REESE>LIBRARY / t UNIVERSITYTDF CALIFORNIA. * c i(ecevved , Occasion No. Sj imi .92418 C/jss r-u-u-ll 1 FIELD MANUAL FOR ENGINEERS. BY PHILETUS H. PHILBRICK, C.E., M.S. M. AM. MATH. Soc., Chief Engineer, Kansas City, Wat\ins and Gulf Railway, .A'. Am. Land and Timber Co., etc., etc.; Sometime Professor of Civil Engineering at the State University of Iowa. FIRST EDITION. FIRST THOUSAND. : extended, and extra topi*, ,10r clearness, considered in connection with the icrai matter of the text. ..xpi<. Table II dispenses with all calculation Dy numerous readings the last figure. \j & 4- .L>aces, has bden in prepara- IV PREFACE. as shown in Chapter IV; Table VII greatly simplifies the finding of the tangent and the external of any curve; and Tables XVIII and XIX, and some others to which the remark can apply, are, it is believed, in terms of the proper arguments and in the best form. In Chapter IV the laws of errors in field-work are demon- strated and illustrated; and the best method of conducting a preliminary survey, introduced by the author a generation ago, is explained. In Chapter V simple and exact formulas for determining the height of a mountain or other object by the dip of the horizon are substituted in place of the approximate formulas in use. For the stadia, as well as for the telemeter, new formulas are found which require no general computation; and the formula for finding the proper elevation on curves, unlike other formulas, involves no large factors. In addition to a very general treatment of Compound Curves, Chapter VI includes the location of such curves of any number of branches (pp. 143-5), as well as easy and symmetrical formulas for finding their tangents (p. 176). The reader interested in the philosophy of mathematics will find, it is hoped, an elegant and fruitful illustration of the prin- ciples of substitutions in determining general curves to fulfill required conditions on pages 145, 146. This is susceptible of general application. There has also been added a general treat- ment of the subject of Curves tangent to Curves, including the "Wye Problems"; and also that of Concentric Curves applic- able to Parallel Turnouts. The finding of the angles between the rails at the crossings of curved tracks is also thought to be a valuable addition. Chapter VII includes a variety of problems in Reversed Curves. The solution of Problems IV, V, and VI were first given to the Senior Class (1869) of the University of Michigan, while the author was in temporary charge. Chapter VIII treats extensively of Turnouts. The distinction between connecting a straight line and a track, and two tracks, and what \^-^ i uired in each case, is shown on pages 199, 200. of certain proposed turnout curves is and a variety of methods of laying out PREFACE. V Chapter IX applies entirely to the author's True Transition Curve. Chapter X shows that the formulas for the computation of earthwork may be abridged one half by supposing that the side slopes are produced in this intersection. A general relation between the " end-area " volume and the " middle-area " volume is shown by means of symbols, and new formulas are given for the volume of a frustrum of a pyramid and the frustrum of a cone. A formula showing the true correction of earthwork for curvature is also deduced, and the best method of computing earthwork tables explained and illustrated. Chapter XI contains only a brief exposition of the subject of approximate and abridged computations, which it is thought may serve to encourage the shortening of computations. Chapter XII describes the processes incident to construction and calls attention to two principles that aid very much in " staking out " earthwork. The logarithmic tables are not reproduced, for the reason that they are but little used and should not be used at all and most emphatically so in this line of work. It is fair to observe that the space required for such tables is replaced by numerous useful tables, applying directly to the matter in hand, and also to enlarging the subject-matter of the book, thus sav- ing greatly in time and labor. Furthermore, it should be stated that there is no problem in the book requiring a computation more complex than to find the cost of 29 oranges (say), sup- posing that 17 oranges cost 43 cents. The author must believe that no person much less an engineer would think of apply- ing logarithms to the above example; and if so, he could not with any propriety apply them to any problem in the Manual, since the nature of the numerical computation to be made, and not the subject-matter of the problem, furnishes the test of methods. The author invites criticism, and, should another edition be called for, will make the best use possible of any suggestion that may in good faith be made to him. The Tables, all but three, were computed expressly for this book, and scrupulous care has been taken, by numerous readings and checks, to make all tables exact to the last figure. The book, as indicated in a few places, has be"en in prepara- VI PREFACE. tion several years and contains matter gathered all along the paths of the author's experience; and is the result of a belief, on his part, of his ability to aid his professional brethren in this direction. While this delay has not been to the advantage of the author, it lias nevertheless given opportunity for due reflection and re- consideration; and therefore, as a work of judgment based on experience, the volume is offered to his brother enquirers. If the book even partially accomplishes the object of the author's aims, lie will feel that the days he has devoted to it, though many, have not been spent in vain. P. H. PHILBKICK. CONTENTS. CHAPTER I. PRELIMINARY OPERATIONS. PAGE The Reconnoissance 5 The Preliminary Survey 5 The Location 5 The Organization of the Transit Party 8 The Compass: What kind to use and when to use it 8 Requirements for a Successful Reconnoissance 9 Train Resistances 10 Total Ascent the Main Test of Gravity Resistance n CHAPTER II. ADJUSTMENTS, USE, AND CARE OF INSTRUMENTS. The Transit. Adjustments 12 Use and Care of the Transit 15 Best Way to Set up the Instrument 15 To Measure the Angle between Two Lines or Objects 16 Hints on the Care of Transits and Other Instruments... . 16 The Level. Adjustments Use of the Level. The Compass. Adjustments 22 Use of the Compass 23 vit Viii CONTENTS. CHAPTER III. PLANE TRIGONOMETRY. PAGE Definitions and Explanations 25 Fundamental Relations 26 Solution of Plane Right Triangles 31 Table for Plane Right Triangles 32 Fundamental Relations for Oblique Triangles 32 Short Solution of " Tangent Problem " 34 Solution of Plane Oblique Triangles 35 Table for Solution of Oblique Triangles 36 List of Fundamental Formulas 37 CHAPTER IV. SIMPLE CURVES CONNECTING RIGHT LINES. Properties Relating to the Circle 39 Some Elementary Relations 41 Notation 41 Degree of Curve Defined and Explained 41 Rational Treatment of Curves, Example Illustrating 42 Difference in Lengths of Arcs and Subtended Chords 43 Huygens' Formula for the Length of an Arc 44 Table Showing Excess of Arcs over Subtended Chords when the Arcs are Aliquot Parts of 100 44 Table Showing Excess of Sub-chords over Aliquot Parts of 100, when the Chord = 100 45 Reason for these Large Excesses Pointed Out 46 Formulas for Radius, Tangent, External Secants, Offsets, etc 47 Proper Course to Pursue in Locating a Curve 49 Long Chords and Ordinates to Long Chords 50 Approximate Value of Ordinates to Short Chords 5 1 Offsets in Terms of the Degree of a Curve 52 Applications of Formula 53 Laying Out Curves. A. By Deflection Angles 53 B. By Tangent Offsets. New Method. Without Calculation 55 C. By Ordinates from a Long Chord. Without Calculation 57 D. By Chord Offsets. Without Calculation 58 E. By Middle Ordinates. Without Calculation 59 F. By Radial Lines. Without Calculation 59 Errors in Field-work The Nature of. "A" Method of Laying Out 60 "B" Method of Laying Out. 64 "C" Method of Laying Out 64 " D " Method of Laying Out 64 CONTENTS. IX PAGE "E" Method of Laying Out 65 Fourteen Problems in Simple Curves - 65 Obstacles in Surveying. To Erect a Perpendicular at Any Point of a Line 76 Table and Formulas giving Sides of Right Triangles 77 To Drop a Perpendicular from a Given Point to a Given Line 77 To Draw a Perpendicular to a Line from an Inaccessible Point.... 77 To Prolong a Line past an Obstacle and to Measure its Length... 78 Obstacles to Measuring a Line. When One End is Inaccessible 79 When Both Ends are Inaccessible 80 When an Inaccessible Space Intervenes 81 Rest Method of Making a Preliminary Survey KJ Table Illustrating the Same 83 To Replace a Broken Line between Two Points by a Straight Line 85 To Find the Angle between Two Straight Lines when the Point of Intersection is Inaccessible 86 To Connect Two Tangents by a Curve when the Vertex is Inac- cessible 87 To Locate a Curve when the Vertex and Both Ends of the Curve are Inaccessible 87 To Pass from Any Point on the Curve to Any Point on the Tangent 87 To Find any Desired Point on the Curve when Obstacles preclude ti.e Use of Ordinary Methods 88 CHAPTER V. LEVELING, STADIA MEASUREMENTS, ETC. Bench, or Bench-mark 90 Form of Field-book for Level Notes 92 Proof for Level Notes 93 Benches: Where to Establish them 93 The Location of a Level Line 94 The Location of a Grade Line 94 Correction for Curvature and Refraction 95 Trigonometric Leveling 97 True Simplified Formulas for Heights by the Dip of the Horizon.. 98 The Stadia. Formulas for the Stadia Simplified 102 The Gradient er. Formulas for the Gradienter Simplified 107 Xll CONTENTS. PAGE To Connect a Straight Track and a Straight Line ................... 199 Two C'onnect Two Straight Tracks by a Curve of Three I tranches, the Turnout Curve having Radii to Suit Given Frogs ............ 199 Formulas Supposing the Switch-rail to be a Part of the Turnout Curve ................................................................ 204 A Simple Curve Cannot Meet the Conditions Required Above ...... 205 Several Methods of Laying Out Turnout Explained ................. 207 Double Turnouts from a Straight Track ............................. 208 To Fit a Curve to a Given Middle Frog ............................. 210 Turnouts from Curves. Turnout from the Inside of a Curve .................................. 212 Turnout from the Outside of a Curve ................................ 213 Double Turnout on Opposite Sides of a Curve ....................... 214 To Find Degree of a Turnout from a Curve ......................... 215 Other Turnouts from a Straight Track ............................... 217 To Fit the Turnout to a Given Middle Frog ........................ 219 CHAPTER IX. THE TRUE TRANSITION CURVE. Reason for the Need of Such Curves Definition and Properties of Such Curves Elementary Relations. To Find the Relative Length of the Offset and the Transition Curve ................................................................ 225 To Find Any Tangent Distance ...................................... 2_>6 To Find Any Offset .................................................... 228 Having the Offset, t, Supposing the Offset Curve of the Same De- gree as the Original Curve, to Find the True Value, /', of the Offset ................................................................ 230 To Find the Offset iri Terms of the Cenrtal Angle and the Radius.. 230 To Find the Angle between the Tangent and Any Chord Drawn from A ............................................................... 23 1 Cubic Parabola Not Suitable for a Transition Curve ................ 232 To Find Point on Curve where the Tangent is Parallel to the Chord of the Curve ................................................. 232 To Find Tangents at the Extremities of the Curve .................. 233 To Find the Length of Any Radius Vector, or Chord, and the Angles between these Chords ...................................... 233 To Find Deflection Angle at Any Point, also Any Chord ............ 235 To Find the Exsec dV ', also TV, etc ................................. 236 To Find the Radius of Curvature at Any Point ..................... 236 To Lay Out the Curve by Offsets from the Tangent AQ ............ 236 Special Problems and Examples. Given Length and Degree of Kd, to Find the Offset AK, Tangent AO, and to Lay Out the Curve ................................... 237 CONTENTS. xiii PAGE Given the Degree of the Offset Curve and Offset AK, to Find the Length of the Transition Curve, etc 239 Given the Degree (Z) 1 ) of Kd, and the Tangent AO, to Find the Length of the Transition Curve and the Offset AK 239 Given the Degree of the Main Curve, and the Length of Kd, to Find the Offset, Tangent, etc 241 Given the Degree of the Main Curve and the Length of Af Re- placed, to Find the Offset, Tangent, etc 242 To Replace Each Half of a Simple Curve AfA 1 by a Transition Curve 243 To Connect Two Tangents by Two Transition Curves, Each of a Given Length 5 244 To Connect Two Tangents by Two Equal Transition Curves having a Common Vertex Distance E 245 The Transition Curve Very General in Use 245 To Lay Out the Curve from Any Point on it 245 To Substitute a Transition Curve for Each End Portion of the Main Curve without Changing the Rest of the Curve 246 The True Transition Curve Compared with Some Others and His- torical Note 251 CHAPTER X. CALCULATION OF EARTHWORK. Prismoid Defined 253 Area of Level Sections 254 Area of Sections not Level 255 Area of Irregular Sections 256 Formulas for Regular Excavations and Embankments 256 Error of the " End Area Volume " Always Twice the Error of the "Middle Area Volume." Demonstrated by Symbols 257 The End-area Method Simplified 258 The End-area Volume Generalized 259 Special Formulas and Cases 259' Formula for the Volume of the Frustum of a Pyramid Sim- plified 261 Loaded Flat Cars, Piles of Stone, etc 263 Ends of Embankments or "Dumps" 264 Ground Irregular Laterally 264 Mixed Work, Excavation, and Embankment 265 Correction of Earthwork for Curvature 267 Overhaul 268 Monthly Estimates 269 Final Estimates : 270 Computation of Prismoids Level Laterally 270 XIV CONTENTS. CHAPTER XT. APPROXIMATE AND ABRIDGED COMPUTATIONS. PAGE Definitions and Notation 273 The Relative Error 273 Addition 274 Subtraction 276 Multiplication and Division 277 Abridged Multiplication 279 Abridged Division 281 CHAPTER XII. CONSTRUCTION. Clearing and Grubbing, How to do it 284 Grade-line 284 Surface Ditches Importance of 285 Cross-sections Proper Places for 285 Staking Out Earthwork when Ground is Level 286 Staking Out Earthwork when Ground is Not Level 287 Two Principles to Aid in Laying Out Earthwork 291 Borrow-pits - 291 Shrinkage of Earthwork 294 Retracing the Line 295 Track-laying 296 Culverts 297 Location of Bridge Piers 298 Tunnels 298 CHAFTEB 'Mil. EXPLANATION OK TARI.KS AM. MISCEI.LAKKOUS TOPICS. To Gauge a Stream Approximately jo i Transverse Strength of Beams 302 Safe Bearing Power of Piles 302 TABLES. Table for Right Triangles 32 Table for Oblique Triangles 36 Table Showing Excess of Arcs over Subtended Chords when the Arcs are Aliquot Parts of 100 44 Table Showing Excess of Sub-cords over Aliquot Parts of 100 when the Chord = 100 45 Table Giving the Sides of Right Triangles 77 Table for Traverse Survey 83 Table for Level Notes 92 R., R, Table Showing the Least Value of T ~-^^-- 120 CONTENTS. PAGE Table Showing Computation of Prismoids 271 1. Degrees, Radii, etc 304 II. Tangent Offsets, i to 100 Feet 310 III. Offsets for Arcs of 100 Feet 312 Ilia. Middle Ordinate Arcs of 100 Feet 313 Illb. Chords of Arcs of 100 Feet 313 IV. Long Chords 314 V. Middle Ordinates 316 VI. Turnouts from a Straight Track 318 VII. Tangents and Externals of a i 1 Curve 319 VIII. Arcs of Degrees, Minutes, and Seconds for Radius = i 3.3 IX. Acres for Various Lengths and Widths 323 X. Total Grades 324 XL Correction for Curvature and Refraction 325 XII. Elevation of Outer Rail 325 XIII. Coefficients for Stadia 3^6 XIV. Coefficients for Gradienter 327 XV. Offsets for Transition Curves 328 XVI. Tangent Distances for Transition Curves 330 XVII. Deflection Angles for Transition Curves 332 XVIII. Earthwork Tables, Different Slopes and Bases 333 XIX. Earthwork Tables, Two Slopes and All Bases 337 XX. Sines and Cosines 33'-$ XXL Tangents and Cotangents 352 XXII. Versines and Exsecants 35<; XXIII. Useful Numbers and Formulas 382 XXIV. Conversion of Feet into Meters and Meters into Feet; also Miles into Kilometers and Kilometers into Miles 383 FIELD-MANUAL FOE ENGINEERS. CHAPTER I. PRELIMINARY OPERATIONS, 1. THE engineering operations preparatory to the construction of a railroad are : The Reconnoissance ; The Preliminary Survey or Surveys ; and The Location. 2. The Reconnoissance is a general but incomplete examination of the country through \vhich the proposed road is to pass, made for the purpose of acquiring data upon which surveys may be made and compared, and the best possible route for the road selected. 3. A Preliminary Survey consists of the measurement of a line, including its angular deflections ; the elevations of various points upon it, the determinations of the topography along it and near it, for the purpose of furnishing the data from which the line may be definitely located; or the survey compared with other sur- veys, for the purpose of selecting one from which the location may be made. 4. The Location consists in placing the line in the exact posi- tion in which it is intended to be. This position is called The Location. 5. It is convenient to carry on these operations concurrently. The main points to consider in the location are the relative cost of grading and bridging, and the relative grades and curva- K 6 ITELD-MAKFAL FOR ENGINEERS, ture of the lines. In grading, the character of the soil for stabil- ity, in both " cut " and " fill," should be considered. It is sometimes important to know the relative value of property- traversed by different lines ; and if the lines are far apart, the probable amount of traffic that the respective lines can command must also be taken into account. 6. It is evident that the best possible location requires the least possible grading, bridging, curvature, etc., taken together, re- garding the cost and the expense of operating the road. We can afford, therefore, to increase the curvature, for example, if by so doing we can at the same time decrease the earthwork, and the line is bettered more by the latter than it is damaged by the former. The field-work of location has for its object to determine the exact position of the selected route on the ground, to establish the grade, to compute the amount of earthwork, decide upon the amount of bridging, etc. 7. A railroad line usually follows the valleys of watercourses or the dividing ridges between watercourses, or crosses valleys and ridges more or less obliquely. 8. The location on dividing ridges is perhaps the simplest of all. In this no bridges and few culverts are required ; and the elements governing the location are mainly the amount of earth- work and the curvature of the line. In this case a sketch of the ridge, especially of its prominent features and governing points, is made while walking over it; and the preliminary line is accurately run, and made into a location, if the route is adopted. 9. The location along the valley of a stream is usually more complex than the former. If the stream is so small that the cost of bridging it would be plainly less than the advantage to the alignment by crossing ; or if, on the contrary, the river is so large that the crossing of it is out of the question, the -cost of bridging is not considered and the problem of location is reduced, in the main, to that of making the best alignment within the limits of the valley in the one case, or upon one side of the river in the other. The reconnoissance and surveys would be made as already de- scribed. Usually the most favorable ground both for alignment and con- struction alternates from one side to the other of the stream, and PRELIMINARY OPERATIONS. 7 only the results of careful and scientific surveys can tell how many and where the crossings rnust be in order to make the align- ment and grades the best possible, and to secure the most favora- ble ground. In this case the ground on both sides of the stream must be carefully examined, and if the stream, in consequence of banks or bottom, is not easily crossed at most points, the most suit- able crossings must be found. When this is done the engineer will mark out and survey one or more lines so as to fit the chosen crossings and other governing points. A comparison between different lines will point out the best, and generally the one from which the best location can be made. 10. In locating a line across valleys and ridges, the engineer must find the best crossings of the streams, and the gaps or notches in the ridges, and must connect such of the former with such of the latter as will furnish the best line. 11. Sometimes a railroad may occupy either valleys or dividing ridges for the greater part of its length ; in which case a choice must be made between the higher and the lower line. The higher line will require very much less drainage than the lower line, which is an important advantage; but, on the contrary, probably the curvature and the length of the higher line will ex- ceed that of the lower. 12. It is not to be supposed that, in general, a railroad will follow either a valley or a ridge; or that it will cross valleys and ridges obliquely throughout its entire length ; but parts of the line will generally do so, and to these, and therefore to the whole line, the preceding principles will apply. 13. The regular " reaches " in a stream furnish the best cross- ings. Throughout these, compared with other points, the flow of the current is most uniform; the wash of the bottom, and the caving and the shifting of the banks, are least; while the security of the foundations of bridges and of approaches is greatest. If ihe bottom of the stream is variable, the best site on some given " reach " must be selected with a view to the kind of foundations f/.ui table to the place. Sharp bends in streams should be studi- ously avoided. .14. The engineer should freely consult the best maps of the ( (Kuitry that he can command, and he should prepare a map on a 8 FIELD-MANUAL FOR ENGINEERS. convenient scale, upon which he should copy the principal features of the country, such as streams and lakes, roads and towns, and fill in the details as he progresses. He should also locate on the map the governing points of the route, such as the best crossings of streams, the "gaps" in the ridges, mountain passes, etc. lie may then sketch the line on the map. In a densely wooded country, the making of a thorough recon- noissance is comparatively difficult. In this case it will often be necessary to cross and recross the country many times before a comprehensive knowledge of it can be gained. 15. The one almost indispensable instrument in making a re- connoissance is a pocket-compass. Field-glasses, hand-levels, telemeters, and other instruments are sometimes used, but are rarely needed, and cannot be used to advantage. Those who cannot make a proper reconnoissance without them would better employ some one who can, and turn their attention to other parts of the work. 16. For preliminary surveys the corps of engineers may be constituted as follows: A chief engineer or engineer in charge, an assistant engineer or transitinan, a levelman, a rodman, a stake- man, a rear flagman, two chainrnen, and one or more axmen accord- ing to needs. The head flag should be carried by the head chain- man. 17. Since a survey can be made more rapidly with the compass than with the transit, the compass may be used in preliminary work to a limited extent. Owing, however, to the inherent in- accuracies of the compass, a line run with it is generally worthless, except as a guide to a transit preliminary, from which the location may be made. The compass is therefore of very little, if any, use in a comparatively level country, but is useful, if at all, for run- ning the first preliminary line in a billy region, where several lines must be run. The compass should be light, should be mounted on a Jacob 's-staff, and should have a narrow slit in one sight, and a fine platinum wire, or its equivalent, stretched along the center of a wide slit in the other sight. A self-reading rod is best for this work because it saves much time over the sliding rod; and because, too, it enables the levelman to do his own reading. The ax for driving stakes should have a broad head. . Stakes shotild be of a generous length, say from 30 to 36 inches, well driven into PRELIMINARY OPERATIONS. 9 the ground, and projecting above the grass and other vegetation, so that the line may be easily followed or recovered, in fields or woods, by the engineer and others. Short stakes occasion the loss of much time, and those shorter than about 30 inches are generally a nuisance, except on streets, well-traveled roads, etc. 18. To make a successful reconnoissance requires a good eye for distances, elevations, etc. , and a quick and clear perception of the salient features of the country. One must be able to form a mental picture or image of the country along the proposed line ; and of a number of lines crossing and recrossing each other within the limits of his mental map. Thus he may be able to sketch a proposed line, or several proposed lines, which at any time may be tested. The comparison of lines, to test their relative economy, is a question of science, aided by mathematics. 19. It is plain that without skill in making the reconnoissance, the surveys would be at first more or less at random, and the reaching of a location roundabout and expensive. On the other hand, the greatest skill without a sufficient knowledge of mathe- matics and without a knowledge of. the principles of engineering bearing upon the question, cannot produce the best location. A happy combination of the qualities necessary to the successful locating engineer is most assuredly found in comparatively few individuals. The two main obstacles to contend with in building a railway are grades and curves. Since both affect the cost of construction and the expense of operating the road, such grades and curves as will render the total cost of construction and operation of road a minimum are, as already suggested, the best. Since the resistances due to grades as well as to curves add to other resistances to the movement of trains, and since it is often possible to lessen grades at the expense of curvature, and vice versa, it becomes necessary to briefly consider the nature of the resistances that a moving train encounters, with a view to compensation. The resistances to overcome are, first, the friction of the moving parts of the engine and train ; the friction of the wheels on the rails ; impacts and oscillations, and the resistance of the air. These resistances vary with the condition of the rolling machinery, the road, and the weather, and are not accurately known. Fric- tion is nearly independent of the speed of the train; but the resist- 10 I'lKI.iJ-MANUAL FOR ENGINEERS. ances due to impact increase with the speed, and those due to the atmosphere increase in a still greater ratio. The sum of these resistances on a level track in fairly good con- dition and average fair weather is, according to Vose, in which r is the resistance in pounds per ton, and v is the velocity in. miles per hour. It is not to be supposed that this formula is accurate, though perhaps it is as nearly so as any; and probably the resistances are no greater than the formula indicates. "With a velocity of 20 miles per hour the formula gives r = 10.34 pounds per ton of the entire weight. Under unfavorable conditions, as a wet, soft, and rough track and high wind, the resistance might be 20 to 40 per cent more, or even greater still. The second resistance is due to grades of the track. The force necessary to overcome this is such a part of the total weight of the train as the vertical rise of the grade is to its length. Let r' resistance per ton, and U = the ascent per 100 feet. Then ?' = ; X 2000 pounds per ton. For a \% grade (52.8 feet per 1 UU mile) h = 1 and r' 20 pounds per ton, the force necessary to overcome the load. A \% grade would of course create a resist- ance of 10 pounds per ton, or approximately the same as the resistance caused by friction, impact, resistance of the atmos- phere, etc., at a speed of 20 miles per hour, as stated above. The third resistance is due to the curves of the track. This re- sistance is not accurately known. On American roads with American rolling stock it is probably about one half a pound per ton for each degree of curvature. Letting ?'' represent the resistance per ton per station, and D the degree of curvature, the resistance per sta- tion is r" = .5D. We observe that the resistance of a 1 curve is but the ^ part of that due to a \% grade, or equivalent to K<) Q -^- = 1 .32 feet per mile. The resistance of a 5 curve is equiva- lent to 6.6 feet per mile, and that of a 10 curve to 13.2 feet per mile, etc. The resistance offered to a train in moving through 100 feet of a 1 curve is one-half pound per ton moved 100 feet, which is equivalent to the total load lifted -^ X 100 = ^ = .025 ft.; and since the resistance varies as the product of the degree of PRELIMINARY OPERATIONS. 11 curvature and length of the curve, or as the total curvature, the resistance on any curve for each degree of change of direc- tion is equal to the lifting of the train ^ of a foot. Since an en- gine can haul over a road only what it can haul over the most unfavorable part, easy curves are necessary where heavy grades must occur, and vice versa ; the object being, of course, to keep at all points the sum of the resistances due to grades and curves be- low the allowable maximum. In equating for grades and curves, however, the ruling element is largely the total ascent and not the slope merely, as is too often considered to be the criterion. No more power is required to make an ascent of 20 feet, for ex- ample, on a 2/c grade 1000 feet long than is required on a \% grade 4000 feet long. To compare grades simply, that is, the rale of ascent or descent especially on short grades, as is usually done, is meaningless at least it is absurd. On light grades no allowance usually need be made for curva- ture, since the momentum of the train would carry it over the ascent even without aid from the engine. Grades situated near stations, however, or other points where a train must stop or " slow up " are greater obstacles than in other situations to the movement of trains, and upon such grades an allowance for cur- vature, as above shown, should be made. Every locating engineer is presumed to make himself familiar with the principles involved in locations before making them; and to aid in this the reader is referred to Vose and others, and especially to the elaborate treatise of Wellington on "Railway Location." Wellington gives in Table 118 the " Total Energy, or Potential Lift, in Vertical Feet in Trains moving at Various Velocities," and discusses at length all subjects connected with location. CHAPTER II. ADJUSTMENTS, USES, AND CARE OF INSTRUMENTS. THE TRANSIT. The following are the usual adjustments of the transit : A. To Adjust the Levels (that is, to place the levels in a plane perpendicular to the upright axis). Set up the instrument upon its tripod as nearly level as may be; unclamp the plates and hring the two levels above and on a line with the two pairs of leveling- screws. Then by means of one pair of screws bring the bubble of the level above them to the middle of the opening. Without moving the instrument bring the other bubble to the middle in the same way. Since in moving one pair of screws very far the other pair is liable to become cramped and the corresponding bubble somewhat disturbed, it is advisable to bring the bubbles in succession near the middle, repeating if necessary, and ending by bringing them exactly to the middle. When both bubbles are in place turn the instrument through about 180; if the bubbles are now in position they need no cor- rection ; but if not, turn the small nuts at the end of the levels until the bubbles are moved over one half of the error. Then bring the bubbles again to the middle by the leveling-screws and repeat the operation just described until the bubbles will remain in the middle during a complete revolution of the telescope. This shows the adjustment to be completed. B. To Set the Cross-wires Vertical and Horizontal. Level the instrument. Move the telescope upward or downward and note whether the vertical wire traverses some fixed point or not. If not, loosen the four cross- wire screws and, by the pressure of the hands on their head outside the tube, move the cross- wire ring around, what seems to be sufficient, and repeat the operation, if necessary, until the vertical wire will traverse some fixed point. The cross-wire screws are near the screws of the centering-ring 12 ADJUSTMENTS, USES, AND CARE OF INSTRUMENTS. 13 of the eyepiece (the beads of which are usually covered by an outside ring), but between them and the axis of the telescope. C. To Make the Line of Collimation Perpendicular to tlie Axis of tlie Wyes, so tliat it will revolve in a plane. Set up the transit at some point near the middle of a level piece of ground and level it carefully. Let AOB be a straight line, AO and BO being nearly equal. Direct the line of sight to A, clamp the instrument and revolve the telescope, and if the line of collimation is not perpendicular to the axis, the line of sight will determine a point 0, say, on one side of J3. To test the mat- ter, loosen the upper clamp and turn the vernier plate almost or quite half-way around, so that the line of sight will again be on FTG. 1. A, and clamp the plates. Again revolve the telescope and note tbe point D, suppose, thus determined. If the point C coincides with B, D will also coincide with B and the line of collimation is in adjustment. If, however, C is on one side of B, as shown, D will be equally far on the other side, showing the line of collimation out of adjustment. To correct the error, use the two capstan-head screws on the side of the telescope, to move the ring to which the wires are fastened laterally, and with it, of course, the intersection of the wires. Having moved the vertical wire until by estimation one fourth of the space DC is passed over, return to A, clamp and revolve the telescope, and if the correction has been carefully made the line of sight will now cut the point B. It will, however, generally be necessary to repeat the operation, the adjustment not being perfected at first. Remember that the eyepiece inverts the position of the wires, and therefore in moving the ring the operator must proceed so as seemingly to increase the error. The wires may now be brought into the center of the field of view by moving the screws of the centering-ring of the eyepiece, which are slackened and tightened in pairs, the movement being now direct until the wires are seen in their proper position. 4 tftELft-MANUAL FOR ENGINEERS. It is proper to observe that the position of the line of collirna- tion depends solely upon that of the objective, so that the eye- piece may be moved in any direction, or replaced by another without at all deranging or in any way affecting the adjustment of the wires. I). To Adjust the Standards to the Same Height so that the Line, of Collima'.ion icill Revrtve in a Vertical Plane. Set the transit as close as convenient to the base of a lofty spire, or other high object; level it carefully and clamp it, and direct the telescope to the top of the spire or other elevated and well-defined point, as A in the figure. B is in the same vertical plane with A and the instrument, that is, with the line of sight. Turn down the telescope to some good point on the ground, either found or marked, as C. Unclainp the plates or spindle, revolve the telescope and turn it half-way around, or far enough to again sight to the high point. Again clamp and turn down the telescope to ~Q some point D opposite C. In setting C the left standard must have been too high, and in setting D the same standard (now the right by revolving the telescope) shows to be equally high, the errors EG and BD being equal. Correct the error by raising or lowering the sliding-piece at one end of the axis by means of the screw beneath and those above, so that when the telescope is directed to A and lowered it will cut B half-way between C and D. If the instrument is in adjustment it will, of course, cut B instead of C and D in the first two trials. E. To Adjust the Vertical Circle. Set up the instrument firmly, level it carefully, bring into line the zeros of the circle and vernier, and with the telescope find some well-defined point, from two to five hundred feet distant, which is cut by the horizontal wire. Turn the instrument half-way around, revolve the tele- scope, fix the wire upon the same point as before, and observe if the zeros are again in line. If not, loosen the capstan-head screws which fasten the vernier, and move the zero of the vernier over half of the error. Bring the zeros again into coincidence and proceed as before, as many times as necessary, until the error is entirely corrected. ADJUSTMENTS, USES, AND CARE OF INSTRUMENTS. 15 F. To Adjust the Level on Telescope. First level carefully and clamp the telescope approximately horizontal by the eye. Then, Laving the line of collhnation previously adjusted, drive a stake, say two or three hundred feet away, and note the height cut by the horizontal wire upon a staff set on top of the stake. Fix another stake in the opposite direction and at the same dis- tance from the instrument, and without disturbing the telescope turn the instrument upon its spindle, set the staff upon the stake, and drive the stake into the ground until the reading is the same as on the first stake. The tops of the two stakes are equally high however much the telescope may be out of level. Now set the instrument some twenty or thirty feet from one of the stakes and on the prolongation of the line joining the two. Level the instrument, clamp the telescope as nearly horizontal as may be, and note the readings of the staff on the two stakes. If they agree, the telescope is level. If they do not agree, then with the tangent-screw move the wire over fully the whole error, as shown at the distant stake; repeat the operation just described a < many times as may be necessary, so that the wire will give the same reading at both stakes, showing that the telescope is truly horizontal. Taking care not to disturb the position of the tele- scope, bring the bubble into the middle by the little leveling-nuts at the ends of the tube, which will render the adjustment com- plete. The above are all the adjustments usually required at the hands ofthe engineer ; for any others see the Manual of W. & L. E. Gurley. USE AND CAKE OP TRANSIT. The instrument should be set up firmJy with the plates nearly level, and thus save much time in turning the leveling-screws, besides the worse than useless wear upon them. To do this : Set up the transit approximately over the desired point. Hold firmly the leg of the tripod on the left with the left hand, and move the foot of the leg on the right with the right hand in any direction, so as to bring the lower plate of the tripod head approximately horizontal, determined by placing the eyes in the plane of its upper surface and sighting as nearly as may be to the distant horizon. It will do no harm to bend the body the trifle necessary to do this. This sight is so long that, according to 16 FIELD-MANUAL FOR ENGINEERS. the principle of the " division of errors," the plate can be readily set very nearly horizontal in almost any position of the instru- ment. Now place the left hip against one leg; grasp the opposite leg with the left hand, and the leg on the right with the right hand; and keeping the relative position of the legs unaltered, move the instrument laterally over the desired point and lower it to the ground. Of course the two movements of tilting and sliding may be done together and in much less time than required to describe the movement. To MEASURE THE ANGLE BETWEEN Two LINES OR OBJECTS. Level the instrument carefully; bring the zeros of the verniers and limb together by means of the upper clamp and tangent-screw, and direct the telescope toward one of the objects by means of the lower clamp and tangent-screw. Upon loosening the upper clamp and directing the telescope toward the second object, the angle desired is then shown upon the limb. Before making an observation with the telescope, the eyepiece should be moved in or out until the cross- wires may be distinctly seen. This may be accomplished with the greatest precision by directing the telescope toward some white object. The sky will serve very well for this. The objective is then adjusted by moving it in or out until the object is seen clear and well defined, and the cross- wires appear as if fastened to its surface. Water and dust are very destructive to instruments ; indeed, the most destructive of all agents is dust. An instrument may be badly bruised, bent, or broken, by a fall or otherwise, and yet it may be repaired. If, however, it is allowed to stand on its tripod in a dusty office and all offices are dusty its life will be short. The author has known of instruments so choice that their owners would allow no one to handle them lest they might become slightly soiled, or some other mishap befall them; and yet in a few months they were ruined, in the way pointed out. A minute quantity of dust on the sockets will probably cause them to grind; and more of it will increase the grinding so that the instrument will soon become useless. Whenever not in actual use the eyepiece should be covered by the lid, and the object-glass by the cap, to protect them from dust, moisture, rain, etc, ADJUSTMENTS, USES, AND CAKE OF INSTRUMENTS. 