21 40 35293 93565 36921 92935 38537 92276 40141 9590o 41734 90875 20 41 35320 93555 36948 92924 38564 92265 4o0168 91578 4176o 90863 19 42 3534/7 93544 36975 92913 38591 92254 40195 91566 41787 90851 18 43 353-75 93534 37002 92902 38617 92243 40221 91555 4i813 90839 17 44 31540o2 93524 37029 92892 38644 92231 40248 91543 418401 90826 I 45 35429 93514 37056 92881 38671 92220 40275 91531 41866 90o814 15 46 35456 93503 37083 92870 38698 92209 4030o 91519 41892 90802 14 47 35484 93493 37110o 92859 38725 921 40328 91 08 41919 9079~0 3 48 35511 93483 37137 92849 38752 92186 4o355 91496 41945 90778 12 493538939 28388 [ 913877 49 35538 93472 37164 92838 38778 92175 40381 914i4 41972 90766 ii 50 35565 93462 37191 92827 388o5 92164 40408 91472 41998 90753 io 5I 35592 93452 37218 92816 38832 92152 40434 91461 42024 90741 9 52 35619 93441 37245 92805 38859 92141 40461 91449 42051 90729 1 53 35647 93431/ 37272 92794 38886 92130 40488 91437 42077 907 17 7 54 35674 93420 37299 92784 38912 92119 40514 91425 42104 90704 6 55 3570o 9341 0 37326 92773 38939 92107 4054 91414 42130o 90692 5 56 35728 93400 37353 92762 38966 92096 40567 91402 42156 90680 4 57 35755 93389 37380 92751 38993 92085 40594 91390 42183 90668 3 58 35782 93379 37407 92740 39020 92073 40621 91378 42209 90655 2 59 3581o 93368 37434 92729 39046 9062' 40647 91366 42235 90643 i 60 35837 93358 37461 92718 39073 9o050 40674 91355 42262 9o631 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 69 680 6 660 65~ TAbLE III. NATURAL SINES AND COSINES. 69 __ -- 25 ~ - 2650 270 28 j 293 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0 42262 90631I 43837 89879 45399 89101 46947 88295 48481 87462 60 420288- 90618 43863 89867 45425 89087 46973 8828I 48506 87448 59 2 42315 90606 43889 89854 45451 89074 46999 88267 48532 87434 58 3 42341 90~594 43916 89841 45477 89o06 47024 88254 48557 87420 57 4 42367 90o582 43942 89828 45503 89048 47050 88240 48583 87406 56 5 42394 90569 43968 89816 45529 89035 47076 88226 48608 87391 55 6 42420 90557 43994 89803 45554 /89021 47o101 8823 48634 87377 54 7 42446 90545 44020 89790 4558o 89008oo 47127 88199 48659 87363 53 8 42473 90532 44046 89777 45606 88995 47153 88i85 48684 87349 52 9 42499 90520 44072 89764 45632 88981 47178 88172 48710 8733 5i io 42525 90507 44098 89752 45658 88968 47204 88158 48735 8732 50o i1 42552 90495 44124 80739 45684 88955 47229 88144 48761 87306 49 12 42578 90483 44151 89726 457IO 88942 47255 8813o 48786 87292 48 13 42604 90470 44177 89713 45736 88928 47281 88117 4881I 87278 47 14 42630 90458 44203 89700 45762 88905 47306 881o3 48837 87264 46 1i5 42657 90446 44229 89687 45787 88902 47332 88089 48862 87250 45 i6 42683 90433 44255 89674 45813 88888 47358 88075 488888 87235 44 17 42709 9042I 4428i 89662 45839 88875 47383 88062 48913 87221 43 18 42736 9g408 44307 89649 45863 88862 47403 88048 48938 87207 42 19 42762 90396 44333 896364 474 88o34 4893648 87793 41 20 42788 90383 44359 89623 45917 88835 47460 88020 48989 87178 40 21 42815 o90370 44385 89610 45942 88822 47486 88oo6 490o4f 87064 3q 22 42841 90358 44410 89597 45968 88808 47511 87993 49040 87150 38 23 42867 90346 44437 89584 }45994 88795 47537 87979 49065 87136 37 24 42894 90~334 44464 89571 46020 88782 47562 87965 49090 87120 36 25 42920 90320 44490 89558 46046 88768 47588 87951 490I6 87107 35 26 42946 90309 44546 89545 46072 88755 47604 87937 4941 i87o93 34 27 42972 90296 44542 89532 46097 88741 47639 87923 49i 66 87079 33 28 42999 90284 44568 89509 i46123 88728 47665 87909 49192 87064 32 29 43025 90271 44594 89506 46149 88715 47690 87896 49217 87050 31 30 430o5 90259 44620 89493 46073 88701 47716 87882 49242 87036 30 31 43077 90246 44646 89480 46201 88688 47741 87868 49268 87021 29 32 43104 90233 44672 89467 46226 88674 47767 87854 49293 87007 28 33 43130 90221 44698 89454 46252 8866i 47793 87840 49318 86993 27 34 43156 90208 44724 89441 46278 88647 47818 87826 49344 86978 206 35 43182 90196 4475.0 89428 46304 88634 47844 87802 49369 86964 25 36 43209 901go83 44776. 89415 46330 88620 47869 87798 49394 86949 2 4 37 43235 90171 44802 89402 46355 88607 47895 87784 49419 86935 23 38 4326i 90158 44828 89389 4638i 88593 47920 87770 4944 86921 22 39 43287 90146 44854 89376 46407 88580 47946 87756 49470 86906 21 Ao 43313 90g33 44880 89363 46433 88566 47971 87743 49495 86892 20 401 4334o 90120 44906 89350 46458 88553 47997 87729 49521 86878 19 42 43366 90o08 44932 89337 46484 88539 48022 87702 49546 86863 18 43 43392 90095 44958 89324 4650o 88526 48048 87701 49571 86849 17 44 43418 90082 44984 89311 46536 88512 48073 87687 49596 86834 16 45 43445 90070 45010o 89298 46561 88499 48099 87673 49622 86820 15 46 4347 900oo57 45o36 89285 46587 88485 48124 87659 49647 86805 14 47 43497 90045 45062 89272 46603 88472 485o0 87645 4986 86790 13 48 43523 90032 45088 89259 46639 88458 48175 87631 49697 86777 2 49 43549 90019 45114 89245 46664 88445 48201 87617 49723 86762 ii 50 43575 90007 4514o 89232 46690 8843 48226 87603 49748 86748/ o 51 43602 89994i 45166 89219 467016 88417 48252 87589 49773 86733 9 52 43628 89981 45092 89206 46742 884o4 48277 87575 49798 86719 8 53 43654 89968 45218 89193 46767 88390 48303 87561 49824 86704 7 54 43680o 89956 45243 89180 46793 88377 48328 87546 49849 86690 6 55 43706 89943 45269 89167 46809 88363 48354 87532 49874 86672 5 56 43733 89930 45295 89153 46844 88349 48379 87518 49899 86661 4 57 43759 89918 45321 89140 46870 88436 484o0 87504 49924 86646 3 43780 89905 45347 89127 46896 88322 48430 87490 49950o 86632 2 59 438i1 89892 45373 89114 46921 883o8 48456 87476 49975 86607 I 60 43837 89879 45399 89101 46947 88295 48480 87462 5oooo 866o3 o Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 640 631~ 620 610 600 i0o NATURAL SINES AND COSINES. TABLE III. 300 31~ 322 330 340 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. 0o 00oooo0 866o3 5150o4 85717 52992 848o5 54464 83867 55919 8290460 1 50025 86588 51529 85702 53017 84789 54488 8385i 55943 82887 59 2 50o050 86573 51554 85687 5304i 84774 54513 83835 55968 82871 58 3 5oo0076 86559 51579 85672 53066 84759 54537 83819 55992 82855 57 4 5oioi 86544 5i6o4 85657 53091 84743 54561 838o4 56016 82839 56 5 50oI26 86530 51628 85642 53ii5 84728 54586 83788 56040 82822 55 6 5oi5i 865i5 51653 85627 5314o 84712 5461o 83772 56064 828o6 54 7 5o016 865o01 5678 85612 53164 84697 54635 83756 56o88 82790 53 8 50o2o 86486 51703 85597 53i89 8468i 54659 83740 56112 82773 52 9 50227 86471 51728 85582 53214 84666 54683 83724 56i36 82757 51 i1 50252 86457 51753 85567 53238 8465o /47o8 837o8 56i6o 82741 5o ii 50277 86442 51778 85551 53263 84635 54'732 83692 56184 82724 4 0 12 50302 86427 5803 85536 53288 84619 54756 83676 562o08 82708 48 13 50327 86413 51828 855215 3312 84604 54781 8366o 56232 82692 47 14 50352 86398 51852 855o06 53337 8488 548o5 83645 56256 82675 46 12 50377 86384 51877 85491 53361 84573 54829 83629 56280 82659 45 i6 504o3 86369 51902 85476 53386 84557 544 83613 56305 82'643 44 17 50428 86354 51927 85461 5341i 84542 54878 83597 56329 82626 43 18 50453 86340 51952 85446 53435 84526 54902 8358i 56353 826o10 42 19 5o478 86325 51977 85431 5346o 8451. i 54927 83565 56377 82593 41 20 5o5o3 863Io 52002 854i6 53484 84495 54951 83549 5640o 82577 40 21 50528 86295 52026 854oi 53509 84480 54975 83533 56425 82561 39 22 50553 86281 52051 85385 53534 84464 54999 83517 56449 82544 38 23 50578 86266 52076 85370 53558 84448 55024 83501 56473 82528 37 24. 506o3 86251 52101 85355 53583 84433 55o48 83485 56497 82511 36 25 50628 86237 52126 8534o 53607 84417 55072 83469 56521 82495 3 26 5o654 86222 52151 85325 53632 84402 55097 83453 56545 82478 34 27 50679 86207 52175 853i1o 53656 84386 55121 83437 56569 82462 33 28 50704 86192 52200 85294 5368i 84370 55145 83421 56593 82446 32 29 50729 86178 52225 85279 53705 84355 55169 834o5 56617 82429 31 30 50754 86163 52250 85264 53730 84339 55194 83389 56641 82413 3o 31 50779 86148 52275 85249 53754 84324 55218 83373 56665 82396 29 32 50oo4 86133 52299 85234 53779 84308 55242 83356 56689 82380 28 33 5o829 86119 52324 85218 53804 84292 55266 83340 56713 82363 27 34 50854 86104 52349 85203 53828 84277 55291 83324 56736 82347 26 35 50879 86089 52374 85i88 53853 84261 553i5 833o8 56760 82330 25 36 50904 86074 52399 85173 53877 84245 55339 83292 56784 82314 24 37 50929 86059 52423 85157 53902 84230 55363 83276 5680o8 82297 23 38 50954 86045 52448 85142 53926 84214 55388 83260 56832 82281 22 39 50979 86030 52473 85127 53951 84198 55412 83244 56856 82264 21 4.0 5100oo4 86o5 52498 8512 53975 84182 55436 83228 56880 82248 20 41 51029 86ooo 52522 85096 54000 84167 5546o 83212 569o4 82231 19 42 5io54 85985 52547 8508i 54024 8415i 55484 83i95 56928 82214 i8 43 5o.79 85970 52572 85o66 54049 84135 55509 83179 56952 82198 17 44 5iio4 85956 52597 50o5 54073 84120 55533 83i63 56976 82181 16 45 51129 85941 5262 850o35 54097 84io4 55557 83147 57000 82165 15 46 51154 85926 52646 85020 54122 84088. 5558i 83i3i 5 024 82148 14 47 5ii79 85911 5267I 85005 54,46 84072 55605 83ii5 57047 82132 13 48 512o4 85896 52696 84989 5417i 84057 5563o 83098 57071 82115 J2 49 51229 8588i 52720 84974 54195 84041 55654 83082 57095 82098 II 50 51254 85866 52745 84959 54220 84025 55678 83066 57119 82082 10 5i 51279 8585i 52770 84943 54244 84009 55702 8305o 57143 82065 9 52 5i304 85836 52794 84928 54269 83994 55726 83o34 57167 82048 8 5.3 51329 85821 52819 84913 54293 83978 55750 83017 57191 82032 7 54 51354 858o6 52844 8489'7 543I7 83962 55775 83oo001 I 57215 82015 6 55 51379 85792 52869 84882 54342 83946 55799 82985 57238 81999 5 ~b 5i4o4 85777 52893 84866 54366 83930 55823 82969 57262 81982 4 58 51429 85762 52918 8485 54391 83915 55847 82953 57286 81965 3 51454 85747 52943 84836 54415 83899 55871 82936 573I 81949 2 59 51479 85732 52967 84820 54440 83883 55895 82920 57334 81932 i 60 51504 85717 52992 84805 54464 83867 55919 82904 57358 81915 o Cosine. Sine. Cosine. Sine. Cosine. SinSine. CCosine. Sine. 9 680 5~0 0 85 6 6 TABLE 11L _ NbATURAL SINES AND COSINES. __ 850 36~ 87^ ~8~ i) 35 398 Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. o 57358 81015 58779 80902 60182 79864 61566 78801 62932 777i5 o0 i 573,81 81899 58802 80885 I60205 79846 6589 78783 62955 77696 59 2 574o5 81882 58826 80867 60228 79829 61612 78765 62977 77678 58 3 57429 8i865 58849 8o85o 60251 79811 6i635 78747 63ooo 77660 57 4 57453 81848 58873 8o833 60274 79793 6i658 78729 63022 77641 56 5 57477 81832 58896 8o0816 60298 79776 61681 78711 63045 77623 55 6 57501 8i815 58920 80799 60321 79758 61704 78694 63o68 77605 54 7 57524 81798 58943 80782 60344/ 79741 61726 78676 63090 77586 53 8 57548 81782 58967 80765 60367 79723 61749 78658 631i3 77568 52 9 57572 81765 58990 80748 639o0 79706 61772 78640 63i35 77550[ 5i 10 57596 81748 59014 80730 60414 79688 61795 78622 63i58 77531 bo i1 57619 81731 59037 80713 60437 79671 6i8i8 78604 6318o 77513 49 12 57643 81714 59061 80696 6046o0 79653 61841 78586 63203 77494 48 13 57667 81698 5o84 80o679 60483 79635 6i864 78568 63225 77476 47 14 57691 81681 59io8 80662 6o5o6 79618 61887 7855o 63248 7745'S 46 15 57715 81664 59131 80644 60529 79600 6go1909 78532 63271 77439 45 i6 57738 81647 59154 80627 60553 79583 61932 78514 63293 77421 44 17 57762 8i63i 59178 80o6o 60576 79565 61955 78496 63316 177402 43 i8 57786 81614 59201 o80593 60599 79547 61978 78478 63338 77384 42 19 578o10 81597 59225 80576! 6o622 79530 62001 78460 6336i 77366 41 20 57833 8i58o 59248 80558 0o645 79512 62024 78442 63383 77347 40 21 57857 81563 59272 8o54 6o668 799494 62046 78424 63406 77329 39 22 57881 81546 59295 80524 60691 79477 62069 78405 63428 77310 38 23 57904 8530o 59318 8o0507 60714 79459 62092 78387 63451 77292 37 24 57928 8i5i3 59342 80489 60738 79441, 62115 78369 63473 77273 30 25 57952 81496 59365 80472 60761 79424 62138 78351 63496 77255 35 26 57976 81479 59389 80455 60784 79406 62160 78333 63518 77236 34 27 57999 81462 59412 80438 60807 79388 62183 78315 63540 772o 8 33 28 58023 81445 59436 80420 6o83o 79371 62206 78297 63563 77199 32 29 58o47 81428 59459 80403o3 60853 79353 62229 78279 63585 77181 31 30 58070 81412 59482 8o386 60876 79335 62251 78261 636o8 77162 30 31 58094 81395 59506 8o368 60899 79318 62274 78243 63630 77144 29 32 58ir8 81378 59529 8035i 60922 79300 62297 78225 63653 77125 28 33 58i4i 8136i 59552 80334 60945 79282 62320 78206 63675 77107 27 34 58i65 81344 59576 8o3i6 60968 79264 62342 78188 63698 77088 26 35 58i89 81327 59599 80299 60991o 79247 62365 78170 63720 77070 225 36 58212 81310 59622 80282 61015 79229 62388' 78152 63742 77o051 24 37 58236 81293 59646 80264 61038 79211 62411 78134 63765/ 77033 23 38 5826o 81276 59669 80247 61o61 79193 62433 78116 63787 77014 22 39 58283 81259 59693 80230 6io84 79176 62456 78098 63810 76996 21 40 58307 8124/2 59716 8o0212 61107 79158 62479 78079 63832 76977 20 41 5833o 81225 59739 8oI95 6ii3o 7914o 62502 78061 63854 76959 19 42 58354 81208 59763 80178 61i53 79122 62524 78o43 63877 76940 18 43 58378 81191 59786 Soi6o 61176 791o5 62547 78025 63899 7692I 17 44 584oi 81174 598o9 8oi43 61199 79087 62570 78007 63922 769o3 i6 45 58425 81157 59832 8o125 61222 79069 62592 77988 63944 76884 15 46 58449 Si140 59856 801oo8 61245 79051 62615 77970 63966 76866 14 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 13 48 58496 811io6 59902 8oo0073 61291 79016 62660 77934 64o11 76828 12 49 58519 81089 59926 80056 61314 78998 62683 77916 64033 76810 ii 50 58543 81072 599491 80038 61337 78980 62706 77897 64056 76791 10 51 58567 810o55 59972 80021 6i360 78962 62728 77879 64078 76772 9 52 5859o 8io38 59995 8ooo3 61383 7 8944 62751 77861 6410oo 76754 8 54 58637 8ioo4 60042 79968 61429 78908 62796 77824 64145 76717 6 55 5866 80987 600o65 79951 61451 78891 62819 77806 64167 76698 5 56 58684 80970 60089 79934 61474 78873 62842 77788 64190 76679 4 57 58708 8o953 6011o2 79916 61497 78855 62864 77769 64212 76661 3 58 5873i 80936 6oi35 79899 61520 78837 62887 77751 64234 76642 2 59 58755 80919 6oi58 79881 6i543 78819 62909 77733 64256 76623 i 6o 58779 80902 60182 79864 6i566 78801 62932 77715 64279 76604 o Cosine Sine. Cosine. Sine. Cosine. S ine. Cosine, Sine. Cosine. Sine. 640 53~ 520 510 60~ 1.5 2 niNATURAL SINES AND COSINES, TABLE III. 400 41~ 42~ - 430 440 Sine. Cosine.. Sine. [Cosine. Sine. Cosine. Sie. Cosine. Sine. Cosine. 64279 76604 656o6 7471 66913 74314 68200 73135 69466 71934 6o I 64301 76586 65628 75452 66935 74295 682221 730I6 69487 71914 50 2 64323 76567 6562o 75433 66956 74276 68242 73096 695087 71894 58 63 64346 76548 65672. 75414 66978 74256 68264 730-76 69529 71873 57 14 64368 76530 65694 75395 66999 74237 68285 73056 69549 71853 56 5 6439o0 7651 65716 75375 67021 74217 68306 73036 69570 71833 55 6 64412 76492 65738 75356 67o43/ 74198 68327 73016 69591 71813 54 7 64435 6443 76473 65759 75337 67064 74178 1683649 72996 695612 71792 53 64457 76455 65781 75318 67o86 74159 68370 o/72976 69633 71772 52 9 64479 76436 65803 75299 67107 74139 68391 72057 69654 71752 5i io 64501 76417 65825 75280 67129 74120 68412 72937 69675 71732 50 11 64524 76398 65847 7526I 67175 740oo 68434 72917 69696 717I1 49 12 64546 76380 65869 7524 6 71 72 671 740 68455 72897 69717 7169 48 3 64568 76361 65891 75222 67194 74061 68476 72877 69737 71671 47 14 64590 76342 65913 75203 67215 74041 68497 72857 69758 71650 46 15 64612 76323 65935 75184 67237 74022 68518 72837 69779 71630 45 i6 64635 76304 65956 75165 67258 74002 68539 72817 69800 71610o 44 17 64657' 76286 65978 75146 67280 73983 68561 72797 69821 71590 43 18 64679 76267 66000oo 75126 6730i -73963 68582 72777 69842 71569 42 9 64701 76248 660221 7507 67323 73044 68603 7 52757 69862 71549 4 20 64723 76229 66o44 75088 67344 73924 68624 72737 69883 71529 40 21 64746 J 76210 66o66 75069 67366 73904 68645 72717 69904 71508 39 22 64768 76192 66088 75050 67387 73885 68666 72697 69925 71488 38 23 64790 76173 66109 75030 67409 73865 68688 72677 69946 746 7168 37 24 64812 76154 6631 75011 67430 73846 68709 722657 69966 71447 36 25 64834 76135 66i53 74992 67452 73826 68730 7 2637 69987 71427 35 26 64856 761716 66175 74973 67473 738o 6 68751 72617 70008 71407 34 27 64878 76097 66107 74953 67495 73787 68772 72597 70029 71386 33 28 6401 o 76o78 66218 74934 67516 73767 68793 72577 70049 7i366 32 209 64923 76059 66240 74915 67538 73747 68384 72557 70070 71 345 31 30 64945 76041 66262 74896 67559 73728 68835 72537 70091 71325 30 3i 64967 76022 66284 74876 6758o 73708 68857 72517 70112 71305 2 32 640989 76oo3 66306 748 57 67602 73688 68878 7 2497 70132 71284 28 33 65o01 75984 66327 74838 67623 73669 68899 72477 70153 71264 27 34 65033 750965'66349 74818 61645 73649 68920 72457 70174 71243 26 35 65o55 72946 66371 74799 67666 73629 68941 72437 7019o 71223 25 36 6077 75927 66393 74780 67688 73610 68962 7247 70215 71203 24 37 65ioc 750o 8 66414 74760 677091 73590 68983 72307 70236 71182 23 38 65122 78890 66436 7474 6773o 73570 69004 72377 70257 71162 22 30 65144 7587o 66458 74722 J 67752 73531 69025 72357 70277 71141 21 40 65166 7585I 66480 747o3 67773 73531 69046 72337 70281 71121 20 4I 65188 75832 665oi 74683 67,795 73511 69o67 72317 70319 71100 19 42 65210 75813 66523 74664 67816 73491 69o088 72297 70339 71080 18 43 65232 757,4 66545 74644 67837 73472 69109 72277 70360 71059 17 44 65254 75775 66566 74625 67859 73452 69130 72257 70381 7I039 i6 45 65276 75756 66388 74606 67880 73432 69151 72236 70401 71019 1 46 65298 75738 666o10 74586 67901 73413. 