17 "When an instrument is exposed to the hot rays of the sun for some time its parts are subjected to very unequal expansion, which throws the instrument more or less out of adjustment, thus to some extent vitiating the work and damaging the finer parts of the in- strument. To prevent this it should be shielded by an umbrella or screen of some kind supported on top of a high stake; or, if nothing better is at hand, a cloth should be placed over the telescope. When an instrument is in the office or in transport it should be clamped and placed in a stable position, upon proper supports, in a tight box, well cushioned if possible. If the box cover does not fit upon the box closely it can be remedied by sticking suitable cloth on the top edge of the box or on the under side of the lid. When handling an instrument it should be supported by plac- ing the hand under the lower plate. Tangent and micrometer screws should be used equally on all portions of their length. Keep the tripod legs tight enough on the tripod head to secure stability by tightening the nuts on the bolts when necessary. Also tighten the shoes of the tripod if they become loose. Neglect of these things sometimes prevents an engineer from keeping his telescope on a point, and causes him to run a zigzag line without knowing the cause of it. Secure the instrument well to the tripod head before using it; and bring all four leveling-screws to a bearing and cover the instrument with an oilcloth hood before carrying it. Use a fine camel- hair brush or a piece of old and soft linen to clean the glasses of the telescope. Dust, moisture, perspiration, etc., will sometimes cause a film to form on the lenses of a telescope which may greatly im- pair the sight through it. To remove the film, the lenses, after being carefully brushed, should be gently wiped with a piece of chamois-skin moistened with alcohol, and the lenses must be wiped dry by using fresh portions of the skin on separate parts of the lens. To remove dampness in the main tube of the telescope, take out the eyepiece, cover the open end with cloth, and leave the instrument in a dry room for some time. The centers of an instrument should aluavs be lubricated with fine watch-oil only, and after a careful cleaning. First wipe off all old grit and oil before applying fresh oil, 18 FIELD-MANUAL FOR ENGINEERS. If dust settles on the cross-wires, unscrew the eyepiece and the object-glass, and gently blow through the telescope tube; cover up both ends ahd wait a few minutes before replacing the eyepiece and object-glass. Be sure to bring the object-glass cell to a firm bearing against its shoulder, and then examine the adjustment of the lines of collimation. To clean the threads of a leveling or tangent screw use a stiff tooth-brush to remove the dust, then apply a little oil and turn the screws in and out with alternate brushing to remove dust and oil, until it moves freely and smoothly. Use such a brush to clean the object-slide. Xo screws should be strained more than necessary to insure a firm bearing, and this applies with special force to the cross-wire screws. All straining of such screws beyond this impairs the accuracy of the instrument and the reliability of adjustment. These remarks in reference to the care of the transit apply substantially, of course, to all instruments. If obliged to work with an instrument of faulty graduation, it is best to read each angle on different parts of the circle, and take the mean value. If Are take the mean result obtained by both verniers, we eliminate the errors due to eccentricity of the vertical axis, and also reduce the errors of graduation. For greater accuracy clamp the vernier-plate to zero and read the angle by both verniers. Then keep the vernier-plate clamped, point the telescope to the first object and proceed as before, any number of times. Then read the verniers, adding 360 for each complete revolution which has been made, and divide this sum by the number of times the angle was read. The quotient is the required angle. The cross-wires can be illuminated easily by placing a piece of white cardboard, with a hole through it for the line of sight, in front of the telescope, and in an oblique positio'n, so as to reflect into the telescope the rays of a lamp or of a lantern placed back of the object-glass and near the telescope. THE LEVEL. The principal adjustments of the level consist in the follow- ing: I, Bringing the cross-wires into the optical axis of (he lele- ADJUSTMENTS, USES, AND CARE OF INSTRUMENTS. ID scope and consequently parallel to the line of bearings of the wye rings. % 2. Causing the line of bearings to be parallel to the plane of the level. 3. flaking either of these lines, and therefore all of them, parallel to the bar and consequently perpendicular to the axis of the instrument. 1. To adjust the line of collimation, that is, to bring the cross-wires into the optical axis, so that their point of inter- section will remain on any given point during an entire revolu- tion of the telescope. Set the tripod firmly, remove the wye-pins from the clips, so as to allow the telescope to turn freely; clamp the instru- ment to the leveling-head, and by the leveling and tangent screws bring cither of the wires upon the clearly defined edge of some object. Then carefully rotate the telescope half-way around. ! f the wire docs not coincide with the line observed, bring it half-way back by means of the capstan-head screws at right angles with it. always remembering the inverting property of the eyepiece. Then by the leveling and tangent screws bring it again upon the "edge," and repeat the above opera- tion if necessary, and continue to do so until the telescope may be rotated without changing- the position of the wires. If both wires are much out of position, it will be well to approximately adjust the wires alternately before attempting to make the adjustment of either one complete, since an error in one somewhat affects the other. It may be advisable to center the eyepiece. To do so unscrew the covering of the eyepiece centering-screws, and move each pair in succession, with a screw-driver, until the wires are brought into the center of the field of view. The inverting property of the eyepiece does not affect this operation, and the screws are moved directly. To test the cen- tering, rotate the telescope, and if an object observed appears to change position the centering is not perfect. In all telescopes the line of collimation is determined by the cross- wires and objective, and is not affected in any way by the eyepiece. 2. To make the line of bearings parallel to the bubble-tube, 20 FIELD-MANUAL FOR ENGINEERS. so as to insure that it is horizontal, when the bubble is in the center. This adjustment* embraces two parts: ' First, to bring the center-line of the bubble and the line of bearings in the same plane. Second, to make these lines parallel. To effect the first: Clamp the level and bring the bubble to the center by the parallel-plate screw*. Now rotate the telescope in the wyes 20, more or less. If the bubble runs toward the end, it shows that the center-line of the bubble and the axis of the telescope are not in the same plane; in other words, the bubble-tube lies crosswise of the telescope. To correct the error, bring the bubble by estimation half- way back, by the capstan-'head screws, which are set in either side of the level-holder. Again bring the level-tube under the telescope, the bubble to the middle, etc., repeating the operation just described until the bubble will keep its position, when the telescope is re- volved. For the second part: Bring the bubble to the middle of the tube by the leveling-screws, and take the telescope out of the wyes carefully and turn it end for end. If the bubble runs toward either end, lower that end, or raise the other by turning the adjusting-nuts on one end of the level until by estimation half the correction is made. Again bring the bubble to the middle by the leveling-screws, and repeat the whole operation just described until the reversion can be made without causing tiny change in the bubble. 3. Having made the previous adjustments, it remains to make the level-bubble (and therefore the line of bearings and line of collimation) parallel to the bar, and therefore perpen- dicular to the vertical axis of the level, so that the bubble will remain in the middle during an entire revolution of the tele- scope. Place the level over a pair of leveling-screws, and by means of them bring the bubble to the middle. Turn the instrument half-way around horizontally. If the bubble runs toward either end, bring it half-way buck by either pair of nuts, at the ends of the bar. Then bring the bubble to the middle again, by the level ing-screws, etc.. repeating the operation just, described, ADJUSTMENTS, USES, AND CARE OF INSTRUMENTS. 21 until the bubble will remain in the middle of the tube when the instrument is revolved. In making this adjustment it is best to use the opposite pairs of leveling-screws alternately, thus bringing the upper parallel plate of the tripod head into a position as nearly hori- zontal as possible, so that the error caused by not revolving the instrument precisely 180 may be the least possible. This adjustment is for convenience and not for accuracy in any appreciable degree. Xow turn the telescope in the wyes until the pin on the clip of the wye will rest in the little recess in the ring to which it is fitted. Apply the horizontal wire to any level line, and in case it does not coincide with it, loosen two cross-wire screws at right angles to each other, and by their heads outside turn the cross- wire ring until the horizontal wire coincides with the level line. The line of collimation must then be adjusted again. In readjusting the line of collimation none of the lines referred to in making the adjustment is disturbed, and the adjustments are complete. ADJUSTING BY THE " PEG " METHOD. By this method the main adjustments are effected at once. Drive two pegs several hundred feet apart, and set the in- strument midway between them. Read the rod on each, keep- ing the bubble in exactly the same position, preferably at the center. The difference of the readings is the difference of the heights of the pegs, no matter how much or in what way the level may be out of adjustment. Then set over either peg and measure the height of cross-wires above top of peg. The difference of heights of pegs, added to this, or taken from it, according as the instrument is over the higher or lower peg, gives the height of the cross-wires above the other peg. Set the rod on that peg, and bring the horizontal wire to that height on the rod by the leveling-screws, keeping them at a bearing. Then bring the bubble to the center by raising or lowering one end of the level-tube. The first part of the second adjustment, namely, to bring 22 FIELD-MANUAL FOR ENGINEERS. the level-bubble and line of collimation in the same plane, also the third adjustment, should be made as heretofore. USE OF THE LEVEL. The instrument should be set up firmly, with the top of the tripod as nearly level as may be, so as to save time in leveling and the wear of the screws, etc. The setting up of the level is, of course, similar to that of the transit already described. The bubble should then be brought over each pair of level- ing-screws successively, and leveled in each position. Bring the wire precisely in focus by the eyepiece, and the object distinctly in view by the objective, so as to avoid all " traveling of the wires " or parallax. It is best, where practicable, to take approximately equal fore and back sights, so as to eliminate any error due to a lack of perfect adjustment, Athich is difficult to secure. For precise reading the rod should not be over 400 or 500 feet from the instrument. If the socket of the instrument sticks in the leveling-head so as to be difficult to remove, be sure that the instrument is un- clamped and the leveling-screws are free. Then place the palms of the hands under the wye-nuts under each end of the bar, and give a sudden upward blow to the bar, and take care also to grasp it the moment it is free. To ADJUST THE COMPASS. The Levels. First bring the level-bubbles into the middle by the pressure of the hands on different parts of the plate: then turn the compass half-way around. If either bubble runs toward one end of its tube, it indicates that that end is too high- Lower it by loosening the screw under the lower end, and tightening the one under the higher end, until the error is, by estimation, half removed. Level the plate again, and repeat the operation until the bubbles will remain in the middle dur- ing an entire revolution of the compass. The Sight-ranes. The sights may next be tested by observ- ing through the slits a fine hair or thread made exactly verti- cal by a plummet. ADJUSTMENTS, USES, AND CAKE OF INSTRUMENTS. 23 If either slit does not coincide in direction with the hair or thread, it must be made to do so by filing its under surface on the higher side. The Needle. Having the eye nearly in the same plane with the graduated rim of the compass circle, observe whether or not the ends of the needle cut opposite points on the rim. If not, bend the center-pin (by means of a small wrench) about an eighth of an inch below the point of the pin, so as to make the ends cut opposite points on the rim. The needle now may be supposed to occupy the position Npti, the pivot p not being in the center 0. Now keeping the needle in the same position, turn the compass half-way around. The needle will now oc- cupy the position N'p'S'. Correct half the error by El bending (that is, straigJit- CH'UH/) the needle, making it cut points half-way be- tween its former positions, so that it occupies the posi- tion Ap'B and is straight; and correct the other half by bending the pin, placing the point of the pin at and giving the needle the position NOS. The operation should be repeated until perfect reversion is secured in the first position. Then try the needle on another quarter of the circle, and if an error is manifested, correct the center-pin only, the needle being already straightened by the previous operation. Do the same on other quarters of the circle until the needle will reverse in any position. To USE THE COMPASS. In using the compass keep the south end toward the person, and read the bearings from the north end of the needle. Mark every station or point at which the compass is set, so 24 FIELD-MANUAL FOll ENGINEERS. that it may be easily found for verification or a resurvey. It is much more important to have the compass level laterally, or crosswise of the sights, than in their direction; since if it is not so, on looking up or down hill through the lower part of one sight and the upper 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 be obtained. A continuous line thus run, to say nothing of other im- perfections of the compass, would be a zigzag line probably very much in error. The compass cannot be leveled by the needle, for the dip of the needle is continually varying. If the needle touches the glass when the compass is leveled, balance it by sliding the coil of wire along it. The vibrations of the needle may be checked by gently rais- ing it off the pivot, so as to touch the glass, and letting it down again, by the screw on the under side of the box. The compass should be smartly tapped after the needle has settled, to destroy the effect of any adhesion to the pivot or friction of dust upon it. The glass sometimes becomes charged with electricity by carrying it against clothing, or wiping it, etc., so that it at- tracts the needle to its under surface, preventing its free move- ment. The difficulty may be remedied by breathing on the glass, or touching it in different places with the moistened finger. Of course the chain and all other metals should be kept away from the needle. CHAPTER III. PLANE TRIGONOMETRY. 1. Plane Trigonometry treats of the relations of the sides and angles, and of the solution of plane triangles. 2. Some French writers divide the right angle into 100 degrees, the degree into 100 minutes, the minute into 100 seconds, etc. This centesimal system, in which reductions 'are made by simply moving the decimal point, is altogether preferable to our sexagesi- mal system. 3. Two angles whose sum is equal to 90 are complementary. Two angles whose sum is 180 are supplementary. 4. Let us consider a series of right triangles ABC, AB'G', etc., having the common angle A. The triangles are equiangular and therefore similar, and we have BC BG' B"G" AB ~ AB' AB" BG AG B'C' B"C' AG AB _ AB' AC ~ AC' AC' AB" FIG. Thus it appears that the ratios of the sides are the same in all right triangles having the same acute angles; and therefore if these ratios are known in any one of these triangles, they will be known in all of them. As any triangle may be divided into two right triangles, it is evident that the solution of oblique triangles may be made to de- pend upon the solution of right triangles. The above ratios, depending upon the angle alone and not at all upon the absolute lengths of the sides, may be considered as indices 25 2G FIELD-MANUAL FOR ENGINEERS. of the angle, and have received special names, which we will ex- plain. 5. Let us represent the sides and angles of a triangle in the usual way, shown in Fig. 5. The side opposite an angle, divided by the hypothenuse, is called the sine of that angle. Thus = sin A, c and - = sin B. c The side opposite an angle, divided by the adjacent side, is called the tangent of that angle. Thus The hypothenuse, divided by a side adjacent to an angle, is called the secant of that angle. Thus v . = sec A; b - = sec 3. a 6. The terms cosine, cotangent, and cosecant are convenient abbreviations for the "sine of the complement," "tangent of the complement," and " secant of the complement, " respectively. The reader must not suppose that there is such a thing or entity as co- sine, cotangent, or cosecant of an angle. Since the acute angles of a right-angled triangle are comple- mentary, that is, A = 90 B and B = 90 A, it follows that cos A means the sine of (90 A), or sin B; cot A means the tangent of (90 A), or tan B; cosec A means the secant of (90 A), or sec B, etc. Hence sin J. = cos 5=; c tan A cot B = > cos A = sin B = ; cot A = tan B = ; a (4) sec A cosec B = r ; cosec A = sec B = - . h a PLANE TRIGONOMETRY. Sin ,4 = cos B is really an identical equation, since cos B is the sine (90 B) = sin A; and so is tan A = cotB, etc. Furthermore, c b vers A = = 1 cos ^L. exsec A = - ; = sec A 1 = cos A 1 cos A vers A cos J. cos A' B 7. Plane trigonometry, applied to plane triangles, is but the ap- plication of the one well-known proposition in geometry: " Equi- angular triangles have their homologous sides proportional and are similar"." Comparing Figs. 5 and 6, we observe that the above ratios -, -, etc., change as the angles A and B change. These ratios have been com- puted, however, for all values of the angles, differing by single minutes or less, and placed in tables under the corresponding headings, sine, tangent, etc. , and opposite the correspond- ing angles. 8. From the above equations we observe that b FIG. 6. Hence (6) (7) sin A cosec A = sec A cos A = tan A cot A = 1. , (8) We also have, by division, sin A a cos A b 3 FIELD-MANUAL FOR ENGINEERS. Again, sin' 4 + cos' ;t=^+i' =<;=!. . . . (11) Again, 6 6 2 and sec A = 4/1 -f tan 2 A ; (12) cosec 2 A = sec 2 jB = = = 1 -f- = !-}- cot 2 ^4, and cosec A = From (9), tan sin A = cos A tan J. = From (10), cos J. = sin J. cot J. cosec . If one of the above functions of the angle A is given, all the others may easily be found. For example, if sin A is given, we have, from (11), cos A = |/1 sin 5 A. Then sm A sin A tan J. = T = - = J . = (16) cos J. /i _ s i n s j. sm A sin = - : ... cos A 4/1 _ sin 2 A (18) PLANE TRIGONOMETRY. If cos A is given, we have, from (11), sin A 4/1 cos* A. Then tan A, etc., as above. Similarly when other functions of the angle are given. 9. The sine and cosine of two angles being given, to find the sine and cosine of their sum, and the sine and cosine of the difference of their angles. In Figs. 7 and 8 let EOF = A, and FOE=B; then, in Fig. 7, EOG=A+B, and, in Fig. 8, EOO = A - B. From any point F in OF draw FE perpen- dicular to OF; also EG and FH per- pendicular to OH, and FK perpendicular to EG. Now the three sides of the tri- angle FEK are perpendicular to the three sides of the triangle FOH, and therefore FEK = FOH. No\v, in Fig. 7, FIG. ! (A EK EO But and FH FH FO Ji/Ji = cos A sin B. . '. sin ( A + B) = sin A cos B -f- cos A sin B, . . Again, . OG OH-KF OH KF . (20) OR OF KF EF = C S A Cos 3 ~ sin A 30 FIELD-MAKUAL FOR ENGINEERS. Then, in Fig. 8, _ EG _ HF-KE_ HF OF _ KE FE sm(A-B)=~gQ- EQ ~ OF' ^EQ~ ~YE'~EO = sin A cos B cos A sin B\ (22) OG cos (A-B} = = FK_OH OF^ FK^ FE 1 ~ OF' E0~^~ FE ' OE cos A cos B + sin A sin B. (23) H G FIG. 8. In (20), make B = A and get sin 2A = 2 sin A cos A. (24) In (21), make B = A and get cos 2 A = cos 2 A sin 2 A = (1 sin 2 A) sin 9 A 1 2 sin 8 A. = cos 2 J. - (1 - cos 2 A) = 2 cos 2 ^. - 1. (20) and (21) give sin A sin B sin (J. -j- B) _ sin .4 cos 7? -f cos AsinB_ cos A cos B cos (A + B) ~ cos .4 cos B sin .A sin B 1 - cos X sin B ' or tan ( A -f B) = tnn tan B I tan A tan Similarly, from (22) and (23), tan (A ~ B) = ~ tan A tan B 1 -|- tan A tan B (27) (28) PLANE TRIGONOMETRY. 31 We note some special values of the trigonometric functions. See Fig. 9. Let COP represent any triangle ; angle COP = 0. Let = 0. Then CP = 0, PO and BO coincide and are equal. Hence S inO = ^ = 0, tanO = JL = , c = = I. Let = 90. Then PC and PO coincide with B'O, and CO = 0. Hence ~)f f\ T> ' /") T>/ /^) sin 90 = r =1, tan 90 = oo , sec 90 = ~- = oo . _O U Let = 180. Then PO and CO coincide with B"0 and PC = 0. Then sin 180 =^L = 0, tan 180 = ^ = 0, sec 180= |^ = 1. Let = 270. Then PO and PC coincide with B'"0 and CO = 0. Hence 7?"'O 7? //7 O Ti" f O sin 270 = 57^=1, tan 270= -- =00 , sec 270 =^r-^= oo . 1> U The values of the functions of 360 are the same as those of 0. We need not consider angles greater than 180. SOLUTION OF PLANE RIGHT TRIANGLES. 16. In order to solve a plane right triangle it is only necessary to select from equations (4) an equation containing the two given parts aside from the right angle and the part sought. By trans- posing, if necessary, so as to express the latter in terms of the former, it becomes known. There are two cases : CASE I. A side and an angle given. Given A and c. See Fig. 5. Example. Let A = 35 23', and c = 874.8. We have B = 54 37'. Also table of sines and cosines gives sin 35 23' = .57904, and cos 35 23' = .81530, Hence a = c sin A = 874.8 x .57904 = 506.54; b = c cos A - 874.8 X .81530 = 713.22. FIELD-MANUAL FOR ENGINEERS. CASE II. Given two sides. Example. -Let a = 184.3, and c = 246. We have sin J. = _= .74919. c We find in the table sin 48 31' = .74915, .. A = 48 31' to the nearest minute; B = 90 - A = 41 29'; & = c cos A = 246 X .6624 = 162.95, b = Vc 2 - a 8 = 162.95. TABLE FOR SOLUTION OF RIGHT TRIANGLES. Give-u. Required. Formulas. 1. a, b A, B, c tan A = -, b B = 90 - Ay c = b sec ^1 =a sec .Z?. 2. a, c A, B, b a sin A = , B = 90 - A, 5 = c cos .4 =a tan B. 3. A, a B, b, c B = 90 - Ay b = a tan B , c _ 2 c cos ^ } or COB A ~ Similarly, or by analogy, cos B = ^ C , and From (29) and (31) we also have* BD c sin A tan G = 7=^ = T- sin 6 - c cos A b cos A tan 5 = tan J. = sin (7 . a - cos U sin B (34) c a 18. Let ABC, Fig. 11, represent a plane triangle, the parts be- ing represented as usual. Take GE = CA, and draw AD and EH perpendicular to AE. We have CAB + OEA = 180 - (7 = 4 + 7?. and 4# = (7^4 - GBA = l(A +B) - B = (A - B). FIELD-MANUAL FOR ENGINEERS. Also GAD = 90 - CAE, and CD A - 90 - (AEC '= CAE). Hence CD = AC = b. Now a + b BD AD AEi&n \(A +B) tan JEET . . (35) FIG 11. From triangle ABE, or BE _ sin BAE AB~ jsin AEB' a b _ sin \(A B) c ~ sin \(A + B) In triangle ABD, BD _ a -f- b sin BAD cos |(^4 B) AB ~ c ~ sin ^IJW? ~~ cos ^(^4 + B)' (36) (37) Eq. (37) divided by (36) also gives tan + B) tan B)' (38) which furnishes another demonstration for (35). A slight variation of the above solution of the tangent problem was given by the author in Vol. I, No. 1, of The American Math- ematical Monthly. It has since found its way into text-books on trigonometry. Still another solution by the author may be seen jn Vol, III, No, 11, of the same journal, FLAKE TRIGONOMETRY. 35 SOLUTION OF PLANE OBLIQUE TRIANGLES. 19. There are three cases. CASE I. In this case two of the given parts are a side, and the angle opposite ; the other part being either a side or an angle. Example 1. Let A, a, and B, Fig. 10, be given. C = 180 - (A + B). From (32), sin B sin C b = a . ; similarly c = a : -T-. sm A sm A Example 2. A, a, and b given. (32) gives sin B = sin A ; then the table gives B. Now C = 180 - (A + B) ; and c = a --. CASE II. Given two sides and the included angle. Let b, c, and A be given. We have, from (34), sin A tan C = Then cos A c = ISO 8 (4 + <7) ; and a = b ~. Or, from (33), a = (&' -f e* - 2bc cos Then sin B = sin A , and C 180 ( A -f- B). CASE III. Given the three sides a, b, and c. Eq. (33) gives COS J. = 36 FIELD-MANUAL FOK ENGINEERS. Then sin B = sin A-, and C = 180 - (A + B). The above formulas are all-sufficient for all practical purposes. This chapter constitutes a complete treatise on trigonometry, though the deductions from it are endless, as the examples in arithmetic are endless, TABLE FOB SOLUTION OF OBLIQUE THIANGLES. (See Fig. 10.) Given. Required. Formulas. 6. A,B,a C, b,c 7. A, , b B,C,c 8. A, b, c B,G, a 9. a, b, c A,B,C asinB a sin (A -f B) sin A b sin A sin A a sin (A + B) a sin A sin A b sin A tan B - cos A b a*-(b-cy ~ sin B ' (a+b-c}(a+c-b) 2bc b sin A 2bc a Use the first form for vers A with a table of squares, the second without ; they are the best formulas known for this case. The following are the best formulas known for the area. 10. Area lab sin C %ac sin B = \bc sin A. It is never necessary to compute but one unknown part, and in the 3d case none at all, to have the required data for one of these equations ; and the computation is shorter than by any other formula. Observe that a sin C is equal to the perpendicular from Bupon b, b sin G is equal to the perpendicular from ^1 upon <(, etc. PLANE TLUGONOMKTRY. 37 TRIGONOMETRIC FORMULAS. tl. sin A = 4/1 cos' J A 2 sin ^A cos A. cosec A 12. cos A \'\ sin* A = - = cos 2 IA - sin' 2 \A sec 4. = 2 cos 2 ^ - 1 = 1 - 2 sin 2 \A. 13. tan A ; = r = cosec 2 A cot 2 A cos A cot .4 / 1 - cos 2.4 sin 2/1 sin 2 A 1 + cos 2^ ' 14. cot ^4. cosec 2 A + cot 24 the reciprocal of any expres- sion for tan A. 15. sec A -- the reciprocal of any expression for cos A. cos A 16. cosec A - - - the reciprocal of any expression for sin A. sin A 17. vers A = 1 cos A 2 sin 2 \A. 18. exsec A sec J. 1 = -p. cos 4 19. sin (.1 B) = sin .1 cos 5 cos A sin 5. 20. cos (A B) = cos A cos .5 T sin A sin Z>. 21. sin A. + sin Z? = 2 sin |(4 + 5) cos \(A B). 22. sin A sin I? 2 cos |(/1 -f B) sin |(.l 7?). 23. cos A + cos B = 2 cos |(A -f .#) cos |(4 - 1?). 24. cos B cos A = 2 sin \(A + 5) sin (J. - B). 25. sin 2 4 - sin 2 B = cos 2 B - cos 2 /I = sin (A + 5) sin (.4 - 1?). 20. cos 2 A sin 2 5= cos (4 -f- B) cos (4 B). 27. tan A tan B= sin 28, cot 4 cot 7? = COS 4. COS 5 si " < A B > 38 FIELD-MANUAL FOR ENGINEERS. The above formulas contain the practical general relations exist- ing among the functions of an angle. By writing \A, 2A, etc., in place of A, by repetitious, etc., the formulas may be greatly multiplied without producing any new relations. This practice is too common in Trigonometries, Field-books, etc. CHAPTER TV. SIMPLE CURVES CONNECTING RIGHT LINES. LET ABODE represent a circular arc joining the straigLt lines A V and E V, which are tangent to the C' curve at A and E. AV and JSV are tan- V t^ents to the curve, A and Zare tangent points, and the angle KVE is the an- gle of intersection, and shows the change of direc- c tion in passing from one tangent to the other. Vis the point of inter- section, or vertex. PROPERTIES RELATING TO THE CIRCLE. The following proposi- tions rest upon elementary geometrical principles, may be regarded, for the most part, as axiomatic. (a) A tangent to a circle is perpendicular to the radius, at the point of contact. (&) Tangents drawn to the circle from the same point are equal, and the angle between these tangents, and the chord join- ing the tangent points, are equal. Thus, AV - EV, and VAE - VEA. (c) The central angle AOE, subtended by a chord, is equal to the exterior angle KVE between two tangents to the curve at the extremities of the chord. (d) The angle between a tangent and chord is equal to the angle subtended at the circumference by the same or by an equal chord, Thus, VAB - BCA = BAG, etc. 39 40 FIELD-MANUAL FOR ENGINEERS. (e) An angle between a tangent and chord, or an angle subtended at the circumference by that chord, is equal to one half the central angle subtended by the same chord. Thus, (/) Equal chords subtend equal angles at the center of a circle, and also at the circumference. Thus, if AB -- BC, etc., ACS = BAG, etc., and AOB = BOG, etc. 7 FIG. 13. radius perpendicular to a chord bisects the chord, and also the angle and the arc subtended by the chord. Thus, if OG is per- Px_^_ xN.T pendicular to'AE, we have AM = ME, AOV = EO V, and AC=GE. (h} Parallel chords, or a tangent and parallel chord, intercept equal arcs. Thus, in Fig. 13, if PT, P'T', and P"I " are parallel, then PP" and T'T" are equal, also PP' and TT' are equal. (i) The exterior or deflection angle KBG, Fig. 14, between any two chords AB and BG is equal to half the central angle AOG, subtended by the chords. This is easily shown as follows : 1. Join AC. BAG = %BOC, / \i and BGA = %AOB, as stated above. But / / \C KBC= BAG+BCA- .-. KBC = ^AOB + iBOC = \AOC. 2. Draw the tangent EBL. Then KBC = KBL + LEG = NBA + LBG Fio. H. SIMPLE CURVES CONNECTING RIGHT LINES. 4l J3. Since the chords ^47? and BC are parallel to tangents drawn at the 'middle of the arcs AB and B C, it is evident that, in passing from one chord to the other, we turn through an angle measured by one half the arc AB -\- one half the arc BC, that is^ through an angle equal to one half AOC. SOME ELEMENTARY RELATIONS. In Fig. 12 drop the perpendiculars Bb, Cc, etc., upon the tangent AY. Ab, be, etc., are called tangent distances, and Bb, Cc, etc., tan- gent offsets. Prolong AB, making Bd = ' AB. dC is called the chord offset. Then, from the preceding article, we have CBd = ^AOC = AOB. Drop the perpendicular Bp upon Cd. This bisects the angle CBd and the base Cd. Hence CBp = dBp = iCBd = \AOB = BAb. Since, therefore, the triangles CBp, dBp, and BAb have an acute angle in each equal, and the hypothenuses also equal, they are equal in all respects ; and therefore Cp = pd = Bb ; also Cd = 2Bb. Since Ab is tangent to the curve at A, Bp is likewise tangent to the curve at B. Represent the radius AO by R, the tangent, or vertex distance, A Fby T, and the angle of intersection K VE, as well as the central angle AOE, by V. Represent the chords AB, BC, etc., by c, a long chord, as AC, by C, the tangent distances Ab, be, etc., by d, di , etc., and the tangent offsets bB, cC, etc., by t, ti , etc. Then the chord offset Cd = 2t. Represent the vertex distance GVby E, and the middle ordinate of a long chord by M. Let L represent the length of the curve, P. C. (Point of Curve) the beginning of the curve, and P.T. (Point of Tangent) the end of the curve. The degree of a curve has been defined as the number of degrees FIELD-MANtJAL FOR subtended at the center by a chord 100 feet in length. This defi- nition of curvature is, however, awkward, arbitrary, and false. It is founded on error; it involves unnecessary labor and ends in anomalous and erroneous results. The degree of a curve may be defined as the change in its di- rection between one point, and another 100 feet from the first, measured on the curve. This is the change in direction which one would make in moving 100 feet on the curve from one point to another. Or, as the angle subtended at the center of the curve liy an arc 100 feet in length. Let D == the degree of the curve. The circumference of a 1 curve is therefore 360 X 100 = 36000 OfjAAA feet, and its radius is ~ 5729.578 feet, almost exactly. Since a 2 curve changes its direction 2 in 100 feet, its circum- ference is only half that of a 1 curve ; and since the radius varies directly with the circumference, the radius, too, is only half that of a 1 curve. For the same reason the radius of a 3 curve is precisely the radius of a 1 curve; and, generally, the radius of a D curve is exactly equal to the radius of a 1 curve divided 57*^9 57S by 1); it is therefore equal to 1 ~ . Referring to Fig. 15, we see that for the same central anglo AOE, the arc, the tangent, the chord, the middle ordinate, the er.- ternal secant, etc., vary directly with the radius or inversely with the de- gree of curvature. For example . If aO = %AO, then ape - \APE, ai = \AY, etc. If APE is a 1 or 2 or 3 curve of length L, then ape is a 2 or 4 or 6 curve of length Hence to compute any function of any curve, the tangent or exter- nal, for example, from the corre- FIG. 15. spending function of another curve, it is only necessary to multiply or divide, as the case may be, by the ratio of their radii or degrees of curvature. Example. Find the tangent of a 7 13' 433' curve, the cen- tral angle being 37 50', SIMPLE CUHVES CONNECTING EIGHT LINES. 43 The tangent of a 1' curve, by Table VII, is 117812. Then T= 117812 H- 433 = 272.1. Using a table giving functions of a 1 curve, we have: Tangent of a 1 curve = 1963.6. Then Tangent of a 1' curve = 1963.6 X 60 = 117816. Finally, :i!7816 + 433 = 272.1. Or, 13' T*| 60 = 0.216, arid therefore 7 13' = 7. 216. Then 1963.6 -4- 7.216 = 272.1. With a table giving functions of a 1 curve there is no escape from dividing by 60, which division is obviated by giving the func- tions of a 1' curve as in Table VII. It is often desirable to know the difference in the lengths of a chord and its subtended arc, and for this purpose we deduce the following formula : d = .001269239249^* - .0000000048329. (8) n A n b IP D 4 = .001269 .0000000048^, very nearly, (9) n s n 5 Z> 2 = .001269, approximately (10) In these equations D = the degree of curvature, d = the differ- ence between any chord and the subtended arc, and n = arc of 100 feet divided by this arc. For an arc of 100 feet n = 1 and d = .001269Z* 2 , nearly; . (11) For an arc of 50 feet n = 2 and d = .000158D 2 , nearly; . (12) For an arc of 25 feet n = 4 and d = .00002Z)' 2 , nearly. . (13) For n sub-chords per station the sum of the difference per station is nd -= .001269 (14) 44 FIELD-MANUAL tOlt ENGINEERS. Representing the central angle of the curve l>y V, and tlie num- ber of stations in the curve by A", we have N ; and hence tin- length of the curve exceeds the sum of the lengths of the sub- chords by 7)2 V VD E= Nnd= .001269=^ X -~ = .001269,.. . (14') Hence, in laying out curves, should be nearly constant. Since in a 4 curve the chord of an arc of 100 feet is 99.98 feet, curves from to 4 can be properly laid out with chords of 100 feet, and with the same degree of accuracy we may lay out curves from 4 to 16 with chords of 50 feet, and those from 16 to 04 with chords of 25 feet. Very sharp curves can be easily laid out by swinging a chain around, while one end is held at the center of the curve. Let s represent any arc, c its chord, and e t the chord of one half of the arc s. Then, from above, This is said to be Huygens' approximation to the length of an arc. The following table shows the differences between arcs of 25 feet and of 50 feet, and the chords of those arcs. (See Fig. 16.) De* D Arc 25 Arc 50 Deg. D Arc s>5 Arc 50 Deg. D Arc 25 Arc 50 1 .000 .000 11 .002 .019 21 .000 .070 .000 .001 13 .003 .023 22 .010 .077 3 .000 .001 13 .003 .027 23 .010 .084 4 .000 .003 14 .004 .031 24 .011 .091 5 .000 .004 15 .004 .036 25 .012 .699 6 .001 .000 16 .005 .041 . 26 .013 .107 7 .001 .008 17 .006 .046 27 .014 .116 8 .001 .010 18 .COS .052 28 .01C .124 9 .002 .013 19 .007 .058 29 .017 .133 10 .002 .010 20 .008 \m 30 .018 .143 bIMI'LE CURVES CONNECTING KKIHT LINES. 45 II. Suppose the chord AB = 100 feet, and AOB = D, (Fig. 16). Then AOM=^, MAtt = , etc. 2 4 Hence AM = AEsec MAE = 50 sec ~i. 4 Therefore AM - 50 = 50 (sec - - 1 j = 50 exsec . Similarly, AK = AFsvc = 25 sec - sec , etc., etc. o 4 o The following table gives the excess of AK over 25 feet, and of AM over 50 feet, when chord AB is 100 feet. Deg. AK AM Deg. AK AM Deg. AK AM D! -25 -50 A -25 -50 DI -25 -50 1 .000 .000 11 .036 .058 21 .131 .211 2 .001 .00-2 1-2 .043 .069 22 .144 .231 3 .003 .004 13 050 .('81 23 .157 258 4 .005 .008 14 .058 .093 24 .171 .275 5 .007 .012 15 .067 .101 25 .186 .29?. 6 .011 .017 16 .076 .122 26 .201 .323 7 .015 .0:23 17 .086 .138 27 .217 .349 8 .019 .030 18 .096 .155 28 .233 .375 9 .024 .039 19 .107 .172 29 .250 .408 10 .030 .048 20 .119 .181 30 .268 .431 Comparing the preceding tables we learn that, when the chord AB is 100 feet, the chord AM differs nearly four times as much from 50 feet as the chord of the arc of 50 feet differs from 50 feet. Furthermore, that the chord AK differs nearly sixteen times as much from 25 feet as the chord of the arc of 25 feet differs from 25 feet. Thus let us first suppose A MB (Fig. 16) to be a 10 curve, and the chord ^47? = 100 feet. Then by the table AK = 25.030 feet, and this would lead to an error of .030 X 40 1.2 feet in laying out a curve 25 X 40 = 1000 feet long, taking chord AK = 25 feet long. 46 FIELD-MANUAL FOR ENGINEERS. Suppose, secondly, that the are AMD 100 feet and therefore arc A K 25 feet, and the chord AK is but .002 of a foot less than 25 feet. Hence in laying out the above curve, taking the chord AK ' 25 feet, the error committed would be only .002 X 40 = .08 of a foot. Thus we see that when the chord of a station is taken = 100 feet, the sub-chords AM, AK, etc., differ so much from 50 feet, 25 feet, etc., as to largely vitiate the results, whereas such is by no means the case when the arc AMB is taken = 100 feet. Indeed, when the chord AB is 100 feet, the shorter the sub- chords used in laying out a curve, the greater the discrepancy in the measurement, on this basis. Thus for a 10 curve the difference between four equal sub-chords and 100 feet is .080 X 4 = .120 of a foot ; whereas the difference between two equal sub-chords and 100 feet is .048 X 2 = .096 of a foot. The reverse is of course the case when the arc AMB is made the standard of measurement. These facts are evident ; for when the chord is made the stand- ard of measurement, the sum of the lengths of the sub-chords ex- ceeds more and more the length of the chord, the shorter they are ; whereas when the arc is the standard of measurement, the sum of the lengths of the sub-chords falls short of the length of the arc less and less the shorter they are. Most recent writers have endeavored to obviate the inconven- iences and inconsistencies above pointed out by inconsistent assump- tions, such as basing the curves of different degrees upon chords of different lengths. This scheme gives the values of some of the radii quite correct ; but it causes sudden breaks in the value? where the changes are made. For example, how can there be two different values (819.02 an<* 818.64) of the radius of -a 7 curve? And why should the radius of a 7 10' curve be given quite cor- rect, while that of a 6 50' curve is quite incorrect? Again, we are told by a recent author that, in practice, it is cus- tomary to take the radius of a 1 curve as 5730 feet, and to assume the radius to vary inversely as the degree. Thus for a 4 curve the radius would be j- = 1432.5 feet. This is rational and, 4 moreover, is precisely what is here advocated, except that the true value of the radius of a 1 curve is used, vi/., 5729.58 feet. SIMPLE CURVES CONNECTING RIGHT LINES. 47 FORMULAS. From the triangle AOV, Fig. 12, we have AV=AOt&nAOV, or T=Rtau^V. . . (15) Transposing, we Lave From the triangle AOF, we find A0 = . Ani ;, or R = -rn = & cosec sin ^107^ sin \J) This value of R in (15) gives Measure equal distances FIT" and VL along the tangents in Fig. 12 and draw HNL. Measure #JVand VN. A V VN VN that is, T=R-. ... ..... (19) If ^4J/ and FJf are measured, then Eqs. (19) and (20) serve to fix geometrically, without measuring angles, the tangent point of a curve of given radius that will unite two straight lines on the ground. In Fig. 12 draw VG' perpendicular to A V to meet A C prolonged. Now the angle AVG \AVE = |(180 - F) = 90 - |F. Hence VCC' = AGO = 90 - BOO = 90 - 4 L F. 48 FIELD-MANUAL FOR EXGJNKEkS. Also, VG'C = 90 - CA V = 90 - i V; .-. VC'C- VCC', and VC' = VG = #. Hence 7^7 /"fr TT ^- = -jy = laniF, or E = T \&\\ V. . . ("21) Substituting T ft tan IF for T, from (15), gives ^= tftan^Ftan F. ...... (21') Since the triangles BCd and 7?0(7are similar, \ve have Gd BC BC* or and Bb = t = ~ ......... (3:.) We also have 56 = AB sin 7?^4&, or i = c sin |D, . . . (24) and 2t = 2c sin \D ........ (25) Example. Given R = 1909.9 feet and c 100 feet, to find the tangent and the chord offsets for 100 feet. By (23), 1 0000 = 2.618, and 2t = 5.236, or Table I shows that 1909.9 is the radius of a 3 curve, and \D = 1 30', and sin \D .02618. Then, by (24), t = .02618 X 100 = 2.618, and 2t = 5.236. The tangent offsets are given in Table I, and the chord offsets are twice the tangent offsets. SIMPLE CURVES CONNECTING RIGHT LINES. 49 In laying out curves, the chain is stretched from point to point >n the curve, and coincides, therefore, with chords of the curve. Since the process is the same whatever the length of chain used, we will assume it to be 100 feet long 1 . FIG. 17. The length of the curve is expressed in chains, in terms of the central angle AOF = V, and the degree of the curve AOB = D. V The number of chains is evidently equal to . Thus, in the figure, if V 23 and D = 5, the curve is - 2 / = 4f chains, or 460 feet long. As the angle AOE = 5 X 4 20, EOF 23 20 = 3 ; and the arc EF is f X 100 = 60 feet long. It is usual, in laying out curves, to assume the radius 11, and to find the degree of the curve D from it ; or to assume D (usually in degrees and minutes), and to find R from it. Neither way is best. To assume a value of R or of D does not aid in the least in properly locating the curve. Generally the surface of the ground does indicate approximately the position of the curve, and the proper course to pursue, therefore, is the following: Divide the tangent of a one-minute curve by the length in round numbers of the desired tangent, and neglect the decimal in the quotient. This give.s the degree of the required curve in minutes. We may ..nzrociTV 11 50 FIELD-MANUAL FOR ENGINEERS. change the quotient to an even number of minutes, or to some multiple of 10 if we wish, if there is sufficient latitude to be taken in the position of the P.C. Then divide the tangent of a one-minute curve by the corrected quotient for the tangent required. Example. V 46 30', and the tangent should be 1100 feet or over. Find the degree of the curve, and the length of the tan- gent. Dividing the tangent of a 1 curve by 1100 gives 147697 + 1100 = 134' = 2 14'. Now 2 10' = 130', and 147697-^- 130 = 1136.13, the tangent re- quired. LONG CHOKDS AND ORDINATES TO LONG CHORDS, Let A, B, C, etc., represent stations upon a curve. Draw the lines as repre- sented, EX being a perpendic- ular from the middle of the curve E upon the tangent AT. Let Jlf = EM, M' = FN, etc. Draw AE, and we have the angle EA M equal to the angle EAX, and therefore the tri- angles EAM and EAX are equal in all respects. From this we learn that the tangent offset EX for any arc AE h equal to the versin EM of that arc, or to the middle or- dinate EM ot the chord AK of twice the arc. Or, draw the tangent Et and the perpendicular Kt upon it. Then the tangent offset Kt of the arc EK = the versin EM of the same arc = the middle ordinal e EM of the arc AEK. To find the middle ordinate, M. (See Fig. 18.) 1. M = EM = EX = AM tangent EAM = \C tan 3. M = EM = EX- ET cos TEX - #eos V, . (26) . (26' ) SIMPLE CURVES CONNECTING RIGHT LINES. 5l 3. M = EO - MO = U - R cos | F = #(1 -cos F) = R vers | F. . . . (26") 4. J/= EO-MO=EO- i/AO*-AM*=R- \/lt r -iC-. . (26'") To find any ordiuate FN distant d from the center of the chord. Prolong FN, to meet OS, drawn parallel to AK, and join FO. Now, FS = - OS" = 4/-R 8 - d*. JVS = MO = R - M . = M+ - - d* - R. Other methods will be given in connection with laying out curves by ordinates from a long chord. Approximate Values of Ordinates to Short Chords, ^ F G, Divide the chord into any number of equal parts, eight for example, at z, k, l y etc. , and erect the ordinates Em, Fn, etc., and prolong them to A meet the curve in 13', F', etc. Let Em = m, Fn = m', etc. We have, from geometry, _ Am X mB __ \ c X $c or, approximately, = 2r-=55- < 27 > 6.K oil E'F FIG. 19. ^G JW Similarly, c X |c 15 c* ^L WSR = m =- = . ~ = m = m - Hq = m'" = (28) (28') (28") For any other equal divisions of the chord we have similar re-, suits. FIELD-MANUAL FOR ENGINEERS. tablisli the points E, F, etc. Set E equally distant from A and 13, and at the distance m from the point m ; then F, equally distant from k and 13. and \im from n ; then G, equally distant from m and B, and w* from p; lastly, H, equally distant from^) and B, and T 7 g w from q. This method involves much less labor than that of drawing sub- chords to find the points F, G, etc. If we draw a tangent at E, the offsets to F, G, H, etc., will be, according to the preceding formulas, fam, yV^, ~i% m > e ^ c - This shows a convenient way of finding the points on the curve. To compute tangent offsets and middle ordinates by means of a series. In a way similar to that pursued in finding the difference be- tween an arc and its chord we find t = .872664625997Z> - . 000022 15240389.D 3 -f.00000000022493360386Z> 5 -, . (29) or t = .872664626,0 -.000022152404 3 -f-.000000000224933604Z) 5 . (30) We may find the tangent offset for m stations by multiplying the successive terms of (30) by m 1 , m 4 , etc. We thus find tang offset for arc of 50 feet, or stations, 2 t 1 = .2181G6156499Z) - . 000001 38452524D 3 + .00000000000351458756Z) 5 -. . (31) For 1 foot m* = (.Ol)' 2 = .0001, m 4 = .00000001, etc. Then, The tang offset for 1 ft. = t, = .0000872664626Z) - .00000000000022152404D 3 + . (32) From (30), we have t = .SID, approximately. For n stations tn = .87/i J X> (33) Wellington in Railway Location, ch. xxx, recommends the equation t = | 8 A , , , (33') SIMPLE CURVES CONNECTING RIGHT LINES. 