69172 72216 70422 70998 14 47 65320 66632 74567 67923 73393 69193 72196 J] 70443 7o978 3 13 48 65342 75700 66653 74548 67944 73373 69214 72176 70463 70957 12 49 65364 75080 66675 74528 67965 73353 69235 72156 70484 70937 I 50 65386 7566i 66697 74509/ g 67987 73333 69256 236 7o0505 70o916 i o 51 65408 55642 66718 7/4 489 68008 73314 69277 7721I16 70525 70896 52 65430 75623 66740/ 74470 680291 73294 69298 72095 70546 70875 53 65452 75604 66762 74451 68o51 73274 69319 72075 70567 70855 7 54 / 474 75585 66783 74431 68072 73254 0340 72055 70587 708341 6 55 65496 75566 668o5 74412 68093 73234 69361 72035 70608 70813 5 56 65518 75547 66827 74392 68i15 73215 69382 72015 70628 70793 4 57 6554o 75528 66848 74373 68i36 73195 69403 71995 70649j 70772 3 58 65562 75509 66870 74353 68157 73175 69424j 71974 70670 70752 2 59 65584 75490 66891 74334 68179 73155 6094__5 7954 70690 70731 I 6o 656o6 75471 66913 74314 68200 73135 69466 710934 70711 70711 0 Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. Cosine. Sine. 490 480 47 460 4-0 TABLE III. NATURAL TANGENTS AND COTANCTENTS. 73 i 00 10 20 80 Tangent. Cotang, Tangent. Cotalng. Tangent. Cotang. Tangeat. Cotang. 0 00000 Infinite. 01746 57 -2900 03492 28-6363 05241 I9 -o8ii 60 I 00029 3437.75 01775 56-3506 03521 28.3994 05270 18 9755 59 2 ooo58 1718.87 oI804 55-4415 o3550 28-1664 0529 188711 58 3 ooo087 145-92 oi833 54-5613 03579 27-9372 05328 18.7678 57 4 ooI06 859.436 01862 53.7086 03609 27-717 05357 x186656 56 5 o0145 687-549 01891 52.8821 o3638 27-4899 05387 I8-5645 55 6 00175 572-957 01920 52-0807 03667 27-2715 o54I6 18.4645 54 7 00204 491*106 0g949 5I13032 03696 27.o566 05445 I8.3655 53 8 00233 429.718 01978 50o5485 03725 26-8450 05474 18-2677 52 9 00262 381 971 02007 49.8157 03754 26-6367 05503 18-1708 5I 10 0029I 343O.74 02036 49.1039 03783 26-4316 o5533 18-0750 50 ii 00320 312 521 02066 48-4121 I 3812 26-2296 05562 I 7 9802 49 12 00349 286-478 02095 47-7395 03842 260o307 05591 17-8863 48 13 00378 264.441I 02124 47.o853 03871 25-8348 o56o2 17.7934 47 14 00407 245-552 02153 46-4489 03900 25-6418 0.5649 17.7015 46 15 00436 229-I82 02182 45-8294 03929 25.4517 05678 17.6106 45 i6 oo465 214-858 02211 45-226I 03958 25-2644 o5708 17.5205 44 17 00495 202-219 02240 44-6386 03987 25-0798 05737 17-4314 43 18 00524 190.984 02269 44.0661 o40o6 24-8978 05766 17-3432 42 c19 oo553 8o0-932 02298 43.5o081 04046 24-7185 05795 17-2558 41 20 00582 1718 85 02328 42.9641 04075 24-5418 05824 171693 40 21 00611 163.700 02357 42-4335 o4104 24-3675 o5854 17.0837 39 22 00640 156-259 02386 419I 58 04133 24-1957 o5883 I6-9990 38 23 oo066 149460 02415 41.4106 04162 24-0263 05912 6.9150 37 24 o006g8 43-237 02444 40-9174 04191 23-8593 05941 16.8319 36 25 00727 137-507 02473 40 4358 04220 23.-6945 05970 16-7496 35 26 oo756 I32-219 02502 39-9655 04250 23-5321 05999 16-6681 34 27 00785 127-321 02531 39.5059 04279 23.3718 06029 16 5874 33 28 oo084 I22.774 02560 39.o568 o4308 23-2137 o6o58 165075 32 29 00844 ii8.540 02589 38.6177 04337 23.o577 06087 I6-4283 3I 30 00873 114.589 02619 38-i885 o4366 22-9038 o0616 16-3499 3 31 00902 110o892 02648 37-7686 04395 22-7519 o0645 16.2722 29 32 00931 I07.426 02677 37-3579 04424 22-6020 06175 16-1952 28 33 00960 104-I71 02706 36-9560 04454 22.4541 06204 16.1190 27 34 o0989 101-107 02735 36-5627 o4483 22.3081 06233 16.o435 26 35 oIo0 8 98-2179 02764 36-1776 04512 22.1640 06262 15.9687 25 36 01047 95-4890 02793 3580oo6 04541 22-0217 06291 15-8945 24 37 01076 92-9085 02822 35.43i3 04570 2I.8813 06321 I158211 23 38 o1105 o0-4633 02851 35-0695 04599 21-7426 o6350 15.7483 22 39 oi035 88.1436 02881I 34.715 I 04628 21.6056 o6379 15.6762 21 40 o1164 85-9398 02910 34-3678 04658 21-4704 06408 15-6048 20 41 01193 83-8435 02939 34-0273 04687 21.3369 06437 15.5340 19 42 01222 81.8470 02968 33.6935 04716 21-2049 06467 15.4638 18 43 01251 779-9434 02997 33-3662 o4745 21-0747 06496 15-3943 17 44 01280 78-1263 03026 33-0452 04774 20-9460 06525'15.3254 16 45 01309 76.3900 o3055 32-7303 04803 20.-888 06554 15-2571 15 46 o1338 74.7292 03084 32.42131 o4832 20-6932 06584 1I5.893 14 47 01367 73-1390 o31i4 32-1181 04862 20-569 o66i3 15-1222 13 48 01396 71615i1 03143 31I8205 04891 20-4465 o6642 i5.0557 12 49 01425 70o.533 03I72 31.5284 01o4920 203253 06671 14-9898 II 50 o0455 68-750 0320I 312416 04949 20.2056 06700. 14-9244 10 51 oI484 67-4019 03230 3o.9599 04978 20-0872 o6730. 14-8526 9 52 oi5i3 66.io55 03259 3o.6833 05007 19-9702 06759 i4-79.34 8 53 01542 64-8580 03288 30o.4i6 05037 19-8546 o6788 14-73,1,7 7 54 0157I 63-6567 03317 30o.446 o5o66 1I97403 06817 14.6685 6 55 01600 62-4992 o3346 29-8823 05095 Ig.6273 o6847 14,60o5 5 56 01629 61-38290 03376 29-6245 o5124 951561 06876 14-5438 4 57 o1658 60-305 03405 29-3711 05i53 1-94051 06905 14.4823 3 58 oi687 5-2659 03434 29.I220 05182 19-2959 06934 14-42I2 2 59 1 01716 58-2612 03463 28-8771 05212 Ig-9879 06963 14-3607 1 60 o01746 57-2900 03492 28-6363 05241 1I9-0811 06993 14-3007 0 i Cotang.l Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 89i 88~ 8 860 L 74 NATURAL TANGENTS AND COTANGENTS. TABLE III. 40 50 60 0 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 06993 l4300oo 08749 11.430o o5io 9.51436 12278 8. 4435 60 I 07022 14-2411 08778 1139I9 10540.9-48781 12308 81248i 59 2 07051 14-1821 08807 i -354o 10569 9-46141 12338 8.10536 58 3 07080 14.1235 08837 i1.3163 10599 9o43515 12367 8.08600 57 4 07110 i4-o655 08866 11-2789 I10628 940904 12397 8.06674 56 5 07139 14oo0079 08895 112417 10657 9.38307 12426 8.0o4756 55 6 07168 1395o07 08925 11-2048 10687 9.35724 12456 8.02848 54 7 07197 13.8940 o0854 11-1681 I0716 9.33154 12485 8.00948 53 8 07227 13.8378 08983 ii —3i6 10746 930599 12515 7-99058 5-2 9 07256 13'7821 09013 1i-o9544 10775 9-28058 12544 7'97176 {I io 07285 13.7267 09042 11.0594 10805 9.25530 12574 7.95302 50 II 07314 13.6719 09071 11-.0237 10834 9.23016 12603 7.93438 49 12 07344 13.6174 09101 10.9882 10o863 9.20o516 12633 7.91582 48 13 07373 13.5634 o913o 10.9529 10893 9.18028 12662 7.89734 47 14 07402 13.5098 09159 10-.9178 10922 -9.15554 12692 7.87895 46 15 07431 13.4566 09189 10o8829 10952 9.13093 12722 7.86o64 45 i6 07461 13.4039 09218 10o.8483 / 1081 09.10646 12751 7-84242 44 17 07490 13.3515 09247 10.8139 iioi 90-o82II 12781 7.82428 43 18 07519 13-2996 09277 IO.7797 11040 9.05789 12810 7.80622 42 19 07548 13-2480 o0306 107457 110o7o 9.o3379 12840 7-78825 41 20 07578 13-1969 09335 I10o-711i 9 9 9o 00983 12869 7.77035 40 21 07607 13.1461 09365 10o6783 11128 8.98598 12899 7.75254 39 22 07636 3.o958 09394 10o645o0 11158 8.96227 12929 7.7348o 38 23 07665 13.0458 o9423 I-o.6i8 11187 8.93867 12958 7-71715 37 24 07695 12-9962 09453 i105789 11217 8.9-520 12988 7.69957 36 25 07724 129469 o09482 i105462 11246 8.89185 13017 7.68208 35 26 07753 12-8981 o0511 1o-5i36 11276 8.86862 13047 7-66466 34 27 07782 12.84Z96 0954I io.48i3 11305 8-84551 13076 7.64732 33 28 07812 12-8014 09570 10o4491 11I335 8-82252 13io6 7.63005 32 29 07841 12-7536 0600oo 10-4172 1364 8.79964 13i36 7.61287 31 30 07870 12-7062 09629 io.3854 11394 8.77?689 13165 7.59575 3:0 31 07899 12.6591 09658 io-3538 11423 8-75425 13195 7.57872 29 32 07929 12-6124 09688 10-3224 11452 8.73172 13224 7.56I76 28 33 07958 12.566o 09717 10o2913 11482 8.70o93 13254 7-54487 27 34 07987 12.5199 09746 10-2602 11511 8.68701 13284 7.52806 26 35 08017 12-4742 09776 10-2294 11541 8-66482 i33i3 7-51132 25 36 08046 12-4288 09805 10-1988 11570 8.64275 13343 7.49465 241 37 080o75 123838 0)0834 io.-683 i60oo 8.62078 13372 7-478o6 23 38 o8io4 12.3390 09864 0o.-38i 11629 8.59893 13402 7.46154 22 39 08134/ 122946 09893 jo.-o8o 11659 8.57718 13432 7-445o0 21 40 o8163 1225o05 09923 10-0780 11688 8.55555 13461 7-42871 20 41 08192 12.2067 09952 o-o483 11718 8.53402 13491 7-41240 19 42 08221 12-1632 o9981 1ioo0187 11747 8.51259 13521 7-39616 18 43 08251 1/2.1201 looi 09.98930 11777 8.49I28 1355o 7.37999 17 44 08280 12-0772 10040 9.96007 11806 8-47007 i3580 7.36389 6 45 08309 12-o346 100oo69 9.93101 836 8.44896 13609 7.34786 5 46 08339 11-9923 iooo0 9.90211 1I865 8-42795 13639 7-33190 14 47 o8368 11-9504 10128 9.87338 11895 8-40705 13669 7.316600 3 48 08397 11.9087 101o58 9.84482 11924 8.38625 13698 7.300o8 12 49 08427 11.8673 10187 9.81641 11,954 8.36555 13728 7.28442 Ii 50 08456' 11.8262 10216 9.78817 11983 8.34496 13758 7-26873 io 51 08485 11-7853 10246 9.76009 12013 8.32446 13787 7.25310 9 52 o85i4 1-.7448 10275 9.73217 12042 8.30406 3817 7.23754 8 53 o8544 11-7045 10o3o5 970441 12072 8.28376 13886 7.-2204 7 54 08573 11-6645 io334 9.67680 12101 8-26355 13876 7-20661 6 55 08602 11-6248 10363 9.64935 12131 8.24345 13906 7.19125 5 56 08632 11-5853 10393 9-62205 12160 8-22344 13935 7'17594 4 57 o866 11.5461I 10422 9.50o49o 1290 8-20352 13965 7-16071 3 58 08690 11-50o72 o10452 9.5679 12219 8-18370 13995 7-14553 2 59 08720 11.4685 10 481 09.54Io6 12249 8-16398 14024 7-13042 1 6o 08749 11 43oi Io51o 9-51436 12278 814435 14o54 7-i1537 0 Cotang. Tangent. Cotang. Tangent. Cotang. anen. Cotang. Tanent. 85~ 840 83~ 820 TABLE III. NATURAL TANGENTS AND COTANGENTS. 75 80~ 90 100 110 l Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 0 14054 7-11537 i5838 6~31375 17633 5-67128 I9438 5. 14455 60 14084 7.1oo38 I5868 6.30oi89 I7663 5.66i65 19468 5.I3658 59 2 i4i13 7 o8546 15898 6- 29007 17693 5.65205 I9498 5-12862 58 3 I4143 7-07059 I5928 6.27829 17723 5.64248 19529 5-12069 57 4 14173 7.05579 15958 6.26655 17753 5.63295 19559 5.11279 56 5 14202 7-04105 I5988 6.25486 17783 5 62344 19589 5.10490 55 6 14232 7o02637 I6017 6.24321 17813 5.6I397 19619 5.09704 54 7 I4262 7.-0174 I6047 6.23160 I7843 5.60452 19649 5.08921 53 8 14291 6.99718 16077 6.22003 17873 5.595II 19680 5.08139 52 9 1432I 6.98268 16107 6.20851 179o3 5.58573 197Io 5.07360 5i 1o I4351 6.96823 16137 6.19703 17933 5.57638 19740 5.06584 50 I 14381 6.95385 16167 6.18559 17963 5.56706 19770 5.o5809 49 12 144 10 6.93952 16196 6.17419 17993 5.55777 19801 5.o5o37 48 13 14440 6.92025 16226 6.16283 8023 5.54851 I983I 5o.4267 47 14 14470 6.9 io4 16256 6.-i5i5i 8053 5.53927 1986I 5.03499 46 15 i4499 6.89688 16286 6.14023 i8083 5.53007 I989g 5.02734 45 I6 14529 6.88278 x63i6 6.12899 I8i 3 5.52090 19921 5.01971 44 17 14559 6.86874 16346 6.11779 I8143 5.5I1176 19952 5o01210 43 18 I4588 6.85475 16376 6.io664 18173 5.50264 19982 500oo45 42 I9 1461i8 6.84082 16405 6.09552 18203 5.49356 20012 4-99695 41 20 I4648 6.82694 I6435 6.o8444 18233 5.48451 20042 4.98940 40 21.I4678 6.81312 16465 6.07340 I8263 5 47548 20073 4-98188 39 22 I4707 6.70936 16495 6.o6240 18293 5.46648 20103 4.97438 38 23 14737 6.78564 1 6525 6.0o543 18323 5.45751 20I33 4-96690 37 24 14767 6-77199 16555 6.o405i 18353 5.44857 o0164 4-95945 36 25 14796 6.75838 16585 6.o0262 I8383 5.43966 20194 4-95201 35 26 14826 6.74483 166I5 6o01878 I8414 5.43077 20224 4-94460 34 27 I4856 6.73133 16645 6.00797 18444 5.42192 20254 4-93721 33 28 I4886 6.71789 16674 5.99720 18474 5.41309 20285 4.92984 32 29 14915 6 70450 16704 5.98646 18504 5.40429 20315 4-92249 31 30, 14945 6-69116 16734 5.97576 18534 5.39552 20345 4.91516 30 31 I4975 6 67787 16764 5-96510 18564 5.38677 20376 4-90785 29 32 15005 6.66463 16794 5-95448 18594 5.37805 20406 4 90056 28 33 15034 6.65144 1 6824 5.94390 18624 5.36936 20436 4.89330 27 34 i5064 6.6383 16854 5.93335 18654 5.36070 20466 4.88605 26 35 15094 6.62523 16884 5-92283 I8684 5.35206 20497 4.87882 25 36 15124 6.61219 16914 5-91235 18714 5-34345 20527 4-87I62 24 37 5153 6.59921 I6944 5-*90191 18745 5.33487 20557 4-86444 23 38 15183 6.58627 16974 5.89151 18775 5-32631 20588 4-85727 22 39 15213 6.57339 17004 5.881I4 I8805 5.31778 20618 4.85013 21 40 I5243 6.56055 17033 5.87080 I8835 5.30928 20648 4-84300 20 41 I5272 6.54777 17063 5.86o5I I8865 5-30080 20679 4-83590 19 42 15302 6-535o3 17093 5-85024 18895 5.29235 20709 4.82882,8 43 15332 6.52234 17123 -58400o 18925 5.28393 20739 4.82175 17 44 15362 6.50970 17153 5.82982 I8955 5.27553 20770 4.81471 i6 45 15391 6.497I10 17I83 5.81966 18986 5.267i5 20800 4.80769 15 46 15421 6-48456 17213 5.80953 19016 5.25880 20830 4.80068 14 47 1545I 6.47206 1 7243 5.79944 19046 525048 20861 4.79370 i3 48 1548I 6.4596I 17273 5.78938 9076 524218 2089I 4-78673 12 49 I55II 6.44720 17303 5.77936 19106 5.23391 20921 4-77978 II 50 I5540 6.43484 17333 5.76937 19136 5.22566 20952 4-77286 io 51 1557o 6-42253 17363 5 75941 19166 5.21744 20982 4-76595 9 52 5600oo 641026 17393 5.74949 19I97 5.20925 21013 4.75906 8 53 i5630 6.39804 17423 5-73960 19227 5.20107 21043 4'15219 7 54 5660 6.38587 1 7453 5.72974 19257 5*19293 21073 4.74534 6 55 15689 6.37374 17483 5.71992 19287 5.-I8480 2110r4 4.-7385 5 56 15719 6.36I65 7513 5.71013 I9317 5.17671 2II34 4-73170 4 57 15749 6.3496I 17543 5.70037 19347 5.16863 21164 4-72490 3 58 15779 6.33761 17573 5.69064 19378 5-i6058 21193 4.71813 2 59 i58o9 6.32566 I7603 5.68094 19408 5.15256 21225 4-71137 1 60 15838 6.31375 17633 5-.6718 19438 5.14455 2I256 4-70463 o Cotang. Tangent. Cotang. Tangent. Cotan g. Tangent. Cotang. Tangent. 810 800~ 90 78 76 NATURAL TANGENTS AND COTANGENTS. TABLE III. 120 130 14~ 150'Tangent. Cotang. Tangent.j Cotang. Tangent. Cotang. Tangent. Cotang. o 21256 4.70463 23087 4-33i48 24933 4-01078 26795 37320o5 6o i 21286 4,69791 23117 4.32573 24964 4.00582 26826 3-7277I 59 2 21316 4.69121 23i48 4.32001 24995 4.ooo86 26857 3-72338 58 3 21347 4-68452 23I79 4.3430o 25026 3.99592 26888 3-71907 57 4 21377 4-67786 23209 4.30860 25056 3.099o099 26920 3-71476 56 5 21408 4-67121 23240 4-30291 25087 3.98607 26951 3710o46 55 6 21438 4.66458 23271 4.29724 25118 3-98i17 26982 3.70o616 54 7 21469 4.65797 23301 4'29159 25149 3-97627 27013 3.70188 53 8 21499 4-65138 23332 4-28595 2518o 3-97139 27044 3-69761 52 9 21529 4.64480 23363 4.28032 25211 3.96651 27076 3-69335 5i 10 21560 4.63825 23393 4-'27471 25242 3-96165 27107 3-68909 50 II 21590 4.63171 23424 4-26911 25273 3 95680 27138 3-68485 4 12 21621 4-62518 23455 4-26352 25304 3-95196 27169 3-68o61 48 13 21651 4.61868 23485 4-25795 25335 3.94713 27201 3-67638 47 15 21712 4.60572 23547 4-24682 25397 3 -93751 27263 3-66796 45 16 21743 4.59927 23578 4.24132 25428 3 93271 27294 3.66376 44 17 21773 4.59283 23608 4-23580 25459 3 92793 27326 3-65957 43 18 28o04 4.58641 23639 4-23030 25490 3-92316 27357 3-65538 42 19 21834 458oo00 23670 4-'22481 25521 3-91839 27388 3-65121 41 20 21864 4-57363 23700 4.21933 25552 3- 9364 27419 3-64705 40 21 2189.5 4.56726 23731 4.21387 25583 3 90890 27451 3.64289 39 22 21925 4.56o91 23762 4-20842 25614 3 -904 17 27482 3.63874 38 23 21956 4-55458 23793 4-20298 25645 3.89945 27513 3-63461 37 24 21986 4.54826 23823 4.19756 25676 3-89474 27545 3-63048 36 25 22017 4.54196 23854 4'192i5 257o7 38900oo4 27576 362636 35 26 22047 4-53568 23885 4'18675 25738 3.88536 27607 3-62224 34 27 22078 4.52941 23916 4'18137 25769 3'88o68 27638 3.618i4 33 26 22108 4-52316 23946 4-176oo00 25800 3-87601 27670 3-61405 32 29 22139 4.51693 23977 4-.7064 25831 3-87136 27701 3-60996 31 30 22169 4-51071 24008 4i.653o 25862 3.86671 27732 3-6o588 30 31 22200 4.50451 24039 4-15997 25893 3-86208 27764 3-6oi8i 29 32 22231 4-49832 24069 4-15465 25924 3.85745 27795 3-59775 28 33 22261 4.49215 24100 4-14934 25955 3-85284 27826 3-59370 27 34 22292 4.48600 24131 4-i44o5 25986 3-84824 27858 3-58966 26 35 22322 4.47986 24162 4-13877 26017 3.84364 27889 3-58562 25 36 22353 4.