53 \vliicli is practically the same as. (33). (33) is a trifle more accurate than (33'), but either is sufficiently accurate for all cases in which ii-D does no* much exceed thirty. The same formuia gives the middle ordinate, n representing the number of stations on either side of the center, or 2n the number of stations in the arc. Any other ordinates desired are then given by eq.-?. (27). (28), etc. Example. Find six offsets of a 3 curve at points 50 feet apart. (See Fig. 18 ) For n n n n n n = 1, 1, t. 2, 2i, 3, ^ 7 t = t = t x X X \s x i 1 9 r 4 L' 5 "T" 6 X x X X x X 3 :j 3 8 3 8 1= 9 5 10 16 2:' .66; .62; 91; .50; .41; .62. These results are of course equal to the middle ordinates for 1, 2, 3 ... 6 stations of the same curve. LAYING OUT CURVES. Since in laying out curves the operation or method is the same whatever the length of the chain or chord, we will here assume it to be 100 feet long. A. By Deflection Angles. Let A in Fig. 17 be the P.C. Set the instrument at A, and turn off from the tangent AVthe given deflection angle VAB = \D, D being the degree of curve. This will give tlie direction, AB, and measuring 100 feet in this direction, the point B will be determined. Turn off the additional angle B AC = \T), the tele- scope being now directed toward C, and set C in the line AC &nd 100 feet from B. Turn off the additional angle CAD = ^D, and set D in the line AD and 100 feet from C. Proceed in the same way for other stations to the end of the curve, or so far as the stations can be seen from A. It is usually impossible, on account of obstructions likely to be met with, to lay out the whole of a curve from the first station. When such is the case, we determine as many stations as conven- ient, remove the instrument to the last station so determined, and 54 FIELD-MANUAL FOR ENGINEERS. proceed from that as from the first station. For example : Suppose B, C, and D to be found with the instrument at A. Remove the instrument to Z), sight to A, turn off the angle ADV ' = DA V \D, and the line of sight will be in the direction of the new tan- gent DV Sit D. Reverse the telescope, and the line of sight will point forward along the same tangent VDG. Now set E, F, etc., from the tangent DG, as B, C, and D were set from the tangent In setting the first stake from the new position of the instru- ment, as E from D, no notice need be taken of the tangent at D, it being necessary simply to turn off from the line ADUihe angle HDE = HDG -f GDE = AD V+ GDE - 4(iZ>). In the new position of the instrument we observe that, in all cases, the deflection from the line pointing to the back station to the line pointing to the forward station is as many times the de- flection angle \D as there are chains in the curve between the back and the forward station. Of course this applies to simple curves only. The beginning of a curve, as well as the end, usually falls between regular stations, giving short chords at the ends. The deflection angle for a short arc is such a part of the full deflection angle as the short arc is of the full arc. Let a represent the arc AB or BC, etc., and a, represent the arc EF. Also EOF =Di. Hence DFE^D, and EDF=$D 1 , and therefore = , or \D a In beginning a curve, the instrument being at the first station, it is convenient to place the zeros of the instrument plates to- gether, and direct the line of collimation along the tangent to the curve. The reading on the limb for any station will then be equal to the total deflection angle for that station. If the vernier is not disturbed while laying out the curve, it is plain that when the instrument is moved to its second position D, Fig. 17, and the line of sight directed to A, the reading will be equal to the total deflection from A to D; and that, an additional angle equal to this deflection being turned off, the reading will be equal to the central angle AOD, and the line of sight will be in the direction of the tangent at D. The same is true for all posi- SIMPLE CURVES CONNECTING EIGHT LINES. ^ tions of the instrument. Since the deflections for the last section of the curve, that is, from the last position of the instrument to the end of the curve, is turned off but once, the reading on the instrument, when the curve is finished, will be equal to the total central angle V, less the deflection for the last section. This fur- nishes a convenient check for the work, B. By Tangent Offsets. New Method. Let ABCDEFGH represent a curve having a short chord AB -- c' subtending an angle AOB = - Di at one end of the curve. Define the tangent A V by set- ting stakes upon it. Draw Be, TX etc., parallel to A V, and BX, iJcX', etc., perpendicular to AV, or suppose such lines to be drawn. Now BX=ABsmA X=c'sm \Di = t^ and is given by Table II. Since the chords BC, CD, etc., are parallel to the tangents at the middle of the arcs BC, CD, etc., FIG. 20, we have CBc = D, + \D, DCd - D, + \D, EDe = A -f- ID, etc. Hence Cc = c sin (D 1 4- |), Dd = c sin (D, + |Z>), Ee = c sin (D, -f |Z>), etc. We observe that Cc, Dd, Ee, etc., are respectively the tan- gent offsets for curves of the degrees (Di -(- ^D), (Z>i -f- %D), (2>, _|_ |/)) > etc., and may be taken from Table III. Then CX' = BX + Cc, DX" = CX' + Dd, EX"' = DX' + Ee, etc. Set B at a distance c' from A, and ti from A V; then C a dis- tance c from B, and CX' from AV; D a distance c from C, and Dx" from A V, etc. FIELD-MANUAL FOR ENGINEERS. be observed that in this method we avoid constructing t at B, in consequence of the short chord AE\ we avoid, secondly, the finding of AX, XX', etc.; thirdly, the errection of perpendiculars, at X, X', etc.; and, lastly, we avoid the use of the radii which are large and fractional. Since c is generally 100, though sometimes an aliquot part of 100, no computation is gener- ally required, and none to mention in any case. Thus we see how simple and short this method is, compared with the method of tangent offsets in general use. When the curve begins at a station there is no short chord. Then A B = c, D! D ; BX = c sin $D, Cc = c sin |Z>, Ed c sin f Z>, etc. It is best to lay out the curve from each end, so as to avoid off- sets inconveniently long. Long offsets may be avoided by drawing a tangent at any station and continuing the curve from that station precisely as from A, when there is no short chord. To draw a tangent at any station, draw a line through that sta- tion and at a perpendicular distance from an adjacent station equal to the tangent offset for a station = c sin \D. Or, draw it parallel to the chord joining adjacent stations. It will be noticed that stations on the curve are not opposite stations originally set on the tangent. The length of the curve gives the number of the station at H. This is perhaps the best of all methods without a transit ; but a combination of methods is sometimes advisable, as will be shown further on. We also have AX CCOS^D! , XX' Be = c cos (D l -f ^D), etc., or XX ', X'X", etc., are respectively equal to the tangent distances for one station of curves of the degrees D\ -f- \D, D\ -f- f Z), etc. These quantities are not needed, however, and it is to be ob- served that the points X, X', etc., are not established or used. SIMPLE CURVES CONNECTING EIGHT LINES. C. To Locate a Curve by Ordinates from a Long Chord. Let Fig-. 21 represent a curve having an odd number of full chords, and a short chord AB c' subtending an angle AOB = D l} and a short chord KL c" sub- tending an angle KOL = D. These quantities may be taken directly from a table of sines and cosines, as already pointed out. It is not necessary, as shown under the preceding method, to compute and lay off Cd, Cl, etc. D. To Locate a Curve by Chord Offsets. Let ABCDEF be the curve, having short chords AB = c' , sub- tending an angle AOB = D\ and EF c", subtending an angle EOF = D.,. Fa Locate B as in the las' T method. Prolong AB, mak ing BCi = c. The angle CBC, = CAB + ACB = \D-\-\Di. Hence CiCis the chord offse 2t' for a curve of \D -)- \D degrees. = 2c sin i(Z>, + D}. Place C, therefore, at a distance c from B, and 2t' from Ci, 2t' beinj v twice the tangent offset which is given by Table III opposite the. SIMPLE CURVES CONNECTING RtGHf LINES. 59 degree $(D + DI). Prolong BC to D, , making <7A = c. Now D 1 D 2t is given by Table III, being twice the tangent offset. Place D, therefore, at a distance c, from (7, and 2t from DI. Place all regular stations similarly. C may be placed, also, as in the preceding method. J^is easily placed from the tangent at E, as B was placed from the tangent at A. It may be observed that by the above method we are able, by means of a table giving offsets for full stations only, to locate any station, as C, by chord (or tangent) offsets, though the chord, as AB, preceding the adjacent chord may be of any length. E. To Locate a Curve by Middle Ordinates. Let ABCDEFG be the curve, having a short chord AB = <', subtending an angle AOB D\ , and a short chord FG = c", sub- tending an angle FOG = D. Locate the stations B and C from the tangent A x, or by some other method as already explained. Then set off on CO the distance Cc = t c sin \D the middle ordinate of a chord of two stations ; and set D in the prolongation of Be, and at a distance c from C, Set off Dd the same as Cc, and set E in the prolongation of Cd, and at a distance c from D. Locate all subsequent stations in the same way. To test the accuracy of the work, measure the per- pendicular Fy from the last regular station upon the tangent at G. This distance ought to be F. To Lay out a Curve by Radial Lines from the Center. Consider the case of a half-mile race-track, having two parallel sides, each 600 feet long, connected at the ends by semicircles, as 204Q 1 9 00 shown in Fig. 24. Now - ' - = 720, the length of each 180 semicircle. Hence the degree of each curve - = 25, and the radius is 229.18 feet. 66 FIELD -MAX UAL FOR ENGINEERS. Set the instrument at 0, and ran radial lines 01, 02, etc., making 6 FIG. 24. angles of 80 with each other, for example, and set stakes at 1, 2, etc., 229.18 feet from 0. These stakes are 100 f on to the end of the curve. 2. Suppose an error to occur in some other chord than the first in the second chord, for example. B is supposed, therefore, to be set correctly. Suppose the next stake set at C' instead of at C, FIELD-MANUAL FOR ENGINE Let BC r = c + e. Now BC -f- CC' > BC', or c + <7C" > c + e. .-. (7(7' > e. From C" the errors will follow the same law as in the former case. With reference to these two cases we remark that if the error occurs in the chord adjacent to the instrument, the errors in the stations will decrease slightly to the end of the run, and from that point remain constant to the end of the curve ; but if the error occurs in a chord not adjacent to the instrument, the error in the station, at the end of that chord, is slightly greater than the error in the chord, though the errors slightly decrease from this point to the end of the run, and remain constant from the end of the run to the end of the curve. In all cases the error at the end of the curve may be regarded as equal to the error in the chord, whether adjacent to the instrument or not. 3. Suppose the chain is in error, in which case all the chords will be in error. Let the curve AB (R = AO = radius, and D de- gree) be run with a chain 100 feet long, and the curve AB'(R' = AO' = radius) with a chain 100 -f- e feet long. The number of V chords is equal to = n. Then for the difference in length of the curves we have AB' (100 + e)n. AB = ICOn, and therefore AB' - AB = en. (34) For the chord we find, since R' lit AB' = and therefore AB' - AB = BB' = ', AB = 21i ZRes'm ^V ~100 ' SIMPLE CURVES CONNECTING RIGHT LINES. But BB' 360tfsin^y 114.60esin"-|-y (35) TtU D Also 11' = (100 + < < 180 180. 57.3, (36) We also have TV 77. -BB'~'m * V 114 - 6 ^( sin * F)2 (37) ul TV, 57.3^ sin y 7? 7? ' cos i y . (38) Example. Let e = .02, T 7 = 60, and D = 6. The error of the curve = .02 X TT = -200. o 114 6 X 1 X 02 The error of the long chord = = .191. Also 11' - R - - ' ~ = -19! 114.6 X .02 X ([ = .0955, B. Suppose that the first stake is set at B', Fig. 27, instead of at B, the error in the angle being BAB' = A. Then it is evident that C f , D', etc., will be set at the same angular distance from the chords p, AC, AD, etc., as B' is set from AB. B f , C', etc., are on the curve, having 0' as center and AO' = AO as radius. OAO' = BAB'. To find the error at the end of the curve. Suppose .Zthe end of the true curve. C4 FIELD-MANUAL FOll ENGINEERS. Now AE' = AE, and the angle EAE' = the angle BAB'. Hence JE7i" = 2^1# sin \EAE' = 2AE sin If the error in the angle is corrected at the end of a run, say at IS', then, for reasons given above, the error in the position of the stations is constant from E r onward to the end of the curve. If the error is corrected during a run, say just before D' is set, then that station will be set at Di on AD and 100 feet from C'. Similarly Ei will be set on JJ^and 100 feet from 2) lf It may be shown that EEi is less than DD l precisely as it was shown that CO' is less than BB ' in Fig. 25. Hence in this case the error will decrease to the end of the run, and remain constant from that point to the end of the curve. II. Curves Laid Out by Tangent Offsets. If in Fig. 22 the tangent at B is swung through an angle A, say, then all stations following B will be misplaced, the error increas- ing to the end of the " run " in the manner shown in Fig. 27. If a new tangent is drawn before reaching the end of the curve, as at E', Fig. 27, it is easy to see that such tangent would make an angle A with the tangent to the true curve at E, and that the error, therefore, would go on from E' forward precisely as from B to E', and hence the error would increase regularly from B to the end of the curve. III. Curves Laid Out by Ordmates from a Long Chord, In this case, if the end C of the chord CK, Fig. 21, is misplaced, then all stations from D to H inclusive will be misplaced in pro- portion to their distances from K\ the greatest displacement being less than that of G. If K is also misplaced, the same stations will likewise share that error, in proportion to their distances from C. IV. Curves Laid Out by Chord Offsets. (Fig. 22.) If in this method any station, as B, is set at one side of its true position, then all subsequent stations will be in error in the same direction ; the errors increasing regularly to the end of the curve, in the manner shown in Fig. 27. If, owing to an error in some chord, some station is set forward or backward from its true position, then all subsequent stations will be in error the same amount and in the same direction. SIMPLE CU-RVES CONNECTING RIGHT LINES. 65 V. Curves Located by Middle Ordinntes. In Fig. 23 suppose G'to be placed a distance a to the right, say, (that is, along CO,} of its true position. Then c will be a distance ft to the right ; D and d will be 2a to the right ; ^and e will be '*i to the right of their true positions, etc. If C is correct, but c a distance a too far to the right, then D .nd d will be 2a to the right, E nud e will be 4 to the right, F ind/ will be $a to the right of their true positions, etc. PROBLEMS IN SIMPLE CURVES. I. Given the tangent distance AB = d and the tangent offset BD = t, to find the radius of a curve that will pass through D ind be tangent to AB at A. Let AD = c. We have From this we have 1 - d\ . . (39) FIG. 28. and d = 4/t(2JZ - t). . . Second Solution. Let AOD = A ; then BAD = ADC = %A. Now the triangle ABD gives d - = cot $A. . . (40) Also and - = vers ^4, = cosec A. a (41) (42) (43) 66 FIELD-MANUAL FOR ENGINEERS. If d and t are given,. find A from (41), then R from (42) or (43) If t and .7? are given, find A from (42), then d from (41) or (43). Finally, if d and R are given, find A from (43), then t from (41) or (42). Example 1. Given t = 12 and d = 171.6, to find It. tan A = p^-g = .06993. . ; . \A = 4, and A = 8. Then R = _ - _!_ - --^_ = 1233.3. vers .4 .00973 Example 2.~ Given 72 = 1233.3 and d = 171.6, to find t. cosec A = -^ 8j |- = 7.187. .-. 4 = 8. Then t = 1233.3 X .00973 = 12. This problem is useful in finding points on a curve beyond an obstacle. II. To find the distance to a curve in a given direction from a given point on a tangent. We have AO = 11, AS = d, and ABP = B. tan ABO = ? . PBO = ABO - ABP; d sin OPI = sin OPB = ~ X sin ji sin sin ABO' Then 1 sm(OP2-PBO) FIG. 29. faTPBO This problem furnishes a general method of finding any desired point on a curve when obstacles preclude the usual methods. (See Problem 12.) III. Having run the curve AD, radius AO = It, Fig. 28, to find the radus R' of a curve that will puss through D', given by angle BDD' = D, and DD' = E. SIMPLE CURVES CONNECTING RIGHT LINES. 67 Let AB = d, and BD = t. Draw D'H parallel to AB. Now HD' = E sin D, and 1W = E cos D. AB' = d+ #sin D = d lt and B'D' = t - E cos D = ti. It will be observed that when AB' < AB, E sin Z> must be subtracted from d to give d* ; and that when B'D' = BH > BD, E cos D must be added to t to give ,. IV. Having run a curve of radius R and tangent T, to find the new tangent T' corresponding to a new radius R' ', or to find a new radius h' corresponding to a new tangent T' ', the central angle re- maining constant. Eq. (15) gives T' = R' tan \V\ T = R tan % V\ .'. T' - T (11' - R) tan * F, . . . . (45) or 72' R = (T' T) cot \V. '..'.. (46) Similarly, from eq. (17), we get C' C = 2(72' R) sin ^F. .... (47) These equations are of advantage for computing the change in one element, T' T for example, from the change in another, R' R for example, when the given change is small, or is an aliquot part of the element changed. Example 1. Having run the curve of radius 72 = 1910, and the central angle V= 46 12', and tangent distance T = R tan \ V = 1910 X- 42654 = 814.7, to find T' when 72 is made equal to 1900. Eq. (45) gives = 814.7 - .42654 X 10 = 814.7 - 4.27 = 810.4. Example 2. Given F = 52 04', 72 = 5730, and T = .48845 X .5730 = 2798.8, to find 72' corresponding to T' T + -- . We have R' = R + ^ = 2798.8 + 233.2 = 8032.0. 68 FIELD-MANUAL FOR ENGINEERS. V. (riven a curve joining' two tangents, to change the position of the P. C. so that with the same radius the curve may end in a given parallel tangent. Let ^47? be the given curve, and IV F'the parallel tangent. W' a shows the distance and direction that all points of the curve are moved. The curve will therefore begin at A ' and end at B f ; A A and BB', as well as 00', being' equal and parallel to YV. It is not necessary to run the tan- gent B' V in order to find the distance VV. To find this FIG. 30. distance run a line, such as BB', parallel to A V from any point on BV to meet B'V. Then make A A' = BB'. If the perpendicular offset Bh 7t is measured, we have AA = BB' = V being the vertex angle. It B'V were on the other side of .BFfrom that shown, the new tangent point A' would fall on the opposite side of A from that shown in the figure. If the new curve is required to end at a given point on B'V, we .have, then, the new tangents, length of the A'V = B'V = T', which gives the position of A (and B'), and the corresponding radius, It' = 7" cot IF, or, by (46), R' =, R + (T' - T) cot *F. FIG. 31. VI. Given a curve AB joining two tangents, to find the radius of a curve that from the same P, C. will end in a given parallel tangent. SIMPLE CURVES CONNECTING RIGHT LINES. 09 Let^F=F = T, AV' = B'V' = T' t AO = R, A0 f = R'. We liave, from the figure, R' T' 7" R=-T> r ll = E T- Also, from eq. (46), R' = R + (T f - T) cotF. Or, prolong AB to B' and measure BB'. Let ^45 = c, and A#' = c', BB' c' - c. Then, from the figure or from (17), If the parallel tangent is defined by a perpendicular offset, as B'p Ji, draw BG parallel to AO. Then Cp = BCcos BCp = (R 1 - R) cos V. .-. CB' = (R'-R) = (R'-K)coz V+U, or (1? -)(!- cos V)=7i, or (R' - R) versin V = h, or R' = R + - A The quantity added to IMn the above equations must be sub- tracted from it to find R' in the case in which V falls between A and V, that is, when T' is less than T 7 . Example 1. V -- 78, R = 954.9. T. 7 may be computed or found by Table VII to be 773.3. Let VV = 20 feet. Then R' =R+ (T' - T) cot F= 954.9 + 20 X 1.2349 = 979.6. Example 2.R 1909.9, F= 46 38'. T may be computed or found by Table VII to be 823.2. It is desirable to move the vertex from Fto V about 100 feet. Find the new radius R'. 823.2 -* 8 = 102.9; 1909.9 H- 8 = 238.7, Hence A V - 823.2 + 102.90 = 926.1, and the new radius AO' = If _ 1909.9 + 238,7 = 2148.6. 70 FIELD-MANUAL FOR ENGINEERS. VII. Given a curve joining two tangents, to find the new tan- gent points, corresponding to the same radius, after eacli tangent Las been moved any distance in the di- rection of the other. Let A Fand .Z?Fbe the given and A' V and B'V the required tan- gents. Let Hbe at the intersection of AV &ul B'V. Let VH = a, VII = b, and VV = c. We observe that B VHV = 180 - F. .'. sin VII V sin V, FIG. 32. and cos VHV = cos F. Chap. Ill, formula No. (8), gives sin F sin F HV r- f -j- COS V cos V Then VV = b sin V sinhVV We observe that the directions of F//and HV are the same as that in which the tangents B V and .4 Fare moved, and there- fore there can be no ambiguity about the direction of these lines or of FF', which is the line joining F and V. Since R and F are not changed, it is evident that all parts of the curve are moved in the direction VV and a distance equal to FF'. Hence make the angle VAA' = HVV, and A A' = VV. The curve will begin at A' and end at B' , BB' as well as 00 r being equal and parallel to W . If the distances the tangents are moved are given by perpendic- ular offsets Vh' = h f , and Vh = h, the triangles Vllh and V ' Hh' are similar and give HV HV sin VHV HV HV' or SIMPLE CURVES CONNECTING RIGHT LINES. sin V ~ h 71 Now the triangle hVV ' gives vv- - 7 ^- smHW VV and Vh are on the same side of B V; also, VV, and Vh> are on the same side of A V. VIII. Given a curve AB joining two tangents AV and BV, to change the curve so as to end at the same point as before, bat in a tan- A. A' V V' D gent inclined at a given angle A with the original tangent. Let A V- BV = T, and the new tangents A'V = BV = T'. Let AO = /?, and A'O' = II'. Draw BD = p perpendicular, and BMN parallel to A V. Let V'=BV'D=BVD+VBV'=V+A. Now An = It versin V, and A'm 11' versin V. But A'm = An', B FIG. 33. . '. R versin V = R' versiu V, R versin V R' = versin V With this value of R' run the curve back from B through the angle V, and it will end at A', tangent to A V\ A' V being equal to BV. If the-length of the new tangent is desired, we have, from the triangle VBV, Then sin V T' sin V ~ sin V P tan sin V tan \V ~ vers V 72 FIELD-MANUAL FOR ENGINEERS. If we wish to run the curve from A, we have D V = p cot V, and D V = p cot V. .-. VV =p(cot V- cot V), and AA' = AV+ VV - A'V = VV -\-T-T'. When V < 90, T increases as V decreases, and vice versa; and when V> 90, I 7 and V increase and decrease together. In all cases R increases as V decreases, and vice versa. IX. Given a curve, radius AO = R, joining the tangents A V and BV, to find the radius AO' = R' of a new curve start- ing from A when the forward tangent VB' takes a new direc- tion from the vertex. "We have and VA = R tan \V t R' = VA cotF. tan FIG. 34. tan | V X. Given a curve AB, radius AO R, joining the tangents A Fand VB, to find the change in the P.C., the radius remain- ing the same, when the forward tangent takes a new direction from the vertex. We have VA = 5 tan F; VA' = Btan^F. FIG- 35. XL Given the angle of tersection V of two tangents 4 Fand BV> tP find the radius SIMPLE CURVES CONNECTING RIGHT LINES. R and tangent distance T of u curve joining the tangents and passing tb rough the point E. 1. Let E be given by VE=l, and angle EVO = A. Let VEO = E, VOE = 0, and AO = 21, AOV = $V. No\v i v- A0 - E0 - ^A. -Jd~'VO~ sin sin E sin A cos \ V This gives E. Then = 180 - (A 4 E). Moreover, EO _ R _ sin A ~EV ~~ T " ~ sin sin A or 7? I -. sin Finally, T= 2. If E is given by VZZ" and HE perpendicular to each other, EH then tan E VII = VII' EVF = FVH - EVH = 90 - F - EVH; VII ~~ cos EVH ' With these values proceed as above. 3. Let Ebe given by VD a, IXfiJ = ft, the angle VDF being rial to FO^l = 4F. Produce i>^to 2^ and (7. Let DF = c. Then c a cos ^F, and AD = \/DE X DO. But DO = DF+FG = DF -\- EF=2DF- DE=2c-b. ~1>), and VA= VD + DA = T. .-, AD= Now It= Tcot \V, or HE = DE sin = b sin i F. 74 FIELD-MANUAL FOR ENGINEERS. D1I = b cos V. Then VH= VD DH = a - b cos \V. With these values of F/iTand HE proceed as above. XII. Given a tangent and curve (Fig. 36), to find the distance from a given point on the tangent to the curve in a given direc- tion. Let V be the point, and suppose the direction defined by the angle EVA = B. Let AV = T. We have T taniF=-. Then EVF = D VF - EVA = (90 -$V)-B = A, say. Now equation under Problem XI gives sin A sin E = -- T^pf. cos^F Then = 180 -(A + E), or, ^, and EVF = FVD - EVD = FDV - B = A. Then find sin J^, then and 7 as before. Example. R = 954.93, T 7 = A V = 350, ^L VE= 40. We have tan AVO = - 2.72837; .'. J[FO = 69 52', and AOV=V= 20 8', and #FF = 29 52' = F. - 180 - 177 50' = 2 10'; _ 954.93 X -03781 _ 70 4 .49798" SIMPLE CURVES CONNECTING RIGHT LIKES. 75 XIII. To locate a tangent to a curve of given radius R from a given point V. (Fig. 37.) v c 1 . If the curve is marked by stakes visible from the given point, a tangent can be sighted in at once. 2. If the curve is not visible from the point, run a trial tangent VB, and FIG. 3T. measure VB A arid the angle VBO = B. Chapter III, formula (8), gives sin B tan^FO = Then OV=OB. Lastly, sin B cos B and sin EVO = sin B VO" BVE = EVO - BVO. EO or sin A - cos A This gives the angle to be laid off from the trial tangent to give the true tan- gent AE. XIV. Given two curves AB and AC, radii AO = R, AOi = R\, subtending the central angles AOB = and AO^C = 0,, to find the length of the line EC. (Fig. 38.) Let A, B, and G represent the angles of the triangle ABC, and , b, and c the sides opposite. We have c = 2R sin \0, and & = 2R } sin |0,. Also, BAO = 90 - \0, CAOv = 90 - 0, .-. BAC= A = 1(0 - 0i). -, (See Chap. Ill, formula (8).) and then , a = b sin A - ~. sin B FIELD-MANUAL Foil EXGIXKEKS. If the curves are run through an integral number of stations, Tables IV and V give at once the tangent distances AK and AH and the tangent offsets BK and CH. Then, drawing CD parallel to A //to meet BK'ni D, we have CD = AH - AK = d, say, and BD = BK - Ctf= t, say. Then BC = f < and BC = BD OBSTACLES IN SURVEYING. It is often necessary to draw lines parallel and perpendicular to other lines. Hence the follow- ing problems : I. To erect a perpendicular at any point cf a line. (Fig. 39.) 1. Let J. be the point, and EC the line. Make AE = AC, and with B -and C as centers and any radius greater than AE de- scribe arcs intersecting at D or at E, or (with a different radium) at F. Any two of the points A, D, E, and F determine the per- pendicular required. "! 2. Fix any two points of the chain at B and at C. Take hold of the point midway between B and G and stretch the chain, the middle point being at D. AD is the perpendicular required. We may find, similarly, other points E, F, etc. Any two of these points A, D, E, F, etc., determine the perpendicular required. 3. Let C be the point. Take any point D as a center, and with a radius DC describe an arc BC. Prolong BD, making DH BD. CIlis the perpendicular required. For DA (A being at the mid- dle of BC) is perpendicular to BC, and, by construction, CH is parallel to AD. 4. A right angle may be obtained by laying off on the ground the three sides of any of the triangles represented in the following table, or any equimultiples of these sides, making one of the FIG. 39. 8IMPLK CURVES CONNECTING RIGHT LINES. 77 sides adjacent to the right angle (a or b) coincide with the line. Let c = the hypothenuse. No. u. b. c. No. a. b. C. 1 3 4 5 6 20 SI 29 2 5 12 13 7 12 33 37 3 8 15 17 8 9 40 41 4 24 25 9 11 60 61 5 10 24 26 10 13 84 85 B Thus, in Fig. 40, using 70 links of the chain, hold the first end, also the end of the 70th link of the chain, at A, the end of the 21st link at B, and the end of the 50th link at C. If in the three expressions ra 2 ir, 2mn, and m 2 -j- ifi we assign to m and n any values at pleasure, in being greater than n, we will have sets of numbers representing the sides of right- angled triangles. In that way the above numbers were found. Equimultiples of any set of the above numbers will represent the sides of a right-angled triangle. II. To let fall a perpendicular from a given point to a given line. Let //in Fig. 39 be the point. Measure any line HB to the given line. At the middle of ///> take D as a center, and with a radius DB describe an arc BC. HO is the required per- pendicular. For continuing the arc to II, we see that the angle BCIIis inscribed in a semicircle. III. To let fall a perpendicular to a line from an inaccessible point. Let BC (Fig. 41) be the line, and P the point. Let p represent the perpen- dicular PK. Then BK p cot B, and CK = p cot C. BK _ cot B '' CK ~ 21 Fm. 40. FIG. 41. and BK cot C' cot B cot B -j- cot C ' 78 FIKLD-MAXUAL FOR EXGTXEKRS. Since BK -\- CK = BC, we have BK = BC cot B -f cot * Tills gives the foot of the perpendicular K. If BC is taken equal to 100 or some small multiple of 100, BK is very easily found. This problem is particularly useful in locating important ob- jects, such as mills, warehouses, bridges, etc., on one side or the other of a railway survey. In this case B and C represent stations or points on the survey, and the angles at B and G can be measured and recorded while the instrument is set at B and at C. The simple divi-ion required to find the position of .STcan be made at any time. Of course the point Pis located graphically by drawing .Z?Pand CP. IV. To prolong a line AB (Fig. 42) past an obstacle and to measure its length. This is easily done by perpendicular offsets, a method to > familiar to need description, but not the best way. B FIG. 42. Or, measure BC in any convenient direction, and at C deflect any angle FCD. Draw BD and the perpendicular CH. The angle CDS = FCD - CBD. Hence sin FCD also = ^^ sin BBC' If the angle BCD is made equal to 90, then CD = BC tan CBD, and BD = BC -f- cos CBD. SIMPLE CURVES CONNECTING RIGHT LINES. If the angle FCT) is made equal to 2CBD, tlien CDS = FCD - CBD = CBD. Hence CD - BC, and BD 2BH = 2BC cos CBH. OH is tlie departure of the line BC, or of DC, from the line ABD. It is also the approach of the line CB, or of CD, to ABD. If necessary more than one course may be run away from the main line ABD, and more than one in returning to it. To recover the main line it is only necessary to make the sum of the approaches equal to the sum of the departures. The distance measured on the main line is obtained as above. V. Obstacles to Measurement. Methods have been pointed out in connection with Fig. 42 for rinding the length of obstructed lines when the ends are accessible. When inaccessible the following- problems apply. A. When one end of the line is inaccessible. (Fig. 43.) 1. Let AB be the line to be measured, across a river for ex- ample. Measure AC in any convenient direction, and the angles at A and C. Then AC sin C AB = B sin B = AC-. sin C sin (A + C)' 2. If the angle ACS is made equal to half of DAC, then OB A = CAD - BCA .'. AB - AC. FIG. 43. 3. Or, in Fig. 43, make the angle BAG = 90. Then AB = AC tan ACB. If, in this, AC = 100, or some simple multiple of 100, which is usually easy to effect, the formula requires no computation whatever. 4. If ACB in Fig. 44 is made equal to 45, AB AC. 80 FIELD-MAXUAL FOR EXGIXEERS. 5. If at G we make the angles ACB and ACD equal, we liavo AB = AD. When the river or other obstruction occurs on a continuous survey, as a railway survey, AD is a measured line, and this method gives AB = AD without any computation whatever. FIG. 45. 6. In Fig. 45, AB being the distance required, run and measure any line AC and measure the angle BAG A. Make Then ACB = 90 - A = C. AB = AC sin C. B. When both ends of the line are inaccessible. (Fig. 46.) Let AB be* the line to be meas- ured. Find the distances from the A- point G to each end of the line A and B by preceding methods, and measure the angle G. Then sin G tan A = AC (see Chapter III, formula (8).) C FIG. 46. . sin A C. To erect, at a given point A (Fig. 47), a line ylZf perpendicu- SIMPLE CURVES CONNECTING RIGHT LINES. 81 \ lar to an inaccessible line BC, and to draw a parallel AH to the same line. Find .47? = c and AC b by pre- ceding methods. Then pin A tan B = - . - cos A I) Now draw A K, making BA K = 90 - B. A K will be the required per- pendicular, and AH, making BAH B, will be the required parallel. D. To find the length and relative position of an inaccessible lino, AB, from an accessible line, CD, separated n from the former by an inaccessible space. - Measure CD and the angles at C and D. Example. Let CD = 4000; BCD = 126 25V; ADC = 47 53 V; ACB= 3 10'; ADB= 3 01'. .-. ACD = 129 35V; SJ)C = 50 54 1'. Also, CAD = 180 - ACD ADC 2 31'; CBD = 180 - BCD - BDC = 2 40'. Hence 40= 400o5!2*a.' = 6781.; BC = 4000 tan CAB = sin 2 40' sin C = 66728.8; 'AC FIG, 48, Finally, CAB = 75 28V. ^sin 3 10' -, - 3807.8. 82 FIELD-MANUAL FOH ENGINEERS. If AD is accessible, it can be measured as a check on the com- putation. Such is the case when it is a tangent of a railway sur- vey adjacent to the inaccessible space. The data of this example are taken from an actual night survey across an inaccessible sea-marsh. Rockets were thrown and lights then exhibited at A. and B. which were observed with transits from the tops of towers at C and D. The computed and the measured length of AB agreed within a few inches. The best method of making a preliminary railway survey through a wooded region is by a suitable adaptation of the method of traversing, which we will now explain. So far as known to the author, this was first devised by him in 1869, and used for him by his assistant, Prof. J. B. Davis (now of Michigan University) in making the preliminary surveys of the Owosso and Northwestern Railway. Suppose we wish to run from A in the direction of ABF, which we will call the base line; and upon which numerous obstacles, such FIG. 49. as trees, occur, making it necessary to run the line AB i CiD i E [ F, called a traverse. The deflection angles at Bi, C\ t D iy pud E^ are supposed to be small. From Bi , d , D\ , and Ei draw perpendiculars to AB, and from Si, C>, and D l draw parallels BiK, dL, and D,P to AB as shown. Prolong ABi to R, and B } Ci to 8. The course of a line is its direction with reference to the base line. The departure of a line is the distance that a point recedes from or approaches to the base line in moving from one end of the line to the other. We have D 1 L CiDi sin DiCiL. Now since the sines of small angles vary nearly with the angles or with the number of minutes in the angles, we see that the departure of a line varies SIMPLE CUHVES CONNECTING 1UGHT LINES. 83 nearly as the product of the length of the line by the number of minutes in the course. The departure of a point is its distance from the base line. Thus the departure of Z>, = DD,. The departure of the end of a line, as B\ C\ , inclining from the base, is equal to the departure of the beginning of the line plus the departure of the line. Thus Cd = #77, + f,7i. The departure of the end of a line, as C\D\ , inclining toward the base, is equal to the departure of the beginning of the line minus the departure of the line. Thus DDi = CC\ DiL. The departure of the end of the line D\E\ which crosses the base line is equal to the departure of the line minus the departure of the beginning of the line. Thus EE l = E,P - DD t . The record of the survey can be conveniently kept, as shown in the following table, the columns of 'the transit-book serving the purpose perfectly. Angles turned. Angles with Main Line. Departures. Stations. Each Course. Total. Left. Right. Left. Right. Left. Right. Lcfr. Right. A = 10 10' 10' 11 12 ,= 13 22' 32' 3000 3000 14 15 Ci = + 20 4'2 X 10' 7040 10040 16 17 18 Z>!= 19 3C' 40' 3800 6240 21 E,= 24 120' 40 20000 137GO 25 26 27 F =+44 13760 .00 .00 The deflection at station 10 is 10' R.> and at station 13 it is 22' R., etc. The course from 10 to 13 is evidently 10' R. ; from 13 to 15 -f- 20 it is 10 -f 22 = 32' I*.; from 15 + 20 to 19 it ig 42 - 32 = 10' L., etc. 84 FIELD-MANUAL FOR ENGINEERS. From 10 to 13 the departure is 300x10 = 3000 foot-minutes. " 13 " 15+20 the departure is 220x32 = 7040 " 15+20 to 19 " " "380X10:= 3800 '< 19 to 24 " "500X40=20000 " etc. The aggregate departures are : At 13 ................................ 3000 R. At 15 + 20 ...... 3000 + 7040 = 10040 R. At 19 ............ 3000 + 7040 - 3800 = 6240 R., etc. The distance necessary to run from a given station on any given course to reach the base line is found by dividing the departure at that station by the course. Thus from station 24 forward the course is 40' R. Then 13760 -4- 40 344 feet, showing that the auxiliary line (E^Fin the figure) will reach the base line 344 feet beyond station 24, or at 27 + 44. The figure represents a main angle at F, the forward tangent being FIT, and which may be followed approximately the same as AF WAS followed. Let I = the length and d = the departure of any line, and n the number of minutes in the course. Then d = I sin n' = In sin 1' = .0002909fo. Since In is given in the last two columns, the departures in feet are found by multiplying the quantities in these columns by .0002909 or .00029 nearly. We observe that 1 4- .0002909 = 3438 nearly. Hence the departures given in the table (in foot-minutes) divided by 3438 will give the departures in feet. A rough approximation for the purpose of keeping sufficiently near the base line on sideling ground is generally all that is needed. This being the case, it is not in general necessary or advisable to find the total departures, except when it is desirable to " run for the base " preparatory to turning a main angle. Thus to find the departure at station 24. The sum of the product to the left is ____ ............. 3800 + 20000 = 23800 and the same to the right is ........... 3000 + 7040 = 10040 The difference is .................................. 13760 L. We have AR = AB, cos ~ 4W - <-s #4#.) = AB vers BA SIMPLE CURVES CONNECTING RIGHT LINES. 85 For BAB l = 2 34' this becomes AB, - AB = .WlAB nearly. This shows that ABi exceeds the true distance measured along the base by only one thousandth part of its length for an angle of 2 34'. If greater accuracy than this is desired, the angles between the auxiliary line and the base line, or the "courses," may usually be 'made smaller than 2 34'. Since the error is approximately as the square of the number of minutes in the angle, for 1 1?' it is nearly .00025, or nearly 1 in 4000; and for 38' " " " .00006, " " 1 " 16000, etc. To find the angle in minutes between the base line and a line joining any two stations. Divide the difference or the sum of their departures, according as they are on the same or on opposite sides of the base line, by their distance apart. Thus the line BiDi makes with the base line the angle PA - BB, _ 6240-3000 _ _ BD WO The line AE, makes with the base an angle In platting, the auxiliary lines are penciled only, so as to plat observed objects in proximity to the line necessarily observed from the auxiliary lines. When these objects and the base lines are mapped the auxiliary lines need not be retained. This method yields quite accurate results when the angles be- tween tbe lines of the survey are 2 or 3, as we have seen. For perfect accuracy, however, use the following method : Problem. Having run a broken line ABCD, to find the angle between the first course and the direct course JIT). Represent the lines run, in their order, by a, &, and c, and the deflection angles at FIG. 50. B and at C by B and C respectively. Draw CR parallel to AB, and (7/7 and KDK perpendicular to AB. Angle DCR = DCE - RCE ~ C - B, 86 FIELD-MANUAL FOR ENGINEERS. Now Now BH=bcosB; HK = c cos (C - B) ; AK - AB + BH+ HK CH = b sin B; DR = c sin (C - B); DK = CH - DR. DK 4jr Also AD = AK -*- cos DAK. Of course the method is applicable whatever the number of lines run. All but the last line could usually be taken equal to a whole number of chains, which would reduce the required computation to a simple mul- tiplication. E. To find the angle of deflection, i", between two straight lines A V and VM, when the point of intersection is inaccessible; and the dis- A M tances of the intersection from FlG - 51 - given points on the lines. 1. Run and measure a perpendicular PK to one of the lines. Measure the angle VPK = P. Then V= 90 + P; KV= A'Ptan F, and PV - KP t- co* P. 2. Run and measure any line PL from one line to the other. Measure also the angles VPL = P' and PL V =i /,. Then V= P' + L. Hence, in the triangle PVL, PL and the angles are known, to find FPand VL. 3. If obstructions prevent the use of the former methods, run and measure any broken line ABCD. Prolong AB and BC to meet VM at M and N. Measure the deflection angles CBM = B, DCN = C, and CDN = D. Let AMV = M, and CNV = N. Then sTiHv ' tto.(0+J5' = BC -f CN. Also M = W - B- C -f D - /?; SIMPI.I-; CURVES CUNXKCTI.NI.T RIGHT LINES. BM = sin N AM = AB + BM. Now we have ^43/arid the angles at A and M, to find vl F, MV, and the angle F = A 4- Jf. A similar explanation will apply to any case. Wlie.n a broken line must he used, the above method involves fewer computations than any other. F. To locate a curve joining two tangents when the vertex is inaccessible. Find by the last problem the distances Va and Vb to convenient points on the tangents, and the / \ \, V angle V. Then assigning or com- puting the tangent T = A For BV from, the radius, we have and aA = T - a V, bB = T- bV. We now have the tangent points and can run in the curve as usual. GK To locate a curve of radius U or tangent T when the vertex, the beginning, and the end of the curve are inaccessible. Find, as shown with Fig. 52, the angle F and the distance Fa to any point, , on A V. Then a A = T - a F, and cd a A FIG. 53. Also ac = Ad = J?versin AOc. Drawing the tangent cb, we have abc = AOc, or acb = 90 - AOc. This gives the direction of the curve at c, and it may be run in each way from c. To pass from any point c on the curve to any point n on the tangent. We have 1an (dcu = an<<) = 88 FIELD-MANUAL FOR ENGINEERS. Set the instrument at c and turn off from the tangent cb an angle ben = bed den = dbc anc, , Us IV and measure en = . cos anc H. To find any desired point on a curve when obstacles precl udo the use of ordinary methods. (1.) In Fig. 28 measure any convenient tangent distance AB = d. Then, as shown in Problem 1, eq. (43), cosec A = ; then t d tan A. d and t give the point D on the curve. It is important to note that if AB is made equal to one half the long chord for any number of stations given by Table IV, BD = AC is the corresponding middle ordinate and may be found in Table V. Example. Let AD be a 4 curve, and AB one half the chord of four stations = ^ = 199.35. Then Table V gives BD = 13.94. (2.) Problems 2 and 12 of this chapter furnish general methods of overcoming obstacles on curves. (3.) We can find points on the curve as follows: Let b be a station near the obstacle. Deflect from the tangent at b some small multiple of the deflection angle for one station ^D, giving the line bd. The length of bd may be taken at once from Table IV and measured off, giving d, a station on the curve beyond the obstacle. Taking bm = \bd, and measuring off the middle ordinate me taken from Table V, gives also a point c on the curve. If more convenient, make bx = cm, and xc = bm, which also gives c. Again, run the tangent b V = d' any jr IG 54 convenient distance. Then SIMPLE CURVES CONNECTING RIGHT LINES. 