-47374 24193 4-I3350 26048 3.83906 27920 3-58i6o 24 37 22383 4-46764 24223 4-12825 26079 3-83449 27952 3-57758 23 38 22414 4-46155 24254 4.-12301 26110 3-82992 27983 3.57357 22 39 22444 4.45548 24285 4.II778 26141 3-82537 28015 3-569571 2 40 22475 4.44942 24316 4-11256 26172 3-82083 28046 3-56557 20 41 22505 4-44338 24347 4o10736 26203 3.8163o 28077 3-5615J9 9 42 22536 4.43735 24377 4-10216 26235 3.81177 28o109 355761 18 43 22567 4-43134 24408 4.09699 26266 3.80726 28140 3-55364 I7 44 22597 4.42534 24439 4-09182 26297 3-80276 28172 3-54968 i6 45 22628 4.41936 294470 4-08666 26328 3-79827 28203 3-54573 15 46 22658 4-4i340 24501 4.08152 26359 3-79378 28234 3-54179'4 47 22689 4-40745 24532 4-07639 26390o 3.78931 28266 3-53785 13 48 22719 4j401oi52 24562 4-07127 26421 3.78485 28297 3-53393 12 49 22750 4.39560 24593 4-o66i6 26452 3.78040 28329 353ooi00 ii 50 22781 4-38969 24624 4.06107 26483 3.77595 2836o 3-5260o io 51 22811 4.38381 24655 4-05599 26515 3-77152 2839i 3.52219 9 52 22842 4-37793 24686 405o092 26546 3.76709 28423 3-51829 8 53 22872 4.'37207 24717 4-04586 26577 3.76268 28454 3-5144i 7 54 22903 4.36623 24747 4-04081 26608 3-75828 28486 3-5io53 6 55 22934 4.36o40 24778 4-o3578 26639 3.75388 28517 3-5o666 5 56 22964 4.35459 24809 4.03075 26670 3-74950 28549 3-50279 4 57 22995 4.34879 24840 4-02574 26701 3-74512 28580 3-49894 3 58 23026 4.34300 24871 4,02074 26733 3.74075 28612 3-49509 2 59 23o56 4-33723 24902 4-01576 26764 3.73640 28643 3.49125 i 6o 23087 4.33148 24933 4oo1078 26795 373205 28675 3-48741 0 Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. I W'~ "1 5 J'' TABLE III. NATURAL TANGENTS AND COTANGENTS. 7 I I0~ 11 ^ 1_ 18~ __ 19~ I Tanlgent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 28675 3.48741 30573 3 27085 32492 3 07768 34433 2 90421 60 i 28706 3.48359 30605 3 26745 32524 3-07464 34465 2 -o0147 5 2 28738 3 47977 30637 3 26406 32556 3.07I60 34498 2 89873 58 3 28769 3.47596 30669 3 26067 32588 3.o6857 34530 2 89600 57 4 28800 3 47216 30700 3 25729 32621 3 06554 34563 2 89327 56 5 28832 3-46837 30732 3 25392 32653 3 o6252 34596 2 8955 55 6 28864 3.46458 30764 3 25055 32685 3 05950 34628 2.88783 54 7 28895 3-46080 30796 3.24719 32717 3 -5649 34661 2.88511 53 8 28927 3.45703 30828 3.24383 32749 3.o05349 34693 2.88240 52 9 28958 3-45327 30860 3 24049 32782 3.o5049 34726 2'87970 5i 10 28990 3-44951 30891 3 23714 32814 3 0o4749 34758 2.87700 50 I 2902 3-44576 30923 3-23381 32846 3 -o4450 34791 2.87430 49 12 20053 3.44202 30955 3.23048 32878 3 -o4152 34824 2 87 61 48 x3 29084 3 -43829 30987 3.22715 32911 3. o3854 34856 2 86892 47 I4 29116 3.43456 31019 3.22384 32943 3 o3556 34889 2.-86624 46 15 29I47 3.43084 31o05 3.22053 32975 3.03260 34922 2 86356 45 i6 29179 3-427i3 3o183 3 21722 33007 3o02g63 34954 2-86089 44 17 29210 3 42343 3 I 5 3-21392 3304o 3o02667 34987 2 85822 43 18 29242 3 41973 31I47 3.2o163 33072 3 -02372 350I9 2.-85555 42 19 29274 3.41604 3414 78 3 20734 3304 302077 35052 2 285289 41 20 29305 3 41236 312io 3-20406 33I36 3 01783 35085 2.85023 40 21 29337 3.40869 31242 3. 20079 33169 3 oi489 35117 2.84758 39 22 29368 3.-40502 31274 3.19752 3320I 3 0o1196 35150 2.84494 38 23 29400 3.40136 3i306 3-19426 33233 3.00ooo3 35 83 2.84229 37 24 29432 3.39771 31338 3.190oo 33266 3-oo006 35216 2.83965 36 25 29463 3.394o6 31370 3.18775 33298 3o00319i 35248 2.83702 35 26 29495 3.39042 31402 3.-I845i 33330 3-00028 35281 2 83439 34 27 29526 3.38679 31434 3 18127 33363 2'99738 35314 2.83176 33 28 29558 3 38317 31466 3. 17804 33395 2 99447 35346 2 -82914 32 29 29590 3 37953 31498 3.1748i 33427 2-99158 35379 2.82653 31 30 29621 3.37594 31530 3.17159 33460 2 98868 35412 2.82391 30 3I 29653 3.37234 31562 3.-6838 33492 2 9858o 35445 2.82130 29 32 29685 3.36875 31594 3.i65i7 33524 2 98292 35477 2'81870 28 33 29716 3.365I6 31626 3. 6197 33557 2-98004 35510 2.8I61o 27 34 29748 3.36158 3i658 3.15877 33589 2'97717 35543 2.81350 26 35 29780 3.358oo 31690 3.i5558 33621 2 97430 35576 2 -8191O 25 36 29811 3 35443 31722 3.15240 33654 2-97I44 35608 2.80833 24 37 29843 3 35087 31754 3. I4922 33686 2 96858 35641 2 80574 23 38 29875 3 34732 31786 3 - I46o5 33718 2 96573 35674 2 803 16 22 39 29906 3.34377 31818 31I4288 33751 2 96288 35707 2 - 80059 2 40 29938 3 34023 3850 3-.13972 33783 2 96004 35740 2'79802 20 41 29970 3'33670 31882 3-13656 33816 2-95721 35772 2' 79545 19 42 300o0 3.33.3I 31914 3.13341 33848 2G95437 35805 2 79289 18 43 30033 3 3296 31946 3.I3027 33881 2-95155 35838 2 79033 17 44 30065 3 32614 31978' 3.12713 33913 2 94872 35871 2 78778 16 45 30097 3.32264 3201o 3.12400 33945 2 94590 35904 2 78523 15 46 30I28 3.31914 32042 3.12087 33978 2.94309 35937 2-78269 I4 47 3oi6o 3.31565 32074 3.11775 34010 2-94028 35969 2-78014 3 48 30192 3-31216 32106 3.1I464 34043 2.93748 36002 2-7776i I2 49 30224 3.30868 32139 3-.II53 340o7 2-93468 36035 2.77507 I1 50 30255 330o52I 32171 3-Io842 334108 2.93189 36068 2-77254 Io 5i 30287 3.3074 32203 3-I0532 34I40 2.92910 36o10 277002 9 52 30319 3.29829 32235 3-I0223 34173 2-92632 36134 2.76750 8 53 3035i 3-29483 32267 3-09914 34205 2-92354 36167 2.76498 7 54 30382 3.29139 32299 3.0o9606 34238 2.92076 36I99 2.76247 6 55 304i4 3.28795 32331 3-09298 34270 2-91799 36232 2.75996 5 56 30446 3.28452 32363 3.08991 34303 2.91523 36265 2.75746 4 57 30478 -3-2819o 32396 3.oS8685 34.335 291246 36298 2.75496 3 5 30509 3-27767 32428 3.08379 34368 2'90971 3633 2-75246 a 59 30541 3-27426 32460 3o08073 34400 290696| 36364 2-74997 i 50 30573 3-27085 32492 3-o7768 34433 2.9042I 36397 2-74748 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. 730 72~ 71! oo 78 NATURAL TANGENTS AND COTANGENTS. TABLE 1II. 200 __ 210 1 220 230 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 36397 2a74748 38386 2960509 40403 2-47509 42447 2 35585 60 i 36430 2-774499 38420 2.60283 40436 2.47302 42482 2 35395 59 2 36463 2 47425I 38453 2.60057 40470 2 47095 425I6 2335205 58 3 36496 2-74004 38487 2-59831 40504 2 46888 42551 2-35015 57 4 36529 2.73756 38520 2.59606 40538 2 46682 42585 2 34825 56 5 36562 2.73509 38553 2.59381 40572 2-46476 42619 2.34636 55 6 36595 2.73263 38587 2.59156 40606 2-46270 42654 2.34447 54 7 36628 2.73017 38620 2-58932 40640 2.46065 42688 2.34258 53 8 36661 2-7277i 38654 2-58708 40674 2-45860 42722 2o34069 52 9 36694 2-72526 38687 2-58484 40707 2.45655 42757 2-33881 51 Io 36727 2-7228I 38721 258261 40741 2.4545 42791 2-33693 50 I 36760 2~72o36 38754 2.58038 40775 2.45246 42826 2.33505 49 12 36793 2-71792 38787 2.57815 40809 2.45043 42860 2.33317 48 13 36826 2.71548 38821 2.57593 40843 2.44839 42894 2-33I30 47 14 36859 2.71305 38854 2-57371 40877 2-44636 42929 2.32943 46 i5 36892 2-71062 38888 2-57 50 40911 2-44433 42963 2.32756 45 I6 36925 2-70819 38921 2.56928 40945 2-44230 42998 2.32570 44 17 36958 2.70577 38955 2.56707 40979 2-44027 43032 2.32383 43 I8 36991I 2-70335 38988 2.56487 41io3 2-43825 43067 2.32I97 42 19 37024 2-70094 39022 2-56266 41047 2-43623 43o10 2-32012 41 20 37057 2.69853 39055 2.56046 41081 2-43422 43136 2.31826 40 21 37090 2.69612 39089 2.55827 4II15 2-43220 43170 2.31641 39 22 37124 2-6937I 39122 2.55608 41149 2.430I9. -3205 2.314.56 38 23 37157 2.6913i 39I56 2.55389 4ii83 2-42819 43239 2-3127I 37 24 37190 2.68892 39190 2.55i70 41217 2.42618 43274 2.3o086 36 25 37223 2.68653 39223 2.54952 41252 2-42418 43308 2.30902 35 26 37256 2.68414] 39257 254734 41285 2-42218 43343 2.30718 34 27 37289 2.68175 39290 2.54516 41319 2-42019 43378 2.30534 33 28 37322 2.67937 39324 2-54299 41353 2-41819 43412 2.3035I 32 29 37355 2-67700 39357 2.54082 41387 2-41620 43447 2-30I67 31 30 37388 2'67462 39391 2.53865 41421 2-41421 43481 2-29984 30 3r 37422 2.67225 39425 2.53648 41455 2-41223 4351 6 2.29801 29 32 37455 2.66989 39458 2.53432 41490 2-41025 43550 2.29619 28 33 37488 2-66752 39492 2-53217 41524 2-40827 43585 2-29437 27 34 37521 2.665I6 39526 2.53001 41558 2.40629 43620 2-29254 26 35 37554 2.66281 39559 2.52786 41592 2-40432 43654 2-29073 25 36 37588 2-66046 39593 2 52571 41626 2.40235 43689 2-28891 24 37 37621 2.65811 39626 2-52357 41660 2-40038 43724 2-28710 23 38 37654 2.65576 39660 2.52142 4I694 2.39841 43758 2-28528 22 39 37687 2.65342 39694 2.5I929 41728 2.39645 43793 2.28348 2 40 37720 2.65I09 39727 2.51715 41763 2.39449 43828 2.28167 20 4I 37754 2.64875 39761 2-5I502 4I797 2.39253 43862 2-27987 19 42 37787 2.64642 39795 2-51289 41831 239058 43897 2-27806 I8 43 37820 2.64410 39829 2.5I076 41865 2.38862 43932 2-27626 17 44 37853 2.64177 39862 2.50864 41899 2.38668 43966 2-27447 16 45 37887 2.635 3945 39896 2-50652 41933 2-38473 4400I 2 -27267 i 46 37920 2.63714 39930 2-50440 41968 2.38279 44036 2-27088 14 47 37953 2.63483 39963 2.50229 42002 2.38084 44071 2-26909 I3 48 37986 2.63252 39997 2.50018 42036 2.37891 44I05 2-26730 12 49 38020 2.63021 4003I 2-49807 42070 2-37697 44140 2-26552 11 50 38053 2.62791 40065 2.49597 42105 2-37504 44175 2.26374 io 51 38086 2-62561 40098 2.49386 42139 2.37311 44210 2.26196 9 52 38120 2.62332 40I32 2-49177 42173 2.37118 44244 2-26018 8 53 38i53 2.62103 40I66 2.48967 42207 2.36925 44279 2-25840 7 54 38i86 2.61874 40200 2.48758 42242 2.36733 44314 2.25663 6 55 38220 2-61646 40234 2.-48549 42276 2.36541 44349 2.25486 5 56 38253 2.6I4I8 40267 2-48340 42310 2.36349 44384 2-25309 4 57 38286 2.61190 40301 2.48132 42345 2.36158 44418 2-25I32 3 58 38320 2.60963 40335 2-47924 42379 2.35967 44453 2.24956 2 59 38353 2.60736 40369 2.47726 42413 2.35776 444488 2.24780 i 60 38386 2.6o5o 9 404o3 2-47509 42447 2.35585 44523 2-24604 O Cotang. Tangnt. an. Taen t. Ctan anent. Coatan. aangent Tan t. 690_________ _ 6 6__ 66_ TABLE IIL NATURAL TANGENTS AND COTANGENTS. 9 240 250 260 2~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. 44523 2.24604 46631 2 14451 48773 2 05330 50953 1-96261 6o I 44558 2.24428'46666 2-14288 48809 2~04879 50989 1.96120 59 2 44593 2 24252 46702 214125 48843 2o04728 50o26 1.9579 58 3 44627 2 24077 46737 12. 3963 48881 2 o4577 5i063 95838 57 4 44662 2 -23902 46772 2 13801 48917 2.04426 51099 1 95698 56 5 44697 2-23727 46808 2.13639 48953 2.o4276 51136 I 95557 55 6 447322 2 23553 46843 2 13477 48989 2 04125 51173 I-95417 54 7 44767 2 23378 46879 2-133 16 49026 2-o3975 51209 I 95277 53 8 44802 2 23204 46914 2 3 54 4906 2 20 3825 5246 -9537 52 9 44837 2 23030 46950 2 12993 49098 2-03675 51283 I 94997 5S Io 44872 2.22857 46985 2 I2832 49134 2.03526 51319 -94858 50 I 44907 2 22683 47021 2 12671- 49170 2-03376 5i356 1 94718 4 12 44942 2 22510 47056 2-12511 49206 2-03227 51393 -94579 48 13 44977 2.22337 47092 2.12350 49242 2-03078 5I430 I 94440 47 14 45012 2 22164 47128 2 12190 49278 2-.02929 51467 -94301 46 15 45047 2 21992 47163 2'12o3o 493I5 2 02780 51503 I 94162 45 i6 45082 2 21819 47199 2-11871 4935I 2 31 263i51540 194023 44 17 45117 2 21647 47234 2 11711 49387 2 02483 51577 1 93885 43 18 45152 221I475 47270 2 I 552 49423 2 02335 51614 1.93746 42 19 45187 2 21 304 47305 2-11392 49459 2o02187 5165 i 193608 41 20 45222 22132 4734 2 1233 49495 2 020&j 51688 193470 40 21 45257 2.20961 47377 2 -11075 49532 2 01891 51724 1.93332 39 22 45292 2 -20790 47412 2 o0916 49568 201743 51761 1 93195 3 23 45327 2-20619 47448 2 10758 49604 2 01596 51798 I 93057 37 24 45362 2-20449 47483 2 -0600 49640 2 01449 51835 1-92920 36 25 45397 2 -20278 47519 2 10o442 49677 2o01302 51872 1 92782 35 26 45432 2-20IO8 47555 2-10284 4973 2-o0155 51909 1-92645 34 27 45467 2-19938 47590 2-10126 49749 2ooo0 8 51946 1-92508 33 28 45502 2.19769 47626 2-0996 49786 2o00862 51983 1.92371 32 29 45537 2.19599 47662 2-I0981 49822 2-007I5 52020 I-92235 31 30 45573 2g19430 47698 2-09654 49858 2-oo00569 52057 1-92o98 30 3i 45608 2-19261 47733 2-09498 49894 2-00423 52094 1.91962 29 32 45643 2-19092 47769 2-09341 49931 2-00277 52131 1-91826 28 33 45678 2-18923 47805 2-09184 49967 2-o0031 52i68 1-91690 27 34 45713 2- 8755 47840 209028 500o04 1.99986 52205 1 g9554 26 35 45748 2-I8587 47876 2-08872 50040 1-99841 52242 Ig1418 25 36 45784 2-18419 4 47912 2-08716 50076 1g99695 52279 1-91282 24 37 45819 2-I8251 47948 2-08560 50o13 1-99550 52316 191147 23 38 45854 2-I8084 47984 208405 50149 1I99406 52353 I 91012 22 39 45889 2-17916 48019 2-08250 50185 -9926I 52390 I~90876 21 40 45924 2-17749 48055 2o8094 50222 19gg916 52427 I-90741 20 4 45960 2-I75'82 480 91 2o079349 50258 1-98972 524 64 1 90607 42 45995 2-17416 48127 2-0778 50295 1I98828 5250i 1.9047 1 43 46030 2-17249 48163 2-07630 50331 1 98684 52538 1o90337 17 44 46065 2-17083 48198 2-07476 50368 1-98540 52575 190203 16 45 46o0 2-16917 448234 2-07321 50404 1-98396 52613 oo90069 15 46 46136 2-16751 48270 2-07167 5044i 1-98253 52650 1-89935 14 47 46171 2.16585 48306 2-07014 50477 1 -9810 52687 1-8980I 13 48 46206 2-I6420 48342 2-o6860 5054 1-97966 52724 1-89667 12 49 46242 26255 48378 2.o6706 50550 197823 52761 I-89533 II 50 46277 2.-6090g 48414 2-o6553 50587 1i97680 52798 189400 10 51 463I2 2-15925 48450 2-o6400 50623 1-97538 52836 -89266 9 52 46348 2-15760 48486 2-o6247 50660 I-97395 52873 I89133 8 53 46383 2-15596 48521 2.06094 50696 1 97203 52910 1-89000 7 54 46418 2-15432 48557 2-05942 50733 1-97111 52947 1-88867 6 55 46454 2-I5268 48593 2-05790 50769 1I96969 52984 1.88734 5 56 46489 2-15104 48629 20-5637 50806 -96827 53022 188602 4 57 46525 2-14940 48665 054855 50843 196685 53059 i-88469 3 58 46560 2-I4777 48701 2-o5333 50879 1 96544 3 53096 -88337 2 59 46595 2.14614 48737 2-05I82 50916.-96402 53134 -88205 1 60 46631 214451 48773 2-o5030 50953 1-96261 53171I i88073 o Cotang. 4 Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. J Tan get. 1 ________ __ __ __ 62/ 65~ 64~ 63~ 62~ 80 NATURAL TANGENTS AND COTANGENTS. TABT,I IIL f 280 290 390 31''Tangent. Cotang. ent.I Cotang. Tangent. Cotang. Tangent. Cotang. o 53171 1.88o73 5543I 1-80405 57735 1o73205 60086 i.66428 6o 1 53208 1.8794i 55469 1-80281 57774'73089 6o0126 i.663i8 59.2 53246 i.87809 55507 1So8058 57813 -.72973 6o0165.66209 58 3 53283 1.87677 55545.800oo34 57851 1.72857 60205 1.66o99 57 4 53320 1.87546 55583 1.7991I 57890'.7274i 60245 11.6599o 56 5 53358:.87415 55621 1.79788 57929 1 72625 60284 1.65881 55 6 53395 I.87283 55659 1.79665 57968'172509 60324.65772 54 7 53432 1.87152 55697 1I79542 58007 1.72393 60364 i.65663 53 8 53470. 87021 55736 1.79419 580o46 72278 60403 I.65554 52 9 53507 1.86891 55774 1-79296 58o85 1.72163 60443 i.65445 51 10o 53545 1.86760 55812 179174 58124 1.72047 60483 i.65337 5S ii 53582 1.86630 5585o 1.790o5 58162 1.7i932 60522 I.65228 49 12 53620 1.86499 55888 1.78929 S82201.71817 60562 I165120 48 13 53657 1.86369 55926 i 78807 58240:.71702 60602.65011 47 14 53694 1.86239 55964 1.78685 58279 1.71588 60642:.64903 46 i5 53732.86o109 56oo3 1.78563 583i8 171473 60681i'64795 45 i6 53769 1-85979 56041 1.7844,1 58357 1-7I358 60721 1.64687 44 17 53807 i.8585o 56079 1.78319 58396 1.71244 60761.'64579 43 18 53844 1.85720 56117 1-78198 58435 1.71129 6o8o1 1.64471 42 19 53882 1.85591 56i56 1.78077 58474 1.71015 60841 1.64363 41 20 53920 1.85462 56194 1.77955 585i3.70901o 60881i.64256 40 21 53957 1.85333 56232 1.77834/ 58552 1.70787 60921 i.64i48 39 22 53995 1852o04 56270 1.77713 58591 I 70673 60~960.64041 38 23 54032 1.85075 56309 1.77592 58631 7056o 61000ooo.63934 37 24 54070 1.84946 56347 1/77471 58670 1.70446 61040 163826 36 25 54107 1.84818 56385 1.77351 587o0 1.70332 6io8o 1.63719 35 26 54i45 1.84689 56424 1.77230 58748 1.70219 61120 i.63612 34 27 54i83 1.84561 56462 1.77110 58787 1-70o106 6ii6o.635o5 33 28 54220 1.84433 565oo i:76990 58826 i.69992 61200 1.63398 32 29 54.258 1.84305 56539 1 76869 58865 1 69879 61240 1.63292 31 30 52296 1.84177 56577 1-76749 58904 1.69766 61280 i'6385 30 31 54333 1.84o49 566i6 1.76630 58944 1.69653 61320 1.63079 29 32 54371 1.83922 56654 1.7651o 58983 1.69541 61360 1.62972 28 33 54409 1.83794 56693 i'7639o 59022 1.69428 614oo 1.62866 27 34 54446 i.83667 56731 1.76271 59061 1I69316 61440 