89 Deflect at V an angle equal to 2bOV, and make V'd = bV. d will be a point on the curve. The number of stations from b is equal to bOd 2bOV bV sliould usually be taken equal to a whole number of chains, 7) in which case ^- f is very readily found. The lines bV" and V"d lying on the inside of bd may be run instead of bV and V'd. CHAPTER V. LEVELING, STADIA MEASUREMENTS, ETC. THE field operations in connection with the level are more simple than those required with the transit, but they require greater skill and facility in manipulation in. order to produce correct results. It is to be observed that the elevation of points is a relative matter. The elevation of some point, from which all others are to be found, is arbitrarily assumed to be 100, or some other number sufficiently large, so that the elevation of all points to be considered will be greater than zero. Near the coast, and in fact wherever practicable, it is im- portant to refer the levels to the mean level of the sea, calling this zero, or 100, or some other number, taking care to estab- lish from it some convenient and permanent reference-point called a bench-mark, or bench. All points having the same height as this bench are some- times said to be on a level surface called the datum. This, however, makes no difference with the work and serves no use- ful purpose, and need not be considered. All elevations thus found become of much importance in determining the relative elevations of the country, and in the construction of physiographical maps, etc. Having established the first bench, and recorded its elevation, the rod man stands squarely on both feet behind the rod, and rests it on the bench as nearly in a vertical position as possi- ble, which is best done by simply steadying it with the thumbs and fingers, taking care not to grasp it. The levelman sets up his level, preferably in the direction that the line extends, in any position from which he can well see the bench, as well as points to be afterwards observed. He then makes sure that the instrument is in adjustment, and is focused; levels it carefully and sights to the rod. He 90 LEVELING, STADIA MEASUREMENTS, ETC. 91 may plumb the rod laterally by means of the vertical cross- wire of the level, and the rod may be waved gently on each side of the vertical toward and from the instrument, the short- est reading being the true reading. The line of sight on the rod covered by the horizontal cross- wire is then on a level with, or at the same height as, the wire itself, and the latter is therefore higher than the bench by the distance intercepted on the rod between the line of sight and the bottom of the rod. This is called the reading of the rod, or simply the reading. Adding this reading to the height of tho bench, we obtain the height of the cross-wire, technically called the height of instrument, and designated by the initials H. I. Having obtained the height of instrument, the elevation of any other point upon which the rod can be read can be found by taking a reading of the rod upon it. Of course the point is below the instrument an amount equal to the reading, which must therefore be subtracted from the height of instrument to give the elevation of the point. The elevations of any number of points may be thus obtained. In order to obtain the elevation of points above the instru- ment, or below it more than the length of the rod, the instru- ment must be moved from its present position to one higher or lower as the case may require. Before the instrument is moved to a new position a temporary henfJt, tailed a turning-point (and designated by T. P. or "Peg"), must be established and its elevation ascertained with care, since any error in its elevation is carried forward through- out the whole line of levels. A turning-point must be firm and definite and not easily disturbed or lost. A small stake or " peg " driven with its upper surface about flush with the surface of the ground is generally used. The top of a rock may well serve the purpose. Benches and turning-points are of course the same in prin- ciple, but the more or less permanent point taken as the basis of the elevations of the Survey, and also those made usually along and near the line, for future reference, whether used as turning-points or not, are usually called benches. From this new turning-point we proceed precisely as before, by getting a new height of instrument, etc., and it is important FIELD-MANUAL FOR ENGINEERS. to note that the operation just described, of obtaining a height of instrument from a bench or turning-point, and then obtain- ing the heights of any number of desired points within range of the instrument, including a new bench or turning-point, includes the whole subject of leveling. Since the cross-wires must be higher than any point upon which a reading is taken it must be remembered that: 1. The reading on a point, added to its elevation, gives the height of instrument. 2. The reading on a point subtracted from the height of in- strument gives the elevation of the point. In other words: We must add a reading (to the height of some point) to get a height of instrument, and must subtract a reading (from a height of instrument) to get the height or elevation of some point. The theory of leveling requires, therefore, only a simple application of addition and subtraction, and it is not easy, it would seem, to go wrong in it. Station. + 8 H. I. - S Elevs. Remarks. EM 1 Peg 2 3.46 203.46 7.29 5.34 0.81 200.00 196.17 198.12 202.65 W. Oak 60 ft. R. of Station + 40 8 4 5 Peg + 60 6 4.17- 6.18 206.82 211.93 1.12 3.16 6.09 4.14 1.07 3.13 205.70 203.66 200.73 202.68 205.75 208.80 The accompanying table shows a convenient form of field- book for keeping the level notes of a railway or other survey. The first column contains the stations and benches. The second the plus readings taken on points whose elevations are assumed or already determined. The third column contains the heights of instrument recorded one line below the elevation of the turn- ing-point (or bench) from which it is calculated. The fourth column contains the minus readings. The fifth column con- tains the elevations of all points observed. The right-hand page is reserved for remarks describing the benches and their location, also objects crossed by (or near) the line, as roads, streams, ditches, etc. LEVELING, STADIA MEASUREMENTS, ETC. 93 It is to be observed that for any series of levels the sum of the plus sights less the sum of the minus sights (omitting those for determining intermediate points on the ground) is equal to the difference between the first and last elevation. Thus to prove station 3, we have 3.46 -f 4.17 0.81 3.16 = 203.66 200 3.66. To prove the H. I., 211.93, we find 3.46 -f- 4.17 4- 6.18 0.81 1.07 = 211.93 _ 200 11.93. In practice it is best to check each page of the field-book by comparing, as above, the first turning-point or height of instru- ment (brought over from the preceding page), with the last turning-point or height of instrument on the page. To facilitate this work some engineers use two columns for the minus sights, placing those which determine the turning- points in a column by themselves. This practice is commendable. Benches should be established at short distances apart along the line, taking care to locate them, so far as possible, near the crossings of roads, streams, railways, etc., and at all points where their need can be foreseen, in the location of cattle- guards, culverts, bridges, etc. Of course an extra-good bench should be established at the end of the survey. An extra-good turning-point or bench should also be established at the end of each day's work. The object of obtaining a line of levels is- to furnish a profile of the line surveyed, showing the undulations of the surface over which it passes. The elevations are platted on profile paper, the horizontal scale being about 400 feet to an inch, and the vertical scale about 25 feet to an inch. This distortion of scale magnifies the vertical measures about ****/.,?, = 16 times, so that the slight changes in the elevation of the surface may be distinctly seen. In running a line of " flying " levels no readings are taken except on turning-points. If the difference of levels of the ex- treme points only is desired, it is necessary to find the differ- ence only between the sum of the plus and of the minus read- ings, as already explained. This is very convenient for testing 94 FIELD-MANUAL FOR ENGINEERS. a line of levels already run; in which ease it is best to touch on the benches only, and if found correct, the intermediate ele- vations may be regarded as correct also. No line of levels should be taken as correct, and so used, without first being carefully checked. The Philadelphia rod is the most convenient and best rod in use. It is plainly lettered and easy to use, and may be read by the levelman when desirable, and at a distance of sev- eral hundred feet. To Locate a Level Line. Set a peg at the desired height, as a starting-point, and take a reading of the rod thereon. Send the rod forward in the desired direction, and have it moved up- ward or downward along the slope of the ground until a point is found which gives the same reading as before. Of course the reading is taken on a peg. This second peg is of the same height as the first. Find in the same way :i third peg from the second, etc. In this way stakes may be set at points on the ground level with the top of a proposed dam, or with the supposed top of water flowing over the dam. Then joining these stakes by lines, the area thus inclosed may be measured. The water behind a dam is not level, but is curved con- cavely upward and so increases in height back of the dam, and sets back farther than if level. For the subject of backwater, Works on Hydraulics must be consulted. Other applications of the level line are to obtain " contour lines " for topographical maps, for levees in irrigated rice- fields, etc. To Run a Grade-line This consists in setting a series of pegs so that their tops shall be points in a line, which shall have any required slope ascending or descending. First drive pegs at each end of a line to the heights required. These heights may differ by a given amount, or this difference may be undetermined. Set the level over one of the pegs and measure the height, a. of the cross-wires above the top of the peg. Set the rod on the other peg, and make the reading on the rod equal to the height a. Without disturbing the level drive any desired number of pegs LEVELING, STADIA MEASUREMENTS, KTC. 95 along the line, so that the reading on each will also be equal to a. A line of uniform grade or slope is not a straight line. Calling the globe spherical, this line when traced in the plane of a great circle would be a logarithmic spiral. On a length of six miles the distance of its middle point from the middle of its straight chord would be six feet almost exactly. CORRECTION FOR THE EARTH'S CURVATURE AND FOR REFRACTION. This is necessary for long distances. Let AB (Fig. 55) represent a portion of a section of the earth's surface. Then if a level be set at A, the line of sight of the level will be the tangent AD, while the true level will be the arc AB. The difference BD between the line of sight and the true level is the cor- rection for the earth's curvature for the distance AB. This must be sub- tracted from the reading of the rod at B, or, what is the same thing, added to the height of B, as given by the reading of the rod. Let AE = R, AB = 1), and BD = E. By geometry, AD* = BD(BD 4- A& BD = FIG 55. BD + Omitting BD in the right-hand member, since it is small com- pared with 2R, and supposing AD AB = J), we obtain 22t 2 X 20913650 r x = .0000000239087) 5 . . (1) This formula gives a result or value for E slightly too small; but the relative error is only about one in 24,000 for a distance of ^ Q = 39.6 miles, or arc of 5 - m = 34' 22".65. 96 FIELD-MANUAL FOR ENGINEERS. In observing distant objects, a ray of light traversing tlie air from an object to the eye or instrument is refracted, and takes a curved path which, for points near the surface of the earth, is practically the arc of a circle, concave downward, and whose radius is 7.K. Thus a point at C (Fig. 55) would appear at D higher than it really is by an amount CD. This may be found from the above formula by substituting 7R for 2L Hence the correction for refraction is E' = ~_ = . 00000000341 5D 2 ..... (2) The correction for curvature and refraction is IP D 2 3 /) 2 E" = C=BD-CD=: - ~ rt = - - = .000000020492D 8 . (3) This must be added to the apparent elevation of the observed object to give the true elevation. Table XI gives the value of the correction for the value of R = 20911790 feet. When it is possible to set the level midway between the points whose heights are required, the corrections will balance each other and may be omitted. The above equations may be put into a form sometimes more convenient as follows : The length of arc on the earth's surface subtending angle of one minute is Then ~ = .12638 (!') = the correction for refraction for distance 6083 ft. or arc of 1'. Also ~~ = .8846 (2') = the correction for curvature for the same distance or arc 1', LEVELING, STADIA MEASUREMENTS, ETC, 97 and i 1 .75828 (3') 7/t = the correction for curvative and refraction. TRIGONOMETRIC LEVELING. First Method. When the point C (Fig. 56) can be seen from two points A and B on the same level, then AD - CD cot CAD, and BD = CD cot CBD, Subtracting gives AB AB=OD(cot CAD- cot CBD), or ^= calCAD _^ cB1) - W Second Method. Let A and B (Fig. 57) occupy any positions C B FIG. 56. except in line with C. Measure AB and the angles at A and B\ also the angle of elevation GAD. C=1SQ-A-B, AC = sin C , and CD = AC sin CAD. (5) If A, B, and C are in the same vertical plane, the solution is in no wise affected. Unless the distance AD = D is short it is necessary to add to. the correction found by the preceding formulas the correction O 7^0 for curvature and refraction, namely, .0000000204927) 2 , FIELD-MANUAL FOR ENGINEERS. To find the height of instrument by an observation of the horizon (Fig. 58). First Method. Let C be the place of the transit, and BAA a portion of a section of the earth's surface. Were there no refraction the line of sight would be the tangent CA\ ACE C would be the angle of de- pression or dip ; and we would have BC = BOX exsec CO A = It exsec C. Owing to refraction, how- ever, the line of sight would be a curve concave down- ward whose radius = 7/; and it would therefore ex- tend from C to a point A', say a distance A A' beyond A. Draw CDF tangent to this curve at C to meet the ra- dius AO prolonged in F. Draw also the tangent A't. Let COA=N, COU=I1', and CO A' = 0. Then FIG. 58. CO = r sec // = cos H' Let E, on A'O prolonged but not shown, be the center of the arc A'C. Now #0V_ CO 9 cos = - cos COE = - 2CO . EO - ___ TsT^eclT^ 12 Clearing, substituting 1 vers for cos 0, 1 vers H for cos II, and 1 -f exsec 7?" for sec H, we find 13 vers 13 vers // + exsec Jf. LEVELING, STADIA MEASUREMENTS, ETC. 99 Or, writing versines for exsecants or vice versa, we have vers = vers II, or, approximately, exsec = exsec H. In the triangle CEO, 49 r 5 -f 36 ?* - ~U6 l _ 85r 2 - r 2 sec 9 H _ 85 - sec 2 H 2~x~ 6r"x~7V * 84?" ~~84~~ Also sin E = sin .-. sin #sin = sin 2 0' ---- = (1 cos' 2 Oj _ 1 70 - 169 cos 2 H - sec 2 H sec H ~l44~ ~~T"~ Now since OA' is tlie prolongation of EO, and JS'C'and OZ) are perpendicular to CD and therefore parallel, we have CO A - 1)0 A' = COD, or - E = II'; .-. cos H' = cos (0 E) = cos cos E -\- sin sin E. Substituting in this the above general values of cos 0, cos E, and sin 7? sin 0, and expanding and reducing, we find 13 cos E + sec H , (sec H cos //) Putting cos H= 1 vers //, sec 7/= 1 + exsec IT, etc., we find (exsec // vers H) vers /f = f vers 77 -- , . . (a) or vers 77' = | vers 77, very nearly, ...... (6) or exsec //' = f exsec //, very nearly ...... (7) exsec H ' vers //' Hence vers //' = f vers H -- . i& From this we have , (exsec H' ver s H'} vers H \ vers //' -f ^ , . (ft) 100 FIELD-MANUAL FOR ENGINEERS. or vers //= | vers //', very nearly, ..... (6') or exsec H= | exsec II', very nearly ...... (7') Since the versines of small angles are very nearly in the ratio of the squares of the magnitudes of the angles, we have, from (6), H 4/JZT = 1.08//', approximately. . . . (?') T Supposing r = 4000 miles and AB = 50 miles, then it is 80 easy to show that the error of eq. (7) is less than .00000002, and the error of (6) is about .0000000004. Ki' |0, or sec 2 a > sec 2 10. Hence the fraction is, by the double approximation, increased sliglitly more than it is decreased. Hence TV W T)' W < cos 8 a ; but cos* a, almost exactly. . (12) o o The total relative error is e sin 2 ^0 (sec 2 a sec 2 4;0) = sin 3 (tan 2 a tan 2 0). 104 FIELD-MANUAL FOR ENGINEERS. Since tan 1 ^Q is very small compared with tan 2 a, we have e = sin 5 -|0 tan- a, very nearly, = .000012 tan 2 a, very nearly. Hence D'E' = cos 5 a(l - .000012 tan 2 a) o =: cos 2 a .000012 sin 8 a (13) Since sin 9 = tan ^0 = = .005. very nearly, the quantity neglected above is only (.005) 4 = .00000000625 and does not come within the range of the table. The above is the coefficient of reduction by which to multiply the observed space DE S in order to produce the true space D'E' which would be observed at the same distance if the line of colliination were horizontal. Hence we have, from (10), I = (1005 + c +/)(cos 2 a - .000012 sin 2 a). By eqs. (25) and (26), Chap. Ill, 1 4- cos 2a cos 2 a = ~ SB . J and Bin' a = Hence I = (100S+ +/)l - ~- - .000013 , . (14) I = (100/8 + c +/) (l - ver * 2(l \ very nearly. . . . (14') \ / These coefficients may be read from a table of versed sines without any computation whatever. The last equation is quite accurate enough, but the coefficients of Table XIII are calculated by the exact formula. JSxample. Find the coefficient for a = 7 20'. We write half the vers 14 40' = .01629 and subtract from unity and find coefficient .98371 Another method of procedure is that in which the rod is held perpendicular to the line of collimation. LEVELING, STADIA MEASUREMENTS, ETC. 105 To secure this position of the rod a bar is attached to it having sights upon it, through which the rodinan watches the instrument during an observation, the line of sights being perpendicular to the rod. FIG. 61. The horizontal distance of the point B from the instiument is IE = IK + KH = 1m cos a + Bm sin a, or IE = (1005 + c +/) cos a + r sin a. (See Fig. 61, in which r is the reading of the rod by the line of collimation.) The elevation of B above /is BH = mK Bm cos a, or BH = (1008 + c -f /) sin a - r cos a. . . . (15) When the distances are sufficiently great correction, must be made for curvature and refraction as already pointed out. THE GRADIENTER. This attachment consists of a screw working against a clamping- arm suspended from the horizontal axis, on the opposite end from the vertical arc. A strong spiral spring presses the arm against the end of the screw. The large silvered head of the screw is usually divided into 100 equal spaces. When the screw is turned the head moves along one division of a small silvered scale for each revolution of the screw. When the regular clamp of the telescope is free the telescope may be revolved; but when the telescope is held by this clamp it - = - r - = 2 _ - - ' - I - . - - . ' Let -i ^ ,----- - T ^ , -* ^ :: - >. . ., j - : - - 108 FIELD-MAKUAL FOR ENGINEERS. F, and the horizontal lines AK and CO to meet BF in .STand 0. Join AB, and drop the perpendiculars aimibi and a 2 ?w a & 2 . Let 7i = the number of stations in AC or CB. Then, since these dis- tances are measured horizontally, we have AH = HK, and there fore AD = DB. The vertical line CD is therefore a diameter of the parabola, and the distances of points on the curve in a vertical direction from corresponding points on the tangent AF are in the ratio of the squares of the distances of these points from A. Also, .: CE=ED. Now, since CF = CA, FO = CH= the rise of AC in n stations = ng. Also, OB = the fall of OB in n stations = ng' ; .'. FB = n(g+g'}. Now, since B is %n stations from A, we have - r - r = ; 9 , 4n* 4n Offset at first station from A a, mi = a = (18.1 The value of a\m,i being determined, the distances of the curv<< at all points from the tangent ^L^are also known. Thus a^m-2 = 4a, Ce = 9a, etc. It is easy to calculate the heights of the points of the curve above AK, though these heights are scarcely needed. Thus = g a ; EH = CH - CE - 9a, etc., etc. LEVELING, STADIA MEASUREMENTS, ETC. 109 Finally, BK = FK- FB = 2ng n*a = 2ng - n(g + g'} = n(g - g'}, or BK= CH- OB = ng - ng' = n(g g'} as before. The successive grades are found by taking the successive differ- ences of the heights just found. Thus mibio = g a ; m^mibi = g 3 ; The change in the grades from station to station, which is the same as the chord deflection of the curve, is, we observe, equal to the constant quantity 2a. Second Proof. The curve must change its direction in its length of 2n stations an amount equal to g g', and therefore in each station it must change an amount equal to ; and 2n this is equal to 2a by eq. (18). Third Proof. The ordinates to the curve are a, 4, Qa, 16, etc. The differences of ordinates are 3a, 5a, 7, etc. .-. the differences of grades are 2a, 2a, etc., a constant. Fourth Proof. Let m 2 represent the nth station, and prolong the chord m^m-i to meet CE in x (not shown). Now awi = (n 1)X a^m* = tfa, and CE (n But Cx + a 1 m l = 2a- t m- l , or Cx = 2a^m y a-^m^ = [(n -j- I) 9 2]a. Hence Ex = CE Cx ~ 2a. In finding the value of a, etc., it is necessary to know when we are to take the sum of the grades, and when the difference; for there may be four combinations of two adjacent grades. It is only necessary to observe : (1) That when the grades are both ascending or descending FB is equal to n(g g'}, g being the steeper of the two grades, and therefore _ n(g - g') _ g_--_jf_ W in ' 110 FIELD-MANUAL FOR ENGINEERS. (2) That when one of the grades is ascending and the other de- scending, FB = n(g + g'), and a = 4n If one of the tangents is horizontal, let g = grade of the other and then a = ~-. 4n In all cases it is necessary to consider only the change of grade at C; which we will represent by G. Then & = ff - ff', or G- = g + /, according as the grades are both rising or falling, or one rising and the other falling. Then FB G rB = JiG, and a = - = 4n? 4n These formulas apply without change to points between stations, as the following examples will illustrate. Example 1. A .9$ up grade joins a .3$ down grade at station 76 at an elevation of 94.0. Find ordinates, etc., for a curve 6 sta- tions long. a = ' 9 ^/ 3 = .1. (See Fig. 64.) Stations on AF 73 74 75 76 77 78 79 Elevations on AF 91.3 92.2 93.1 94.0 94.9 95.8 96.7 Corrections 00 .1 .4 .9 1.6 2.5 3.6 Elevations on curve.... 91.3 92.1 92.7 93.1 93.3 93.3 93.1 It is necessary to compute the ordinates a, 4a, etc., for one half the curve only, since they are the same for corresponding points on each tangent. Thus we have : Stations on AC and CB 73 74 75 76 77 78 79 Elevations on AC and CB 91.3 92.2 93.1 94.0 93.7 93.4 93.1 Corrections 0.0 .1 .4 0.9 .4 .1 0.0 Elevations on curve 91.3 92.1 92.7 93.1 93.3 93.3 93.1 LEVELING, STADIA MEASUREMENTS, ETC, 111 Example 2. A .2$ up grade is continued beyond station 17, whose elevation is 46.4, by an .8$ up grade. Find ordinates, etc., for a curve 6 stations long. Stations on A C and CS. . 14 15 16 17 18 19 20 Elevations on AC and CB 45.8 46.0 46.2 46.4 47.? 48.0 48.8 Ordinates 0.0 .05 .20 .45 .20 .05 0.0 Elevations on curve 45.8 46.05 46.4 46.85 47.4 48.05 48.8 To find the grade of the parabola at any point. Let the curve be that of Example 1, and let the point be at a\ -f 40, or 140 feet from A. Elevation at a, -f 40 = 92.2 -f .40X.9 = 92.2+ .36 = 92.56. Also, (1.4} 2 a = I Ma = .196 = .20, nearly. Therefore elevation of curve at a, -f 40 = 92. 86. The special need of these curves is in sags in the grades to pre- vent the breaking of the train. According to Wellington, Railway Location, page 365, vertical curves in sags should be at least 200 feet long, or 100 feet on each side of the vertex, for each tenth in the rate of change of grade. This would call for a curve 1200 feet long in the last example. It is not always practicable to meet this requirement, but such curves could l:e and should be much longer than they are gener- ally made. 112 FIELD-MAXUAL FOR ELEVATION OP THE OUTER RAIL ON CURVES. A car of weight w moving on a curve of radius R with a velocity of v feet per second develops a centrifugal force in the direction ab (Fig. 66) expressed by FIG. To counteract thig force the outer rail on a curve is raised | c above the inner rail an amount be = e, so that the car may rest on an inclined plane. Let ac = g, the gauge of the track. The component of /in the direction of ac is ab ab f wtf ab f~m = f~g~f = 32.15J? ' 7* The component of w in the direction of cq is be e w' = w = w. ac g Now/' and w' are opposite in direction, and in order to satisfy the mechanical conditions they must be equal. Equating their values, therefore, we find 32.157? (19) But ac distance between rail centers = gauge -f one rail head = 4.708 -f- .202 = 4.91. For an elevation of 6 inches ab = ac .03 = 4.88, nearly. This is a good average value for ab. Again, if V = the velocity in miles per hour, we have _ 5280 22 ~ 3600 15 ' LEVELING, STADIA MEASUREMENTS, ETC. 113 73 Furthermore, R = -, in which D is the degree of a curve of radius R, and 7?i '= the'radius of a one-degree curve = 5729.58. Substituting these values in (19), we have 4.88X484DF* bs.io X 225 X 5729.58 = .000057Z)F 2 , almost exactly (20) 22 If we substitute in (19) ab = g, and v = V, we have, approx- 15 iniately, _ . 06688,7 F 2 This is the formula in general use. To give the most favorable view possible of this formula, s-ub- stitute in it g 4.708 and find .3149** ~~ Example. Find the elevation, e, for a 7 curve and a velocity of 40 miles per hour. Eq. (20) gives e = .000057 X 7 X 1600 = .638 feet - 7.66 inches. Eq. (22) gives = - 31 * 9 * 1600 = .616 feet = 7.39 inches. 81o.O It will be observed that the two large factors appearing in (22) do not occur in (20) ; and, moreover, eq. (20) is very much more accurate than the formula (22) generally used. The author has elsewhere pointed out the evil of elevating the outer rail on sharp curves sufficiently to meet the requirements of too high velocities. The elevation should be much less than required for the speed of the fastest passenger-coaches ; for it is better, on such a curve, for a coach to hug the outer rail somewhat or to slacken speed, or both, than to pull all the freight-cars against the inner rail. The best conditions are realized when the speeds of fast trains 114 FIELD-MANUAL FOB ENGINEERS. are lessened so as to exceed only slightly the speed for \vlucli the elevation of the curve is suited. Thus a train moving at a speed of 50 miles per hour should be decreased to a 35-mile rate or less in passing over a curve elevated for a 30-mile rate. Under such conditions the cars would slightly press the outer rail around the curve ; their motion would.thus be steadied, and the movement would be the steadiest possible. The maximum elevation should probably not exceed eight inches, except, possibly, at some special point, under peculiar conditions. CHAPTER VI. COMPOUND CURVES. A COMPOUND curve consists of two or more consecutive circular arcs of different radii; any two adjacent ones having a common tangent at their point of meeting, and their centers on the same side of that tangent. This common point of tangency is called a point of compound curve, or P. C. C. Compound curves are used to bring the line of the road upon more favorable ground than any simple curve could occupy. Let AP and PB (Fig. 67) be the two branches rf a compound curve uniting the two tangents AV&nA BV. Continue APto k, where the tangent kD is parallel to BV, and draw the chords AP, Pk, and PB. To prove that the chords Pk and PB coincide. Since the radii PO, and P0 a are each perpendicular to the < common tangent at P, they co- incide ; and since the radii Oik and 0. 2 B are perpendicular to parallel tangents Dk and BV, they are parallel. Hence the vertex angles of the isosceles triangles PO^k and PO^B are equal, and the angles at the bases must be equal also. Hence OiPk = O^PB, and therefore Pk and PB coincide. Prolong Ak to meet BV\n E. Let R, the first radius A0 l} and .#3 = the second radius POiO*. Let Oi = AO^P, the central 115 116 FIELD-MANUAL FOR ENGINEERS. angle, subtended by the arc of radius .ZJ,, and 0* = FO^B, the central angle subtended by the arc of radius 7? 3 . Now AkP(=BkE) = 0,, by (), Chapter IV. Also, kAP - PkD = PBV - \0* , by (d), Chapter IV, In the triangle ABVlet the tangent A V = T lt and the tangent BV = TI. We observe that the shorter tangent is adjacent to the shorter radius, and the longer tangent is adjacent to the longer radius. Let AB = c, BA V = A, AB V = B. .-. BVF = V = A + B = 0, + a . Draw kag parallel to A V. aB+ak _ tan \(akB -f aBk) _ cot | -T4-T l ri ~colTF' aB -ak- B V- A V = T* - T,. Also aBk = DkP = |0 2 , and akB = F- }0 2 . Hence A;2? aBk V - 0* = O l , and akB -f aBk = V. Substituting these values in the above equation gives T*+Ti 2R, cot JO, 2\ - r l\ (T 9 Ti) cot \V cot Hence ' i) cot IF- l(T, - 7\) cot 40, ) (1) = Ti cot ^F- COMPOUND CURVES. 117 Similarly, y,) cot JF- cot 2 = - ; =- - * or ^2 = $(T a + 7 7 i) cot I V \(I\ Ti) cot L - (3) (3) From (1) and (2) we get (ft - ft) =a |(7 T 2 - IVXcot ^0 + cot We observe that either radius can be found by two multiplica- tions and both by three multiplications, since -|(Ja-f- TI] cot \V is common. This is one multiplication less than is necessary by any other formulas. The first equations of groups (1) and (2) are preferable when both radii are to be found, but the last equation of either group is best when only one radius is required. Since |F= 0, -f i0 2 < 90, i0j < 90, and a < 90, Supposing the triangle AB V unchanged ; then, since 0j -j- a is constant, 2 varies inversely with 0,. Eq. (] ) shows that ft varies directly with Oi , and eq. (2) shows that Rv varies inversely with 2 and therefore directly with 0,. Hence the radii vary directly with the central angle corresponding to the shorter radius. This shows the advantage of making Oi as large as possible. Prolong the tangent Dk to meet .Z?0 2 in H, and draw the per- pendicular PbL. Now BH = BL kb = ft vers 2 ft vers 2 = (ft ft) vers a<> Then ^ , . T _ sin V sin V 118 FIELD-MANUAL FOR ENGINEERS. Hence (AD-\-DV)smV = Then vers 2 = and 7? 2 = Similarly 7 i sin F ?i (vers F vers 2 ) ~f- R u vers 2 . r, sin V-R, vers F i sin F - /*i (vers V- vers O a ) vers 3 sin F= /A vers O l (vers V vers y 2 sin F 7? 3 (vers F vers 0,) - - or sin V- R* vers F It should be noticed that the foregoing equations are symmetrical with reference to the radii, the tangents, and the central angles, and hence eq. (2) may be written at once from eq. (1) or nee versa, by simply ex- changing the radii, the tan- gents, and the central angles; and the same is true of eqs. (4) and (5). Let Fig. 68 represent a compound curve, in which B A 0, = R t , BO* = # 2 , P = the / / P.C.C. Drop the perpendiculars Oiin and O-iii upon the chord AB, and draw Oi /^parallel to AB. Let I be the intersection of AB and P0, , and I the an- gle AlP. We have Oyll Oi 2 sin I = (R? Rj) sin I. Also, 2 7/ = 2 n 0, w = 72 2 cos B Ri cos A. Equating these gives -R 2 cos B Ri cos A COMPOUND CURVES. 119 Again, Am = R! sin A, mn = 0i#= 0i a cos 2 = (R Si) cos I, nB = 7? a sia 5. Adding gives (7 = M! sin .1 -f ^ 2 sin B -f (.Kg J?j) cos . . . (6) Now r , = o ^? , and r. = (7 4"-4, sm F sin F When 72 1 , .Ra, A, and Bare given, ', C, and the tangents J^ and Ta may be found from the above equations. We have I = 0,^ + AOil = (90 - .4) -f 0! , or 0, = -A + J - 90. Also, J + J50 a + 10*B = 180, or I -f (90 - 1?) + 3 = 180, or Oi = B -1 + 90. When /?, , 7? 2 , J., and 5 are given, we may find I, 0, , and 2 from these equations, and then Ti and r l\ from (4) and (5). To find the limit in one direction of each radius of a compound curve. We observe that a curve run from A (Fig. 67) with a radius I? = T 7 ! cot ^F would be tangent to B Fat E t VE being equal to VA. A curve run from A or from any other point of this curve with a radius greater than r l\ cot \V would lie outside of the above curve and would intersect VE. Hence the smaller radius must be less than T t cot \ V. Again, a curve run from B with a radius 11 "= 7 T 3 cot \ V would be tangent to VA prolonged at B' ( VB' being equal to VB) and would pass inside of A. A curve run from B or from any other point of this curve with a radius less than Ti cot F would lie inside of the above curve and would pass still further from A, Hence the greater radius must be greater than 7\ cot \ F. From trigonometry we have cot (45 - 9) -f cot (45 -f 2) = 2 sec 2z > 2. 120 FIELD-MANUAL FOR ENGINEERS. Therefore cot (45 - -a ) + cot [(45 a -f z)] > 2 sec 2s > 2. Hence if the sum of two angles < 90, the sum of their co- tangents > 2, and the more unequal are the angles the greater is the sum. of their cotangents. Now since V < 180, 0, -f 40 a = \V < 90. Therefore cot \Q\-\- cot -|0 a > 2 in all cases, and eq. (3) shows that HI Ri > TI TI in all cases (e) Again from eqs. (1) and (2), R* > TI cot 4 V, and l?i cot Therefore JR, These equations show that the radii differ more than the tan- gents do, both relatively and absolutely. n 71 Putting Ox = a in (3) gives ' ^ _ ' = cot |0 X = cot iK. The following table shows the value of this ratio for some values of V. For the more common values of V, say 40 or less, we see that the difference between the radii is several times greater than the difference between the tangents; and we may remember that when Oi and a are unequal 7p Tf the ratios of -= ;=- are somewhat larger -fa -Li than those given in the table. But the tabular results are sufficiently accurate, and they will aid much in assigning practical values to the radii when the tangents are known. Of the quantities 7?,, Jf a , 0i, and 2 , and the sides and angles of the triangle ABV, four independent quantities must be given in order to find the others. Or three besides V must be given. Of course the parts of the triangle must be consistent, and all of the given quantities must be con- sistent with the necessary relations existing among them. For example, we must have A -f- B = 0j -j- a = V, COMPOUND CURVES. 1 Vis supposed to be known in all cases. If either Oi or 2 is known, the other is known from the relation above. PROBLEMS. I. Given Ti and J' 2 , also either one of the quantities Ri , It?, 0i, and 02, to find the others. Example l.T* = 447.32, r l\ = 510.84, F = 15, and R, - 3000. .-. jP 2 + T, = 958.1, T 8 - 1\ -= 63.52, %(T* - T^ = 31.76, and cot F= 7.596. (1) gives cot j 0l = mix* -6000 = 787764-6000 = 2fl ^ Do. O^; uo.O^ and |0, = 2 50' 45". Now ^0 2 - %V - i-Oi = 4 39' 15", and cot i0 2 = 12.283. .-. cot 0i -f cot 2 = 32.398. From (3), R, = 31.76 X 32.398 + 3000 = 4028.96. If 7? 2 is given, find in the same way 2 , 0i, and 7?!. If 0! is given, 3 = F- 0,. Then find #1 from (1), and R* from (2). Similarly if 2 is given. Example 2 (see Henck, Prob. 48). Let 7\ 480, T* 500, F=18, and 1 =F=9. .-. 2 = 9. In this case the common tangent at the P.C.C. makes equal angles with the tangents A Fand BV. We have KZ; + i\) = 490, i(r t - ro = 10, cot F- 6.31375, cot 0x = cot -i0 9 = 12.7062. (1) gives R l = 490 X 6.31375 - 10 X 12.7062 = 2966.67. (3) gives R, = 10(12.7062 + 12.7062) -f 2966.67 = 3220.79. FIELD-MANUAL FOR ENGINEERS. II. Given A, B, and (7 and any one of the quantities jR, , 7? 2 , 0, , and 3 , to find the others. Example 1. Let A = 8, B = 7, and C7= 950. Let R, = 3000. We have sin V T a = ^ y = 8670.5 X 13917 = 510.84. Eq. (1) gives .12187 = 3670.5 X .12187 = 447.32; _ 958.16 X 7.59575 - 6000 _ 1277.94 CO fro ~ /'> K0~ == 63.52 03.52 .. 40, = 2 50' 45"; . % ^0, = 4 39' 15", and cot a = 12.2836. Eq. (3) gives & = T *~ Ti (cot |0r+ cot |0 a ) + Ifc = 31.76 X 32.4024 + 3000 = 4028.96. If Hi is given, find T\ and T* as above, and then 2 , Oj , and Jib, in order. If 0, is given, we have 2 = (A -f #) - 0,. Find Ti and T 2 as shown, then ^ and /?, from (1) and (2). Similarly if 2 is gi\en. Example 2. Let Oi = A = %\ then 0*= B = 7, and (7 950. Let ,R = 3000. In this case the common tangent at the P.C.C. is parallel to the chord AB. Ti and Ti are found as above. Then eq. (1) gives Si = K^ + T,) cot ^F- (T 9 - TO cot -1(9, = 479.08 X 7.59575 - 31.76 X 14.3007 = 3638.97 - 454.19 = 3184 8, and # a = -~^ (cot 0, -f- cot 1 O a ) + 5, = 31.76(14.3007 -f 16.35) + 3184.8 = 4158.26. COMPOUND CURVES. 1^3 One of the most useful problems in compound curves is that of fitting such a curve to tlie ground. The lengths of the branches give the corresponding central angles, the sum of which is equal to the deflection angle V. We select a problem proposed by " L. E. Vel " in Engineering News, March 5, 1881. III. "Two tangents A V and BV intersect at an angle of 60. It is required to begin at the point A and locate 2000 feet of 2 curve, followed by 2000 feet of 1 curve. " Required the distance to bo measured back from the intersec- tion V to the point of curve A by the shortest and simplest method." A solution of this problem, marvelous for length and labor involved, can be seen in Engineering News of March 19, 1881. (See also Searh s Field Engineering, Art. 164.) Being challenged by the contriver of tlie above solution, the author gave, in Engineering News of April 2 ; 1881, in substance the following as the "shortest and simplest" solution possible: We have R* = 2JB,, or 7? 2 - R, = /?,. Substituting R l for # 2 R l in (4), observing that 2 = 20, we have T, = R, (vers 20 -f vers 60) sin 60 = 2864.8 X .56031 -f- 86603 = 1853.5. Q.E.D. If, for any reason, T 2 is desired, we have, from (5), _ ZRi^^y & vers _ Ri(i vers 0) _ Ri cos 0i "sliTF sin V sin V~ - 2864.8 X .76604 -4- .86603 = 2534.04. IV. (liven R l and 7? 2 and either tangent, to find the other tangent and the central angles Oi and 2 . Example. Let V - 72, 3\ =1091, R l 954.9, and R* 3437.7. sin V = sin 72 = .95106, vers V = .69098, and R, - R, = 2482.8. (4) gives vers 0, * .-. O a = 32 1' 20" and 0, = F- O a = 39 58' 40"; vers 0, - .23371. 124 FIELD-MANUAL FOR Now (5) gives _ 954. 9 X. 23371+ 3437.7 X .45727 .95106 V. Given either tangent and the adjacent or the opposite radius, also either central angle 0, or 2 , to find the other tangent and radius. Example 1. Given AV '= T> = 1091, R, = 954.9, V= 72, and 0, = 39 58' 20". Hence 2 = 32 1' 40". (4) gives 1091 X .95106 - 954 9(.69098 - .15216) .15216 Then T* is found from (5) as above. Example 2. Given 2\ = 1091, # 2 = 3437.7, and the angles as above. From (4), n _ 1091 X .95106 - 3437.7 X. 15216 _ .53882 " Then TV is found from (5) as above. VI. Given the tangents and a central angle Oi or 2 , to find the radii. Iti and 7? 2 are found from (1) and (2). Example. Lei Ti = 480, I\ 500, V 18 and Oi = 9. Then 2 = 18 - 9 = 9. Now cot \V 6.31375 ; cot \0, = cot ^0 2 = 12.7062. Now, from (1), R l = 240 X 19.02 - 250 X 6.39245 = 2966.7 ; and from (2), Rv - 250 X 19.02 - 240 X 6.39245 = 3220.8. SPECIAL PROBLEMS. It frequently happens that a curve already located must be changed to meet certain conditions. In this case the equation must be solved anew or the results already obtained must be modified. COMPOUND CURVES. 125 If the required changes are small, and especially if some latitude is allowed in making them, the solution can sometimes be readily modified to conform to the new conditions ; but it is generally best to solve the equation anew. This will be illustrated in the following problems. Having located a simple curve APB, radius A0\ = Ri , ending in the tangent BV, it is required to endwithagiven shorter radius, B'O' = It*, in a tangent RV\ parallel to BV, and at a distance within BV equal to BH = p. The distance BP back to the P.C.C., and the distance HB'=d, that the end of the new curve is back of the old, are required. Let PGfB = P0 1 B= 0. Now cos = KO, BO,-HK-BH 0,0' 0,0' fa -fa vers = z> z_> * Zfcl Jtf Q , vers P vers 0" (7) Or proceed as follows: Describe the arc 0' C with center lt Now CK vers = 0,0'' But CK =. BK - BC = BK - PO' = BK - fa , and BH = BK - HK = BK - fa. = BH=p. Hence, as before, = Also, d = Hff = KO' = (R, - St) sin 0. (8) 126 FIELD-MANUAL FOR ENGINEERS. From (7) and (8), d sin p vrs d = p cot | = cot $0, (9) Equation (9) is easily proved as follows : Draw PB'B, which we have seen is a straight line. We have BB'H = B'BV = \PO,B = 10. Hence HB' = d = HB cot BB'H = p cot ^0. Example L Having located a 3 curve ^ PI? (Fig. 69), radius 1909.9, required the distance back from B to the beginning of a 5 curve (radius 1145.9) that will end in a tangent 27 feet within the tangent BV, also the distance HB'. (7) gives vers (PO'B' = PO,B) = ,y ? = .03534; .-. = 15 17'. Then 15 *' = 5.09 chains = 509 feet = arc BP. 3 Also, JIB' p cot %0 .-= 27 X 7.453 = 201.2 feet. II. Suppose the curve APB is located, and it is desired to end with a longer radius, PO' = R?, on a parallel tangent B V' , at a V , distance HB = p without BV. Similar equations to the above apply. Let 7*0,5 =PO'B'=0. Then, as in Fig. 09, p = BI1 = B'K - BD Rt vers Hi vers = (R 9 -J^) vers 0. (!') Also, d = RH=KD = (R* - Jv'Osin 0, (8') and d =pcottO. . . (9') COMPOUND CURVES. 127 Example 2. Having located a 5 curve APS, required the dis- tance back from B to the beginning of a 3 curve that will end in a tangent 27 feet without BV. PO,B = PO'B' = 0. We have vers = T Vr = .03534. .-. 0-15 17'. Then 15 5<> 1 ' = 3.06 chains = 306 feet = arc BP. Also, B'H p cot $0 = 27 X 7.453 = 201.2 feet, the distance that the new tangent point B' is in advance of the original tangent point B. With reference to the preceding problems, we observe that if any two of the following quantities : the new radius, the offset , the tangent distance HB' = (1, the arc BP (or BP) or the equivalent, the angle 0, are given, the other two may be readily found by the use of the preceding' equations. Example 3. Let p = 1, and let PB = 150 feet, the degree of the curve APB (Fig. 69) being 7 10'. Hence = 10 45', B l = 800. Eq. (7) gives & = 800 - = 800 - 57 = 743. . Ul t OO Also, HB' = cot 5 22' = 10.63. Example 4- Having run two given curves, PB and PB' (Fig. 69), subtending equal central angles, PO,B = PO'B' = 0, to find the distance between the terminal tangents BV&ud B'V and the distance RE'. (7) gives p = (#! R. z ) vers 0. Then, from (8), we have SB* = (#1 - 7 a ) sin 0. 128 Fl ELD-MANUAL FOR ENGINEERS. These equations and Figs. 69 and 70 snow that if n curve is sharpened or flattened, so as to end on a tangent parallel to the original tangent, the new curve does not end at a point opposite to the original, but is moved backward or forward according as the curve is sharpened or flattened. Problem. Given a compound curve APE' (Fig. 71) ending in the tangent B'V, the first radius, A0 l = POi = R t , being shorter than the second radius, P0*= R*\ to change the P.C.C. and also the last radius, so that the new curve radius OP' = 1? shall end in a given par- allel tangent B"V". It is evident that the new P.C.C. must be on the first branch AP, since if on the second branch the problem would be the same as the FIG. 71. preceding. The P.C.C. (at P') must be found so that the arc P'B' will be tangent to B" V" . Suppose the first branch AP of the curve extended to B, where tangent BV is parallel to B'V. Let A0 1 P=0 1 , PO*B' = 3 , and P'OB" = 0. Let B'H=a, B"C = p, BH = d, BK=d'. I. Suppose R' < Ri < Ri. Then B"V" is necessarily with- in B'V. From the preceding problem we have directly B'H = (B, - Ri) vers 2 = a; ....... (10) BH= OjOa sin 0, = (R 9 - R,) sin 2 = d. . . (11) Dividing gives vers a _, or d = a cot 2 . . . (12) d COMPOUND CURVES. 129 a and d are expressed 'in terms of known quantities, and are always known. Again, B"K= (#, - .R') vers = a', or vers = _ a . (13) Mi H BK 00 l sin = (#, R') sin = d' (14) .-. tan \0 a -,, or d' = a' cot ^0. ... (15) Since the position of the tangent is given, a' is known and eq. (13) determines 'the beginning of the curve. Also, HK = d + c V a cot ^0 2 -f a' cot 0, . . (16) which determines the end of the curve. 2. If the curve is to begin at a certain point, is known, and we have, from the above eq. (13), radius = 11' = ^ --'-. versO Then the end of the curve is found as above. 3. If the curve is to end at a given distance back (or front) of 13', d' is known, and we have tan 10 = ~, which determines the beginning of the curve. Then the radius, R', is given as above. II. Suppose R' > 2?,. (Fig. 72.) (a) Let B"V" be within B' V. Then R < 7? 2 . We have B'II= (R* - R t ) vers 0, = a; .... (10') BH= (R-2 - A'Osin O a = d (11') .-. tail 40, = ' (12') 130 Also, and FIELD-MANUAL FOIi ENGINEERS. B"K= (R' - R,) vers = a'; . . . . (13') BK= (R 1 - #0 sin = d'; . . . . (14') tan 10 = -. d (15') FIG. 72. Also> Finally, HK = # - BK = d - d 1 . (16') 2. If tlie beginning of the curve is fixed, is known and the radius a' = R' = R 1 -f vers 0' HK = d - d' = a cot |0 a a cot 0, giving end of the curve. 3. If the end of the curve is tixed, d' is known and _ d'' Then R' = R, vers 0' (1)) Suppose B" V" is without E' V. (Fig. 73.) COMPOUND CURVES. In this case we may have R' = JB a . > We have, as already explained, B'H = (# - Et) vers O a = a, . . BH = (R* - R,) sin 2 = d, and C B FIG. 73. Also, and ^ = (R' _ ^) V e r s - ', BK = (R' - ft) sin = d', and By transposition, tan $0 = -,. vers = -,- R' - RS which determines the beginning of the curve. By subtraction, HK=d'-d, . which determines the end of the curve. 132 FIELD-MANUAL FOE ENGINEERS. A special case of this problem is that in which the P. ('. C. only is changed. Then 11' = # 2 , as illustrated in Fig. 73. B''V" is without B'V. (Fig. 73.) (a) Substituting R^ for R' in the value of a' , we have of a = p = (R. 2 J?i) (vers vers 3 ). .'. vers = vers 2 -{- -. Then # 2 = R t vers 0' P'OiP =00? shows the angular distance that P' is back of P. (b) We may consider that the curve AP*B" is first located, and that APB' is the new curve, the new tangent B' V being within B" V". Then P'OB" = 3 , and PO*B' = 0, and we have 7) vers = vers 2 - =-. a = #1 4 -7;, as before. vers P'OiP shows the angular distance that the new P.C.C. (P) is in advance of P' . 2. If, in this problem, the beginning of the curve is fixed, is known and we have S = ft + , vers The end of the curve is given by UK = d' - d = ^' cot ^0 - a cot ^0 2 . 3. If the end of the curve is fixed, d' is known and tan 10 = ^,. This determines the beginning of the curve, COM PO U N D C r R V ES. 13; Then the radius R' 4. A special case of the general problem is that in which the new curve ends on the same radial line as the original curve. (a) OB" coincides with OiB' ', and // with K, as shown in the figure. Hence BK=BH, ord' = d. We have from equa- vers H B' B" FIG. 74. tions preceding, by subtracting and transposing, tan 10 tan \0^ -f , ^ = tan ^0 2 + ^ ~ = tan i0 2 + ^. d d d a This gives the beginning of the curve. We have } L = OiO* sin OtOiL 0^0 sin O^OL, or (It* - Hi) sin 2 = (I? - 2^) sin 0. E' = t sn sin P'O t P= Oi shows the angular distance that P' is buck of P. (b) We may suppose the curve AP'B" first located, and thai, APB' is the new curve, the new tangent B' V being within B"V". Then "=0^ PO^E'-O. tan \0 tan iO, - ~, 134 FIELD-MANUAL FOR ENGINEERS. and sin as before found. In this case the new P. C. C. (P) is on the arc AP 1 prolonged. From the above value of Ri we have sin _ R* R, sin 3 R' RI' which shows that for and 3 , each less than 90, we have = 02 accordingly as 7? 2 = R'. Hence the radius and the P. C. C. must both be changed. EXAMPLES. 1. A 10 4' curve, AP, is followed by 100 feet of 7 44' curve, PB'. It is required to find the beginning, the end, and the length of a 17 48' curve that will end in a parallel tangent, B"V J ', 58 feet inside of B' V. (Fig. 71.) We Lave R, = 569.2, R, = 740.9, and R' = 321.9; also, 0a = 7 44', and p = 58 feet. Hence, eq. (10), B'H = (It* - 7*,) vers 3 = 171.7 X .00909 = 1.56 = a, and, eq. (13), B"K = 58 - 1.56 = 56.44 = a', and therefore vers = ll^l = .22822. .-. = 39 29'. Now 39 29' - 7 44' = 31 45', and 100^-^' = 315.4 ft. Hence the beginning of the new curve is 315.4 ft. back of P. COMPOUND CURVES. 135 Also, eqs. (11) and (14), HK = (B 9 - 7?,) sin 3 -f (R t - R) sin = 171.7 X -1346 -f 247.3 x .6358 = 180.3 -f, or HK = 1.56 X H.79 -f 56.44 X 2.788 = 180.4. The length of P'B" is 100?^-|^ = 221.8 feet. 2. Suppose the original curve and offsets a and a' as above, and that the new curve is to begin at P', 315.4 feet back of P. To find It' and the end of the curve. We have P'O^P- 10 04' X 3.154 = 31 45'. .-. FOB" = 7 44' + 31 45' = 39 29' = 0. Then, eq. (13), R' = R, - vers = 569.2 - .22822 = 321.9. The distance HK is now found as above. 3. Suppose that it is required to end the curve a given distance, HK, back of B'. Then d' = UK d tan = ; and #' = vers 0' Problem. Given a com- pound curve, APS', ending in the tangent B' V , the first radius, AO l = Ri ) being longer than the second radius, P0 2 Hi, to change the P. C. C., and also the last radius, so that the new curve shall end in a given parallel tangent B" V". I. Suppose R' > Ri >R t . Then B" V" is without.B'F 7 . Let the notation be as in the (5 preceding problems. Then FIG. To. 136 FIELD-MANUAL FOR ENGINEERS. B'H= (Ri - Ri) vers 3 = a; ...... (10a) BH= 0,0, sin 3 = (R l - 1? 3 ) sin 3 = d. . (11) a and d are expressed in terms of known quantities and are always known. Dividing gives sm O 2 Similarly, * = tan A0 2 = ff - or d = a tan 40,. (12) B"K = (R' - R } ) vers = ', or vers = _. , (13a) ji jii ^^= OOj sin = (72' - J?,) sin = ^', .... (14a) and tan |0 = , or rf' = a' cotan 0i. . . (15) Since tbe position of tLe tangent is given, a' is known, and then vers determines the position of P' or the beginning of the new curve. Again, HK = d 4 d' = a cot iO a + a' cot {0. . . (16a) This determines the end of the curve. 2. If the beginning of the curve is assigned, is known and we have J.* -*-*l s\9 vers The end of the curve is given as above. 8. If the curve is to end at a given distance in front of B', d' known and we have tan \0 = -,. This determines the beginning of the curve. Then the radius = R' = /?, -)- COMPOUND CURVES. 13 r IT. Suppose R' < Ri. (a) Let the terminal tan- ^ gent B" V" be without B' V , then R' > R*. Now =(U,-R^ vers a =a; (106) BH0^0-, sin O a tan i0, = , or d = a cot a . . (126) Again, =i- vers = ' Oi or vers U 77 (136) FIG. 76. Ri-R' and BK= 00, sin - (JR, - J2') sin = d'; . . (146) tan |0 = or d' = a' cot |0. ... (156) a vers = determines the beginning of the curve, and HK= BII- BK^d- d' = a cot 2 - a' cot (166) determines the end of the curve. 2. If the beginning of the curve is given, find R' ^= Mi =, vers as before. The end of the curve is given by UK = d d', as above. 3. If the end of the curve is given, d' is known, and then tan \0 = ~. This determines the beginning of the curve, i Then the radius is given as above. 138 FIELD-MANUAL FOR ENGINEERS. (b) Suppose B"V" to 'be within B'V. We may have < Now B'H (Ri R*) vers a a; . . . . (We) BH= 0i0 2 sin 2 ' W-Y = /P' ^ tan |0 a = ( L t "J5r = (#! - /*') vers 0=0', K or vers = ^__. (13c) ^T= (90, sin i? / )sinO = d'; (14c) FIG. 77. or d' = a' co (15c) The position of the tangent gives a f , and then eq. (13r) deter- mines the beginning of the new curve. HK ~ d' - d = a' cot \0 - a cot 2 . . (16c) determines the end of the curve. 1. A special case of the above is that in which R' = fa. Sub- stituting /? 2 for R' in the values of ft' and a, and subtracting, we have CB" = p = a' a = (fit - R 9 ) (vers - vers 2 ). .'. vers - vers O a -f and --^. vers COMPOUND CURVES. 139 We may regard AP'B" as the original curve, APE' as the new curve, the new tangent B'V being without B"V". Then and P-OB" = 2 , vers = vers 2 = 0; P vers 0' as before, 2. If in this problem the beginning of the curve is fixed, is known and R' = R,~ vers 0' The end of the curve is given as above. 3. If the end of the curve is fixed, d' is known and tan = . Then the radius is found as above. 4. A special case of the last is that in which the new curve ends on the same radial line as the original curve. (a) OB" coincides with OiB' ', and .fiTwith K. Hence d' = d. Then tan 7 = ; tan O a = . d d d .'. tan tan 2 = d '' or tan 40 = tan + = tan |0, + J-. 0, FIG. 78. This gives the beginning of the curve. Now BII=OiOs sin 2 a < =(#,-#) sin 0^(R,-R') sin 0. 140 FIELD-MANUAL FOU ENGINEERS. From this we have (R, R-,} sin 0o R' = (b) We may regard AP'B" as the original curve, and APB' as the new curve, the targent B' V being without B" V" '. Then P'OB" = PO,B' = 0. tan W = tan a -, and , w (B t - R 2 ) sin 2 H = R i ; ^r . sin In laying out compound curves it is generally best to lay out each branch from the station at the beginning of the branch, and subsequent stations if necessary. Thus, in Fig. 67, AP would be laid out from A, and other sta- tions on AP if necessary. Then the transit would be moved to P, directed along the tangent at that point, and then deflected to stations on PB, etc. We will show further on, however, how to lay out one branch of a compound curve from a station on another branch. Problem. Given a compound curve APB, ending in a tangent BV, to change the curve so as to end on a new tangent BV, but at the same tangent point as before. I. Suppose the shorter radius to precede the longer. (Fig. 79.) Let B = the known angle VBV',AV = Ti,BV= f l\. LetAV'= T f , BV' = T" and VV = a. We have, from the figure, V=V-B. F '- 79 - COMPOUND CURVES. In the triangle BVV we have T" sin V Also, T* sin 7" _o_ _ sin B 1\ ~ sin V" in 7'' 141 Then r = r i\ - a. With these values of V, T' , and T" the tangents may be con- nected by assuming a value for one of the quantities Oi, 2 , I\\, or Ri, and using eq. (1) or (2) and (3). When V is on the prolongation of AV, we have V = 7 + B, and 2" = Ti + a. Otherwise the solution is just the same as the above. 'II. Suppose the longer radius to precede the shorter. (Fig. 80.) We have A AV=T lt BV=T*. AV'=T f , and BV = T". Also, VV = a, and, in the figure, the angle V = V+ B. sin 7 Then Also, = _ sia B ^ T' = T, + a. FIG. 80. With these values of V, T', and T" a compound curve may be located, as already explained. Remark. A special solution of this problem in a voluminous work on Field Engineering, with an example worked out, occupies six pages. 14-3 FIELD-MANUAL FOR KXO1NKKUS. Problem. Having found the radius AO = R of a curve, to substitute for it two radii, AOi Ri, and P,0 2 = 7?a, the longer of which, -40i, is to be used for a certain dis- tance only, at each end of the curve. FIG. 81. Let A and Pi = BCP, = 0,, Supposing the arc AP\ described with the longer radius, the intersection of P,0, with the central radius MO of the simple curve in 0* gives 2 as the center, and Pi0 9 as the radius, of the central branch of the compound curve. In the triangle 00,0.., 0,003 = 180 - 00,0 2 = 0,, and 00,0, =P 1 0*M= \V - 0, ; 00, = Si - R, and 0,0 2 = & - R,. Hence 0.0, 0,0 sin \V 'R, - R ~ sin (f7- 0,)' (17) From this we find sin I V 'sin ('7- 0,)' (18) We may assume values for any two of the quantities li lf lit, and 0i, and find the other from either of the above equa- tions. We observe that 0, -f a = 7, which is known. Hence if a value of 0i is assumed, or if 0, is computed from Hi and ll, 2 also becomes known. There are then but three independent quantities, I?,, R*, and 0, (or 2 ). COMPOUND CURVES. 143 When this is used as a transition curve, R : is much greater than It; the first branch is short, or Oj. is small; hence R, - R ~ sin(iF-Oi) does not much exceed unity ; or Hi J? 2 does not much exceed Ri R, or JB a is not much less than R. Hence, by sharpening the main portion of the curve a small amount, the ends of the curve may be much flattened. To determine the distance MM' between the middle points of a simple curve and a three-centered compound curve joining the same tangent points. (Fig. 81.) From the triangle OO^O*, Then MM' = 00? + 0*M' - OM sin (19) Problem. To connect two tangents by a compound curve of four branches, the first two consuming one half of the vertex angle V. Let A'PitnPiB' represent the curve. Let A'Ot = #, , P,0 a = R,. The vertex distance Vm = E, arid A'OiPi = Oi. Draw Z H and 0*K per- pendicular to A'V and to A'Oi respectively. Then 144 Hence cos Oi and FOR EXGlXKERS. K0t 'i - A ' K 0, 2 . vers Oi = 1 - cos 0, = - Transposing, we have i v_ # vers ( 20) and E cos I V - 7/ a (cos 0, - cos j F) vers 1 E cos t F - g. vers Q, cos Oj cos i F Also, .ffF = F0 3 sin ^ F = (E + #) sin | F; A'H = KO., = Q!*}, sin 0, (/?i - # a ) sin OL Hence ^' F= T, = (^ + 7^ 2 ) sin |F+ (/A - tf a ) sin 0,. (21) Since cot iOi = - , we have, from (20) and (21), vers 0, cot cos F ,Bn 74 vers F' In the figure the curve is flattened at the ends, but either or both ends may be sharpened, the equations applying without change. Supposing the vertex distance 2i7to be known, we may assume It t and 7? 2 and find <9i from (20), then T l from (21). Again, we may assume /, and 2\ , and find Oi from (22), then II, from (20). Again, we may assume 72 3 and H and find 7?j from (20), then 7 T , from (21). Finally, we may assume 2\ and Oi, and find 7i j a from (32), tlu-n 7?! from (20). After finding the unknown quantities, we may lay out the curve A'Piin, consuming out 1 half of the vertex angle F. Then, COMPOUND CURVES. 145 letting B'E = R l , P*C - 7? 2 , B'V= T, , and B'EP* = 0, , we may assume values without reference to the corresponding values in the curve A'P\in\ and find and lay out a curve mP^B', differ- ing to any possible extent from the curve A' Pitn. Otherwise, drawing the tangent LmN (Fig. 82), we have mL V - mNV = k V, mL = T" = E cot \V \ L V = E and A'L = T = A'V - LV - T, - sin \V' E sin \V With the vertex angle mLV = \V and the tangents A'L and Lm a compound curve A'P t m of two branches may be located as usual, using eqs. (1) and (2). Similar!}' for the curve viB. It is evident that a compound curve of any number of branches and having in general any radii may connect two fixed tangent points. We may, for example, lay out all but two of the proposed number of branches ending, we will suppose, at m, Fig. 82, Then run out and measure or compute mN\ likewise NB' . Then, observing that the vertex angle at .ZVis equal to V less the central angles already run off, we may connect the tangent points m and B' as usual, thus completing the curve A'mB'. Let us suppose the compound curve to pass through the middle point of the simple curve AmB, Fig. 82, of radius AO = R, con- necting the tangents AV and BV. Then mV = E = 7?(sec \V - 1), or Ecoa \V R(l cos $V). Multiplying by - i- = tan $V, we obtain Jsin %V= 2?(tan \V sin F). Now since these equations express the relations existing among" the elements R, V, and E of a simple curve, and since, too, eqs. (20) and (21) express the relations among the elements of a com- pound curve, it is plain that if we substitute in eqs. (20) and (21) the value of E (or of ^sin Fand of ^cos |F) drawn from these equations, the resulting equations will express the relations among the elements of a compound curve that vi ill pass through the middle point m of a simple curve of radius R and central angle V* 146 FIELJKXTANU.'.L FOR ENGINEERS. We thus find 1 (72 versO, = - _ . . (33) vers O t 7?versF- ft vers 0, or ft, = - TTF ! vers \V vers 0j Also, 7', = 7^ tan \ F+ (ft - 7* a ) sin O t (72 - 72 3 ) sin \ V. (24) But T, = FA + ^.4' = R tan F-f ^^1'. Hence AA' = d - (R l - 72 3 ) sin 0, - (R - 7? a ) sin i F. . . (25) These results are of course easily deduced from the figure in the same way as results were derived from Fig. 69 and others. For example, and r^^y-M^l'^tf tan |F+(ft ft,) sin 0,-(ll-R z , sin \V. Example. Let 0, = 24 15', ft = 905, and F= 81. It is required to find the radius PiO? ft of the second branch of the compound curve passing through the middle point of a simple curve of radius H = 630 (Fig. 82). We have V=K (sec -|F-1) cos F=.K(l-cos F) = 21 vers = 630 X .23959 = 150.94; J2 vers 0, = 905 X .08824 = 79.86. Also, cos d - cos -|F = .15136. COMPOUND CURVES, 147 Then R 9 = 71.08^.15136 = 469.6; 2. If the radius 7? a is com- mon to both central bran- ches, we have a carve of three branches shown in Fig. 83. The same equa- tions, (20), (21), and (2'2), apply. Example. Let V = 80 16', 0, = 12 40', and 7? 3 =860. Then - Ri = 151.1. From the first of group (20) we find FIG. 83. - IBM X .9756 ,7o4o .-. #=1129.8-860 = 269.8. (21) gives T, = (E+ It,) sin -|F-f (7?, - Rj sin 0, = 1129.8 X .6446 + 151.1 X .2193 = 728.27 + 33.14 = 761.4. In the curve MP^B we have E = 269.8, 7V> 3 = R* = 860. BCP? = d = 17 50'. Let (20) gives 269.8 X .7645 - 860 x .1874 45.10 .04805" 148 FIELD-MANUAL FOR ENGINEERS, Also, from (21), J3V = 2\ - 1129.8 X .6446 -f- 151.1 X .3062 = 728.27 -f 46.27 = 774.54. 3. If the end branches have a common radius, R., and central- angle, Oi , and the central branches a common radius, /? 3 , we have a symmetrical curve of three branches shown in Fig. 84. Equations (20), B ' (21), and (22) apply. I. Suppose the curve is flat- tened at the ends. Example. Let V - 67, R l = 3437.7, and := 1432.4. It is required that the com- pound curve shall be tangent at the middle point of the simple curve whose radius is R - 2291.8. We have FIG. 84. E = R exsec \V - 2291.8 X 0.1992 = 456.53. Eq. (20) gives cos O l = Bl - 3437.7 - 1575.1 _ _,) 2005.3 ... o = 21 45' and O a = V - 20 1 = 23 30'. Now, from (21), T, = (E + 7? 2 ) sin %V+(Ri - R*) sin 0, = 1888.9 X .55194 + 2005.3 X .37056 = 1042.56 -f 743.08 = 1785.64. The tangent of the simple curve is COMPOUND CURVES. 149 Hence tbe tangent point of the compound curve is back of tLe tangent point of the simple curve 268.72 feet. Example 2. Let V = 60, R = 2000, R, = 1200, and A A 1 = d = 240. To find E, r i\ , 0, , and S t . W- lessee IV - 2000 X .1547 = 309.4. .'. E + R* = 1509.4. Ti = rf + fltan $ F = d -f 2000 X .57735 = 240 + 1154.7= 1394.7. " l l = 1394.7 - 1509.4 X .5000 640 1077T8 _ .-. 1 = 19. Now (20) gives 267.94 - 1200 X .0795 172.54 .05448 ,05448 _ - OlOl.U. Observe that E cos F= 267.94 is computed in finding cot If Oi is given, we have, from (23), vers 0, Then find T l (and E if desired) as before. II. The curve sharpened at the tangents. This will occur when the ', compound curve is a moder- ately easy connecting curve between two tracks, the curves being sharpened at the ends to fit the frogs. Example 1. Let V 27 40', and E = 45. Assume R } = 744, and O l = 6 34'. Eq. (20) gives - 43.695 - 4.881 .022444 = 1729,4. 150 FIELD-MANUAL FOH ENGINEERS. Eq. (21) gives r, = 1774.4 x .2391 - 985.4 X .11436 = 424.26 - 112.69 = 311.6. Example 2. Let V = 63, and E 270. Assume ^ = 744, and 0, = 20" 31'. Eq. (20) gives _ 270 X .85264 - 744 X .06333 _ 183.10 _ .08403 " .0840^ ~ Eq. (21) gives AV - T, = 2448.9 X .5225 -f 1434.9 X .35021 = 1279.6 -f 502.5 = 1782.1. Problem. To draw a tangent to two given curves. Let AP\ and J3P 3 be the curves, AO = #,, and BO' = It*. FIG. 86. I. When the curves are visible from each other. In this case a point Pi may be found by trial on one of the curves, through which if a tangent line be drawn, it will be tan-: gent to the other curve also. II. "When the curves are not visible from each other. (A) Suppose the distance between the centers 00' = c, and the direction of 00' to be known. Let DOO' 0. Run any convenient radius AO. Then the angle AOO' is known. Let Pil\ a represent the tangent line, and suppose O'D to be drawn parallel to it. Then DO K, - fa COMPOUND CURVES. 151 Also, 1\1\ = DO 1 = c siu 0. , Then A01\ = AOO' - 2)00', which gives the length of the arc APi, and consequently the posi- tion of Pi. P,Pi makes an angle with the tangent at A equal to AOP t , and therefore PiP 2 is determined in every respect. If more conven- ient, run any line AB, and measure the angle OAB A. Suppose BK to be drawn parallel to 00' , and let ABK B. We have c Then AOO' = AKB = 180 - (A + B). Now find 0, AOPi, and P,P 2 = a as before. (B) Suppose that 00' is not known. Run and measure any line AB = d, and the angle BAK = A. Then, in the triangle ABK, we have AB = d, AK = #1 - # , and JOS = 4. Hence, Chap. Ill, formula (8), sin A sin J. tan B = AB d -j-= - cos A - cos A A A lii - a Then i*- T = ^^ = (, - A)"4=4 sm 5 sin Z> Now proceed as before. We have already considered generally the problem of connect- ing two straight lines by a compound curve of two or more branches. We now come to consider the problem of connecting two curves by a third curve, thus forming a compound curve of three branches. Problem. To locate a curve tangent internally to two given curves, as shown in Fig. 87. 152 FIELD-MANUAL FOR ENGINEERS. Let AOi Ri, and (70 2 12 2 , represent the radii of the curves. I. Suppose the tangent point on either curve to be given. Let A be the point. By construction. Make An (70s = It?. Draw nO-^ and at the middle point of w0 2 erect a perpendicular to it to meet AOi in 0. is the center and A0= JR is the radius of the connecting curve. Calculation. Run and measure a line from A to any point (7 on the other curve. Let AC a. Measure the angles PC A = C, and CAO A (P is on CO* pro- longed). Draw the perpendiculars CH and 0*K to ^40i. Evidently the angle between (70 2 and AOi is equal to ^1 6 y . Therefore AK = AH + J7# = cos J. + It* ccs ( J. - 6 Y ). Similarly, J?0 3 = sin A + J? 3 sin (A C}. Now Kn = AK An = AK - It?. Let KnG* - 71. Then Kn ; cos 11 tan n = -jf\ n Finally, AO = DO = Kn Kn cos n 2 c< 4- nO = II ., . r-f- r c6s,2n 153 If ACOi can be made a straight line, (70, AK (a -f 74) cos A, and KO* = (a -f R^) sin A. We Lave 7i00 2 = 180 2n, which gives the central angle, and consequently the length of the connecting curve. To find OiOi c : Find KO* and AK as above. Then d,0 1 r=:I J and If the tangent point D on the curve of shorter radius is given, lay off on Z>0 3 prolonged a distance Dn =AOi = Hi, and proceed precisely as above. II. Suppose the radius of the connecting curve to be given. Consider any radius of either of the curves that will, when pro- longed, intersect the other curve. Let O^C represent such a radius, and O^CP the same line prolonged. Measure CP = b, and OjP0 2 = P. Let P0,0^ = O lt sin P sin P tan Oi -7777 = - (See Chap. Ill, formula (8).) Then P0*0 l =m* -(P+Oj. Also, Ok'& The triangle 00, 0* gives cos Then AOi P = 180 - (PO.O^ -f 00,0^. This gives the distance from P to the tangent point A 154 L FOR ENGINEERS. Also, tan 0,00., = sin 00! a sin 00,0 2 - cos 00,0, 0,0 a cos 00, a which gives the length of the required curve. If C is at the intersection of the curves, b = 0; otherwise the same formulas apply. The above solution applies whatever may be the relative posi- tion of the given curves. When, however, the given curves are tangent internally there is no connecting curve. Problem. To locate a given curve tangent externally to two given curves, as shown in Fig. 88. Q, FIG 88. Let A be the given tangent point. Let A0 t = Hi, and 7?0- 2 Make An _Z?0 2 , and at the middle point of ?i0 a erect a per- pendicular to w0 a to meet AOi, prolonged in 0. is the center, and AO It is the radius, of the required curve. COMPOUND CURVES. 155 ion. Run the line AC = a, as in tlie preceding prob- lem, and suppose the perpendiculars (7/7 and 0*K to be dra\vn. Let AOL = C, and CA0 1 = A. Then AK = AH + HK = a cos A + R* cos (J. - C), and KOi = a sin A + R* sin (^4 (7). Now A"// = ^l?z - AK = # 3 - J7f. Let jfiT0 a = w. Then Kn tan 7i = _ ~ Kn ' cos ?i ' A^ Kn cos /<- ~ 2 cos 2 7* ~~ 1 4- CT>S 2n' Finally, AO = - An = 7J-0 - # 2 . Suppose tlie radius of the connecting curve AO = J? to be given, and the position of the tangent points and the length of the curve required. Draw COi intersecting the curve AD in P. Measure CP />, and the angle ; <70 2 = (7. Let tfO.C^ = OL Now sin C sin (7 ' ( p ' ' eq- ()) ' ___ COS C -g- -cosC Then (70,0, = 180 - 7 + 0.). Also, 0,0, = . = ft*^, The triangle OOiOt gives Then AOiP= OOiO* - COiO 9 . 156 FIELD-MANUAL FOR ENGINEERS. (X This gives the position of the tangent point A with reference to P. The above solution applies whatever the relative position of the given curves may be. When, however, the given curves are tangent internally, or one of them, prolonged at pleas- ure, lies entirely within the other, there is no connect- ing curve. Problem. To locate a curve tangent to two given curves, externally to one and internally to the other, thus forming a compound and a reverse curve. 'LetAO l =R l> CO* --=R*, A0= E. Let A be the given tan- gent point. By Construction. On the radius 0\A prolonged make An COi = R y . Draw nOi, and at the middle point of nO? erect a perpendicular mO to meet AOi in 0. is the center and AO = DO = R is the radius of the required curve. Calculation. Run the line AC a, as in the preceding prob- lems, and suppose the perpendiculars CH and 0*Kto be drawn. Let ACL = C, and CAO, = A. Then AK = AH -j- HK = a cos A -f .# a cos (A - C), and. KO* a sin A -f # 3 sin (A - C). Now Kn = An -f AK = R* -f AK. Let KnO* - n. Then KO* Kn tan n = -7^; nO* Jfrj, cos n .-. Tgii, \j% Kn nO = cos n 2 cos 2 n Kn 1 + cos 2n Finally, AO - nO -"An = nO - Suppose the radius of the connecting curve AO R to be given, and the position of the tangent points and the length of the connecting curve required. Draw CO, intersecting the curvu COMPOUND CURVES. 157 AD in P. Measure CP = b and the angle 0,CO^ = C. Let COiO* = #,. Now, Chap. Ill, formula (8), tan 0, = 77- sin sin C Xgi _ cos L^1 J _ cos c Then <70 a 0, = 180 - (C + t ). Also, 0i a = C = J? 2 - 7T. sin #i The triangle 00i0 2 gives cos 00102 = or cos 00! 3 = { ~~ Then AO,P = 180 - ((70, 2 + 00, a ) This gives the position of the tangent point ^L with reference to P. The above solution applies for all positions of the given curves. When, however, the curves are tangent externally there is no connecting curve. Problem. To find a curve tangent to a given curve, and to straight line, which intersects the curve. (Fig. 90.) 158 FIELD-MANUAL FOR ENGINEERS. A. Let AO = 11' = the radius of the given curve A Vm, and let BD be the line. The radius Hoi the required curve may be given, or the tangent point A on the curve, or the tangent point B on the line. I. Given the radius AC = R. Draw VO. Suppose FAKL and CH drawn parallel to BD. Measure the angle PVD (which is equal to DOV) and represent it by A. Let ACB ~ AOD 0, and Dm 11' vers A - a. Now mL BK = Dm. That is, 11' vers It vers = a, or vers = . VB = VD - CH = 11' sin A - (R r - E) sin 0. This gives the position of B. Also, AOV = AOD - VOD = - A. This gives the position of A. From A or from B the curve may be run. We have, from above, R' sin A - VB sin = Whence tan = R' - U vers a sin 6 ~ Wsm A - VB' II. Suppose the tangent point A to be assumed. Let AOV = B. Then AOD = A -f B = is known. We have, from above, R = R' - vers 0' BV is found as above. III. Suppose the tangent point 13 to be assumed. Let VB = b. Substituting this value of VB in the value of tan \0 t we have . . R' sin A Then R = It' -. vers COMPOUND CU FIVES. 5 Xow and R give the length of the arc AB and the tangent point A. Draw Afim, which is a straight line as shown above. Now DBm = AB V = _ mD R ' vers A = , and ED YD VB = R'sh\A- VB. mD a Plien tan mBD = -^^, or tan \O ; ^^, as already shown. B. Suppose the required curve to be tangent to the given curve externally, as also shown in Fig. 90. Let the central angle AC'S' = 180 0, and AC' = r. Then ACB - AOD = 0. I. Suppose the radius AC' r given. We have mL - B'C' - C'F = Dm. That is, R' vers r r cos = a, or R' vers 2r -\-r vers = a. Hence This gives the position of B '. Also, AOV= AOD -VOD = - A. This gives the position of A. From A or B' the curve may be run. B' is also given by the angle AC'B' = 180 0. We have, from above, sin = R -f r Whence tan iO = ^ sin R' sin 160 FIELD-MANUAL FOR ENGINEERS. II. Suppose the tangent point A to be assumed. B. Then A+B=0, and AC'B 1 = 180 - (A + is known. Then, from above, _ K' vers a 1 -}- cos ' B'Vis found as above. III. Suppose the tangent point B' to be assumed. Let Then VB' = V; r = R' vers a 1 -f cos ' = 180 -0 r and 0. Give the length of the arc B 'A and the tangent point A. C. Suppose the required curve to be tangent to the given curve FIG. 91. internally, the centers of the two curves being on opposite sides of the line. (Fig. 91.) COMPOUND CURVES. 161 Let DOV = DVP = A, and Dm = 72' vers J. = a, as before, Let the central angle ACS 180 0. Then 407) = 0. 1. Let the radius 4(7 = r be given. Suppose FAKL, CH, and C'U' drawn parallel to #7). We have mL-\- BC + <7A'= Dm. That is, 72' vers + r~\- r cos = a, or 72' vers -f 2;- ? vers = a. Hence vers = - . -ii T Then yiOF= P T OZ> - which gives the position of the tangent point A. Also, F# = VD - CH= R 1 sin A - (R' - r) sin 0, which gives the position of B, From A or from B the curve may be run. R'sinA-VB From above, sin -- W^r - ' vers a 2r tan = -- II. Suppose the tangent point A is assumed. Let AOV = B. Then ^OD = 4-#- Ois known. We have, from the above equation, a R' vers 1 + cos ' Then FT? as given above. III. Suppose the tangent point B to be assumed. Let VB = b. Eliminating the functions of from the equa- tions involving sin and vers 0, we find _ A _ ( R> sin A ~ b ? T ~ ~2 Ut' - 2a ' 162 FIELD-MANUAL FOtt ENGINEERS, vers a 2r Then tan \0 sin R' sin A b' r and give the center C and arc BA and the tangent point A. D. Suppose the required curve to be tangent to the given curve externally, as also shown in Fig. 91. Let the central angle AC' B' = 0. I. Suppose the radius AC' = r is given. We have mL + B'F = Dm. That is, R' vers -\- r vers = a. Hence vers = R' + r Then AO V = VOD - AOD = A - 0, which gives the position of the tangent point A. Also, VB' & C'H' - VD = (R r -f -r) sin - R' sin A, which gives the position of B r . From A or from B' the curve may be located. R'smA+ VB' From above, sin = - ^7 - . R -\- T vers a Then tan = - = sin It 'sin A + VB'' II. Suppose the tangent point A is assumed. Let AO V B. Then = AC'B' = AOD = A B is known. Then equation above gives -. vers VB' is found as above. III. Suppose the tangent point B' is assumed. Let VB' b. We have, from equation above, tan 10 = ^ry-r R' sm COMPOUND CUBVES. 163 Then r is found as above. r and give the center C", the arc B'A, and the tangent point A. Also, 70.1 = VOD - AOD = A - E L D gives the arc VA and the tangent point A. Problem. To find a curve tangent to a given curve internally, and to a straight line, that does not in- tersect the curve. A. Let AO = R' radius of given curve Am, and ED = given line. I. Suppose the radius, A C= BC=E, of tlie required curve AB to be given. Draw OD perpendicular, and sup- pose AFKdr&vrii. parallel to BD. Let Dm = a, and AOF=ACK = 0. Now BK=Dm + mF, K Fm. 92. or R vers = a -(- #' vers 0, or vers = - It - R'' This gives the length of the arc in A and the tangent point A, Also, DB = FK = (# - R') sin 0. This determines the position of B by giving its distance from D., the point on the line opposite to m. We have, from above, sin = BD R - R'' Whence vers a tan = - = ' II. If the tangent point A is given, we have AOm = given, and from the equation above we find vers (? R'. 164 FIELD-MANUAL FOR ENGINEERS. Also, AK = R sin 0, which gives the position of B. III. If the tangent point B is assumed, let BD = b, and we have, from above, We also have, as explained above, Dm a Also, vers 0* and R give the length of the arc BA and the tangent point A. If Dm is unknown, run and measure any line EP = b and draw PH. Measure EPL (between EP and OP prolonged) and PEL = E. Then OLD = E + P, and POm = - 90 - (# -f P). Hence DO = DH ~f- HO = b sin E + #' cos 0, and Z>ra = J9# inH b sin E R' vers wOP. B. Let the curve be tangent to a given curve externally, and to a straight line that does not B intersect the curve. Let AO = R' =the ra- dius of the given curve Am, and let BD be the line. K I. Suppose the radius, AG-BG-r, of the re- quired curve to be given. Suppose GB and OD drawn perpendicular, and Cll and FAK parallel, to BD, The notation is uniform throughout this class of problems. COMPOUND CURVES. 165 Now DO ~ Dm + mO = a + R r . Also, DO = BC-+ 110 = r + (#' + r) cos 0. Therefore # _|_ .R' r -f (.R' + r) cos 0, a + R' r 2r a or cos = jj, , and vers = -jfTTT This gives the length of the arc Am and the tangent point A. Also, BD = Oil = (R' + r) sin 0. This gives the position of the tangent point B. 2r a Also, tan |0 = vers -r- sin =-=- . x>// II. If the tangent point ^4 is assumed, AOH = is known, and the above equation gives _ - cos 0} 1 + cos Then #Z) = (R' -f r) sin 0, as before. III. Suppose the tangent point B is assumed. Let BD = &. We have 2r - a b tan \0 = 2R' -f a' Observing that HO DO BC = a -f 72' r, CH = b, and CO = R' + 7-, we deduce, from the triangle CHO, or from the above, and / give the length of the arc ^17? and ihe tangent point A. " WYE " PROBLEMS. The preceding equations apply at once to "wye " problems. In this case a straight line and two curves are tangent to each, other, or three curves are tangent to each other, 166 1TELD-MANUAL FOR ENGINEERS. .A 4 curve, radius PjO, = It = 1432.4, ends in a tangent PiP-j. It is required to locate a curve PP a tangent to tbe straight line and curve. Draw 2 /i parallel to PiP a . Let FIG. 94. a = O l and 0,0 3 P a = a . I. Assume the radius 2 P 2 r = 954.9 corresponding to a 6 curve. AT w! Now cos O l = - = 01 02 R-r 477.5 and Hence 2387.3 .-. 0x = 78. 46, 09 = 180 - 78.46 = 101.54. = .2000. arc fP l = - = 1962 feet, and PP 2 = arc 2 f ee t. Also, PiPa = w0 2 = (R + r) sin 0! = 2339.1 feet. II. Assume the tangent point P on the curve P,P. Then Oi is given, and, from equation above, we have R(l cos 0,) P,P 2 = (R+ r)sin O t . Also, This determines the tangent point P 2 . III. Assume the tangent point 1\ on the line PjP a . COMPOUND CURVES. 167 Let Pi-Pa = c. Then, in the triangle n0i0 a , or 4Rr = c 2 , or r = . 0i and 2 and the position of Pi are now found as in case I. Observing that the radii are inversely as the degree of the curve, we may substitute the reciprocals of the degrees of the curves in place of the radii in the formulas. Thus cos 0, = = * = - 2000 - '' 0,= 78.46. Example . A 2 curve, radius P 1 l = R t = 2864.8, is tangent at P to a 3 curve, radius PO., = It* = 1909.9. It is required to locate P,. a 55' curve, radius P,0 R = 6250.4, tan- gent to both curves. In the triangle OiO z O, 0,0. = R, + R,, 0:0 = R - R lf and 2 = R - R y . Hence the angles may be found by the general formula fts _U c 2 - a 9 cos a = 2bc Or, observing that 55' = \% of 1, we have, as explained above, (12. _ 1\2 J_ /12. _ 1\2 _ fi J_ r\V cos = V -i 2(if ~ i)(if - 1) Or, multiplying through by 2 X 3 X 11, we have .-. = 75%2. 168 FIELD-MANUAL FOR ENGINEERS. By simply transposing this equation, we have -0. = +^- = .*,.^., s. Or 2 could be found by the familiar "sine proportion." Now O l = 180 - (0 + 2 ) = 61. 52, 180 - 2 = 136.72, and 180 - 0, = 118.48. Hence P.P., = 7520 -*- H = 8204 feet J PPi = 11848 -f- 2 = 5924 feet; Pl\ = 13672 -H 3 = 4557 feet. II. Suppose PxOjP is assumed. This gives the position of PI. Then 00, O a = Oj is known. Draw 9 n, making P t n P a a . Let 0?iO a = C>0 2 n = n. Then sin 0i sin tan n = -^- = Now nOzOi = Oi n, 00 a 7i = n. .'. OOvP=2)i-0 lt and P0 2 P 2 =180 -OO a P=180-(2>i-0 1 ). This gives the length of the arc PP 2 and the position of P 2 . OjOO,, = 180 2-/z. This gives the length of the arc PiP 2 . To find R, we have sin 0, ,sin O l 7i0 2 = Oi0 2 = (H\ + /la); 1 Sill /i Sin 7i cos ?i sin 2n Finally, P^O = ^ = -K, + : COMPOUND CURVES. 169 If the tangent point P 2 is assumed, draw 0i/h, making I\n l = P. Oi , and proceed as above. Example 3. A 4 curve, radius POi = R\, is tangent externally to a 5 curve, radius P0 2 = If a . It is required to locate an 8 curve, radius PiO = R, tan- gent to both curves. We have cos _ (i _225+169-324_7_ 2X15X13 ~39~ Therefore = 79 40' = 79. 67. - 225 + 824 ~ 169 - - 2 X 15 X 18 ~~ 27 ~ .-. 0x=:45 17' = 45.28. Or 0! may be found by the " sine proportion." Then 2 = 180 - (0 -f- 0j) = 55 03' = 55. 05. II. Suppose 0, or 2 is assumed; l5 for example. This gives the position of PI. Draw 2 n, making P,, = P 2 2 . Let 0n0 2 00 2 n = n. Then tan ;i = COS 0t Then n0^0 1 =n-0 i , and 00 2 0! = 2 = nO^O l + w = 2n - 0,. This gives the arc PP U and the position of P a . Also, = 180 - (0, + a ), which gives the length of the arc PjP a . 170 FIELD-MANUAL FOR ENGINEERS. To find R, we have sm n sin n nO = a) sin 0, Then os /i 2 sin ?i cos ?i sin 2n OP l = R = nO - nP t = nO - 7? 2 . Problem. Having located a simple curve APB, of given radius A0 = E, tangent to a given line AG at A, it is requi.ed to run a compound curve, A'P'B', of two branches exterior to the simple curve, the first branch having the shorter radii s ai d ending in a given parallel tangent A'O', the second branch b 'ing concentric with the simple curve and at a given distance from it. Fw. 97. To find the angle of the first branch. Let A'0 l = J2i, BE' PP' = D, and AE = d. Let AOP = A'0 1 P = 0. Draw 0,K parallel to AG. COMPOUND CURVES. Now KO + R, = R -f d, or KO = R - R, -f d. Also, 0,0= OP' - 0,P' = R + D-Ri. KO R - R, + d Hence cos O -^-^r- = == -^ ; =r, 171 or vers = D-d R - R, -f Z>* This solves the problem fully. The formula shows that d must be less than D, since vers must be greater than zero; and that the less is the difference between d and D, the shorter is the first branch, A'P', of the curve. When d = D, vers = zero; therefore = zero, the branch A'P' vanishes, and the exterior curve becomes a simple curve, concentric with the curve APB. Also, EA' = KO, = 00, sin = (R - R, -f- D) sin 0, which gives the position of the tangent point A', and the distance apart, measured on AG, of the head-blocks at A and A'. When d = zero, AG and A'G' coincide, and the two curves are tangent to the same A A' G straight line, at A and at A', as shown in Fig. 98. The formula for this case becomes Ters0 =___. B B' The formula shows that increases and decreases as R, increases and de- creases. Hence, in order that the parallel branch B'P' may be comparatively long, and therefore the branch A'P' correspondingly short, the radius 7?, must be correspondingly short, and the frog at A' sharp. FIG. 98. 172 FIELD-MANUAL FOR ENGINEERS. Example. Suppose a number 9 frog placed at A, and a num- ber 7 frog at A'. Let BE' = D = 22 feet. Required the angle AOP = 0, and the distance A A' between the frogs. In this case AO = R = 744.0, and A'O, = B l = 439.4. .-. R - It, + D = 326.6, 22 whence vers = .= .06736, and 21 09' = 21. 15. o26.o AA' = (R - R, + D) sin = 326.6 X .36081 = 117.84. The degree of the curve A'P' is 13. 04. = 162 feet. If AA is given, we have, from the preceding equations, sin r> r 77, and vers ~ - - vers _ -P = sin ~ AA'' If any two of the quantities 0, D, or AA' are known or assumed, we may find the other from this equation. We observe that the head-block A' must be placed where the main track, AG, and the first turnout curve, AP are sufficiently fur apart to give room for it. This limits the distance AA' . Problem. Having located a simple curve, A'PB', of given radius, A'0\ = Ri, tangent to a given line, A'G' ', at A' , it is required to run a compound curve, APB, of two branches exterior to the simple curve, the first branch having the longer radius and ending in a given parallel tangent, AO; the second branch being concen- tric with the simple curve and at a given distance from it. Let AO = R, A'Oi = R, , etc., as before. We have R KO Ri -4- d, or KO R Ri d , COMPOUND CURVES. 173 and 00i OP - OiP = R - Ri - D. Hence KO R- R,-d d- D cos -=-^r = ^ ^ . vers 00! ~ R - ji - FIG. 99. Since vers must be greater than zero, d must be greater than D. In case D zero, the arcs BP and J9'P' coincide, and the problem is the same as that given under Figs. 69 and 70, and we have as before shown. In case the simple curve, in the two preceding problems, is exterior to the compound curve, the same solution applies. These problems are useful about yards and switches when it is desirable or necessary to have turnouts from different switches merge into parallel curves. The radii R and Hi are determined by the numbers of the frogs used at A and at A'. 174 FIELD-MANUAL FOR EXGIXKEftS. To locate the second branch of a compound or reversed curve from a station A on the first ^*r branch of the curve. T P T' AP and PB are the two bran- ches of two curves of DI and 7> 2 >B degrees; TPT' the common tan- gent, and A V a tangent at A. Let HI and w 2 be the number of chains in AP and PB respec- tively. Let li and / a represent the long chords AP and PB. O n Then the angle APT=$n l D l , and the angle BPT' = $n,I),. FIG. 100. Hence, by formula (8), Chap. Ill, sin APB sin tan PAB= - -f n BP-^ APB + C Si BAP + ABP = 180 - APB = . (26) Hence, if HiDi + W 2 Z) 2 is small, we have, from the above the approximate formula, PAB = n l D l For reversed curves 7) 2 is negative and the formulas become sin Un-iDi tan PAB ; and . . (28) COMPOUND CURVES. The above formula is easily deduced as follows: BAP + A BP = 180 - APE = APT + BPT f = 175 In the triangle ABP the angles are proportional to the opposite sides nearly. Hence BAP = BP InJJ^ Hi 4- Having run a curve with the center and radius R through a given arc AB, it is required to find the y\ radius of a curve with center 0' on AO that will end at C on B'J at a given distance A from B. Solution. In the triangle ABC we have BC= a, AB = c = 2tt sin \0, and Then tan BA C = sin B cos B a FIG. 101. Now 0' + WAV = 180, and + 2.40' = 180. Equating and transposing gives 0' - = 2J5^0' - 2(L!0' = 2BA C - 2A, or 0' = + 2A. Hence AO' = <70' * (70^ = (fi - )( 8 A sin Oi \sm Oi/ Example. Let # = 640, J5 ^ 70. - a = 100, = 40. Then Now c = 1280 X -3420 = 437.8; - = 4.378; cos B -- .3420, 176 FIELD-MANUAL FOR ENGINEERS. Then .-. ^ = 13 06', and Finally, CO = 0' = 4- %A = 66 12'. 540 X .6428 .9205 = 377.1. Problem. Given a compound curve of any number of branches, to find the length of the tangents. A' A Vi Vaxi y 3 a?2 x 3 Let APi P 2 P 3 (Fig. 102) represent a compound curve of three branches, the central angles being Oi, 0< 2 , and 03, and the radii 7^, 7^ 2 , and J? 3 . Draw through the centers p 3 parallels, and through the tangent points perpendiculars, to the tan , gent at A. 2 P 2 c = P 2 F 2 z 3 = 0i+ 2 , and 02/ 1 d 6 FIG. 102. 3 P 3 e = P 3 V 3 x 3 = 0!+ 2 4- 03. Let PiXi = h, P*x* = U, and P 3 x, t. We have t t R> vers Oi ; t, - ti + PJ - P a c = <, + 7? a [cos 0, - cos (0, + 2 )] = Bi. vers 0, + J? 2 [vers (0, -f- 0,) - vers 0,], or U = (J?i - J2) vers 0, + 7^ vers (0, + 2 ); , = t* + P 3 (Z - P 3 e = 1?, vers Oi + R 2 [vers (0x -\- 2 ) - vers 0i] 4- 12,[vers (0i + 2 -f- 3 ) - vers (0, 4- 2 )], or t t = (Ri - R*} vers 0j -{- (J? 2 - J 3 ) vers (0, -f 2 ) 4- 1? 3 vers (0, 4- 2 4- Ot). The law connecting the terms of these expressions is very simple, and therefore the expression for any tangent may be written at once. Now as before found. P, V, = = sm 0, = Bl tan COMPOUND CURVES. 177 _ #2 _#, ver80i-J-7Z a [vers(0|+0 a )-vers0i] t 2 2 ~ sin(0!-|-0 2 ) yin ( 72 1 vers 0,+^ 2 [vers (0, + a ) - vers 0,] - () __ +# 3 [vers(0.4-0 2 + 3 )-vers(0 1 + 2 )] I sin (0! -f a + O a ) The tangents P! V\ and P 3 F 3 are scarcely needed. Again: Considering any pair of tangents, AVs and P a F 3 for example, ami noting that their relations to the radii and central angles are identical, excep.t that they stand at the opposite ends of the curve, it is plain that the formula for AV* may be derived from that of P 2 Vy or w'ctf versa, by simply interchanging 7?i and 7 a , as well as Oi and 2 . For the same reason, the formula for A V* may be derived from that of P 3 V 3 , or vice versa, by substituting K 3 , J? 2 , and R } , in order, in place of Hi, 7? 2 , and Jtf 3> in order, and doing the same with the central angles. This amounts to interchanging 7?j and M 3 as well as 0j and 3 . Thus we have _R, vers 0, -f ^.[vers (0, + 9 ) - vers Q a ] sin (.0 + a ) .Kg vers 03 + Jf? a [vers (0 2 + 3 ) vers 3 ] _ +ft[vers(0 i + 2 + 3 )-vers(0 2 +0 3 sin(0 1 + a + 3 ) Example. "Let A0 1 P 1 = Oi 38, PjOaPa = 2 = 18, A0 l = fa = 600, and P 2 2 = It, = 900. Tofind^F 2 andP 2 F 2 . From the above formula we have 600 X .2120 -f 900 X .2288 333.1 ~~ = ~ = L8> _ 900 X .048~94 -f 600 X .39187 _ 279.17 _ , .82904 ~ .82904 ~ If we draw lines parallel and perpendicular to I\V 3 instead of to AV 3t as in Fig. 102, \ve may compute AVy, AV 3 , etc., directly. 178 FIELD-MAKUAL FOK ENGINEERS. Problem. To determine a simple curve tangent to a compound curve at any point. Let P 2 (Fig. 102) be the point. Make A A P 2 F 2 ^LF 2 . We readily find P V - AV - (R * ~ Iil ^ VerS ( Ol + Og ) ~( vers Qi + vers Q)] sin (0, + 2 ) This gives the tangent point A'. Then A'E = R--A'V* cot ^(0, -f 2 ). Observe that P 2 represents any point on the arc PiP 2 . In the same manner the elements of a simple curve may be found tangent to a compound curve at any point. Problem. Two curved railway tracks cross each other. To find the angles at the crossings of the rails, and the relation-; between them. Let be the center of the track BA and DC, and X the center of the track AC and BD. Draw the radii OA, OB, etc., OiA, OJ1, etc. OD and O^D not drawn on the figure. Let AO --= R, A0 1 = R,, BO, = R, + g, CO .