1.62760 26 35 54484'.8354o 56769 1.76151 59101 1.692o3 61480 1.62654 25 36 54522 I.83413 56808 1.76032 59140 169o091 61520 1.62548 24 37 54560 1.83286 56846 1.75913 59179 1-68979 61561 1.62442 23 38 54597 i.83159 56885 1.75794 59218 i.68866 6i6o0i.62336 22 39 54635 i.83o33 56923 1.75675 59258 1.68754 6 641 1.62230 21 40 54673 1.82906 56962 1.75556 59297 1.68643 6i68i 1.62125 20 41 54-7i 1.82780 57000 1.75437 59336 1-68531 6I721 i.62019 19 42 54748 1.82654 57o39 1.75319 59376 1.68419 61761.61914 8 43 54786 1.82528 57078 1.75200 59415 1.683o8 6i8oi i.6i8o8 17 44 54824 1.82402 57116 1-75082 59454 1.68i 6 61842.61703 16 45 54862 1.82276 57i55 1.74964 59494 i.68so5 61882.61598 15 46 54900 1.82150 57193 1.74846 59533 1.67974 61922 1.61493 14 47 54938 1.82025 57232 1.74728 59573 1.67863 61962 1.6i388 13 48 54975 1.81899 57271 1.74610 59612 1.67752 62003 1.61283 12 49 55oi3 1.81774 573o9 I'74492 59651 1.67641 62043 1.61179 ii 50 55051 1-81649 57348.74375 59691 1.67530 62083 1.61074 io 51 55089 1-81524 57386 1-74257 59730 1.67419 62124 1.60970 9 52 55127 1.81399 57425 1.74140 59770 1.67309 62164 i.60865 8 53 55165 -I81274 57464.74022 59809 1.67198 62264 1.60761 7 54 55203 1.8ii5o 57503 1.73905 59849 1.67o88 62245,.60657 6 55 55241 1.81025 57541.73788 59888 1.66978 62285.60o553 5 56 55279 1-80901 57580 1.73671 59928 1.66867 62325.60o449 4 57 55317 1.80777 57619 1.73555 59967 1.66757 62366 1.6o345 3 58 55355 1.80653 57657 1.73438 60007 1.66647 62406 1.60241 2 59 55393 1.80529 57606 1.73321 6oo46 i.66538 62446.60137 1 60 55431 i.8o4o5 57735 1.73205 6oo86 1.66428 62487 i-60033 o Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang'angent. 610 600 l 90 580 TABLE III. NATURAL TANGENTS AND COTANGENTS. 81 / 30 330 34~ 35~ Tangent. Cotang. Tangent. Cotang. Taent. Cotang. Target. Cota. 0 62487.6oo33 64941 1I53986 6745I 1-48256 7C021 1 42815 60 I 62527 I 59930 64982 1-53888 67493 1-48163 70064 1-42726 59 2 62568 I.59826 65023.53791 67536 1.48070 70107 1-42(38 58 3 62608 I 59723 65065 I 53693 67578 1.47977 7o0151.42550 57 4 62649 1.59620 651o6 I.53595 67620 1-47885 70194 1-42462 56 5 62689 1.59517 65148.53497 1 67663 1.47792 70238 1.42374 55 6 62730 I.59414 65189 -534u00 67705 147699 70281 1.42286 54 7 62770 1.59311 65231 I-5330;2 67748 1-47607 70325 I 42198 53 8 62811. I 59208 65272 - 53205 67790 o47514 70368 1.421 1 52 9 62852 i 59105 65314 1. 53o07 67832 1 47422 70412 I-42022 51 X0 62892 1.59002 65355 I.530Io 67875 1.47330 70455 1.41934 50 11 62933 1i58900 65397 1-52913 67917 1.47238 70499 1.41847 49 12 62973 1 58797 654.38 1.52816 67960 1.47146 70542 1.41759 48 13 63o04 I.58695 65480 1.52719 68002 1.47053 70586 1.41672 47 I4 63055 1.58593 65521 I1.52622 68045 1.46962 70629 1.4I584 46 15 63095.o584o9 65563 1 525255 68o88 1.46870 70673 1 41497 45 i6 63136.58388 65604 I 524291 68I30 1.46778 70717 1 41409 44 17 63177 1.58286 65646 152332 68173 1.46686 7o760 1-41322 43 18 63217 1.58I84 65688 1.52235 68215 1-46595 70804 1.41235 42 19 63258.58083 65729 I 52139 68258 1.46503 70848 1-41148 41 20 63299 157981 65771 I 52043 6830I: 46411 70891 14106I 40 21 63340 1.57879 658i3 I.51946 68343 1.46320 70935 I 40974 39 22 63380 1.577778 65854 I.51850 68386 1.46229 70979 140887 38 23 6342I 1.57676 65896 151754 68429 I 46137 71023 I.40800 37 24 63462 1.57575 65938 I 51658 68471 I.46046 71o66 1 40714 36 25 63503 I1.57474 65980 1.5562 68514.45955 71110 I 40627 35 26 63544 1.57372 66021 I.51466 98557 1.45864 71I54 1.40540 34 27 63584 1.57271 66063 1.51370 686o00 145773 71198 1 40454 33 28 63625 I.57170o 66o5 1-51275 68642 1.45682 71242 1.40367 32 29 63666 1.57069.66147 1.51179 68685.45592 71285.40281 31 30 63707 I-56969| 66189 -.5Io84/ 68728 1 45501 71329 -40195 30 31 63748 I.56868 66230 1.5o988 68771 I.45410 71373 1I40109 29 32 63789 I 56767 66272 i.50893 68814 1.45320 71417 1 40022 28 33 63830.56667 66314 i50797 68857 1. 45229 71461.39936 27 34 63871 I.56566 66356 I-50702 68900 1-45139 7105 i 39850 26 35 63912 i 56466 66398 I 50607 68942. 45o49 71549 1I39764 25 36 63953 i.56366 66440. I50512 68985 1.44058 71593 1.39679 24 37 63994 1.56265 66482 I-50417 69028 1.44868 71637 1.39593 23 38 64035 I.56i65 66524 I.50322 69071 1.44/778 7168I I.39507 22 39 64076 1.56065 66566 -50228 69114 1-44688 71725 -39421 21 40 64117 i.55966 66608 I.50133 69157 1.44598 71769.39336 20 4 64158.55866 66650 i.5oo38 69200 1.44508 71813 1.39250 19 42 64199 / -55766 66692.-49944 69243 I144418 71857 1.39I65 18 43 64240 I.55666 66734.49849 69286 1.-44329 71901 1.39079 17 44/ 64281 1.55567 66776 1.49755 69329 1.44239 71946.38994 16 45 64322 1.55467 668I8.49661 69372 1.44I49 71990 1.38909 i5 46 64363 i.55368 66860 1.49566 69416 I 44060 72034 1.38824/ I4 47 64404 1.55269 66902 1. 49472 69459 1-43970 72078 1.38738 13 48 64446 I-.55170 66944 1.49378 69502 143881 72122 1.38653 12 49 64487 i. 55071 66986 j 149284 69545 I'.43792 72166 1.38568 ii 50 64528 1-54972 67028 1.49190 69588 1.43703 72211 1. 38484 o 51 64569. 54873 67071 1.49o97 69631 1.43614 72255 1.38399 52 646o10 154774 67113 1.4900o3 69675.43525 72299 I-38314 53 64652 1-54675 67155 1-48909 69718 I-43436 72344 1.38229 7 54 64693 1.54576 67197 1-488i6 69761 143347 72388.138145 6 55 64734.54478 67239 1.48722 69804 1.43258 72432 I-38o6o 5 56 64775 I.54379 67282 1.48629 69847 1.43169 72477.37976 4 57 64817 1.54281 67324 1.48536 69891 1-43080 72521 I137891 3 58 64858 I.54i83 67366 I1.48442 69934 1.42992 72565 -.37807 2 59 64899 1 54085 67409 I.48349 69977 I42903 72610 1.37722 1 60 64941 1-53986 67451 I-48256 70021 1.42815 72654 1-37638 o Cotag.' Tangent. Cotang. Tangent. Ccang. Tangent. Cotang. Tangent. 51 0 L 56 550 540 1 82 NATURAL TANGENTS AND COTANGENTS. TABLE IIL 1 360 ____3_____1 88~ ____ ~ 39~ Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotang. o 72654 1.37638 75355!.327o4^ 78129 1.27994 80978.23490o 60 i 72699 1.37554 75401 -.32624 /8175 1 -27917 81027 1-23416 59 2 72743 1.37470o 75447 I.32544 78222 1-27841 81075.23343 58 3 72788 1.37386- 75492 I.32464 78269 1'o27764 81123 1.23270 57 4 72832 1.37302 75538 1.32384 78316 1.27688 81171 123196 56 5 72877 1.37218 75584 1.32304 78363 1.27611 81220 1.23123 55 6 72921 1.37134 75629'.32224 78410 1.27535 81268 1.230o5o 54 7 72966.37050 75675.32144 78457 127458 8136 122977 53 8 73010 i.36967 75721 1.32064 78504'I27382.8364 122904 52 9 73055 x.36883 75767 1.31984 78551 I.27306 8i413 1:22831 51 o 73oo i.368oo00 75812 1.31904 78598 1.27230 81461'.22758 50 Ii 73144 1.36716 75858 11.31825 78645 1i27153 8i5o.22685 49 12 73189 x'36633 75904 1.31745 78692 1.27077 8i558 I-22612 48 13 73234'.36549 759o0 i31i666 78739 1.27001 8i6o6 122539 47 14 73278 i.36466 75996 i.3i586 78786 1.26925 81655 1-22467 46 15 73323 i.36383 76042 1.31507 78834 1.26849 81703 I-22394 45 i6 73368 i.36300 76088 1.31427 78881 1.26774 81752 1-22321 44 17 73413 1.36217 76134 r.31348 78928 1-26698 81800 1-22249 43 iS 73457 i.36i33. 76180 1.31269 78975 1.26622 81849 1.22176 42 19 73502 i.36o5I 76226 1.31190 79022 1.26546 81898 1-22104 41 20 73547 i.35968 76272 1.31i o 79070 1.26471 81946 1.22o03 40 21 73592 i.35885 76318 i.3io3i 79117 1.26395 81995 1-21959 39 22 73637 1.35802 76364 1.30952 79164 1.26319 82044 1.21886 38 23 73681.135719 76410 1.30873 79212 1.26244 82092 1.21814 37 24 73.726 1.35637 76456 1.30795 79259 1-26169 82141 1.21742 36 25 73771 i.35554 76502 1.30716 79306 1.26o93 82100 1.21670 35 26 73816 1.35472 76548 1.30637 79354 1.26018 82238 1.21598 34 27 73861 i.35389 76594 i.30558 79401 1.25943 82287 1-21526 33 28 73906 1.35307 76640 1.30480 79449 1-25867 82336 1.21454 3. 29 73951 1.35224 76686 i 3040o 79496.-25792 82385 1-21382 31 30 73996 1-35142 76733 I130323 79544 1.25717 82434 1.21310 30 31 74041 i.35o6o 76779 1-30244 79591 1.25642 82483 121238 29 32 74086 1.34978 76825 30oi66 79639 1-25567 82531 r21166 28 33 74131 1.34896 76871 i 30087 79686 1.25492 82580 1.21094 27 34 74176 I.34814 76918 oo3000 79734 1.25417 82629 121023 26 35 74221 1-34732 76964 129931 79781 1.25343 82678 1.2095 25 36 74267 1.34650 77010 I129853 79829 1.25268 82727 120879 24 37 74312 1.34568 77057 1.29775 79877 1.25193 82776 1.208o8 23 38 74357 1.34487 77103 I129696 79924 1.25118 82825 -120736 22 39 74402 1.34405 77149 1-29618 79972 1.25044 82874 1.20665 21 40 74447 1-34323 77196 1.29541 80020 1.24969 82923 1.20593 20 41 74492 1.34242 77242 129463 80067 1[.24895 82972 I.20522 19 42 74538 1.3416o 77289 1-29385 80115 1.24820 83022 -120451 18 43 74583 1.34079 77335 I-29307 8o0163 1.24746 83071 1-20379 17 44 74628 1.33998 77382 I29229 80211 I.24672 83120 1.20308 16 45 74674 1.33916 77428 1-29152 80258 1.24597 83169 120237 15 46 74719 i.33835 77475 1 29074 80306 1.24523 83218 1.20166 14 47 74764 1.33754 77521 1-28997 80354 1-24449 83268 1.20095 13 48 74810 1.33673 77568 1-28919 80402 1.243723 83317 1-20024 12 49 74855 1.33592 77615 1.28842 8o450 1.24301 83366 1.19953 ii 50 74900 i.335ii 77661.28764 80498 1.24227 834i5 -.19882 10 5i 74946 1.33430 777c8 -28687 o0546 1-24153 83465.1.9811 9 52 74991 1.33349 77754 I2861io 80594 1-.24o07 83514 1.19740 8 53 75037 ~.133268 77801 I-28533 80642 1.24o00 83564.I9669 7 54 75082 1.33187 77848 1-28456 80690 1.23931 836i3 -.19599 6 55 7128.3310o7 77895 I2'8379 80738 1.23858 83662.19528 5 56 75173.133026 77941 I128302 80786 1.23784 83712 1.19457 4 57 75219 1.32946 77988 1.28225 80834 1.23710 83761.19387 3 58 75264 i.32865 78035 1.28148 80882 1.23637 838ii.1i9316 2 59 75310 1.32785 78082 1-28071 80930.23563 83860 o.19246 60 75355 1.32704 78129 1-27994 80978.23490 83910o.19175 0 Cotang. Tangent. Cota T gent. Cotang. Tangent. Cotang. Tangent. 5032 i o52 510 o00 TABLE III. NATURAL TANGENTS AND COTANGENTS. 88 40o0 410 420 43[ iTangent. Cotang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotan,. 0 83910.119175 86929 I.15037 90040 1 I Io 61 93252 1.07237 60 i 8396o 1.19105 86980 1.14969 90093 xa10996 93306'o7174 59 2 840oo 1-1903 87o031 I-14902 90146 Io10931 93360 ~-07112 5 3 84059 1I18964 87082 114834 90199 1 10867 93415 o70o49 57 4 84108 I.18894/ 87133 1.14767 90251 1.1o802 93469 IoO6987 56 841 58 1-18824 8784 1I-4699 90304 Io10737 93524 I 0o6925 55 6 84208 I.18754 87236 1.I4632 90357 -10o672 93578 i o6862 54 7 84258 1I18684 87287 1-14565 90410 1.1 0607 93633 i.o6800 53 8 84307 1-I8614 87338 11I4498 90463 I -10543 93688 o06738 52 9 84357 I - 18544 87389 1-14430 90516 I -1o478 93742 I o6676 5S 1o 84407 -18474 87441 1I14363 90569 1o10414 93797 i-o66I3 50 II 84457 I.i8404 87492 1-14296 go621 1-10349 93852 106551 40 12 84507 i.I8334 87543 II4229 90674 I-10285 93906 I-o6489 48 13 84556 I.I8264 87595 11I4I62 90727 1-10220 93961 1o6427 47 14 84606 1 18194 87646 1-14095 90781 I o056 94016 106365 46 15 84656 1-18125 87698 1-14028 90834 1-I10091 94071 1.o6303 45 I6 84706 I180o55 87749 1I13961 90887 I.10027 94125 -06241 44 17 84756 1-17986 87801 I.-3894 90940 I-o9963 94180 10o6179 43 18 84806 1r. 7916 87852 113828 90993 1.o9899 94235 1.06117 42 I9 84856 1.17846 87904 1-13761 91046 1.o9834 94290 I.o6o56 41 2o 84906.17777 87955 1-13694 91099 I ~09770 94345 1-05994 40 21 84956 I-17708 88007 1I-3627 91153 1-09706 94400 1.05932 39 22 85006 -1I7638 88059 Ii356I 91206 I 09642 94455.o5870 38 23 85057 I -7569 88110 1.13494 91259 109578 94510 0o589g 37 24 85107 I'17500 88162 -13428 91313 i-o9514 94565 1.05747 36 25 85157 I-17430 88214 1-1336I 91366 1.o9450 94620.-05685 35 26 85207 r-1736I 88265 I1I3295 91419 1og9386 94676 10o5624 34 27 85257 I 17292 88317 1i13228 91473 o09322 9473I I105562 33 28 85307 1-17223 88369 1-13162 91526 1-09258 94786 i.o550o 32 29 85358 I 17154 88421 1.13096 91580 gg09195 94841 I o5439 31 30 85408 1-17085 |88473 113029 91633 Iog0931 94896 Io5378 30 3i 85458 1.17016 88524 1-12963 91687 o090o67 94952 1-053i17 29 32 85509 1i.6947 88576 1-12897 1740 -o09003 95007 10.5255 28 33 85559 -1687 88628 I-12831 91794 10 o8940 95062 I 05194 27 34 85609 1 16809 88680 1-12765 91847 1o08876 95118 105133 26 35 85660 I -6741 88732 I-I2699 91901 I o8813 95173 I-05072 25 36 85710 I16672 88784 1-12633 91955 1-08749 95229 I-ODOIO 24 37 85761 i-66o3 88836 I-12567 92008 io-8686 95284 1-04949 23 38 858II I.16535 88888 I-12501 92062 I1o8622 95340 Io04888 22 39 85862 I-16466 88940 I12435 92116 1-08559 95395 -04827 21 40 85912 I1I6398 88992 1-12369 92I70 1-o8496 95431 I-04766 20 41 85963 -I16329 89045 1-12303 92223 1-08432 95506 I-04705 19 42 86014 I-16261 89097 112238 92277 1.08369 95562 04644 i8 43 86064 I1i6192 8g9149 I-12172 92331 I.o8306 95618 -04583 17 44 86Ii5.16124 89201 -12106 92385 108243 95673 1-04522 i6 43 86i66 1 I6o56 89253 1-12041 92439 108179 95729 1.04461 15 46 86216 i.15987 89306 1-11975 92493 1 io8i16 95785 1.0440 14 47 86267 1-15919 89358 1I 1109 92547 1-o8o53 9584I 1o04340 13 48 86318 1-15851 89410 1- I844 92601 1-07990 95897 -04279 12 49 86368 [115783t 89463 1-11778 92655 1-07927 95952 1 04218 ii 5o 86419 1135715 89515 1-1I713 92709 1-07864 96008 |.04158 io 5i 86470 1.15647 89567 - III648 92763 1-07801 96064 04097 9 52 86521 1 5579 89620 I 11582 92817 10o7738 96120 o4036 8 53 86572 1-15511 89672 I11517 92872 1-07676 96176 I.o3976 7 54 86623 1 I5443 89725 I 11452 92926 10o7613 96232 03915 6 55 86674 1.15375 89777 I-11387 92980 1.07550 96288 i.o3855 5 56 86725 1i153o8 89830 I-1132I 93034 Io07487 96344 I-03794 4 57 86776 1.15240 89883 I-11256 93088 107425 96400 I03734 3 58' 86827 I-15172 89935 I -1191 93143 1-07362 96457 0o3674 2 59 86878 ii5io4 89988 1 11126 93197 1-07299 96513 i-036i3 60 86929 i 15037 o9004 i -1o6i 93252 1-07237 96569 I-03553 o otang. Tangent. Cotang. Tangent. Cotang. Tangent. Cotailg. Tangent. 490 48a 4 1.46~ _ _ _ _ _ _ _ I 450~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~72~ TIr 84 NATURAL TANGENTS AND COTANGENTS. TABLE Il1. 440 440 T/n gt / / I Tangent. Cotang. Tangent. Cotang. 0o 96569 I o3553 60 31 98327 1 01702 29. 1 96625 I 03493 3 98384 Io01642 28 2 9668 I -03433 58 33 9844I i.o583 27 3 96738 1.03372 57 3 98499.01524 26 4 96794 1-o3312 56 35 98556 I -o465 25 5 96850 1.03252 55 36 98613 I.oi406 24 6 96907 I 03192 54 37 98671 Io01347 23 7 96963 I 03132 53 38 98728.01288 22 8 97020 I 03072 52 39 98786 I.01229 21 9 97076 -03012 51 40 98843 1I01170 20 10 97133 1.02952 50 41 98901 I01112 19 I1 97189 1-02892 49 42 98958 I.oo053 18 12 97246 -.02832 48 43 99016 1.00994 17 I3 97302 1'02772 47 44 99073 1-00935 6 14 97359 -02713 46 45 9913I 1-00876 15 15 97416 1-02653 45 46 99189 Io8 4 16 97472 1-02593 44 47 99247 1-00759 3 17 97529 I.2533 43 48 99304.oo00701 12 I8 97586 1-0 2474 42 49 99362 1 00642 I 19 97643 1-02414 41 50 99420 100o583 10 20 97700 1.02355 40 51 99478 1.00525 9 21 97756 1 02295 39 52 99536 I1oo467 8 22 97813 1.02236 38 53 99594 00oo408 7 23 97870 1-02176 37 54 99652 00oo350 6 24 97927 -02117 36 55 99710 1.0029 5 25 97984 I0o2057 35 56 99768 oo0233 26 98041.01998 34 57 99826 100oo75 3 27 98098 o01939 33 58 99884 i.ooii6 2 28 98155.o01879 32 59 99942.ooo58 I 29 98213 1.01820 31 6o Unit. Unit. 0 30 98270.0176i 30 Cotang. Tangent. Cotang. Tangent. 450 45~ TABLE OF CONSTANTS. Base of Napier's system of logarithms =.................e = 2-718281828459 Mod. of common syst. of logarithms.... com. log. E = = 0-434294481903 Ratio of circumference to diameter of a circle =.............r. = 3. 41592653590 log. 04 - o 97497872694 7r. = 9-8696o 44o0189i.... 0 O / wr= 1772453850906 Arc of same length as radius =.......... I80~ ~=- 1800o' x = 64800oo" 1800 -.- = 57~ 2957795 30 o,......................... log.o - = 1758122632409 10800' + = 3437'.7467707849.........................log. = 3.536273882793 648000" - 7r = 2o6264".8o62470964,.....................log. - 3 5.314425 "3176 Tropical year= 365d. 5h. 48m. 47s..588 = 365d..242217456, log. = 2.56258o1 Sidereal year =365d. 6h. 9m. 1os..742 = 365d. o256374332, log. = 2.5625978 24h. sol. t.=24h. 3m. 56s. 555335 sid. t.=24h. XI 00273791 log. I o002= -00o 11874 24h. sid.t.=24h. - (3m.55s. 90944) sol. t.=24h. xo 9972696, log.o -997-9 9988126 British imperial gallon = 277.274 cubic inches,..............log. = 2 442909 Length of sec. pend., in inches, at London, 39. 3929; Paris, 39. 1285; New York, 39 1285. French metre = 3 2808992 Englishfeet = 39-3707904 inc7ies. i cubio inch of water (bar. 30 inches, Fahr. therm. 620) = 252 458 Troy grains.