- R -f g, and OOi = c. g is the gauge of the tracks. The triangle ^100, gives cos A + RJ - c\ (c + R- vers A Similarly, from the triangles 7?00i, <700i, and 7900,, we have 7^ 2 + (7?, -f fir) 2 - c,* cos B = vers J3 = _ (c -f R - R, - f/Yc - H -f- A', -4- g) . COMPOUND CURVES. 179 72, 2 + (R + ffY - ver y nearly. (35) Let d the degree of the curve of the rail AB, D that of CD, dr. that of AC, and D l that of BD. In case the angles OAOi, OBO lt exceed 90. Let A, B, etc., FIG. 104. represent their supplements, which are the angles of intersection of the rails. Then vers A = cos B = Ri +c}(R+ R, -c) - H* -(B, +.9)* and cos C cos J) (R + g) g)* - (B, Since these values are tlie same as in the former case, with a negative sigu, it is evident that if we eliuiiuate c froin them we COMPOUND CURVES. 181 will obtain equations the same as equations (30) to (35), with the sign of the last or absolute term changed. Thus we have cos B = -^ - cos A - , very nearly; ...... (36) -tii + g & cos C = ^ - cos A ~, very nearly; ....... (37) R + g Mi ras D = cos A ~ ~ ' very nearly ; (38) cosD = - - cos B ; , very nearly; ..... (40) R -f- g A! -f- g l cos 6' -^ -- , very nearly ...... (41) R, -f- ^ We may find results practically true by the following short and simple method. Let L in Fig. 103 mark the intersection of AD with the radius CO. Now the sum of the angles in the triangles A00 t being equal, we have B+J30A + AGO* + 00,B = A + A .-. B = A + AO^B - AOB. Now BH = g, and BAH = A, nearly. Let JBTbe the intersection of the arc CA and the radius B0\. Then All = -$ nearly, and AB = -r^-r, nearly. tan A sin A ._ AB X d dg and the angle AOB = ----- 100 100 sin Hence 5 = A + ---- - - . . (42) 4 sin 18x5 FIELD-MANUAL FOE ENGINEERS. Similarly, C = A +/--(-?- -^ l \ . . . (43) 100\tan A sm AJ Example. Let R = 1228, ft, = 764. .-. d = 4 40', and di T 30'. Also, g = 4.708, and A = 20, to find B and C. Eq. (30) gives cos B= .993875 X .93969 + .003834 = .937869. .-. =20 18'-|-. Or eq. (42) gives B = 20 + .04708(20.606 - 13.644) = 20 + 0.32777 = 20 19' -f . Eq. (31) gives cos 0= .99618 X .93969 + .006162 = .94226 .'. C = 19 34'. Or eq. (43) gives cos = 20 + .04708(12.82 21.93) = 19. 57 = 19 34'. CHAPTER VII. REVERSED CURVES. REVERSED curves are mostly used in connection with turnouts where the velocity is slow and space an object. They are espe- cially objectionable on the main line since, adjacent to the revers- ing point, the outer rails of the two branches are on-opposite sides of the track. This calls for the elevation of both sides at once, which is impossible, and so the track must be level at that point and cannot therefore be properly elevated at any point near it. Theorem. The reversing point of a reversed curve between FIG. 105. parallel tangents is in the line joining the tangent points. Let AMB be such a curve reversing at M, the radii of the two branches being equal or unequal. Draw the chords AM and BM\ and C7/ parallel to AD, to meet OB prolonged ; also the radii as shown. The radii CA and BO, being perpendicular to the parallel tangents, are parallel, and the 183 184 FIELD-MANUAL FOE ENGINEERS, radii CM and MO, being perpendicular to the common tangent at Jfare in the same straight line. Hence ACM = BOM. .'. 90 - %ACM = 90 - IBOM, or A MG = BMO. Hence AM and MB are in the same straight line, that is, M is on the line AB. I. To connect two parallel tangents by a reversed curve having equal radii. Given the perpendicular distance between two parallel tan- gents, BD = b (Fig. 105), and the common radius CM = MO, to find AD = a and the chords AM and BM. Let AB=c. Since the radii are equal and the angle A C'M=BOM, the triangles ACM and BMO are equal, and AM = BM = \c. Hence when the radii are equal the chords AM and BM are equal. Draw CE perpendicular to AM. Then AE = A M = c, angle BAD = ACM = ACE, and hence the right triangles BAD and ACE are similar and give AC-t-AE= AB-^-BD, or R -+- ic = c -T- 6. or J2==, or 6 = . . . (1) But c 2 = a 2 + Z> ........ (2) -b), or b=2Jt - l/4IF-a>. (3) When any two of the quantities , Z>, c, and 7v* are given the other two are readily found by these equations. REVERSED CURVES. 1B5 Again, OH = OB + DH - ED = 2E - b, and CO = 2R. Let ACM = BOM = A. Then cos A = , or vers A = ^ ..... (4) AD = CH=2R sin .4, AM = 2.4# = 2# sin A, ) / ( 5 ) and ^15 = 2.4 3f = 472 sin |^L ) Example. Let b = 20,- and # = 741.7. .'. JD = 7.725. To find ^D, JLJIf, etc. vers .4 = - = .01348. .'. ^1 = 9 25' = 9.417. 1483.4 Then AD =*2^ sin ^4. = 1483.4 X .163613 = 242.70; AM- 2R sin %A = 1483.4 X .08209 = 121.77. 9.417 Also, arc AM = 100,^^^ - 121.90. L i/f/Q If AD and BD are given, then BD BD AB AB tan I A = -p^; AB - -\ and R = ' sin ^A 4 sin Example. Let BD = 13, and AD = 200. Then 1 tan ^=355 = - 065 - .-. ^1 = 3 43' 08"; sin %A = .06486; AB = = 20 ' 43; 186 FIELD-MANUAL FOR ENGINEERS. II. To connect two parallel tangents by a reversed curve hav- ing unequal radii. (Fig. 106.) We Lave /I CO = CM + MO = R+r, and HO = OB+AC-BD=lt+r-b. r-b .'. cos A - It r vers A = ^r . R -f r FIG. 106. Then AD = CH = (R + r) sin A; AM = 2R sin \A ; BM = 2r sin \A ; AS = AM+ BM = 2(R -f r} sin ^A. Again, let AM = c', BM = c", ^5 = c' + c" = c. Then, as above, from triangles J. (7,57 and ABD, and, from triangles BFO and Adding gives a 2 + This shows that for the same distances AD and BD the o/" the radii is constant. 1. Given the perpendicular distance BD between tangents, the radii CM - R and MO - r, to find AD, the chords AM and J/5, and the central angles A. REVERSED CURVES. IS? Example. Let BD = 28, It = 955, r = 574. .-. vers A = T f| T = .01831. .-. 4 = 10 59'. sin J[ = .19052, and sin ^ = .09570. AD = 1529 X .19052 = 291.50; AM = 1910 X .09570 = 182.79; BM = 1148 X .09570 = 109.86. 2. Let AD, BD, and R be given, to find the other radius, r, the chords, and the central angles. Example. Let AD = 400, BD = 56, and R = 900. We have tan \A = ^ = .14. .-. \A - 7 58'. AB = BD -f- sin iJ. = 56 -^ .13802 = 404.04; AM-2R sin $4 = 1800 X .13860 = 249.48; and BM = AB AM = 154.56. r = EM-^ sin 4 = 77.28 -s- .13860 = 557.6. III. To unite two tangents not parallel by a reversed curve of common radius. (Fig.107.) Given the angles of in- tersection at D and E, and the length of the common tangent DE, to find the common radius CM= MO = R to unite the tangents AD and BE. "We have DM R tan fa D; C EM = R tan fa FIG. 107. ' DE = R(t&n fa D -f tan %E), )E tan %D + tan %& Example. Let DE = 250, D = AOM= 24, ^= BOM= 16. tan 12 -f tan 8 = .35310. ,-. R = 250 -* .35310 = 708.02. FIELD-MAKUAL FOR ENGINEERS. IV. To connect two given points on two diverging tangents by a reversed curve of com- mon radius. (Fig. 108.) Given the angles BAD = A, ABE = B, also AB a, to find the radius R of the curve A MB. Draw CL and ONH per- pendicular, and CII paral- lel, to AB. Let HCO=BKO=CKA=K. Then HO = CO sin K= 2B sin K. H FIG. 108. Also, HO = CL + NO = B cos A + R cos B. .-. 2^ sin K=R cos .1+72 cos J5, cos J. 4- cos B T ^ : sin K 4 Again, AL = R sin A ; En = R sin 5. = Off = CO cos K=2RcosK', Adding gives a = R(sm A + sin B + 2 cos IT), sin A -f- sin B -f- 2 cos 1C ACK = ACL + Z(7# = J. + 90 - + 90 - .fiT. or Also, and The foregoing solution requiring barely a single division, ex- cept by the factor 2, may be compared with Henck Problem 40. Example. Let a = 1500, A = 18, B = 6, to find R. cos .4 = .95106 cos B = .99452 sin A = .30902 2)1.94558 sin B = .10453 sin K = .97279 .'. 2 cos K = .46340 Then R = 1500 -f- .87695 = 1710.47. REVERSED CURVES. 189 V. To connect two diverging tangents by a reversed curve, starting at a given point. FIG. 109 (a] Advancing toward the intersection of tangents. (Fig. 109.) Given the angle of intersection AKB = CAL = K, the initial point A, the distance AK = , and the radii CP = R and OP = r, to find the central angles C and of the reversed curve APB. First Solution. We have and Then NO cos =co = : BN = AL AH = R cos K a sin K. BN+r and C = DCP - ACD - ~ K. Now having the central angles C and 0, run the branch AP subtending the angle G ; then reverse and run the branch PB subtending the angle 0. Example.' Let K- 24 30', AK= 854, R - 1440, and r = 1146 Then AL = 1440 X .91 = 1310.4 AH= 854 X .4147 = 354.1 BN= 956.3 r - 1146 (Bn + r = 2102.3) -f- 2586 = .81291. .*. = 35 37', and C = K = 11 07'. If convenient to measure AH instead of AK, the solution is shortened. 190 FIELD-MANUAL FOR ENGINEERS. Second Solution. Suppose the curve run backward to D where the tangent is parallel to BK, and draw lines as shown. Then DC = R vers 0; PM = r vers 0. Hence AE -f PM = AH = b = But Da = R vers ACD = R vers .#. If r = R, + Da-. + vers _ZT= 5, or .4^ = (b - r vers K ) R Now measure ^.^ as found, making angle GAL = JfT. Run ^P perpendicular to AE, ar.d the carve J.Pto intersect _ZPin P. Reverse and run the curve PB. This method does not require the central angles to be known, and neither method "equires the curve to be actually run backward to D. The problem is thus much shortened. (6) Receding from the intersection of tangents. (Fig. 110.) Given the angle of intersection AKB = GAL K, the distance AK a, and the radii CA = 72 and OB = r, to find the central angles G and of the reversed curve APB. REVERSED CURVES. 191 Draw Aa parallel to BK. Then Da = R vers K-, AH = AST sin K\ Da + Aff = DE= &; Otf= OB + DC - ED = R + r - b. r -b b ' COS = r Evidently, C K + 0. The solution here given is, of course, applicable to the former case (Fig. 109) ; and the first solution there given is likewise applicable here. Example. Let K = 18 23', AK = 920, R = 955, and r = 819. Da = R vers K = 955 X .05103 = 48.73 AH= AKs\nK= 920 X .31537 = 290.14 338.87 338.87-- 1774 = .19101. .-. = 36 00' K = 18 23' C = 54 23' Second Solution. DC R vers DCP R vers 0; PM r vers 0, .-. PM^Dc- = Da^ + AE~. Hence ^ JE' + PJf = AH =b = AE(^-^^\ -f Da~. But Da = 72 vers ACD = R vers K. B . 7? AT? - 7' R, AIL = 192 FIELD-MANUAL FOR EKOIKEEES. Now run AE and EP to meet the curve ADP, etc., as in a former case. VI. To connect two given points on two diverging tangents by a reversed curve. Given the chord AB = a, the angles BAL = A and ABK = B, and the radius AC ft, to find the radius BO r of the re- versed curve APB. (Fig. 111.) FIG. 111. Draw CT parallel and CM and ONT perpendicular to AB. GT = MN =AB- AM- BN = a - RsinA - r sin B, and OT = CM -f ON It cos A -f r cos B. Also, CO - E -f r. . .. (R -f r ) 2 = (a #sin J. r sin 5)* + (# cos A -f- r cos ) 2 . Expanding and observing that 1 - (sin A sin B + cos ^L cos B) = I cos (A B) = vers (45) we readily find a R sin A T = 73 vers (A B) + sin B a Compare with Henck Problem 38. REVERSED CURVES. 193 Example. I*t a = 1500, fl = 1600, A = 18, and B = 6, to find r. Then a = 750 tfsinA^ 494.43 255.57 - vers (A-B) = ~ .02185 = .02331 a 1^ sin B = .10453 .12784 r = 255.57 * .12784 = 1999.14. VII. Having located a reversed curve, it is required to shift the P.R.C. so that the termi- ~ ^1 nal branch may end in a tangent parallel to the orig- \ inal tangent. (Fig. 112.) Let APB be the curve, CP=R, OP=r, and BK~ D, the distance ;; tween the original tangent BL and the new tangent B'K, to find the angle of retreat PGP'. Draw Cmn parallel to BL. We have FIG. 112. Oim OiB' + B'H-\- Hm = r + D + Hm, and On = OB + Hm '= r + Hm; .-. Oim On = D. That is, (R + ?) cos Oi-(R+ r) cos D; D . . cos 0i = cos -\- - 7 , , . Then 06*0, = A = 0,Cn - OCn - (90 - 0.) - (90 - 0) = 0- Oi. Example.- -Let =_34 20', D = 94, 7v J = 1433, and r = 879, 94 FIELD-MANUAL FOR ENGINEERS. We have cos = .82577 D-t-(K + r) = .04174 .-. cos 0, - .86751 .-. 0, =29 50'; .'. A = Oi = 4 30'. CHAPTER VIII. TURNOUTS. A TURNOUT is a curved track leading out from a main track. At the place where the outer rail or "lead" crosses the rail of the main track a frog is placed, which allows the flanges of the car-wheels to pass the rails. A frog is a piece of metal, usually steel, having two straight channels crossing each other on the upper surface through which the flanges of the wheels pass. The triangular part of the upper surface formed by the channels is called the tongue of the frog (Fig. 113). Every railway company should own a variety of frogs suited to different turnouts that are likely to be required. Frogs are usually designated by numbers, the number of a, frog being the number of times the bisecting line CF contains the base flffl line AB. Denoting the number by n, we have 11 = - =, and riff 2n = - . Designating the frog angle AFB by F, we have cot F = GF AC' = 2n (1) Standard frogs are made so that n is a whole number or a whole number plus ^, and therefore 2n is a whole number. Let Fig. 114 represent a " split-switch " turnout. AB and JIF represent the rails of the main track, and AC one of the switch- rails. In the figure the switch is set for the turnout, but when set for the main track A takes the position of A'. Prolong the curve backward to c, where the radius cLO is per- pendicular to the main track, 195 196 FIELD-MANUAL FOE ENGINEERS. Draw also the radii GO and FO ; Ca parallel to the main track ; and the tangents (7F7T and FV. Then LFVLOF the frog angle, which we will denote by F. Let 8 = the switch angle BAG = FKV = LOG, then COF = -F- 8. Then Also, DFC = LFV - CFV = F - - 8) = %(F + FIG. 114. Or, since the tangent CV makes an angle 8, and the tangent FVa.ii angle F with the main track, angle FVK = COF = F - 8. Hence CFV = %(F - S), and DFC = DFV -CFV = F - %(F - 8) = (F + 8). Let R = R + \g = CO = the radius of the outer or "lead" rail, and If the degree of the curve CF. Let d = A A' BC, which is properly the throw of the switch, and I = the length of the switch-rail. Suppose d = 5 inches = .416 of a foot, and I = 15 feet. Then TURNOUTS. 197 .'. S 1 35' 30", and vers S .0003858. Now cL =- R vers F, and ca = R vers 8. .'. R' vers F R vers 8 = cL ca aL = CD = g d. vers F vers S cos S cos V or vers F = vers S -f- : f (2) We observe that vers F vers S = (1 cos F) (1 cos 8) = cos S cos F, so that in all such cases either the cosines or the versines may be used as is most convenient. DF = CD cot CFD = (fj - d) cot i(F -\- 8), . (4) HF=l + DF=l + ( HF accordingly as R > 153.6. Example. Find the lead and radius for a number 9 frog, switch- rail 15 feet and throw 5 inches. We have F = 6 9 21' 35", and 8 = 1 35' 30". .-, vers F - vers S = ,005768, and g - d = 4.2917. 4 2Q1 7 and jjp> = l + (g., d) cot #F+ 8) = 15 + 4.2917 X 14.3885 = 76.75. A turnout curve between head-block and frog is in general not the same as the main part of the curve beyond the frog. When, therefore^ two intersecting tracks or a straight line and TURNOUTS. 199 track, for example, are to be connected by a compound curve, the tangent points must be found so tbat witb given frogs tbe con- necting curve may be properly located. Problem. To connect a straigbt track and straight line by a compound curve of two branches, the first branch being a turnout curve fitting a given frog. Eqs. (1) to (5) of compound curves apply to this case without change. Let A V represent the center line of the track, BV the straight line, and APB the center line of the compound curve. Example. Let AO V U v = 437 to fit a No. 7 frog, 11* = BO* = 537, F - 50, and Oi = 10, which is sufficient to clear the frog. It is required to find the tangent point A. Eq. (4) gives A V - T - A * '- sin V 437 X- 357 4- 100 X .234 .766 = 234.2. The tangent VB is not generally needed, but may be found in the same way from eq. (5), Problem. To connect two intersecting straight tracks by a com- pound curve of three branches, the two turnout curves having radii to suit given frogs. Let Fig. 116 represent the case, APi and BP* being the center lines of the turnout curves. The tangents are required. First Method. The equation under Fig. 102, compound CUIVLS, applies directly, and we have BVs'in V=(Ri Z^ 2 ) vers 0j-f-(7? 2 R 3 ) vers(0!4-0 2 4- R* vers (0, 4 2 4- 0.), or, j^Fsin V=Ri vers Oi4-7? a [vers(0,4-0 a ) versO,] + J R 3 [vers(Oi 4- 2 + 3 ) - vers(0i 4- O a ) 200 FIELD-MANUAL FOK ENGINEERS. To find A V we may interchange 7^ and R* , also Oi and 3 in the above formulas, as explained under Fig. 102, p. 177. Thus, observing that Oi -{- 0^ -}- 3 V, we have V (#3 - P 2 ) vers 3 + (R> - J2.) vers (0 a + 3 ) + /?, vers F, and .4 Fsin F= jf? 3 vers 3 + 7? a [vers (O a + 3 ) - vers 0*] - vers (0 2 + 8 )]. Example 1. Let A0 1 P 1 = 14 30', Pi0 2 P 2 = 31, and P y 3 B = 14 30'. Also A0 t = 437 (to fit a number 7 frog). P 2 2 = 490, and P 2 3 400. To find AV. Substituting in first of the above equations, we have A Fsin F= - 90 X .03185 + 53 X .29909 + 437 X = - 2.87+15.85 + 218.5 = 231.48; ~ 231,48 -t- ,886 == 3C6.15, TURNOUTS. 201 Example 2. Let the values be the same as in example 1, ex- cept Hi , which we will suppose equal to J? 3 = 400. Then _ R y = 90, and R 3 = 90. Therefore AVsin V= 90(. 29909 - .03185) -f 400 X .500 . = 90 X .26724 -f- 200 = 224.05. Therefore A V = 224.05 +- = 258.72. Eqs. (a) apply without change whatever may be the order of the radii, but the first form, which usually involves the least labor, may contain some negative terms which must be subtracted from the sum of the positive terms. Second Method. Let A'PiP 9 B' (Fig. 117) represent a symmetri- cal compound curve. Pro- long the middle arc'PjP a to A and B as shown. We will illustrate the method by the following Example. Let the vertex angle = V = 98, A'0,P t = B'OiP* = 9 50', which is sufficient to clear the frogs. .' Let A'Oi = B'O, = 72, = 437.06 corresponding to number 7 frogs, and P, 0% = /2, = 358.1. To find the tan- gents VA' and VB'. We have, from eqs. (7), (8), etc., of compound curves, AH= (R, - #,) vers 0, = 79 X .01469 = 1.16 = o; A'H = (R, - l? a ) sin 0, = 79 X .17078 = 13.49 = d. 202 FiELb-MAJSTtTAL FOR ENGINEERS. .'. HO* = R= 358.1 + 1.16 = 359.26 = radius of simple curve joining IT and K, but not used except to find VH and VA' . Now VH= R tan V= 413.29, and VA' = 426.78. This gives the tangent point A' (or Z>') from which the com- pound curve may be located. The simple curve joining 77 and K is not shown. Let Fig. 118 represent the case in which the curve is sharpened at the ends. Let V= 84, A'OiPi = B'0,P A = 29, A'0, = R t = 437.06, and P, 2 = ft* = 877.06. Then AH=(K* - #,) vers 0, =440 X .12538 = 55.16; A'J7 = (J* 2 - #0 sin 0! = 440 X .48481 = 213.32. .-. H0 y = 7? = 877.06 - 55.16 = 821.9. Now VII = tan F = 740.04. .-. KA' = 740.04 - 213.32 = 526.72. The curve A'PiPiB' may now be located. The simple curve connecting the tangent points //and /Tis not drawn. TURNOUTS. 203 Third Method. We may easily fit a given turnout curve to a different curve beyond the frog by a simple but direct application of Fig. 81 and corresponding formulas of compound curves. Thus let A'm'B', Fig. 119, be a simple curve of radius A'O R con- necting the center lines, A'V and B'V, of two tracks ; to substitute for it a compound curve A'mB' of two different radii, A'O^-B'O^- R, and 7 > 1 O a =A' fl , the longer of which is to be used for a distance adjacent to each tangent, for the purpose of fitting a given frog. The tangents and the elements of the curves A and B'Pi are given. Example. Let V 98, and A'V= B'V = 4.6.78. Then R = A'O = BO' . A' Fcot V = 426.78 X .8693 = 371. .-. D = 15 26' 40". Let A Pi and B'P* be 13 06' 34" curves for number 7 frogs. Then A'0 l = B'0 l = R, = 437.06, and R l - R = 66.06. Suppose A' Pi = B'P* = 75 feet, then 0, = |(13 06' 34") = 9 50'. Eq. (17) p. 142 gives FIG. 119. = 66.06 X 1.193 = 78.94. 7?, = 437.06 - 78.94 = 358.12. .-. Z> 2 = 16. iP^B r , is the same as A'PiPy Observe that this curve, Fig. 117 or of Fig. 118. of 204 FIELD-MANUAL FOR KXGIXKERS. We observe that the degrees of the two branches of ihe com- pound curve (taking A'P, and Ji'l\ together as one) differ approximately from the degree of the simple curve inversely as their lengths. It is, of course, not necessary to locate any simple curve, but simply to make use in the formulas of the radius (A'O, for ex- ample) of a simple curve corresponding to any desired tangent A'V. If we suppose the switch-rail to be a part of the turnout curve, . such curve being tangent to the main track at A (Fig. 120), we have AO" = AH \ersAO"F' and representing A 0" FO" by 11" to distinguish it from the true radius R' = OF (Fig. 114), given by eq. (2), we have R" = r- - - (?) FIG. 120 vers F' The triangle FHO" gives HF = II" sin F. (8) The triangle AFH gives HF=gcot$F, . .,' : . . . . . (80 and AF- . ^ ljr ^V ^ & v . . (9) These equations may be deduced at once from those for Fig. 114 by putting d = 0, 8 = 0, and I = 0, as they are in this case. Introducing the number of the frog, we have (8") AFH = and HF = AH cot AFII = g cot \F = 2gn. Or, make HA' = II A ; then AFA' = 2AFH = F, TtTRKOtTTS. 205 find the triangle A FA' may represent the frog, and we have, by definition, J1F HF n -r-r, -.-, or HF = 2gn. AA' *g (8") Again. JJF* = AH(2AO" - All) = 2AII(AO" - $AH) = 2gli l . (S'") Equating values of HF 9 from (8") and (8"'), we find the radius of the center line of turnout = 7?, 2ff)i*. . (10) A F* = 7IF* + tf = 4g*n* -f cf- .: AF = The foregoing equations are not applicable, however, to straight switch-rails, as we will explain. Let ACiF(Fig. 121) represent the turnout curve actually located under the supposition that the switch- rail is a part of the turnout curve, which is tangent to the rail of the main track at the head-block A, AC being the switch-rail. Eq. (10) gives, for a number 9 frog, 72=2X4.708X81=762.7. The tangent offset of the curve AC\F from the rail A V, at a point opposite C, is = 1.77 inches. Hence the curve AdF is 5-1.77 = 8.23 inches i from C, the end of switch- FIG. 121. rail. But the real turnout curve must connect (7 and T^and be tan- -.MM) FIELD-MANUAL FOR ENGINKKRS. gent to the switch-Tail and frog. Prolong AC to meet the tan- FK in K. Now IlF=2gn; angle FA K = $F-8; AFK = I F. Also, sin ^A^ = sin (FAK + AFK) = sin (F - 8). . . . sin (F - ) sin (F 8) For a number 9 frog, switch-rail 15 feet and throw 5 inches, we have AK = 56.64, or CK = 41.64 = T* , and FK 28.30 = T. Since the tangents are unequal a simple curve cannot meet the required conditions, and we will suppose a compound curve of two branches. Since, moreover, Ihe tangents are quite unequal and the central angles small, we know from Chapter VI, eq. (3), and eqs. (e).and (/) following, that the radii will differ very much. We will suppose the central angles equal, for this will make the radii the most nearly equal possible, as pointed out in Chapter VI, and will result in the best possible curve. The vertex angle CKV = V = F - 8 = 6 21' - 1 35' = 4 46', and therefore 0, = 2 = 2 23'. Eqs. (1) and (2), Chapter VI, give (70, = #! = 1164.13, and TV), = # a = 513.85. We note that the offset from the tangent AV to the curve AC\F is 5 inches at about 25 feet from A, and that the curve has the direction of the switch-rail at about 21 feet from A. By moving the switch-rail forward from .4 (7 to AiCi about 8 feet, making A C about 23 feet, a close approximation to the part of the curve C\F could be used. This would make the lead nearly equal to the true lead given in Table VI. This would also involve some com- promise, and would be somewhat troublesome in practice. Thus is proved ihe impossibility of locating a simple curve tan- gent to a switch-rail and to a frog placed in the position deter- mined by the above formulas. When so placed the result is a turnout difficult and even dan- TURNOUTS. 207 gerous to operate, unless the trackman is able to put in by the eye a compound curve answering more or less well the purpose of the turnout. To LAY OUT THE TURNOUT. (Fig. 114.) When it can be done, which is usually the case, first locate the frog with reference to a joint in the main track. Knowing the length from point of frog to either end of it, we thus find and mark the position of point of frog, which is Fin Fig. 114. Measure off FH, giving place of head-block so as to avoid cut- ting rails if possible. For example, a number 9 frog calls for a lead HF = 76.75. Sup- pose the length of frog from its point to the switch end, which is the part that applies on the "lead," is 3.35 feet, the switch-rail being 15 feet, This will use up 3.35 -f 15.0 = 18.35 feet, and we will need 76.75 - 18.35 = 58.40 feet of rails. We may well use two 30-foot rails, making FH = 3.35 + 15 + 60 78.35. The arc CF is a trifle longer than DF (about 0.1 for a No. 12 and 0.2 for a No. 7 frog), and this will practically allow for expansion and therefore no allowance need be made in measuring FH. Note that FD = FH I. Remembering that the curve CFis tangent to the switch-rail at C, the curve of the outer rail CF may be located by any methods heretofore explained. Drawing CF, the curve CFis easily located by ordinates, noting that CF = DF nearly, and that radius CO R' = E -f %g. The turnout can be conveniently located by setting the instru- ment at C, turning off from the line of the track the angle S, which gives the tangent CV, and locating the curve of the outer rail CF. The degree D' of the curve CF, corresponding to li' = R+lg, may be taken from Table I. Or set at ^and turn off from FD the angle F, which gives the tangent FV, and run the curve both ways from F. This is the handiest and best method. We may locate the center line as follows: Make CE perpendicular to AC, equal to \g, then run the curve from E, with radius EO R given in Table I, the tangent at E being, of course, parallel to AC. Or make Fm = \g and perpendicular to FV t that is, make angle DFm = 90 F, and run the curve both ways from m, the tan- gent at m being parallel to "FY. 208 FIELD-MANUAL FOR ENGINEERS. DOUBLE TURNOUTS FROM A STRAIGHT TRACK. In case of a double turnout (Fig. 122) three frogs are required; the frogs F and F' being the same as F in Fig. 114, the third frog being at F", where the lead-rails cross each other. Fro. 122. Draw the line II' F" parallel to the main track; then it is plain that one half the frog-angle F" is equal to the angle made between fie line H'F" and a tangent at F" to either arc AF or A'F'. Therefore the formulas for this case may be written at once from eqs. (2), (3), (4), (5), and (6) by putting CD' = \g - d instead of CD = g d, and \F" instead of F. Hence rfer -o> or vers p= vers S-f vers 47^ vers $* TURNOUTS. 209 H'F"=l + D'F". . (13) L'F"=R' smlF" ....... (14) Table VI gives the numbers (M"). angles (F"), and distances f rom head-block of middle frogs corresponding to frogs of different numbers at 2^ and F'. Example. Let the frogs at F &nd F' be number 8|. Then F = F' = 6 44', and vers F - vers 8 = .006511. Eq. (2) gives Then (11) gives vers 7<" = .0003859 + ~~~ = .0035253. .-. \" = 4 40' 26", and F" = 9 20' 52". From (4), D'F" = (\g - d) cot %$F" + 8} = 1.9375 X 18.2709 = 35,3998, and H'F" = 50.3998. 12 Finally, " = ^ cot ^" = = 6.11. 6 Hence a number 6 frog at F", the curves being slightly flattened at the frog F", will answer the requirement. To find an approximate relation between JFand F". Comparing eqs. (2) and (11), omitting d and vers S f rom both, we have vers \F" = $ vers F, or 4 vers \F" - 2 vers F. But versines of small angles are nearly as the squares of the angles, Hence, approximately, vers F" = 4 vers F", 210 FIELD-MANUAL FOR ENGINEERS, These equations give vers F" = 2 vers F, or (F")* = '2(F)\ or F" = If F = F' = 6 44', or n = w' = 8|, tlien n" - = 6.01 = 6.0, nearly, or t/2 " = 9 32', nearly Hence in practice take U from Table VI opposite F, and F" = ^2Fas nearly as possible. In case of a double turnout, where no frog f" is at hand of the number given by eq. (11) or by F" = |/2 F, we may locate the turnout as a com- pound curve, and thus fit the frogs, provided \F" is less than F. Let OF" = O t F" = R', and EF" = E^" = ML (Fig. 123.) R' and the lead H'F' are found as in the preceding problem. We observe that in Fig. 114 the inclination to the main track of the arc at is 8; while in Fig. 123 the inclination to the main track of the arc at F" is \V" . Moreover, CD = g d ; while F"K = %g. Hence it is evident that we can write FIG. 123. at once t h e relation betweei F" and F from the relation between 8 and F by substituting \~F" for 8, and \y f or g - d We thus find or B = . (IS) Since the versine of the angle of the inclination of the curve cCF&t any point to the main track is equal to the distance of TURNOUTS. 211 that point below c divided by tbe radius R, it follows that the difference between the versines of such angles at any two points is equal to the difference between their distances below c (or from the rail AS or DF) divided by R>. See eqs. (2) and (15). Thus the relation between the versines at any two points may be written at once. When F = \F" , li\. =00, showing that the turnout beyond F" is the straight line F" L. This is apparent from the figure, since the angle KLl" = LF" M %F". When F < \~F" ', R t is negative, showing that the turnout must reverse at or beyond F" in order to cross the main track at the required angle. By prolonging the arcs F^'C as in Fig. 114, the equations for the last two cases may readily be deduced in the same \vay as were those for Fig. 114. Example. F = 6 44', F" = 8 48'. From (11) we find ~ d 19875 - 756 45 ~ ~ versfF"-versfl ~ .0025613 and from (13) H'F" = I + (i? - rf) cot \(\F" + ) = 15 + 1.9375 X 19.0550 = 51,9191. (15) gives - 2 ' 35416 - 595 96 ~ = and, from (4), KF = \g cot ^(F+ IF") = 2.35416 X 11.2789 = 26.552. FF" may be found by (3), but is not needed. Supposing F" < 2F, then we have approximately as follows : When F" = F \/2, Ri = R, as above pointed out (see Fig. 122). When F" < F f/2, R 1 < R', as shown in Fig. 123. When F" > F \/2, R> > R'. 2L2 FIELD-MANUAL 1 126. truck or on opposite sides of a curved .track. Such frogs should therefore be kept on hand. TURNOUTS. 215 Thus if F corresponds to a number 8| frog, then F" corre- sponds to a number 6 frog 1 , nearly. Problem. To rind an approximate value of the degree of a turnout from a curve in terms of the degree of a turnout from a straight line, the turnouts to fit the same frog, F. Let d = the degrete of the center line of the main track, D ~ the degree of the center line of the turnout from the main track, and D\ = the degree of the center line of the turnout from a straight track to fit the frog F, as in Fig. 114. Let I =. the length of either arc CF or DF in Figs. 114, 124, and 125. Ignoring the difference in curvature between the center line and either rail of a track, we have the following equations : In Fig. 114 the difference in the directions of the arc at C and at F is evidently equal to F 8. But it is also equal to r7^D\- Hence ^D, = F - 8. ...... (21) In Fig. 124, since the arcs CF and DFwake an angle of 8 with each other at C and D, and an angle of F with each other at F, it is evident that the change in direction of the arc CF from C to F, less the change in direction of the arc DFfrorn D to F, is equal to F - 8. But the change in CF= -^D, and that of DF= ~d. 100 100 Hence D ~ d=F - s - Equating with the above and dividing by , we find 100 (22) In Fig 125 the change in direction from C to F plus the change f rom D to F is equal to F 8. Hence I I m D ~^"m d = F ~ s ' Equating, we find 216 FIELD-MANUAL FOR ENGINEEKS. Hence, to find the degree of a turnout for a given frog from the inside of a curve: To the degree of a Turnout for the same frog from a straight track, add the degree of the yiccn track. To find the degree of a turnout for a given frog from the outside of- a curve: From the degree of a turnout for the same frog from a straight track, subtract "the degree of the given track. Example, 1. Required the degree of a turnout from the inside, as well as from the outside, of a 4 curve for a No. 9 frog. The degree of a turnout from a straight track for a No. 9 frog is 7 44'. Hence the turnout from the inside is 11 44', and from the outside is 3 44'. (See Figs. 124 and 125 ) Example 2. Required the degree of a turnout from the outside of a 7 44' curve for a No. 9 frog. In this case the degree of the turnout is 7 44' less 7 44', or 0. Hence the turnout is a straight track. In this case the turnout may be regarded as the main track, and the main track as the turnout. (See Fig. 114.) Example 3. Required the degree of a turnout from the outside of a 11 curve for a No. 10 frog. In this case the degree of the main track exceeds the degree of a turnout from a straight track for the given frog by 11 - (6 09') = 4 51'. Hence, referring to Fig. 127, we see that in this case the center Ei of the turnout, instead of being on the outside of the main track as in Ex- ample 1, is on the inside, that is, on the same side as K. Example 4. Let F (in Fig. 124) = 5 43'. Main track an 8 curve, or r = 716.197. 5=1 35'. Re- quired the lead CF, and the radius of the center line of the turnout. We have FIG. 127. CF= v -^~ 4 2916 The degree of turnout curve from a straight track, correspond- ing to the frpg F, is, by Table VI, 6 09' -J-. TURNOUTS. Adding the degree of the main track (8), we have degree of turnout = 14 09' + and the radius is It = 404.92 . Or (18) gives (2R d) tan tan KDF g - a .-. KDF=8T1S' Then (19) gives GF = Then, by (20), we find (g d) cos 1431.98 X .06879 _ . 4.2916 = ' = 67.34. 33.67 = 406.01, sin(F+#-,8) - .08293 and 11' = 403.66, and D = 14 11' + . The above approximate results differ but little from the true results and show that the approximate formulas are practically all-sufficient. OTHER TURNOUTS FROM A STRAIGHT TRACK. From the same point in a straight track it is required to locate two turnouts on the same side. (Fig. 128.) We will assume that F= F', and that these frogs are opposite to each oilier, and that the curves are tangent to the main track at A. Let KA = r -f %g, and EA = r' + \g. The angle AKF=F, and the angle KF'E-F' - F. Hence the triangle KEF' is isosceles. Therefore EK = EF' = AE, or AE=\AK. That is. FIG. 128. (24) 18 tELt>-MAttUAL FOR ENGINEERS. This relation is easily proved as follows : Let d = the degree of the curve AF, and d' that of AF'. Now we have seen that d' = d -f- the degree of a turnout from a straight track for the frog F, i.e., d' = d -f d, or ' Or again : Draw the tangents F7^' and DF ''; also 777^' parallel to LF. Evidently, VF'D = F' = F, and DF'H = F. .-. VF'H=2F. Hence AH AH vers FF'lf = vers AEF' = vers 2F = AE r' %g' Also, * **' + $# = U r + %9\ as before. In this we assume 77 = ^IZ = F'F- but 1/77 = F'F cos Jf^. We also assume vers 2F = 4 vers F ; but vers 27^ < 4 vers 7^. The two small errors exactly balance, however, and result in a true formula. We have vers F' = vers F= TT> .... (25) and vers F" -^- But r> + ft = fr + ig). .'. vers jp"' = 2 vers 7^, or F" = V^F, nearly. This is the same relation between F and F" as was above shown to exist between the frogs in the case of turnouts to the opposite sides of a straight or of a curved track. (Figs. 122 and TURNOUTS. 219 126.) Hence a set of frogs adapted to a double turnout on oppo- site sides of a straight or a curved track is also adapted to sucli a turnout on the same side of a straight track. In case of the double turnout on the same side of a straight track, the longer radius is equal to twice the shorter radius -j~ \g. In case no frog is at hand equal to F" given by the relation vers F" = 2 vers F, we may select one, which call F", as near the same angle as possible, and find, as already shown (see Fig. 129), ^" = ,' +k = _ and BF" = (r' -f |#) sin F". Then compound the curve at F", and find the radius OF' to suit the frog F', whether it is equal or un- equal to F. Let AK = r -f- \g as before. Then KF' = r - %g. Assume OF' = r" + \g. FIG. 12&. Then EO r' - r", and BK = r - r'. BEF" = F", and OF'K = F'. Let F'KF" = K. We have **>' BK ~ r - \q ..... Then OF"K = BEF" - BKF" = F" - BKF" 1= A, Again, F"F'K = F"F'0 + OF'K= F"F'0 -f F', and F'F"K = F'F"0 - OF"K= F"F'0 - A, Subtracting gives F"F'K - F'F"K = A + F'. (27) 220 FIELD-MANUAL FOR ENGINEERS. Also, TUP" KF " = ^KF'" and UF " = KF " ~ (r ~ W = '' Say> (28) Hence KF" + KF = 2(r - $g) -f ) tan \(A + F 7 ). . (29) Now = BKF" + iT, and 5^" = 2(r - ^) sin ^^F'. (30) This gives the position of F'. We have \(KF'F" + KF"f") = 90 - fZT, and \(KF'F" - KF"F') ~ ftA + F 7 ) J ust found. Subtracting gives ^/?"'F' = 90 - {(A + V + JT). Adding OF"K = A to the above, we have OF"F' =-- 90 + i-4 - i(F' -f A"). Taking OF"F' + OF'F" 20F''F' from 180, we have F'OF" = F' + K- A. Finally, TURNOUTS. 221 Suppose the turnout straight beyond F", the frog being at F lf In the triangle Fi angle F,V"R = 90 - OF"K = 90 - A. .-. sin FiF"K cos A. Also, KFi = r - \g, and KF" = r - ig + e. Hence sin KF,F" = '' r - iff " = 180 - (KFiF" + KF"F,\ Then F"* = KF, "= ~ *>. (33) ' cos A This gives the position of the frog F t . Example. F = 6 43' 59", F' = 6 01' 32", .F" = 8 47' 51". By Table VI, BF = 80.036, r = 680.306; ^" = 61.204, r' = 397.826. By (27), tan BKF" = J^4 = .0902777. .'. BKF" = 5 C 07' 31". o M .95^ vl = F" - BKF" - 3 38' 20"; A -f F' = 9 39' 52"; i(^. -f -F") = 4 49' 56". By (28), fi1 904. ~ .089913 Eq. (29) gives = 680.702. .-. e- 680.702 - 677.952 = 2.75. >ot lK= tan -. \K = I 22' 42", and K = 2 45' 24". "^' = 90 - (4 + 17" -f 7T) = 83 47' 22". 22 FIELD-MANUAL FOR ENGINEERS. Then, by (31), v ,, v , (r ~ \g] sin K 677.952 X .048943 sin KF"F' -^94181- \(F' + K - A) = 2 34' 18". Then (32) gives and r" = 363.132. CHAPTER IX. THE TRUE TRANSITION CURVE. THE object of tliis chapter is to make known the true transi- tion curve, to show its need, to furnish sin, pic foru.ulas for its use, and to show its ready application in practice. FUNDAMENTAL PRINCIPLES. When a car passes from a straight line upon a curve, or vice versa, it receives a shock more or less severe, in proportion to the sharpness of curvature and the rate of speed. This, it is well known, is due to a tendency of any body in motion to persist in its direction of motion at every point of its path. This shock is damaging to rolling stock and track, causes dis- comfort to travelers, and is, moreover, a source of danger. It is evident that the elevation of the outer rail cannot obviate this difficulty to any appreciable extent ; for it is plain that such elevation, which must vary directly with the degree of curvature (see Elevation of Rail, Chap. V), can correspond to a gradual change of curvature only, and not to a sudden change, like pass- ing from a straight line upon a curve or the reverse. Whatever, therefore, the elevation might be, a car passing from a tangent upon a sharp curve, or conversely, would receive a violent shock. The only way possible to do away with such shocks and the resulting evils is to interpose between the tangent and the main curve a "transition curve." A transition curve, as its name indicates, is a curve placed between a tangent and a main curve. A theoretically perfect transition curve must have the follow- ing properties : 1. It must end on the curve with a radius equal to that of the curve, and on the tangent in a straight line. 2. The degrees of curvature of the transition curve at different points must vary directly as their distances from its junction with the tangent. 224 FIELD-MANUAL FOR ENGINEERS. To these must be added the following practical requisites: 3. The curve should be one easy to understand and especially easy to lay out. 4. It should be flexible, so as to accommodate itself to the con- figuration of the ground and to other conditions. Since the curvature of the transition curve is less than that of the main curve with which it connects except at and near the point of tangency, it must lie outside of the latter prolonged. Hence, when a curve is located ending in a tangent, it is neces- sary either to move the curve inward or the tangent outward, in order to interpose a transition curve between them. We w r ill suppose the curve to be moved. Thus, referring to Fig. 130, AO represents a tangent and A the tangent point of the curve Afi, which is replaced by the slightly sharper concentric curve Ti.de. This latter curve is connected with the original tangent at by the transition curve Omd, m being the intersection of that curve with AK, which is the distance be- tween the concentric curves, and is called the offset. Since the curves Kd and Omd subtend the same central angle, while the latter is flatter than the former it must be longer also, and therefore must be back of A. Again, since the radius of curvature of the transition curve at id is equal to dE, which is less ih&nfS, we see that the curvature of that curve at and near to d is a little sharper than that of Hie original curve Afi. Now the degrees of curvature of different arcs are as the angles turned in passing over equal lengths of those arcs. Hence, in THE TltrE TRANSITION CURVE. 225 order that the degrees of curvature at different points of Omd ma.- vary as their distances from 0, the angles turned in passing over arcs of the same length, supposed to be very short, must varv as the distances of their centers from 0; and the total angles turned between and different points must therefore vary as the squares of the distances of those points from 0. Hence if is the total angle dgT turned in passing over any length of this curve, as Omd = S, we must have /S' 2 = cO, c being a constant (1) A transition curve should be long enough to allow the outer rail to gain the elevation at d, proper for the main curve without inclining too abruptly. Froude, Rankine, and others allow an incline of one foot in 300; and, since the maximum elevation of the outer rail should not exceed 8 inches or of a foot, transition curves need not exceed $ x 300 = 200 feet in length. Curves of any length, however, may be readily used by the formulas to be given.* ELEMENTARY RELATIONS. By elementary relations is meant, those pertaining to lines and angles of the figure. 1. To find the length of the offset curve in terms of the length of the transition curve. Let dEK = dgT = 0, and 8 = the length of the transition curve Omd. Then Kd = |, exactly (2) R The figure shows that md is a trifle greater than Kd = -, and 2 o therefore mO is a trifle less than Kd = -. QJ Also, AO is a trifle less than mO, hence a trifle less than -. 2 Let dE= R', then AT = dh = R' sin (3) * The author hopes to give the proof of the following formulas in a treat- ise on the True Transition Curve, tq appear later. 226 FIELD-MANUAL FOR ENGINEERS. 2. To find any tangent distance. We find Or=fl (l _ + *._), ..... (4) or OT = S cos 2 ~0 ......... (5) 4 or OT S cos -0, very nearly ..... (6) 9 Example. Let - i of the unit angle = 19.09859. Then (4) gives OT= i-+-- =5(1 -.011 +. 00005716-) =.9889465. Eq. (5) gives OT= .9889365. Eq. (6) gives OT= .9890465. Hence (5) gives result too small by only .000015, and (6) a result too large by only .00015. iu this example is very large and the errors are therefore comparatively very large. Hence (6) is sufficiently exact for all cases. Let R> = KE, Now Kd = KB X KEd. That is, Again we find AO - /1 _j_ ) (9\ U ~ 2\ 30 + 1080~r 18J AO = - cos y^rO, almost exactly, ... (9) Q 1 f AO = -cos-0, very nearly (10) For = 19. 09859, (8) gives AO = |(1-.00370+.00001143) = .498155, (9) gives AO .498155, and (10) gives AQ = .498265. THE TRUE TRANSITION CURVE. 227 We observe that (9), even for this very large value of 0, gives a result true to five places; and that (10) is also practically exact. Letting KE = R', we also find AO - fR' sin f ....... (11) To find the tangent distance corresponding to any part of the curve it is only necessary to substitute in (4), (5), or (6) the corre- sponding values of the arc and of tie angle. Thus suppose Gab = $8. Then the spiral angle for 6 is , and we have OA '=li( l ~ 160 + 2565^6-)= C -1 = g- cos -0, very nearly. (13) No\*, from (10) and (13), O A A' = OA' - OA = -^-(cos - cos J0). . (14) For = 19.09859, AA' = .00278?-= .00139& A This shows the distance of the middle of the curve 6 to the right of AmK. For all ordinary values of this distance can be neglected and the middle of the curve be taken as at m. From (9) and (10) we have o o - AO = vers ^0 = d, suppose, . . . (15) Q Q and - AO = vers = d, suppose. . . . (16) These values are very easily computed, since the versines of the angles contain few significant figures. 228 FIELD-MANUAL FOR ENGINEERS. o Then, d being very small, AO = d may be easily computed 4 mentally. 3. To find any offset, as dT. We find GO ^ dT = sin -j- = sin $0, very nearly. | Also, dTdVcosO. For = 19.09859, (17) gives dT= .11033$ and (18) gives which is sufficiently accurate even for this very large value of 0. The offset for any point of the curve is readily found by sub- stituting in (1?) or (18) the arc and the angle corresponding to the point. Thus let c = arc Om, and E = the corresponding spiral angle. Then A = (f ** + ^ n -I (19) 9 BT or Am = t = -^ sin ~ (20) 6 O Eq. (1) shows that E varies as c 2 . Also, sin \E increases less rapidly than E, and therefore less rapidly than c 2 . Hence t increases less rapidly than c 3 . 8 Again, let b be the middle point of the curve. Then Ob = , and the angle subtended by the arc Oab . THE TRUE TRANSITION CURVE. 229 Substituting these values for S and in (17), or for c and E in (19), we have Q /^ or -A'& = T sin , almost exactly. . . . (22) 4 o The error of (22) is less than -.0000005 for - 19.09857. Eq. (7) gives 0*f O 4 O 6 \ = -^ - ^ + ^ -J. Also, rr /i-fiT = sin 0, very nearly, ...... (24) o 7iA" = y\/S sin T 7 0, almost exactly ..... (25) Now from (17) and (23) we get < 26) Hence Q S} Q /~l AK - ti = y sin + = sin --, very nearly, . (27) or AK = ti = T 2 T 5sin ^{0, with great accuracy. . . . (28) Comparing (18), (22), and (27), we see that A'b > iAK, A'b > \dT, and AK > However, when is not very large, we have 230 FIELD-MANUAL FOR ENGINEERS. A'b = \AK = IdT, very nearly ; | . . . (29) Am = \AK \dT, very nearly. ) . . . (30) From (10) and (28) we have |siniO i += sm -, very nearly. (31) cos^O cosiO The above values are all very simply expressed in terms of S and 0, the offset curve being known. We now require the following : 4. Having an offset t corresponding to an offset curve Kd, of rr length - and degree D, to find the change in the offset, or the a new offset, t', for the same length of offset curve, when the original Afi is a D degree curve. O r\ We have, from (28), AK= t = sin , when Kd is a D degree 8 curve of length -. the central angle being 0. a When, however, Afi is a D degree curve, Kd is a curve of A W 7? degree Dj^, == D* f D', and, its length remaining the JiHi 41 t n same, the central angle = = - / = 0' ', and the offset t' > t. H t We find ' (83) Or we may find t' as follows : Suppose, for a moment, Kd, instead of Af, to be a D degree curve of radius R, and compute t or take it from Table XV. Then we have KE = R t R', and of course D', with which we may compute t' very accurately, or we can take it from Table XV. 5. To find the offset A K = t in terms of the central angle and the radius KE = R' or of AE = R. Substituting 20R' for S in (27) and dividing by R' gives AK _ (0* _ O 4 G ~$~ ~ IT " 108 + 4680 THE TRUE TRANSITION CURVE. 231 Represent the series by e ; then -=e, or t = R'e. . ' ..... (34) Substituting R t for R r in (34), we find (35) But e = l vers f -f f vers f 0, very nearly. . . (36) Hence, from (34) and (36), t = R$ vers f 0, . . ..... (37) and, from (35) and (36), vers - ~ f + vers f 6. To find the angle between the tangent AO and any radius vector or chord drawn from 0. We find dl d s s ~- = - + 05 6000 ' representing any arc. .'. AOd < , but ^0^ = ~, very nearly. . (40) The angle between AO and the radius vector Od may be called the polar angle. Again, - = tan AOd = f tan -faO with great accuracy. . (41) Example. Let = 6 48'; then ~ V 50', 232 FIELD-MANUAL FOR ENGINEERS, The formula gives tan AOd ^-. f tan 1 59' - .0395766. /. AOd = 2 15' 59", which is but a single second less than -. a The series (39) would give precisely the same result. Furthermore, AOd varies as s , nearly, o Hence for = 13 36' AOd = - - 08", nearly. o Referring to (39), we see that - increases more rapidly than 0, and therefore more rapidly than S-, and still more rapidly than d*. Hence t increases more rapidly than d*. For all values of J or not very large, it is plain, however, that t increases nearly with d 3 . In the cubic parabola whose equation is x z = a?y, AO in the figure being the axis of x, the offsets do increase us the cubes of their distances from 0, measured along AO, increase. For all values of 0, not large, such curve fulfills the theoretical requirements of a transition curve with reasonable exactness. As increases, however, the curve departs more and more from the true transition curve and becomes more and more unsuitable for a transition curve. Moreover, this curve has a minimum radius of curvature which cannot be passed in using it as a transition curve, and which would be troublesome to many, Furthermore, the deflection angles are fractional, are subject to no simple law, and therefore require special computation, which is not the case, as we shall see, with the true transition curve. 7. To find the point of the curve Omd where the curve is parallel to the chord Od. Let p be the point. Since at the curve makes the angle $0 with Od, it changes direction %0 between and p. But it changes direction an amount between and d. Now since these changes are proportional to the squares of the distances Op and Od, we have THE TRUE TRANSilioiST CUKVE. _ 05' ~ 3' .-. Op = VlOd = .577350^ or, Op = %0d, nearly. 8. To find the tangents dg and Og. We have dOg = 0, and hence .'. 0^ f 0, very nearly. o Also, from (18) , dT = - sin Sin * (42, * \* sin 2 sin 0' ' 9. To find the length of any radius vector or chord Od and the angles between these chords. We have angle AOd = , o and . . (43) Then 8 - Od = S vers ~. . (44) o Hence any radius vector is equal to the arc it subtends multi- plied by the cosine of its polar angle. Let c = arc Oa = arc ab, etc.; these arcs being sufficiently short not to differ sensibly from their chords. Let = spiral angle of Oa, 49 = spiral angle of Ob, etc. Then chord Oa c cos , or arc Oa chord Oa = c vers --, "o o chord Ob = 2c cos , or arc Ob chord Ob = 2c vers ~ r -, etc. o o 234 FIELD-MANUAL FOR ENGINEERS. Example. "Lei 30' and c = 100. Since Oa subtends an angle 0, it would subtend an angle 20 if the curvature was everywhere the same as at a; and since Oa = 100, the curvature at a is therefore 1. Since Ob subtends an angle of 46, and would subtend an angle of 80 if the curvature were everywhere the same as at b, and since Ob = 200, the curvature at b is Similarly we find curvature at c = 3, etc. Or, since at a the curvature = i, at b, c, etc., it is 2, 3, etc. Then the chord Oa 100 cos 10' = 99.99958; the chord Ob = 200 cos 40' = 199.986; the chord Oc = 300 cos 90' = 299.897; the chord Od 400 cos 160' = 399.567, etc. We observe that, as 0, 40, 90, etc., increase as I 2 , 2*, 3 s , etc., the versines of these angles increase as I 4 , 2 4 , 3 4 , nearly; and 40 c vers , 2c vers , etc., increase as I 5 , 2 5 , 3 5 , etc., nearly. o o We can also find the long chords Ob, Oc, etc., from Table IV opposite the degrees of curvature of the arcs Ob, Oc, etc., at the points where they are parallel to their chords. Thus the degree of curvature for arc Ob is 120' X .57735 = 69'.28; then chord Ob 199.986. For arc Oc it is 180' X .57735 = 103'.92; then chord Oc = 299.897. For arc Od it is 240' X .57735 - 138'. 56; then chord Od = 399.568. We have 4; o THE TRUE TRANSITION CURVE. 235 Then, by subtraction, 40-0 30 90 - 49 ' 56 166 - 99 70 Hence these angles between the chords are in the ratio of the odd numbers 1, 3, 5, 7, etc. 10. To find any deflection angle, as dbl, at any point b\ also any chord bd. U being the prolongation of Ob, we have sin bOd tan dbl = - COS bOd r-r dO Then Odb = dbl bOd and bd may be found by the sine propor- tion. Or as follows : Let c and Ci represent the chords Ob and bd, and 8 and Si the corresponding arcs. Let Oa = 1, and the spiral angle for Oa. Then AOd - and hence bOd 8i(28 -\- 8,)-^- (45) Now sin &d0 = ^ sin &tfd = sin &(2S -f 8^. bd c\ o c, S Now = , nearly. ... bdO = -. 8^28 + &) = /8(25 + &), nearly. (46) Oi o This amounts to assuming that the angles bOd and bdO are in the ratio of the arcs bd and bO, which is very nearly true. Now dbl = bOd + &eZ0 = (5 + S,)(2 + -S r 1 )--. . . (47) o 236 FIELD-MANUAL FOR We may find any chord, as bd, with sufficient accuracy from Table IV, opposite the degree of curvature of the middle of the arc bd. 11. To find the exsec dV, also TV, etc. dV = dTsec TdV = ^ sin |0 sec 0, or d V = EV - Ed - R sec - K'- TV - dTt&n - f- sin %0 tan 0; OV = 01 -f TV. 12. To find the radius of curvature at any point of the curve Omd. We have K = = ........ (48: Since H' is a radius of the transition curve expressed in terms of the arc Omd and the angle dgT, the relation is perfectly general. Eliminating between (1) and (7) and omitting the accent, we have (49) This shows that the degree of curvature varies directly with the length of arc measured from 0, which agrees with the definition of the curve. 13. To lay out the curve by ordinates, or offsets from the tan- gent AO. We have, by (18), ap = \0a sin 2AOa; A'b = \0b sin 2A Ob\ cC - ^Ocs'm 2AOc, etc. THE TRUE TRANSITION CURVE. Zot Let the spiral angle of Oa = 30'. Then AOa = 10', AOb = 40', AOc = 90', etc. Hence ap = 50 sin 20' = .291; bA' = 100 sin 1 20' = 2.33; cC 150 sin 3 00' = 7.85, etc. Now set a 100 feet from and .29 of a foot from A0\ b 100 feet from a and 2.33 feet from AO; c 100 feet from b and 7.85 feet from AO, etc. We also have, from (6), Op = Oa cos AOa; OA' = ObcosAOb, etc. But these quantities are not needed. SPECIAL PROBLEMS AND EXAMPLES. The reader will observe that since Kd - , when either Kd re or Omd is known the other is known also. Problem 1. Given the length and degree of Kd, Fig. 130, to find the offset AK &nd tangent distance AO, and to lay out the curve. Example 1. Let Kd be a 9 36' curve 100 feet long. . '. Omd = 200 feet. Eq. (16) gives .-. AO = tL - .088 = 99.912, Eq. (27) gives = 2 .789. . Oi O O Or AST and J.O may be taken directly from Tables XV and XVI. 238 FIELD-MANUAL FOR ENGINEERS. To LAY OUT THE CURVE. Use four 50-feet chords, for example. dOg = 3 12' = 192'. o Set the transit at and turn off from AO. = 12' for station (a); 16 12' x 4 = 48' for station (5); 12' X 9 = 1 48' for station (c); and 12' X 16 = 3 12' = ^ for station (d). o Or, having computed the first deflection angle (12'), find it in column 1 of Table XVII, and opposite it, in columns 2, 3, etc., are the deflection angles for stations 2, 3, etc. Proceed in precisely the same way in any case. That is, divide dOg by the square of the number of equal chords to be used, o vfhich gives the deflection angle for station 1; then multiply this angle by 2 2 = 4, 3 2 = 9, etc. , for succeeding stations. Or take the angles, after the first, from Table XVII. Measure the chords Oa, ab, etc., as for a uniform curve. Example 2. Let the degree of Kd be D' = 8 20', and 8 = 300 feet. Then OAA -J Kf) Kd = ~ = 150 feet, and dEK = J^ X 8 20' = 12 30' = 0. A 100 5. = 6 15', ^ = 4 10' = 250', and ~ = 3 7f. 0.9 4 Compute as above, or Table XV gives AK = 5.44, and Table XVI gives AO = 149.78. 2^0 Using chords of 50 feet, the first deflection angle = = 06'. 95 or 07', and the succeeding angles to the nearest minute are 28', 1 03', 1 51', 2 54', and 4 10'. THE TRUE TRANSITION CURVED 239 It is not necessary to know ^lA'iu order to lay out the curve; but AO must be known to give the beginning of the curve. Problem 2. Given the degree of the offset curve Kd and the offset AK, to find the length of the transition curve, etc. Example. Let radius EK = R' = 698.73, deg. L>' = 8 12'--= 492'. Offset AK 4 feet. From (37), 4AK 16 3#' 2096.19 .-. =10 37*', ^ = 3 32V = 212y, and ~ = 2 89$'. o 4 Then Kd-^8= 100, = 100 . ---f = 100 ^-^ = 129.60. Table XVI gives d = .14, and therefore AO 129.46. Use five chords, each "''.. = 51.8 feet in length. 5 212 The first deflection angle is --^- = 8.5, and the successive an- gles are 34', 1 16', 2 16', and 3 32'. In case it is desired to use an offset of an approximate given length, it is generally sufficient and best to take Shi round num- bers so as to correspond with the desired offset sufficiently near, and then proceed as in Problem 1. Table XV very much facili- tates this operation. Thus for D' 8 12' and S = 250, we find (by interpolating for the .02' above 8 10') offset = 3.72 feet. Also, = D' - 10 15', ~ = 205', and - = 2 34'. Then Table XVI give's d = .12, and therefore AO = ~ - d = 124.88. Problem 3. Given the degree (D') of Kd and the tangent dis- tance AO, to find the length of the transition curve Omd, and the offset AK. 'MO FtKLD-MAXUAL FOE ENGINEERS. Example. Let D'=9 36', .'. #'=596.83, andlet ^0 = 99.912. From (11), sin f = 99.912 -5- 1342.87 = .07440; .-. = 9 36'. From (10), ~ = 99.912 -^ .99912 = 100; .-. =200. Now (27) or Table XV gives AK= 2.789. c SO Remark. If $(or Kd = -) and ^10 = cos are given, we have cos?- = AO + ~, which gives 0. Then D' = ~. 4 -c Art cr Or, under 8 in Table XVI, find d = ^10, opposite which, in column 1, is 7)'. Then Then ^4^ = ^ sin ^-; or, with D' and S, Table XV gives AK. ___,,. T _/O,0 . A -rr $ s\ If 5 and ^4 A = sin are given, sm = AK -. -- . Or o 2 26 under S in Table XV find AK, opposite which, in column 1, is D' '. Then as before. If AO and AK are given, (31) gives S Then (10) gives = AO sec . Then as above. THE TRUE TRANSITION CURVE. 241 Problem 4. Given tbe degree D of the main curve Af, and tbe length of Kd or of Omd to find the offset AK, the tangent dis- tance AO, etc. Example I.-Let ^#-#=599.62, ,', D = 9 33*'. Let 5 = 200. Then J5Td = 100, Af> 100, and = degree of Kd = D' > D. Assume = D = 9 33*'. Then | = 4 46f , 4. 4fia/ and, from (27), ^^ = 100 sin - = 2.776. o Then R' = R - t = 596.844. .-. D' = 9 36' = 0,; ~ = 4 48'; ^ = 192', etc. Now AO = 100 cos 2 34' = 99,912. Or these values of AK and AO may be taken from the tables. The above value of offset is only approximate ; but AO. and not AK. is used in laying out the curve. To show the trifling effect upon Kd and AO of an error in the offset we now compute corrected offset. We have AK = t' = 100 sin *li' = 2.789. o Let R" and. D" represent the new values of R' and jy. Then R" = R t' = 596.831, and D" = 9 36' = IT. Or Table XV gives opposite 9 33*' t = 2.776. Then R' = R - 2.776 = 596.844, and D' = 9 36'. Now opposite 9 36' we find t' = 2.789, etc. It is seen that t', the corrected value of t, exceeds t by only .013, and therefore R", the corrected value of R' , falls short of R' by the same .amount; so that D" exceeds D' by only of a second and does not appear in the result. This f of a second in I)' or de- creases AO by less than two units of the fifth decimal place. 242 FIELD-MAXUAL FOR ENGINEERS. We thus see that the assumption,/*??' the purpose of computing the offset, that the degree of Kd is the same as that of Af, leads to only a small error in the offset and to an exceedingly small error in the essential quantities involved; so that it is useless to recom- pute the offset, etc. We also have, from (32), R-2t' Example 2. Let AE = R= 67407, .-. D = 8 30'. Let S = 300, then Kd = 150. Af > 150, and > |Z>. Assume = \D = 12 45', and ^- = 6 22f . S , 100 sin 6 224' Then AK = t = sm = - - - = 5.55. o /& & Hence R' = 674.07 - 5.55 = 668.52. .-. D' = 8 34'; = \D' = 12 51'; = 6 254'; a ?- = 257', and ~ = 3 12f. o 4 It is useless, as already explained, to compute a corrected offset t' , since it would not increase D' but little over 2", nor decrease AO by a unit of the fourth decimal place. Now AO = 150 cos 3 12f = 149.76. Problem 5. Given the degree of the main curve and the length of the part Af replaced, to find the offset AK, the tangent dis- tance AO, etc. Example. Let AE R 599.62, .'. D - 9 33i'. Let Af = 100. Then = D ; f = 6 22'|; - = 191', and ~ = 2 THE TRUE TRANSITION CURVE. 243 By (38), Hence and Now R vers 599.63 X .0061746 | + vers 1.3395 = 2.764. EK=R' = 599.62 - 2.764 = 596.87, D' 9 36'. ?-= Ed= 100^-, = 99.54. JJ Table XVI gives for D' = 936' and 8 = 200, |^-40 = .09; .-. AO = 99.45. Use four chords each - = 49.77 feet long. The deflection angle for the first station is J T 9 ^ = 12', nearly, and others are 48 X , 10?V, and 191'. Problem 6. To replace each half of a simple curve AfA' y Fig. 131, of radius R and degree D by a transition curve. Example. Let etc. = 716.2, .-. D = 8, AEV~ %V = = 19 e 246 FIELD-MANUAL FOR ENGINEERS. Then, see eqs. (45), (46), (47), ceO = S(2S+ Si}~\ o and eel = (S + 8^(28 . o Now the transit being at c, sight to 0, reverse on I, and turn off successive angles found by putting 8, =1, Si =2, etc., in the last equation. These angles are (8 + 1)(28 + I) 6 -, (S + 2)(2S + 2)L etc. o o Then measure cd, de, etc., as usual. The curve may be easily retraced from e toward 0. We have id - OcK' = 2AOc = 28 V- o . (50) o w l >)-. . (51) Let 8' = 8+8,. Then, in (51), for the first station from e put 8 = #' 1, and Si = 1; for the second station from e put 8 = 8' 2, and S t 2, etc., giving the angles (8' - i)9, (ff f )20, etc. Turn off these angles in succession from the tangent et, and locate d, c, etc.; otherwise as usual. Problem 9. To substitute a transition curve for an end portion of a main curve without moving the rest of the curve. Again, cet = Oet Oec = 2AOe - ceO THE TRUE TRANSITION CURVE. 247 Let Aide (Fig. 133) be a part of the main curve tangent to T A.i. Suppose the curve deK, -- R, to be run from some points of tliecurve Aide, through an angle KE'd-AiEd = 0,, and draw dHHi parallel to OT. Suppose a transition curve emO, having a central angle KE'e 0, to connect this curve with the tangent OT at FlG - 133 - 0. The point d must be taken far enough from Ai to give room for the transition curve between d and the tangent AO; and ft' cannot be < %R, for, if so, KH would be < %AH, or AK would be > AH, and no transition curve could connect Ked and the tangent AO. Draw en parallel to OT. Then Kn \An. In general we have AH AiH! = R vers 0,, HK = R' vers 0,. KH AH (52) If R' = f R, KH = %AH&ud the transition curve begins at d. If R' > |/?, KH > \AH> or AK < \AH, and the transition curve begins at some point between JTand d. We have, from the above, AK = AH - KH =(R- R') vers 0,. Also, nK = SAK = 3(R - R') vers t . HK R' " nK ~ 3(7? - R'f 246 FIELD-MANUAL FOR ENGINEERS. Then, see eqs. (45), (46), (47), cOe =?& ceO = 82 and eel = (8+ 8i)(28 +&),y. o Now the transit being at c, sight to 0, reverse on I, and turn off successive angles found by putting 81 = 1, Si 2, etc., in the last equation. These angles are (8 + 1}(28 + 1)-. (8 + 2)(2S -f 2)-, etc. o o Then measure cd, de, etc., as usual. The curve may be easily retraced from e toward 0. We have td = OcE' = 2AOc = 28^~. o . (50) Again, Get = Oet - Oec = 2AOe - ceO - - 8(28+ St)- = 8^8+ 28 l . . (51) Let 8' = 8 + 8.. Then, in (51), for the first station from e put = S r 1, and & = 1; for the second station from e put 8 = 8' 2, and Si = 2, etc., giving the angles (8 f - i)0, (' - $)29, etc. Turn off these angles in succession from the tangent et, and locate d, c, etc.; otherwise as usual. Problem 9, To substitute a transition curve for an end portion of a main curve without moving the rest of the curve. THE TRUE TRANSITION CURVE. 247 Let Aide (Fig. 133) be a part of the main curve tangent to OT &tAi. Suppose the curve deK, with dJ'=R' and draw dHHi parallel to OT. Suppose a transition curve emO, having a central angle KE'e = 0, to connect this curve with the tangent OT at 0. The point d must be taken far enough from Ai to give room for the transition curve between d and the tangent AO; and R' cannot be < $R, for, if so, KH would be < $AH, or AK would be > \AH t and no transition curve could connect Ked and the tangent AO. Draw en parallel to OT. Then FIG. 133. In general we have AH - Kn = \An. T l= R vers 0,, HK = R' vers 0,. KH AH K R' (52) If R' = f R, KH = ^AH&nd the transition curve begins at d. If R' > #, KH > \AH> or AK < AH, and the transition curve begins at some point between .ZTand d. We have, from the above, AK = AH - KH = (R - R') vers 0,. Also, nK = 3AK = 3(72 - R') vers 0^ HK R' nK B(R 248 FIELD-MANUAL FOR ENGINEERS. But IIK R f vers 0, vers nK R' vers vers ' vers Oi _ I? vers -'- 3(7? .#')' or vers O t = ^ vers vers 0. (53) Since .fiTcZ and Aid subtend the same angle, Kd _ R __ D Aid = R ~!y' We have seen that if from any point d of a curve Aide of radius R a curve deK be described with radius R' $R, then a transi- tion curve connecting Aide with the tangent AOi will begin at d. When, however, A\de is a sharp curve, the change from radius R to radius |J? would be rather too sudden, and therefore it is best to take R' > ^R and commence the curve at a corresponding point 6 given by eq. (53). R' = *R, or R' = IR, or R' = -f^R are practical values for R'. 7-7 7f TJt The ratio ^ = ^ =7 shows that the larger R' is the nJ\. o(.K -K ) J-TTT larger = is, and therefore the longer the connecting branch de. It is not necessary to run the curve deK much beyond e; but if run to jfif, we thus determine A, which is opposite to K. In this case the position of T is not needed. A short solution of this problem sufficiently accurate, unless is very large, is the following : We have approximately nK' "' """ ~ "M.,,E>- / THE TtiuE TRAHsmoN CURVE. ^49 This formula gives only slightly too small a value for Kd. Example 1. Let R = 954.9 (D 6), and Ome = 200. Let R f = T y?, then D' = ^D = 6 40'. To find d, e, and 0, and to locate the curves. We have KE'e = = D' = D' = 6 40'; 7?' vers O l = 57^ - vers = 3 vers = .02028; .-. O l = 11 56'. Hence A,d = 100^-^ = 192.7; D Kd = ^A,d = &Aid = 173.4; de = Kd- Ke = 73.4. Or, by (54), Kd = 100 \ft = 173.2; de-Kd~Ke~ 73.2; Aid = V X 173.2 = 192.4, etc. Now AO = - cos t= 100 cos 1 40' = 99.96, or TO = S cos |0 = 200 cos 3 18' = 199.70 Also, - s= 133'. o For four equal chords the first deflection angle is Other deflection angles are found as heretofore, FIELD-MANUAL FOR ENGINEERS. Measure off AO = 99.96 feet or TO = 199.70 feet ; set the tran- sit at 0. and locate the transition curve Ome as usual. Then observing that Oeg' f 0, giving the position of the tan- gent at e, locate the branch de. Example 2. Given R = 716.2 (D = 8), R' = ftf, .-. D' = |Z> = 9, and S = 300 feet, to find the tangent points d, e and 0, and to locate the curves. o Since Ke = r- = 150 feet, B KE'e = \V = 13 30' = 0. vers Oi = 0773 JJT vers ^ = I vers ^ = '07368. ... Ox = 22 08' = 22.13. Then Kd = 100^- = 245.9, and de = Kd- Ke = 95.9 ft. ; J^ = %Kd = 276.6. Or, first find A^d, then JT(Z and de as in the preceding example. 8 300 cos 3 22' Now ^0 = -cos- = =149.74. Also, - = 27 - a' ....... (4) Similarly for another section a, = itf,D, - a'. ....... (5) These are the simplest formulas possible for this case, since each requires but a single multiplication, while any other formula FIELD-MANUAL FOR ENGINEERS. requires two multiplications. These formulas have been used by the author for more than thirty years. See correspondence in the Engineering News, vol. 32, page 73. When the surface of the ground is irregular between the center and side stakes it is best to divide the cross- section into trapezoids by dropping perpendiculars to the roadbed at those stakes, and at each break in the surface between them, thus forming a number of trapezoids as shown in Fig. 137. Then the area of the section is equal to the area F b H FIG. 137. of the trapezoids less the area of the triangle bBK. FORMULAS FOR REGULAR EXCAVATIONS AND EMBANKMENTS. A. The volume of a prismoid may be exactly calculated by the prismoidal formula,* which is = -( -f 4m (6) in which V= volume, I = length ; a t a, = the areas at the ends, and m = the area of the middle section. To find the volume in cubic yards, divide the above by 27. Since the linear dimensions of the middle section are arith- metical means of the corresponding dimensions of the end sections, we have in which a! = the area of the triangle under the roadbed. * The late Prof. W. M. Gillespie proved, many years ago, that the pris- moidal formula is applicable to a prismoid having one warped surface ; and the late Prof. De Volson Wood afterwards proved that the same formula is applicable to a prismoid having two warped surfaces. The author proved (in 1869) that the formula in question applies directly to a prismoid having any number of lateral surfaces, any number or all of which are warped. Mr. A. M. Bannister, C.E., of Lansing, Mich., Mr. Cornelius Donovan, C.E., of Port Eads, La., and others are acquainted with the facts. An elegant proof was since given by Prof. George Bruce Halsted of Austin, Texas. CALCULATION OF EARTHWORK. 257 B. Substituting the values of a, a\ , and in in (6), we find vol = j(JL + ) - ~(C - C&D - A); . (8) vol = lm +g|(0- C,}(D - A) (9) We observe that C Ci = c Ci, -| -j is called the ''end area volume," and lin the 2 / middle area volume. The above equations show that the former differs from the true volume twice as much as the latter and in the opposite sense. This may be shown symbolically as follows : (a + 4m - a,) ^( + = gr(2w - a - ^); (a + 4m + oi) J/w. = -^-(a -f i - 2w). 6 b The last of these equations is one half of the first with a contrary sign. In case the prismoid is level laterally we have, from Fig. 135, D = AB = ab + 2Aa = b + 2cS. Similarly A = & + 2dS. .: D D! 2S(c c,). = J (| + f) - ^( e - ,)'. Hence vol = I - + - 9 - - - (c - c,)\ . . . (10) 70 or vol = lm+ j^(c - Cl )' (11) To generalize equation (8): Let a, a\ , - a . . . a rt represent areas of sections ; c, Ci , Ci ... c n represent center heights ; and d, d\ , di . . . d n represent tl e extreme width of sections, as AB, Fig. 136, 258 FIELD-MANUAL FOR ENGINEERS. Then expressing 1 volumes of successive prismoids, and adding, we get vol = l(\a -f ai + l f n-! - d n )]. (12) The areas of one half of the end sections appear, and the whole areas of other sections. The correction for each prismoid must be taken separately. Let Fig. 138 represent a longitudinal section of a "cut" through the center line. Drop the perpendiculars CD, C\D\, etc., and at the middle points of DD\, DiD?, etc., erect perpendiculars i, MJ?, etc., to meet horizontal lines through C, (7,, etc. D M D a M! Do M 2 D 3 M 4 D 4 FIG. 138. Then the first part of formula (12) gives the volume represented by GDME -f E 1 MM 1 F + E^M^N^Q -f etc., and the second part furnishes the required corrections. Example. Let b = 20 and 8 = 1. Then (see Fig. 136) DE = 10 and area abE 100. Commencing at C, let the center heights be 3.0, 6.0, 7.0, 5.0, and 2.5, and distances out be 26.0, 33.2, 34.8, 31.6, and 26.0. Now, by (2) or (4), \a = i X 13 X 26 - 50 = 34.5 0,1 = | X 16 X 33.2 - 100 = 165.6 a a = $ X 17 X 34.8 - 100 = 195.8 a 3 = | X 15 X 31.6 - 100 = 137.0 $04 = i X 12.5 X 26 - 50 = 31.25 Then ~wTs X 100 = 56410, CALCULATION OF EARTHWORK. 259 Also, (c - c,)(d - d.) = 3.0 X 7.2 = 21.6 ( Cl _ c 9 )(d l - d a ) = 1.0 X 1.6 = 1.6 (c- 9 - c 3 )(d* - d a ) = 2.0 X 3.2 = 6.4 (c, - c,)(d s - d<) = 2.5 X 5.6 = 14.0 43.6 X -W- = 363. Therefore volume = 56052 cu. feet. These quantities may be taken at once from suitable tables, thus requiring only a few additions to give volumes of about as many prismoids. The prismoids ending at and 0' are best computed separately. To compute CDO. Let DO I' 75. The distance out at = b = 20. Then \ol' = 34.5 X 75 = 2587.5 and f|(c - 0)(d - 6) = 6.25 X 3 X 6 = 112.5 .', volume = 2475.0 If the ground is not level laterally in the region of (or 0'}, we must take cross-sections where the center line or either edge of the roadbed comes to grade and compute the volumes between the cross-sections separately. Generalizing eq. (10), we have ~ [( - *)* + (C - , (16) becomes I . vol = -(cd -&). . . (20) We observe with reference to all prismoids that the bases may occupy any relative position in the parallel planes without in the least affecting their volumes. LOADED FLAT CARS, PILES OF STONE, ETC. Eq. (20) gives the volume of a fiat car loaded with earth, gravel, etc., or a pile of stone, etc., c and d being the dimension of either base, and c\ and d\ those of the other base. Usually the side slopes in such cases are nearly uniform, but the formula applies, as we have seen, if each slope is uniform, whether they are equal or not, and also when the side slopes are warped surfaces. The volume of a borrow-pit divided into prisms having equal or equivalent bases, of area = A, is readily found by a single calcu- lation. Let Fig. 140 represent the base of such an excavation, the heights at the corners being denoted by the letters there placed. The total volume is equal to A multiplied by the sum total of the sum of the corner heights of the several prisms. Into this total sum the corner heights a, cij, b, 3 , d, d 3 will enter but once, being found in but one prism; c, c 3 , d^ and c? 2 will enter twice, being common to two prisms; ?>i and 6 2 will enter three times, being com- mon to three prisms. Finally, c being common to four prisms. C] d 3 FIG. 140. and c-2 will enter four times, 264 FIELD-MANUAL FOB, ENGINEERS. Representing these sums in order by S )} S^, 8 3 , and $ 4 , we Lave, for all the prisms, vol = $A(Si + & + 3 + &) ..... (21) ENDS OF EMBANKMENTS OR "DUMPS." Let Fig. 141 represent an embankment. Let c and Ci represent the center heights and d and di the hori- zontal widths of HK and IfiKi. FIG. 141. Draw a#i and bid. Then, by (8), vol abHCK . . . Hi GiKi = I- - c(d di since Ci = 0, and tii 0. ltd- d lt vol of dump = I- (23) GROUND IRREGULAR LATERALLY. Let Fig. 142 represent the part of a prismoid on the right of the center line CO' for the case supposed, which requires the nitev- mediate heights at D, E, D 1 ', and R'. The toiai volmt e is equal to the volume of the prismoids whose bases are ntFH/i, FGLII, and GBB'L, less the volume of the sub-slope prismoid bBK . . . b'B'K', CALCULATION OF EARTHWORK. 265 In general the volumes of the former are found by (16). When, however, the widths at the two ends of the prismoid are equal, (17) applies. When more heights are taken at one end than at the other, the width of one or more prismoids at one end is and (19) applies. The volume of the sub-slope prismoid is found by any equation of group (15). MIXED WORK, EXCAVATION AND EMBANKMENT. Suppose, for example, there is excavation on both sides and em- bankment on one side only. Let Fig. 143 represent the cross-section at one end of the pris- FIG. 143. FIELD-MANUAL FOR ENGINEERS. moid. CDb K shows the cut on the right, CDM that on the left and allM the fill on the left. Db is the half- width of road- way in the cut, and aD the half- width in the fill and cut. 11CK shows the surface of the ground, and a H said bK the side slopes. The volume of the cut on the right is found at once by (8), ob- serving that in this case a and . represent the areas on one side only of the center line, and Z>aud DI tue distances out on one side only. The volume of the cut on the left is found by (14), by substitut- ing the values of CD and DM for c and d, and making similar substitutions for c\ and di at the other end of the prismoid. The volume of the embankment is found by means of the same equations by substituting the values of aM and All in place of c and d, etc. In case a fill is on both sides of the center line and a cut on one side only, the same formulas of course apply. A cross-section must be taken at the point where the surface of the ground intersects the center of the roadbed. At this point the triangle CDM vanishes, CDbK becomes DbK, and HaM be- comes HaD. Then the volume of the cut, of which CDbK shows one end and DbK may represent the other, is given by (8), in which Ci = 0. The volume of the fill, of which allM shows one end and aHD may represent the other, is found by (14), as before. The volume of the pyramid, of which CDM shows one end and the point D represents the apex, is given by an equation follow- ing group (15), and is IKd vol = , in which K = CD, and d = DM. The prisrnoids adjacent to the cross-section where the change considered occurs, but on the opposite side of it to those just con- sidered, are, it is evident, computed in the same way and by means of the same formulas. A cross-section must be taken, also, where the surface of the ground intersects the side of the roadbed. In this case the same formulas apply to the prismoids on either side of the section in question, and adjacent to it, in precisely the same way as in the case just considered. CALCULATION OF EARTHWORK. 267 CORRECTION OF EARTHWORK FOR CURVATURE. Suppose the center line of the prismoid (Fig. 139) is an arc of a circle of radius R. Let e = the length, measured on the line CC', of a very short prismoid, at CDK or C'D'K', or at the middle section. Then the lengths measured through the centers of gravity of these prismoids are as follows : The length at CDK is The length at C'D'K' is and that at the middle section is e R It is evident, however, that we may consider the lengths through the centers of gravity of the elementary prismoids to he the same as their lengths along CC' if we, at the same time, increase the areas of the middle and of the end sections accord- ingly. We must write, therefore, -t A in place of A, H R+Sdi AI in place of Ai , R + tfm and m in place of m. K Hence I (R + Id . R + Wt . .11 + dm \ truevol=r_l _ A-\ -^ A + 4 ~ mj, . (25) and the excess of vol = _[_*_ j[ _j_ L-l^ -j- 4*L-j|A (26) I d 1 A 1 + 4dm X m). (27) FIELD-MAXUAL FOK ENGINEERS. OVERHATTL. No allowance is made for moving excavated material when the haul does not exceed a certain specified distance culled the " limit of free haul." But when the material is carried beyond this limit the extra labor involved is paid for at a stipulated price per cubic yard, per each 100 feet in excess of the tree-haul limit. b f/ FIG. 144. Let us suppose the material in the cut cO, Fig. 144, just sufficient to make the fill Od. First find on the profile two points a and b such that the cut aO will just make the fill Ob, and that the distance ab is equal to the limit of free haul. These points a and b are found by means of the cross-sections and calculated quantities ; though since a, as well as b, usually falls between regular stations, it is generally necessary to find the point a by one or more trials, so that ab is equal to the required limit. Having found ab, it is to be remembered that the contractor is entitled to pay for moving every cubic yard of material from the cut ca to the fill bd' for the whole distance it is moved, less the distance ab. This is equivalent to moving the whole cut a distance gg r ab = ga -f bg' ; g and g' being respectively the centers of gravity of the cut ca and of the fill bd made from it. But the volume of ac multiplied by the distance ag is equal to the sum of the products obtained by multiplying the volume of each prismoid in ac by the distance of its own center of gravity from . It is usually sufficiently accurate to consider the center of grav- ity of a prismoid as being at its mid -section; but if greater accu- racy is required, we have for the distance from the mid-section x . -, very nearly. , . . o (28) 6 A -\- A. CALCULATION OF EARTHWORK. 2G9 This is, of course, to he added to or subtracted from the dis- tance of the mid-section from the point a accordingly as the larger end area A is the farthest from or nearest to the point a. In the same manner we find the sum of the products obtained by multiplying the volume of each prisinoid in bd by the distance of its center of gravity from b. Summing the products for both cut and fill, the distances be. ing expressed in chains of 100 feet, and multiplying the result by the stipulated allowance, we have the amount to be paid for haul. Parts of a cut may be carried in opposite directions, in which case each part must be figured separately. It is evident that no allowance is made on material wasted. MONTHLY ESTIMATES. Monthly estimates are usually made by the resident engineer near the end of each month, though it is sometimes necessary to make them at other times. For this purpose calculated quanti- ties in the field-books are used, supplemented by such measure- ments as may be necessary. The monthly estimate is only approximate, and should not be above the real amount of work done; but it should be as definite and complete as the nature of the case will permit, and should include a detailed statement of all the work done and material delivered. A special field-book is devoted to monthly estimates. Cross- sections must be taken for the purpose of computing the amount of excavation completed ; and notes of everything done must be made. Where work is completed, the corresponding quantities may be taken from the field-books or other books containing them. An allowance somewhat below the actual value should be made for all material delivered but not yet used, as well as for all labor performed and expenditures made in properly forwarding the work. Estimate sheets should be used, the sheet for each month show- ing, for different parts of the work and for each kind of material, the total of the previous estimates, the present estimate, and the total including the present estimate. The total estimate for any month becomes the total "previous estimate" for the succeeding month. 270 FIELD-MANUAL FOB ENGINEERS. The division engineer, or the chief engineer in case there is no division engineer, reviews these reports, copies them on other sheets; attaches the prices to the items ; computes and sums up the amounts. The railway company pays the contractor, each month, about 85$ of the estimate for that month, and retains the rest till the completion of the contract. FINAL ESTIMATE. The final estimate is a complete statement in detail of all the work done and of all the material furnished by the contractor, and furnishes the basis of final settlement between the company and the contractor. This statement is completed, in detail, as the work progresses. As soon as the data of any part or subdivision of the work, or of any structure, are supplied, a complete statement in regard to it should be written out in detail in a book for that purpose. The number of cubic yards in each prismoid, extra for cutting sur- face ditches, overhaul, etc., should be given. A complete state- ment in regard to each bridge or other structure should be cure- fully written out. The notes should be made especially full, while the work is in progress, in regard to all parts of it which are inaccessible after completion, such as foundation-pits and foundations of all kinds, and all works under water. COMPUTATION OF PRISMOIDS LEVEL LATERALLY. Let abmn (Fig. 145) represent one end of an embankment, 100 feet in length, level laterally. FIG. 115. Let cd, ef, etc., represent horizontal planes which divide the embankment into layers 0.1 of a foot in height. CALCULATION OF EARTHWORK. 271 Let the base ab = b, and tlie side slope ==. Then a I) ab = b; cd b + 0.2*; /= & + 0.4s, etc. ** Also, Hence + ** = 5 + = & + 0.3*. etc. the vol. in yards, of = - + 0.1.) = 27 and vol of cdef = ^(b + 0.3*) = etc. It is plain tliat the volumes of tlie successive layers are in aritli- 2.9 metical progression, Laving a common difference of . Hence to find the total volumes corresponding to different heights of the embankment, differing by 0.1, it is only necessary to write down for the volume of abed: add - - to it for the volume of abef, then to the last result for the volume of abgh, etc., omitting, for convenience, the denominator 27. Example 1. Let b = 18, and 8=1. Then ~- = 6 + if Hence we have as follows : 27 etc. Heights. Volumes. Heights. Volumes. 0.1 6 19 6 21 0.4 27. 7 7 0.2 13 13 6 23 0.5 34 7 7 2 0.8 20 9 6 25 0.6 41 9 etc., etc. ElELD-MANUAL FOR EXGIXEE!^. This may easily be carried 20 or 30 feet in an hour, or less, Example 2. Let b = 14, and s = f . Then 27 Then as follows : Heights. Volumes. Heights. Volumes. 0.1 5 13 5 19 0.5 27 17 5 43 0.2 10 32 5 25 0.6 33 6 5 49 0.3 16 3 5 31 0.7 39 1 6 1 0.4 21 34 5 37 0.8 45 2 etc., etc. These results can be copied to the nearest yard very quickly. The author hopes to show the best processes of computing- all earthwork and other tables, and of making all numerical com- putations, in the near future. CHAPTER XI. APPROXIMATE AND ABRIDGED COMPUTATIONS. THIS chapter will treat of the subject of approximate -rind abridged computations to a very limited extent only, dealing mainly, too, with the practical side of the question. Approximate computations are those that lead to results not strictly accurate. Abridged computations are those that reach certain resuliswith lef--s labor than the ordinary computations involve. Abridged computations may be either accurate or approximate, but are usually the latter. The absolute error of a result is the difference between that result and the true result for which it stands. The relative error of a result is the absolute error dividf d by the true result, and it is therefore equal to the absolute error of each unit of the result. Thus if a true result is 84 and the approximate value 77 is taken for it, the absolute error is 84 77 = 7, and the absolute 7 1 error of each unit of the result, or the relative error, is-^- = . o4 1/i It is important to notice that the relative error of a result is not changed by multiplying that result by any number; since ths error in the product or quotient will be increased or decreased in the same ratio as the result is increased or decreased. Thus if n be the true and n -f- e the approximate result, then r - is the relative error. Multiplying by m, we have the Ti approximate product m(n -j- e} mn -f me, whereas the true product is mn. Hence the absolute error of the product is me, and the relative error is = as before. mn n If an approximate quantity is given, the absolute error of which is a given number of units of a given order counting from 373 274 FIELD-MANUAL FOR ENGINEERS. the highest, it follows from tlie preceding that the relative error is independent of the decimal point. Thus suppose the absolute error of each of the numbers 7.073, .07073, and 707.3 to be two units of the fourth order. The left-hand significant figure expresses units of the first order, and therefore the fourth order is express .-d in the above by the figure 3. The relative errors of the numbers .002 .00002 .2 2 are T073~' :07073' an a " d S =PL' CONSTRUCTION. Dividing, we have - CL - d ~ di ~8 ~ "d t ~ di ' or d t =LP f =- ....... (5) Subtracting this from CP' = d, we have - d8 ' Dividing (5) by (6), we have LP' 8 Dividing (5) by CP' = d, we have LP 1 S CP> - s 1 4- 8 ^ These equations show that the subtractive distance LP' is to the distance CP', for a level section, as 8 is to S' + 8, and LP' is to CL as /S is to 8'. It is evident that these formulas apply at once to the upper side of a fill. Example. Let & 20, c = 10, S f 4, and 8=1. To find CL, the distance of P from the center C. We have gj = 4 <' + " = 16. 4-f-l Also, Let Fig. 148 represent a side-hill section. 290 FIELD-MANUAL FOK ENGINEERS. I. To find by trial the position of H. Let e 10.4 and is = 1. It will be convenient to consider the height of the roadbed equal to 0. Distance out for level section d = CH' = CP' 10 -f 10.4 = 20.4. Suppose we judge the ground near H' to be 5 feet higher than at G, or 15.4 feet. The distance out corresponding to that height would then be 5 feet beyond H' ', or 10 i5.4 = 25.4 feet. But the ground 5 feet beyond H' is still higher than at //', re- quiring a still further distance out. Let us test it at e, 26.5 feet out. Suppose we find the ground 6.6 feet higher than at C. This JS, FIG. 148. requires a distance out of 20.4 + 6.6 = 27 feet, or 0.5 of a foot be- yond e. But, as before, at 27 feet the ground is a little higher than at 26.5 feet, so we will try it at 77, 27.2 feet out, for example. Suppose we find this point 0.2 feet above e, or 6.8 above G, This requires a distance out of 27.0 -f 0.2 = 27.2. Hence the height and distance out correspond and are therefore correct. On the lower side of a fill we would, for reasons already given, proceed in precisely the same way. II. To find the position of P. Let us suppose the ground near P 1 to be 6 feet lower than at 0. This would require a distance out of 20.4 6 14.4 feet. But 6 feet inside of P' the ground is higher tlian at P , and, according to the supposition, less than 6 feet below C, and re- quires, therefore, a distance out greater than 14.4. Suppose we find the ground at e', 17.4 feet out, 4.8 feet below C', CONSTRUCTION". 291 or 5.6 feet. This requires a distance out of 20.4 4.8 15.6 feet, or 10 -f- 5.6 = 15.6 feet. But at 15.6 feet out the ground is higher than at c', and there- fore the distance out must be greater than 15.6 feet. Suppose we now find the elevation at P, 16 feet out, to be 0.4 above e', or 6.0 feet. This height and the distance out now corre- spond and are correct. We are now prepared to state two important principles which will greatly facilitate the process of cross-sectioning : I. 1 f, on the upper side of a cut or lower side of a fill, any height found calls for a given change, e, in the distance out, either inward or outward, the real change required is always greater than e, and the excess increases with the lateral steepness of the ground. Thus, referring to Fig. 149 and the preceding notation, ob- serving that H"e' is the additional distance out (beyond H') cor- responding to the additional height 11" H' above C, we have the real addi- ) nv- o> tional dis- [ = 11' K = H"e = H"c' - = #"<_- (9) tance out ) II. If, on the lower side of a cut or upper side of a fill, the height found calls for a given change, e, in the distance out, either inward or outward, the real change required is always less than e, and the deficiency increases with the lateral steepness of the ground. We have as above, observing that P"d' is the distance to come in corresponding to the fall P' P" below 0, l ) e U in ) the real distance PK' = P'd = P"d' X =- =P"d -- (10) to come in The author has used these principles for thirty years, but has not seen them stated. Example. Let ab 24, c = 10.5, 8 f, to find the position of H. (Fig. 149.) CH' = 12 -f |(10.5) = 27.8. Suppose we judge the ground near H' (&iH") to be 6 feet higher than at G. Then we know that the distance out is greater than 27 g -f- 9 36.8. The additional distance corresponding to H"H' i U"c', and eV s= 36.8. Suppose we try it at e, 39.0 feet out, O O, O FI ELD-M A X I' A L FO It EKGIK E ERS. and find e 8.6 feet higher than C, or 19.1 feet, calling for a dis- tance of 27.8 + 12.9 = 40.7 feet, or 1.7 feet beyond e. We will therefore try it at H, 41.7 feet out, and suppose we find the ground 0.7 feet higher than at e. This would call for an additional foot beyond 40.7, or 41. 7 as we have it. The cut is therefore 19.1 -f 0.7 = 19.8, and distance out 41.7 ft. To find the position of P. We have CP' = 27.8. Suppose we judge the ground near P' (at P") to be 6 feet lower than at C '; then we know that P is inward from P" less than 9 feet, or P"d'', and the distance out is therefore greater than 27.8 9.0 = 18.8 feet d'd". Suppose we try it at e f , 22.0 feet out, and find e' 5.2 feet lower than C, or 5.3 feet, calling for a distance out of 27.8 8.0 = 19.8 feet, or 2.2 inside of e'. Suppose we come in 1.2 feet or 20.8 feet out to P, and find it 0.6 feet higher than at e' and requiring a dis- tance out of 19.8 -f 0.9 = 20.7. As this is but 0.1 inside of the point of observation, we can probably set Pat 20.7 feet out with sufficient accuracy. It will be seen that the first point of observation is chosen with reference to the known distance CII' for a level section, and that subsequent points are chosen with reference to the figures ob- tained at the last point observed. A table which can be computed in a few minutes, giving dis- tances out for the given side slope corresponding to different heights, will greatly facilitate the work and likewise conduce to accuracy. If the ground is irregular laterally, the heights must be found at all points along the cross-section where the slope changes, and these heights, as well as their distances out, must be carefully recorded. When the surface of the ground intersects the roadbed as shown in Fig. 150, we have what is called side-hill work. CONSTRUCTION. 293 Let POGH represent the surface of the ground. H is found f.s already explained, and is found by simply finding a point on FIG. 150. HC prolonged, whose height is equal to the height of the roadbed or aero. To find P we have in a = 8, and we must find a point Pon CO prolonged that will satisfy this equation. To do this it is only necessary to observe that ma increases faster than mP ; and ma hence to increase - mP we must move outward, and to decrease the same we must move inward. When two materials are found in the same section, as rock overlaid with earth, it is neces c ary to give each material its proper slope. Since the upper layrr is usually of varying thickness, it will be necessary to excavate a trench along the cross-section PCH in FIG. 151. order to expose the rock, so as to set the stakes PI and Then Pand //can be located as usuaj. 294 FIELD-MANUAL FOR ENGINEERS. It is easily shown, however, that when the layer of earth is uniform + f> + ., . . (11) In these equations S = side slope of the rock, 8 t = that of the earth, and S' = the slope of the ground ; also & = width of the roadbed, c = center depth of rock, and d = that of the earth. BORROW-PITS. When the cuts are insufficient to make the fills, or are too far away, material taken from borrow-pits is used. These should be carefully staked out by the engineer so that their contents can be calculated when completed. It is usual to lay out the area to be used in squares or rectangles, extending one or both sets of lines, so that when the excavation is completed the lines can be readily reproduced in the bottom of the borrow-pit, and heights taken there under the original heights taken upon the surface, thus giving the depths of the excavation. If the sides of the rectangles in one direction are 10 feet or some multiple of 10 feet, and in the other direction 27 feet or some multiple of 27 feet, the computation is very much facilitated. Borrow-pits should be regularly excavated so that they can be easily computed and will not present an unsightly appear- ance when abandoned. Material, when suitable, may be obtained by widening the cuts, provided the fills are accessible and not too far away. Any surplus material in the cuts should be used in widening the adjacent fills if possible, otherwise it should be deposited where the engineer directs, and is said to be wasted. It should in no case be deposited on the upper side of a cut, unless \veli removed from the edge, for otherwise it would greatly interfere with the surface ditches, which should always extend along the upper side of a cut, a few feet from the edge, to prevent the surface water from pouring into them. SHRINKAGE. In estimating the volumes of cuts or - 252 indl < or fully one quarter of an inch. At 80 Fahr., or 50 below the maximum, it need be only half as much. The space required is, of course, proportional to the length of the rail. The engineer should provide, or see that the contractor pro- vides, wedges suitable to different temperatures in which the rails are laid, and also see that thev are used, far too small a space would result in the rails being forced up by expansion, and a space too large would result in a rough track. CONSTRUCTION. 297 Where sidings are required the necessary frogs and switch- ties should be provided in advance, so that they may be put in place as the main, track is laid. Heavy plank for road-crossings should be laid as soon as the rails are spiked so that the highway travel may not be inter- rupted. CULVERTS. For small openings piping answers an excellent purpose. The ground must be brought to grade and well tamped so as to form a firm and homogeneous bed for the pipe. If the natural soil is unsuitable to form a firm bed, a quantity of clay should be provided for it and tamped as above. The til>l>< r surface of the pipe should be at least 2 feet below grade. It is generally bad practice to use stone in any way about these culverts, since it destroys the homogeneity of the struc- ture and thus invites scouring and destruction of the cul- vert. The author has constructed many pipe culverts without the use of a stone, and without, so far as known to him, the loss of a single culvert. Open wooden culverts are well ad^ted for somewhat larger openings in low embankments. Any defects in them are easily seen, and they are easily renewed or replaced if desired. Cov- ered culverts are sometimes used in higher embankments, the covers being square timbers 12 or more inches in thickness. The walls of the culvert should extend 1% or 2 feet below the surface of the ground, and should be well tied into the bank by timbers embedded for that purpose. The bed of the culvert should be left undisturbed, the open- ings should be as nearly of a uniform cross-section as possible, having a moderate uniform slope, and their approaches should be as straight and uniform in slope as practicable. Where these essentials are fully observed and the culverts are properly constructed there is no danger of their washing out. Culverts are often destroyed in consequence of being so small as to partly block up the stream, and thus disturb its easy and natural flow, causing whirls, eddies, washing of banks, etc., and finally undermining and perhaps washing out the culvert. It is not enough that a culvert is able to discharge all the water that approaches it; for safety it must do so without too much disturbing the natural and normal flow of the water. FIELD-MANUAL FOR ENGINEER!?. Culverts of less than 8 or 10 feet opening should not be con- structed in any case. For the foundation of large culverts piles can often bo used to advantage. Such a foundation is comparatively inexpensive; it will last indefinitely and cannot be washed out. Moreover, the structure itself can be so fastened together and to the piles as to be immovable. Where good stone is plentiful, arched masonry culverts, though costly, are in some respects the best of all, since if properly built they are very durable and therefore not likely to fail and cause disaster. The location of a culvert depends somewhat upon the con- formation of the ground, it being necessary sometimes (but should be avoided when practicable) to set the culvert on a " skew," in order to avoid too much excavation or to fit, to the best advantage, the thread of the stream. A stake is set at each corner of the area to be occupied by the culvert, and the cut to be made is marked upon it. The general principles of locating points and lines under various conditions having been fully explained in preceding chapters, including Chapter IV, it is only necessary to apply them to any and every case that may arise. The location of bridge piers on a curve requires, however, something of a compromise; and what that is, and a handy way of doing it, will therefore be explained in this, connection. Let CD, Fig. 152, represent the curve between two adjacent piers. Draw the chord CD. The middle ordinate MN = CE vers CEM . Take CA = DB = y 2 MN. A and B are the centers of the piers. We observe that AB is on a line half-way between the chord CD and tangent to the curve at .!/. Proceed in the same way for other piers. TUNNELS, Tunnels, when possible, should be on a tangent throughout, so as to be easily laid out, and to freely admit the light. The location of Tunnels, like that of Bridges and other struc- CONSTRUCTION'. 209 tures should of course be such as to render the total cost a minimum for a given service rendered. The material to be encountered may in some cases be deter- mined with tolerable accuracy by a study of the geology of the adjacent region, but for more accurate information it is neces- sary to resort to borings. The alignment of a tunnel is more or less elaborate according to its length and surroundings, and great care should be be- stowed upon it. Indeed in all cases where errors v/ould be more or less costly every effort should be made to avoid them by using the best tapes and rods and instruments, in perfect adustment, and by repeating, several times, all observations with transit, level, etc., and all computations. It is of the greatest importance to have the transit revolve in a vertical plane, and to secure this a sensitive bubble-tube should be attached to the horizontal axis of the telescope. It will usually be necessary to find the distance through the tunnel by triangulation. By triangulation also, and by direct surveying, the highest point and other high points on the line of the tunnel are found, as well as other points on the prolongation of the line in both directions. These points determine the line, and stations are established at them, by means of which any desired point may be located and the work easily laid out. As the excavation of the tunnel proceeds successive points on the center line are determined, usually on the roof, from which plumb-lines are suspended, thus plainly and constantly defining the line. Whether tunnels are on tangents or curves the surveying operations connected with them are simple and have been fully explained. As already stated, great care, repetitions, and numerous and varied checks on the work are important. For detailed and full information regarding tunnels the reader is referred to Sims' and Drinker's books on the subject, and to the current engineering journals. CHAPTER XIII. EXPLANATION OF TABLES- AND MISCELLANEOUS TOPICS. Table I. The radius of a 1 curve = ^^-A^. 1 80 V 1 00 v 60 The radius of a 1' curve = A A = 343774.677078. To find the radius of any curve divide the ladius of a 1' curve by the number of minutes in the degree of the given curve. Table II contains tangent offsets for all arcs up to 25 feet, 50 feet, or 100 feet for curves of different degrees. Tables III, III, and lllb contain tangential offsets, middle 01 dinates, and chords for arcs of 100 feet, differing in degree by 10'. Tables IV and V contain the value of long chords and middle ordinates to long chords for curves from to 20, differing by 10 . These tables are very useful for passing obstacles, for finding tho middle point of any arc, for laying out curves, etc., etc. Table VI contains the various elements of frogs and turnout curves. Table VII contains the true values of the tangents and externa ' of a V curve. To find these functions for any other curve divide the tabular numbers corresponding to the given intersection anglu by the number of minutes in the degree of the curve. Thus for a 2 14' curve and V = 28 we have tangent = 85712.7 -*- 134 = 639.7, and external = 10524.2 -s- 134 = 78.5. Thus the true result is given by a single operation, whereas tbw usual tables require three operations to find an approximat result. EXPLANATION" OF TABLES; MISCELLANEOUS TOPICS. 301 Tables VIII, IX, and X are supposed to be in the most conven- ient form for use. Table XI contains the correction e, in feet, corresponding to any distance, D, due to curvature and refraction. Tables XII, XIII, and XIV need no explanation. Tables XV and XVI contain, respectively, the offsets and tan- gent distances of transition curves, in terms of the degree of the offset curve, and the length of the offset or transition curve. Thus for an offset curve of 6 and transition curve 200 feet long the offset by Table XV is 1.74. Also, by Table XVI, - - AO = .03, or AO = 100 - .03 = 99.97. Table XVII contains deflection angles for transition curves, for one to five chords, iu terms of the angle for the first chord. Thus if the deflection angle for the first chord is 2 f . 3, then the angles for the second, third, fourth, and fifth chords are 9'. 2, 20'. 7, 36'. 8, and 57'. 5, respectively. Table XVIII contains the volumes oi earth, etc., for different slopes and bases. Table XIX applies to all bases having slopes of 1 to 1 or 1^ to 1. Example. Let base ab (Fig. 135) = 20, slopes 1 to 1. Let CD = e = 6.4. Then DE 10, and CE ~ 6.4 -f 10 = 16.4. Now for 16.4 the table gives, volume for HKE 996 and for 10 the table gives, volume for cibE 370 Therefore the desired volume for abllK = 626 The subgrade volume abEJ being constant for any given base and slope can be taken from the table once for all. A table designed specially for any base can be used as above ;'or any other base having the same slopes. The remaining tables perhaps require no explanation. MISCELLANEOUS. To Gauge a Stream Approximately. Take some body, a partly filled bottle, for example, that will float nearly submerged, and allow it to do so down a uniform and open stretch of the stream 100 feet in length, and note the time 1 in seconds. 502 FIELt)-MAKUAL FOR ENGINEERS. Measure tlie cross-section A in square feet, then the cubic feet of water that passes per minute is 5000 J. This assumes that the average velocity is f of the observed velocity. (See Bowser's Hydromechanics, p. 217.) Horse-power of falling water = .Q0189QH, in which Q = cubic feet per minute falling H feet. Transverse Strength of Rectangular Beams. Let L length in feet, b = the breadth, and d = depth in inches ; w = load at center in pounds, and R = modulus of rupture. Then Kbd* w = 1SL ' R is taken from 1000 to 15,000 pounds for wood, and from 10,000 to 15,000 pounds for wrought iron. If the load is distributed, it will carry twice the above amount. This is useful during construction in deciding whether or not working trains can safely pass over unfinished bridges, culverts, etc. JSafe Bearing-power of Piles. Let w = weight in pounds of the hammer used in driving a pile, h the fall in feet, 8 = the penetration in inches, and R = the bearing-power sought. Then, assuming a factor of safety of 6, +r This is the formula of the late Mr. A. M. Wellington, formerly editor of the Engineering News. It has the merit of being as trustworthy as any, combined with grea* simplicity. 363 TABLES. 304 TABLE I RADII. CHORDS, OFFSETS, AND ORDINATES. Degree D. Radius R. Chord Sta. Tang. Off. t. Mid. Ord. m. Degree D. Radius B. Chord 1 Sta. Tang Off. t. Mid. Ord. in. 0' No rad. 100.00 1 0' 5729.58 .873 .218 1 343775. .015 .004 1 5635.65 .887 .2-22 a 171887. .029 .007 2 5544.75 .90:. .828 JJ 11459;!. .044 .011 3 5456.74 .916 . xixJ'J 4 85943.7 .058 .015 4 5371.48 .931 .233 5 68754.9 .073 .018 5 5288.84 .945 .236 (i 57295.8 .087 .022 6 5208.71 99.998 .900 .240 7 49110.7 .102 .025 7 5130.97 .974 .244 8 42971.8 .116 .029 8 5055.51 .989 .247 9 38197.2 .131 .033 9 4982.24 1.004 .251 10 34377.5 .145 .036 10 4911.07 1.018 .255 11 31252.2 .160 .040 11 4841 .90 1.033 .258 12 28647.9 .175 .044 12 4774.05 .047 .2(52 13 26444.2 .189 .'047 13 4709.24 .060 .265 14 24555.3 .204 .051 14 4645.60 .076 .269 15 22918.3 .218 .055 15 4583.66 .091 .273 16 21485.9 .'.'33 .058 16 4523.35 .105 .276 17 202-22.0 .247 .062 17 4464.61 .120 .280 18 19098.6 .262 .065 18 4407.37 .134 .284 19 18093.4 .276 .0(59 19 4351.58 .149 .287 20 17188.7 .291 .073 20 4297.18 .164 .291 21 10370.2 .305 .0;.; 21 4244.13 .178 .295 22 15626.1 .320 .080 22 410-2.37 .193 MB 23 14946.7 .33"^ . 50 6875.49 .727 .182 50 3125.22 .600 .400 51 6740.68 .742 .185 51 3097.07 .614 .404 52 6611.05 .756 .189 52 3069.42 .629 .407 53 6486.31 .771 .193 53 3042.25 99.995 .643 .411 54 6366.20 .785 .196 54 3015.57 .658 .415 55 6250.45 .800 .200 55 2989.35 .672 .418 56 6138.83 .814 .204 56 2963.57 .687 .42-2 57 6031.13 .829 .207 57 2938.25 . 702 .425 58 59-27.15 .844 .211 58 2913.34 716 .429 59 5826.69 .858 .215 59 2888.86 .731 .433 CO 5729.58 .673 .218 60 2864.79 745 .436 1 TABLE I.-RAD1I, CHORDS, OFFSETS, AND ORDINATES. 305 ; Degree D. Radius K. Chord 1 Sta. T 0.2( 99.994 .818 .454 5 1858.24 2.690 .673 6 2728-37 .83-.' .457 6 1848.25 2.705 .676 7 2706.89 .847 .462 7 1838.37 2.719 .680 8 2685.74 .861 .465 8 1828.59 2.734 .684 9 2664.92 .87o .468 9 1818.91 99.987 2.74H .687 10 2044.42 .891 .473 10 1809.34 2.763 .691 11 2624.23 .905 .476 11 1799.87 2.777 .694 12 2604.35 .920 .480 12 1790.49 2.792 .698 13 2584.77 .934 .484 13 1781.22 2.806 .702 14 2565.48 .9-19 .487 14 772.03 2.821 .705 15 2546.48 .983 .491 15 762.95 2.835 .709 16 2527 75 99.993 1.978 .494 16 753.95 99.986 2.850 .713 17 2509 .'30 1.992 .498 17 745.05 2.864 .716 18 2491.12 2.007 .502 18 736.24 2.879 .720 19 2473.20 2.021 .505 19 727.51 2.894 .724 20 2455.53 2.036 .509 20 718.87 2.908 727 21 2438.12 2.050 .513 21 1710.32 2.923 .731 2-2 2420.95 2.065 .516 22 1701.85 2.937 .734 23 2404.0'J 2.080 .520 23 1693.47 99.985 2.952 .738 24 2387.32 2.094 .524 24 1685.17 2.96'i .743 . ^5 2370.86 2.109 .527 25 1676.95 2.981 .745 26 2354.62 99.992 2.123 .531 26 1668 81 2.995 .749 27 2338.60 2.138 .534 27 1660.75 3.010 .753 28 2322.80 2.152 .538 28 1652.76 3.024 .756 29 2307.21 2.167 .542 29 1644.85 3.039 .760 30 2291.83 2.181 .545 30 1637.02 99.984 3.053 .764 31 2276.65 2.196 .549 31 1629.26 3.068 .767 32 2261.68 2.210 .553 32 1621.58 3.082 .771 33 2246.89 2.225 .556 33 1613.97 3.097 .774 34 2232.30 2 239 .560 34 1606.42 3.112 778 35 2217 90 2.254 .564 35 1598.95 3.126 .783 36 2203.68 99.991 2.269 .567 36 1591.55 3.141 .785 37 2189.65 2.283 .571 37 1584.22 99.983 3.155 789 38 2175.79 2.298 .574 38 1576.95 3.170 .793 39 2162.10 2.312 .578 39 1569.75 3.184 .796 40 2148.59 2.327 .582 40 1562.61 3.199 .800 41 2135.25 2.341 .585 41 1555.54 3.213 .804 4-2 2122.07 2.356 .589 42 1548.53 3.2^8 .807 43 2109.05 2.370 .593 43 1541.59 99.982 3.242 .811 44 2096.19 2.385 .596 44 1534.71 3.257 .814 45 2083.48 99.990 2.399 .600 45 1527.89 3. -271 .cSKS 46 2070.93 2.414 .604 46 1521.13 3.28(1 .822 47 2058.53 2.428 .607 47 1514.43 3.300 .825 48 2046.28 2.443 .611 48 1507.78 3.315 .8.9 49 2034.17 2. 458 .614 49 1501.20 3.329 .P33 50 2022.20 2 472 .618 50 1494.67 99.981 3.344 .830 51 2010.38 2.487 .622 51 1488.20 3.358 .840 52 1998.69 2.501 .625 52 1481.79 3.373 .843 53 1987.14 99.989 2.516 .629 53 1475.43 3.388 .847 54 1975.72 2.530 .633 54 1469.12 3.402 851 55 1964.43 2.545 .636 55 1462.87 3.417 .854 56 1953.27 2.559 .640 56 1456.67 99.980 3.43i .858 57 1942.23 2.574 .644 57 1450.53 3 446 .8f2 58 1931.32 2.588 .647 58 1444. 41 3.460 .865 59 1920.53 2.603 .651 59 1438.39 3.475 869 60 1909.86 2.617 .654 60 1432.39 1 3.489 '.873 306 TABLE I. -RADII, CHORDS, OFFSETS, AND ORDINATES. Degree p. Radius K. Chord 1 Sta. Tang. Off. t. Mid. Ord. m. Degree Radius R. Chord 1 Sta. T sr t. Mid. Ord. m. 4 0' 1432.39 3.489 .873 5 0' 1145.92 4.361 .091 1 1420.45 3.504 .876 1 1142.11 4.375 .094 2 1420.50 99.979 3.518 .880 2 1138.33 4.390 .098 3 1414 71 3.533 .883 3 1134.57 4.404 .102 4 1408.91 3.547 .887 4 1130.84 99.967 4.419 .105 5 1103.10 3.562 .891' 5 1127.13 4.433 1.109 6 1397.40 3.570 .894 6 1123.45 4.448 1.112 7 1391.80 99.978 3.591 .898 7 1119.79 4.462 1.116 8 1386.19 3.605 .902 8 1116.15 4.477 1.120 9 1380.62 3.020 .905 9 1112.54 99.960 4.491 1.123 10 isrs.io 3.635 .909 10 1108.95 4.506 1.127 11 1339.62 3.649 .913 11 1105.38 4.520 .131 IsJ 1304.19 3.664 .916 12 1101.84 4.535 134 13 13:8.79 99.977 3.678 .920 13 1098.32 99.965 4.549 .138 14 1353.44 3.693 .923 14 1094.82 .564 .142 15 1348.14 3 707 .927 15 1091.35 .578 .145 16 1342.87 3.722 .931 16 1087.89 .593 .149 17 1337.64 3.736 .934 17 1084.46 .607 .152 18 1332.46 3.751 .938 38 1081.05 09.964 . 622 .156 19 1327.32 99.976 3.765 .942 19 1077.00 .636 .160 20 1322.21 3.780 .945 20 1074.30 4.651 .163 521 1317.14 3.794 .949 21 1070.95 4.R65 .167 22 1312.12 3.809 .953 22 1067.02 99.963 4.680 .171 23 1307.13 3.823 .956 23 1064.32 4.694 .174 24 1302.18 99.975 3.838 .960 24 1061.03 4.709 .178 25 1297.26 3.852 .963 25 1057.77 4.723 .182 26 1292.39 3.867 .967 26 1054.52 4.738 ' .185 27 1287.55 3.881 .971! 27 1051.30 99.962 4.752 .189 28 1282.74 3.890 .974' 28 1048.09 4.767 .192 29 1277.97 99.97-) 3.910 .978 29 1044.91 4.781 .196 30 1273.24 3.925 .982' 30 1041.74 4.796 .200 31 1268.51 3.939 .985 31 1038.59 99.961 4.810 .203 88 1263.88 3.954 .989! 32 1035.47 4.825 .207 33 1259.25 3.969 .993 33 1032.36 4.840 .211 34 1254.65 3.983 .996 34 1029.27 4.854 .214 35 1250.09 90.973 3.998 1.000 35 1026.19 99.960 4.869 .218 36 1245.56 4.012 .003 36 1023 14 4.883 .221 37 1241.06 4.027 .007 37 1020.10 4.898 .225 38 1236.60 4.041 .011 38 1017.08 4.912 .229 39 1232.17 4.056 .014 39 1014.08 99.i>59 4.927 .232 40 1227.77 99.972 4.070 .018 40 1011.10 4.941 .236 41 1223.40 4.085 .022 41 1008.14 4.956 .240 42 1219.06 4.099 .025 42 1005.19 4.970 .243 43 1214.75 4.114 .029 43 1002.26 4.985 .247 44 1210.47 4.128 .033 44 999.345 99.958 4.1)99 .251 45 1206.23 99.971 4.143 .036 45 996.448 5.014 .254 46 1202.01 4.157 .040 46 993.569 5.028 .258 47 1197.8,' 4.172 .043 47 990.7'05 5.043 .261 48 1193.66 4 186 .047 48 987.858 99.957 5.057 .265 49 1 189.53 4.201 .051 49 985.028 5 072 .269 50 1185.43 99.970 4.215 .054 50 982.213 5.080 .272 51 1181.36 4.230 .058 51 979.415 5.101 .276 52 1177.31 4.244 .062 52 176.633 99.956 5.115 .280 53 1173.29 4.259 .005 53 973.866 5.130 .283 54 1109.30 4.273 .069 54 971.115 5.144 .287 55 1165.34 99.069 4.288 .072 55 968.379 5.159 .291 56 1161.40 4 302 .076 56 965.659 99.955 5.173 .294 57 1157.49 4.317 .080 57 962.954 5.188 .298 58 1153.61 4.332 .083 58 900.264 5.20^ .301 f>9 1149.75 99.9G8 4.346 .087 59 957.590 5.217 .305 00 1145.92 4.301 .091 60 954.930 99.954 5.231 .3091 TABLE I. RADII, CHORDS, OFFSETS, AND ORDINATES. Degree D. Radius R. Chord 1 Sta. T oT- t. Mid. Ord. m. Degree D. Radius R. Chord 1 Sta. Tang. Off t. Mid. Ord. m. 6 00' 954.930 5.231 .309 7 00' 818.511 6.101 .527 1 952.284 5.246 .312 1 816.567 6.116 .530 94:t. 654 5.260 .316 2 814.632 99.937 6.130 .534 a 947.038 5.275 .320 3 812.706 6.145 .538 4 944.436 99.953 5.289 .323 4 810.789 6.159 .541 5 941.848 5.304 .327 5 808.882 99.936 6.174 .545 6 939.275 5.318 .330 6 806.983 6.188 .548 7 936.716 5.333 .334 7 805.093 6.202 .552 8 934.170 99.952 5.347 .338 8 803.212 99.935 6.217 .556 9 931 .639 5.362 .341 9 801 .340 6.231 .559 10 929.121 5.376 .345 10 793.476 6.246 .503 11 926.616 99.951 5.391 .349 11 797.621 6.260 .567 19 924.125 5.405 .352 12 795.775 99.934 6.275 .570 18 921.648 5.420 .356! 13 793.937 6.289 .574 14 919.184 5.434 .360: 14 792.108 6.304 .578 15 916.732 99.950 5.449 3S3 15 790.287 99.933 6.318 .581 16 914.294 5.463 .367 16 788.474 6.333 .585 17 911.869 5.478 .370 17 786.670 6.347 .588 18 909.457 5.492 .374 18 784.874 99.932 6. 302 .592 19 907.057 99.949 5.507 .378 19 783.058 6.376 .590 20 904.070 5.521 .381 20 781.306 0.391 .599 21 902.296 5.536 .3851 21 779.534 99.931 6.405 .603 22 899.934 5.550 .389 ; 22 777.771 6.420 .607 28 897.584 99.948 5.565 .392! 23 776.015 6.434 .610 24 895.247 5.579 .396' 24 774.267 6.449 614 25 892.921 5.594 .400i 25 772.527 99.930 6.463 .618 26 890.608 99.947 5.608 .403! 26 770.795 6.478 .621 888.307 5.623 .407 27 769.071 6.492 .625 28 8S6.017 5.637 .410 28 767.354 99.929 6 507 .628 29 S83.740 5.652 .414, 29 765.645 6.521 .632 30 881.474 99.946 5 666 .418, 30 763.944 6.536 .036 31 879.219 5.681 .421! 31 762.250 99.928 6.550 .639 32 876.976 5.695 .4251 32 760. 56S 6.565 .643 ?a 874.745 5.710 .429 33 758.885 6.579 .047 84 872.525 99.945 5.724 .432; 34 757.213 99.927 6.594 .651 as 870.316 5.739 .436i 35 755.549 6. 60S .054 36 808.118 5.753 .439 36 753.892 6.623 .657 37 865.931 99.944 5.768 .443| 37 752.242 99.926 6.637 .061 38 863.755 5.782 .4471 38 750.600 6.651 .605 3!) 861.591 5.797 .451 39 748.964 6.060 .668 40 859.437 5.811 .454! 40 747.336 99.925 6.080 .67;: 41 857.293 99.943 5.826 .458 41 745.715 6.695 .676 42 855.161 5.840 .461 42 744.101 6.709 .079 43 853.039 5.855 .465 43 742.494 99.924 6.724 .683 44 850.927 99.942 5.869 .469 44 740.894 0.738 .087 45 848.826 5.884 .472 45 739 . SOO 0.753 .090 46 846.736 5.898 .476 46 737.714 99.923 0.707 .694 47 844.655 5.913 .479 47 736.134 6.782 .097 48 842.585 99.941 5.927 .483 48 734.561 0.790 .701 49 840.525 5.942 .487 49 732.997 99.922 6.811 .705 50 838.475 5.956 .490 50 731.435 0.825 .708 51 836.435 99.940 5.971 1.494 51 729.883 6.840 .712 52 834.405 5.985 1.498 52 728.336 99.921 6.851 .716 53 832.384 6.000 1.501 53 726.796 6. 869 .719 54 830.374 6.014 1.505 54 725.263 6.883 .723 55 828.373 99.939 6.029 1.509 55 723 730 99.920 6.898 .726 56 826.381 6.043 .512 56 722.216 6.912 .730 57 824.400 6 058 .516 57 720.702 0.927 734 58 822.427 99.938 6.072 .519 58 719.194 99.919 0.941 .737 59 820.465 6.087 .523 59 717.692 6.056 .741 60 818.511 6.101 .527 60 716.197 0.970 .745 308 TABLE I.-RADII, CHORDS, OFFSETS, AND ORDINATES. Degree D. Radius R. Chord 1 Sta. Tang. Off t. Mid. 1 Ord. \ HI. Degree D. Radius R. Chord 1 Sta. Taug. Off. t. Mid. Ord. 8 00' 716 197 6.970 T745J 9 00' 630.620 .838 1.963 1 714.708 99.918 6.984 .748 1 635.443 .852 1.966 2 713.2-25 6.999 . 4 52 o 634.271 99.896 .867 1.970 3 711.749 7.013 .756 3 63 5. 103 .881 1 973 4 710.278 99.917 7.028 .759' 4 631.939 .890 1 977 5 ?08.814 7.012 .763 5 630.779 99.895 .910 1.981| 6 707.355 7.037 .766! 6 629.624 .9','t 1.9841 7 705.903 99.916 7.071 .770 n 028.473 .939 1.988 8 704.456 7.086 .774 8 627.326 99.894 .953 1.992 9 703.016 7.100 9 626.183 .868 1.995 10 701.581 99.915 7.115 '.m 10 625.045 99.893 .982 1.999 11 700.152 7.129 .785 11 623.910 7.997 2.002 18 098.729 7.144 .788 12 622.780 8.011 2.H06 13 697.312 99.914 7.158 .792 13 621.654 99.892 8.026 2.010 14 695.900 7.173 .795J 14 620.532 8.040 2.013 15 694.494 7.187 .799 15 619.414 99.891 8.055 2.017 16 693.094 99.913 7.202 .803, 16 61 S. 800 8.069 2.T21 17 691.700 7.216 .806 17 617.190 8.084 2.024 18 690.311 7.230 .810 18 616.084 99.890 8.098 2.028 19 688.927 99.912 7.245 .814' 19 614.982 8.112 2.031 20 687.541 7.259 .817 20 613.883 99.889 8.127 2.035 21 686.177 7.274 .820 21 612.789 8.141 2.039 22 684.810 99.911 7.288 .825 22 611.699 8.156 2.042 23 683.449 7.303 .828 23 610.612 99.888 8.170 2.046 24 682.093 99.910 7.317 .832 24 609.530 8.185 2.050 25 680.742 7.3J-2 .835 25 608.451 99.887 8.199 2.053 2G 679.397 7.346 .839 2d 607.376 8.214 2.057 27 678.057 99.909 7.361 .843 27 606.305 8.228 2.061 28 676.722 7.375 .846 28 605.237 99.886 8.243 2.064 29 675.392 7.390 .850 29 604.173 8.257 2.068 30 G74.0H8 93.908 7.401 .854 30 603.113 99.885 8.271 2.071 31 672.749 7.419 .8)7 31 602.057 8.286 2.075 3 071.430 7.433 .861 32 001.005 K.300 2.079 33 670.126 99.907 7.447 .864 33 599.956 99.884 8.315 2 082 31 60S. 822 7.462 .868 34 598.911 8.3-29 2.086 35 667.524 7.476 .872 35 597.8(i9 99.883 8.344 2.01)0 3G 660.230 99.906 7.491 .875 36 596.831 8.358 2 093 37 6(54.941 7.505 .879, 37 595.797 8.372 2.097 88 663.658 99.905 7.520 .883' 38 594.760 99.882 8.387 2.100 39 602.379 7.534 .886 39 593 739 8.401 2.104 40 661.105 7.541) .890 40 592.71 5 8.416 2.108 41 659.836 99.904 7.563 .894 41 591.695 99.881 8.430 2.111 42 658.572 7.578 .897 42 590.678 8.445 2.115 43 057.313 7.592 .901 43 589.665 99.880 8.459 2.119 44 656.059 99.903 7.607 .904 44 588.u55 8.474 2.122 45 654.809 7.621 .908 45 587.649 99.87'9 8.488 2.126 46 653-564 99.902 7.685 .912 46 586.64ii 8.502 2.1 -21* 47 05J.324 7.650 .915 47 585.647 8 517 2.133 48 651 .089 7.664 .919 48 584.651 99.878 8.53! 2.13; 49 649.858 99.901 7.679 .923 49 583.658 8.546 2.140 50 648.631 7.693 .926 50 582.669 99.877 8.560 2.144 51 647.410 7.708 .930 51 581. 683 8.575 2.118 52 646.193 99.900 7.722 .933 52 580.700 8.58!) 2.151 53 644.980 7 737 .937 53 579.721 99.876 8.603 2.155 54 643.773 99.899 7.751 .941 54 578.745 8.618 2.159 55 642.569 7.766 .944 55 577.773 99.875 S.<532 3.102 56 641.371 7.780 .948 56 576.803 8.647 2.166 640.176 99.898 7.794 .952 57 575.837 99.874 8.6til 2. l(i!> 58 638.986 7.809 .955 | 58 674.874 8.676 2.173 59 637.801 7.623 .959 51) 5 ;'.;.;>i i 8. duo 2.177 GO 636.620 99.897 7.838 1.963 60 572.958 99.873 8.705 2. ISO TABLE I. RADII, CHORDS, OFFSETS, AND ORDINATES. 309 Degree Radius Chord 1 Tang. Off. Mid. Orel. Degree Radius E> Chord 1 Tang. Off. Mid. Ord. D. R. Sta. t. me, D. 1. Sta. t. m. 10 OK 572.958 99.873 8.705 2.180 20 00' 286.479 99.493 17. -277 4.352 10 563.5(55 99.869 8.849 2.217 10 284.111 99.485 1 .418 4.388 20 554.475 99.864 8.993 2.253 20 281.783 99.476 1 .559 4.424 30 5-15.674 99.860 9.137 2.289 30 279.492 99.467 .700 4.460 40 537.148 99.856 9.28-2 2.325 40 277.238 99.459 1 .840 4.497 50 528.884 99.851 9.426 2.362 50 275.020 99.450 1 .981 4.533 11 00 520.871 99.847 9.570 2.398 21 00 272.837 99.441 18.122 4.509 10 513.097 99.842 9.714 2.434 10 270.689 99.432 18.262 4. 605 20 505.551 99.837 9.858 2.471 20 268.574 99.423 18.403 4.641 30 498.224 99.832 10.002 2.T07 30 266.49-2 99.414 18.543 4.677 40 491.107 99.827 10.146 2.543 40 264.442 99.405 18.68J 4.713 50 484.190 99.822 10.290 2.579 50 262.423 99.396 18.824 4.749 12 00 477.465 99.817 10.434 2.616 22 00 260.435 99.3*7 18.964 4.785 10 470.924 99.812 10.578 2.652 10 258.477 99.378 19.104 4.821 20 464.560 99.807 10.721 2.688 20 256.548 99.36S 19.244 4.857 30 458.8158 99.802 10.865 2.724 30 254.648 99.359 19.384 4.8 3 40 -152.335 99.797 11.009 2.761 40 252.775 99.349 19.524 4.929 ro 446.461 99.791 11.152 2.797 50 250.930 99.340 19.664 4.965 13 00 440.737 99.786 11.296 2.833 23 00 249.112 99.330 19.803 5.001 10 435.158 99.780 11.440 2.869 10 247.320 99.3:0 19.943 5.037 20 429.718 99.775 11.583 2.906 20 245.553 99.310 20.082 5.073 30 424.413 99.769 11.727 2.942 30 243.812 90.301 20.222 5.100 40 419.237 99.763 11.870 2.978 40 242.095 99.291 20.361 5.145 50 414.186 99.757 12.013 3.014 50 240.402 99.281 20.500 5.181 14 00 409 256 99.751 12.157 3.051 24 00 238.732 99.271 20.639 5.217 10 404.441 99.745 12.300 3.087 10 237.0815 99.260 20.779 5.253 20 31(9.738 99.739 12.443 3.123 20 235.462 99.250 20.918 5.289 30 395.143 99.733 12.586 3.159 30 233.860 99.240 21.056 5.325 40 390.653 99.727 12.729 3.195 40 232.280 99.230 21.195 5.361 50 386.264 99.721 12.872 3.232 50 230.121 99.219 21.334 5.391 15 00 381.972 99.715 13.015 3.268 25 00 229. IbS 99.208 21.473 5.433 10 377.774 99.108 13.158 3.304 10 227.665 99.198 21.611 5.468 20 373.6t>8 99.702 13.301 3.340 20 226.168 99.187 21.750 5.504 30 369.650 99.695 13.444 3.376 30 224.689 99.177 21.888 5.540 40 365.718 99.689 13.587 3.413 40 223.230 09.1C6 22.026 5.516 50 361.868 99.682 13.729 3.449 50 221.790 99.155 22.164 5.612 16 00 358.099 99.675 13.872 3.485 26 00 220.368 99.144 22.303 5.648 10 354.407 99.669 14.015 3.521 10 218.965 99.133 22.441 5.684 20 3.50.790 99.662 14.157 3.557 20 217.579 99.122 22.578 5.720 30 347.247 99.655 14.300 3.504 30 216.210 99.111 22 716 5.756 40 343.775 99.648 14.442 3.630 40 214.859 99.100 22.854 5.792 50 340.371 99.641 14.584 3.667 50 213.525 99.089 22.992 5.827 17 00 337.034 99.634 14.727 3.702 27 00 212.207 99.077 23.129 5.863 10 333.762 99.626 14.869 3.738 10 210.905 99.066 23.267 5. 899 20 330.553 99.619 15.011 3.774 20 209.619 99.054 23.404 5.935 30 327.404 99.612 15.153 3.810 30 208.348 99.043 23.541 5.971 40 324.316 99.604 15.295 3.847 40 207.093 99.031 23.678 6.007 50 321.285 99.597 15.437 3.883 50 205.853 99.020 23.815 6.042 18 00 318.310 99.589 15.579 3.919 28 00 204.628 99.008 23.952 6.078 10 315.390 99.582 15.721 3.955 10 203.417 98.996 24.089 6.114 20 312. 52. ' 99.5^4 15.863 3.991 20 202.220 98.984 24*226 6.150 30 309.707 99.566 16.005 4.027 30 201.038 98.972 24.362 6.186 40 306.942 99.558 16.146 4.063 40 199.869 98.060 24.499 6.222 50 304.225 99.550 16.288 4.100 50 198.714 98.948 24 . 635 6.257 19 00 801.557 99.542 16.429 4.136 29 00 197.572 98.936 24.772 6.293 10 298.935 99,534 16.571 4.172 10 196.443 98.924 24.908 6.329 20 296.357 99.526 16.712 4.208 20 195 327 98.911 25.044 6.365 30 298.825 99.518 16.853 4.244 30 194.223 98.899 25.180 6.400 40 291 .334 99.510 16.995 4.280 40 193.13', 98.887 25.316 6.436 50 288.886 99.501 17.136 4.316 50 192.053 98.874 25.452 6.472 20 00 286.479 99.493 17.277 4.352 30 00 190.986 98.862 25.587 6.508 310 TABLE II. TANGENT OFFSETS. Curve. g Curve. .002 .002 .002 .003 .003 .004 .004 .005 .J005 .005 3 .005 .005 .006 .007 .008 .009 .008 .010 .011 .011 4 .008 .009 .011 .013 .014 .015 .017 .018 .019 .020 5 .013 .015 .017 .020 .022 .024 .020 .028 .031 .032 6 .019 .022 .025 .028 .03! .035 .038 .041 .044 .047 ,026 .030 .034 .038 .043 .047 .051 .056 .oeo .064 8 .0:54 .03:) .045 .050 .056 .061 .067! .073 .078 .084 9 .042 .049 .057 .064 '.071 .078 .085 .092 .099 .106 10 .05,! .001 .070 .079 .087 .090 .105 .113 .122 .131 11 .063 .074 .084 .095 .106 .116 .127 .137 .148 .158 12 .075 .088 .101 .113 .120 .138 .151 .163 .176 .188 13 .088 .103 .118 .133 .147 .162 .177 .192 .206 .221 14 .103 .120 .137 .154 .171 .188 .205 .222 .239 .257 15 .118 .137 .157 .177 .196 .216 .236 .255 .275 .294 16 .134 .156 .179 .901 .223 .246 .268 .290 .313 .335 1? .151 .177 .202 .227 .252 .277 .303 .328 .353 .378 18 .170 .198 .220 .254 .283 .311 .339 .368 .396 .424 19 .189 .221 .252 .284 .315 .346 .378 .409 .441 .472 20 .209 .244 .279 .314 .349 .384 .419 .454 . .489 .523 21 .231 .269 .308 .346 .385 .423 .462 .500 .539 .577 22 .253 .296 .338 .380 .422 .465 .507 .549 .591 .633 n .277 .323 .369 .415 .462 .508 .554 .600 .646 .692 24 .302 .352 .402 .452 .503 .553 .603 .653 .704 .754 25 .327 .382 .436 .491 .545 .600 .654 .709 .763 .818 2G .354 .413 .472 .531 .590 .649: .708 .767 .826 .885 27 .382 .445 .. r 09 .572 . 636 .7001 .763 807 .890 .954 28 .410 .479 .547 .016 .684 .752 .821 !889 .957 .026 29 .440 .514 .587 .660 .734 .P07 .880 .954 1.027 .100 30 .471 .550 .628 .707 .785 .864 .942 1.021 1.099 .177 31 .503 .587 .671 .755 .838 .922 .006 1.090 1.174 .257 32 .530 .625 .715 .804 .893 .983 .072 1.101 1 250 340 .33 .570 .665 .700} .855 .950 .045 .140 1.235 1.330 .424 34 .005 .706 .807 .908 .009 .1C9 .210 1.311 1.412 .512 35 .641 .748 .855 .962 .069 .174 .282 1.389 1.496 .002 36 .679 .792 .904 1.018 .131 .244 .357 1.469 1.582 .695 37 .717 .836 .955 1.075 .194 .314 .433 1.552 1.671 .790 38 .756 .882 1.008 1.134 .260 .386 1.511 1.637 1.763 .P89 39 .796 .929 1.061 1.194 .327 .459 1.592 1.724 1.857 .989 40 .838 .977 1.117 1.256 .396 .535 1.675 1.814 1.953 2.092 41 .880 .027 1.173 1.320 .466 .613 1.759 1.906 2.052 2.198 42 .923 .077 1.231 1.385 .539 .692 1.846 2.000 2.153 2.307 43 .968 .129 1.290 1.452 .613! .774 1.935 2.096 2.257 2.417 44 .014 .182 1.351 1.520 1.689 .857 2.026 2.194 2.363 2.531 45 .oeo .237 1.413 1.590 1.766 .943 2.119 2.295 2.472 2.647 46 .108 .292 1.477 1.661 1.8 1C. 2.030 2.214 2.398 2.582 2.766 47 .15(5 .349 1.541 1.734 1.9--7 2.119 2.311 2.504 2.696 2.888 48 .206 .407 1.608 1.809 2.0(10 2.210 2.411 2.611 2.812 3.012 49 .257 .466J 1.675 1.885 2.094 2.303 2.512 2.721 2.930 3.138 50 .309 .527 1 .745 1.962 2.180 2.398 2.616J 2.833 3.051 3.268 312 TABLE II. TANGENT OFFSETS. Curve. Arc. 16 17 18 19 20 21 2-2 23 24 25 1 .001 .001 .002 .002 .002 .002 .002 .002 .002 .002 2 .006 .006 .006 .OC7 .007 .007 .008 .008 .008 .009 3 .013 .013 .014 .015 .016 .016 .017 .018 .019 .020 4 .022 .024 .025 .027 .028 .029 .031 .032 .034 .035 5 .035 .037 .039 .041 .044 .046 .048 .050 .052 .055 6 .050 .053 .057 .060 .063 .066 .069 .072 .075 .078 7 .068 .073 .077 .081 .086 .090 .094 .098 .103 .107 8 .089 .093 .101 .106 .112 .117 .123 .128 .134 .140 9 .113 .120 .127 .134 .141 .148 .155 .163 .170 .177 10 .140 .148 .157 .165 .175 .183 .192 .201 .209 .218 11 .169 .179 .190 .201 J211 .222 .232 .243 .253 .264 12 .201 .211 .2-26 .239 .251 .264| .276 .289 .302 .314 13 .236 .251 .265 .280 .294 .310 .324 .339 .354 .369 14 .274 .891 .308 .325 .342 .360 .376 .393 .410 .427 15 .314 .334 .353 .373 .393 .412 .432 .451 .471 .491 16 .357 .379 .402 .421 .447 .469 .491 .514 .536 .558 17 .403 .429 .454 .179 .504 .530 .555 .580 .605 .630 18 .452 .480 .509 .537 .565 .594 .622 .650 .678 .706 19 .501 .535 .567 598 .630 .661 .693 .724 .756 .787 20 .558 .593 .628 .663 .698 .733 .768 .802 .837 .872 21 .616 .654 .692 .731 .769 .808 .846 .885 .9-,'3 .961 22 .676 .718 .760 .802 .844 .886 .929 .971 1.013 1.055 23 .738 .784 .831 .877 .923 .969 1.015 1.061 1.107 1.153 24 .804 .854 .904 .955 1.005 1.055 1.105 1.155 1.205 1.255 25 .872 .927 .981 1.036 1 .090 1.145 1.199 1.253 1.307 1.362 TABLE III. TANGENT OFFSETS ARC 100 FEET. Degree of Curve. 0' 10' 20' 30' 40' 60' .000 .145 .291 .436 .582 .727 1 .873 1.018 1.164 1.309 1.454 1.600 2 1.745 1.891 2.036 2.181 2.327 2.472 3 2.617 2.763 2.908 3.053 3.199 3.344 4 3.489 3.635 3.780 3.925 4.070 4.215 5 4.361 4.506 4.651 4.796 4.941 5.086 6 5.231 5.376 5.521 5.666 5.811 5.956 7* 6.101 6.S46 6.391 6.536 6.680 0.825 8 6.970 7.115 7.259 7.401 7.549 7.693 9 7.838 7.982 8.127 8.271 8.416 8.560 10 8.705 8.849 8.993 9.137 9.282 9.426 11 9.570 9.714 9.858 10.01 10.15 10.29 12 10.43 10.58 10.72 10.86 11.01 11.15 13 11.30 11.44 11.58 11.73 11.87 12.01 14 12.16 12.30 12.44 12.59 12.73 12.87 15 13.01 13.16 13.30 13.44 13.5!) 13.73 16 13.87 14.01 14.16 14.30 14.44 14.58 17 14.73 14.87 15.01 15.15 15.30 15.44 18 15.58 15.72 15.86 16.00 16.15 16.29 19 16.43 16.57 16.71 16.85 17.00 17.14 20 17.28 17.42 17.56 17.70 17.84 17.98 21 18.12 18.26 18.40 18.54 18.68 18.82 22 18.96 19.10 19.24 19.38 19.52 19.66 23 19.80 19.94 20.08 20.22 20.36 20.50 24 20.64 20.78 20.02 21.06 21.20 21.33 25 21.47 21.61 81.75 21.89 22.03 22.16 26 22.30 22.44 2i!fc8 22.72 22.85 22. '.(9 27 23.13 23.27 23.40 23.51 28.68 23.82 28 23.95 24.09 24.23 24.30 24.50 24.04 29 24.77 24.91 25.04 25 J8 25.32 25.45 TABLE IIIA.-MIDDLE ORDINATES ARC 100 FEET. 313 Degree of Curve. 0' 10' 20' 30' 40' 50' .000 .036 .073 .109 .145 .182 1 .218 .55 .291 .327 .364 .400 2 .436 .473 .509 .545 .582 .618 3 .654 .691 .764 .800 .836 4 .873 .909 !945 .9S2 1.018 1.054 5 .091 1.127 1.163 1.200 1.236 1.272 6 .309 1.345 1.381 1.418 1.454 1.490 .527 1.563 1.599 1.636 1.672 1.708 8 .745 1.781 1.817 1.854 1.890 l.!26 9 .963 1.999 2.035 2.071 2.108 2.144 10 2.180 2.217 2.253 2.289 2.325 2 362 11 2.398 2.434 2.471 2.507 2.543 2.579 12 2.616 2. 052 2.688 2.724 2.761 2.797 13 2.833 2.869 2.906 2.942 2.978 3.014 14 3.051 3.087 3.1?3 3.159 3.195 3.232 15 3.268 3.304 8.840 3.376 3.413 3.449 16 3.485 3.521 3 557 3.594 3.630 3.G67 17 3.702 3.738 3.774 3.810 3.847 3.8S:5 18 3.919 3.955 3.991 4.027 4.063 4.100 19 4.136 4.172 4.208 4.244 4.280 4.316 20 4.352 4.388 4.42J 4.460 4.497 4.533 21 4. 509 4.605 4.641 4.677 4.713 4.749 22 4.785 4.821 4.857 4.893 4.929 4.965 23 5.001 5.037 5.073 5.109 5.145 5.181 24 5.217 5.253 5.289 5.325 5.361 5.897 25 5.433 5.468 5.504 5.540 5.576 5.612 26 5.648 5.684 5.720 5.756 5.792 5.827 27 5.863 ' 5.899 5.935 5.971 8.007 6.042 28 6.078 6.114 6.150 6.186 6.222 6.257 29 6.293 0.329 6.305 6.400 6.486 6.472 TABLE IIlB.-CHORDS FOR ARCS 100 FEET. Degree of Curve. 0' 10' 20' 80' 40 50' 100.000 100.000 100.000 iro.ooo 99.999 99.999 1 99.999 99.998 99.998 99.997 99.996 99.996 2 99.995 99.994 99.993 99.992 99.991 99.990 3 99.989 T.9.987 99.9H6 99 984 09.983 99.981 4 99.980 99.978 99.976 99.974 99.972 99 970 5 99.968 99.966 99.964 99.962 99.959 99.957 6 99.954 99.952 99.949 99.946 99 . 944 99.941 99.938 99.935 99.932 99,9-29 25 99.922 8 99.919 99.915 99.912 99.908 519 . HO.") 99.901 9 99.897 99.893 99.889 99.885 99.882 90.877 10 99.873 99.869 99.864 99.860 99.856 99.851 11 99.847 99.842 99.836 99.832 99.827 99.822 12 99.817 99.812 99.807 99.802 99.797 99.791 13 99.786 99.780 99.775 99.769 99.763 99.757 14 99.751 99.745 99.739 99.733 99.727 99.721 15 99.715 99.708 99.702 99.695 99.689 99.682 16 99.675 99.669 99.66.' 99.655 99.648 99.641 17 99.634 99.626 99.619 99.612 99.604 99.597 18 99.589 99.582 99.574 99 566 99.558 99.550 19 99.542 99.534 99.526 99.518 9!). 510 99.501 20 99.493 99.485 99.476 91). 467 > 99.459 99.450 21 99.441 99.432 99.4-23 99.414 99.405 99.396 22 99.387 99.378 99.368 99.359 99.349 99 340 23 99.330 99.320 99.310 93.301 99.291 99.281 24 99.271 99.260 99.250 99.240 99.230 99.219 25 99.208 99.198 99.187 99.177 99.166 99.155 26 99.144 99.133 99.122 99.111 99.100 99.088 27 99.077 99.066 99.054 99.043 99.031 99.020 28 99.008 98.996 98.984 98.972 98.960 98.948 29 98.936 98.924 98.911 98.898 98.887 98.874 314 TABLE IV. LONG CHORDS. Decree 2 3 ! 4 5 6 n 8 9 10 ofCurve. Sta. Sta. Sta. Sta. Sta. Sta. Sta. Sta. Sta. 10' 200.00 300.00 400.00 499.99 599.99 C9J.99 799. Of 699.97 999.90 20 200.00 300.00 899. 9'J 499.98 599.97 699.95 799.93 899.90 99'.). MS 30 200.00 299.99, 399.98 499.96 599.93 699.89 799.84 899.77 999.68 40 200.00 299. 98 ! 399.96 499.93 599.88 699.81 799.71 8119.59 99!). 44 50 199.99 299. 98 | 399.94 499.89 599.81 699.70 799.55 899.36 999.1'.' 1 00 199.99 299.97 1 399.92 409.84 599.73 699.56 799.35 899.08 99S.73 10 199.99 299.1:5 399.89 499.78 599.63 699.41 799.12 898.74 908.27 20 199.98 299.94 399.86 499.72 599.51 699.23 798.85 898.36 997.75 30 199.98 299.92 399.82 499.64 599.38 699.02 798.54 897.93 997.15 40 199.97 299 90 399.77 499.56 599.24 6P8 79 798 20 897.43 996.48 50 199.97 299.88 399.73 499.47 599.08 698.54 797.82 896.89 995.74 2 00 199.96 299. 86 ! 399.68 499.37 598.90 698.26 797.40 896.30 994.93 10 199.95 29!). 84 399.62 499.26 598.71 697.96 796.% 895.66 994.05 20 199.94 299.81; 399.56 499.14 598.51 697. K3 796.47 894.97 993.10 30 199.94 299.79 399. '49 499.01 598.29 097.28 795.94 894.23 992.09 40 199.93 299.76 399.42 498.87 598.05 696.91 795.39 893.43 991.00 50 199.92 299. 72 ! 399.35 498.73 597.80 696.51 794.78 892.59 989.84 3 00 199.91 299.69 399.27 498.57 597.54 696.09 794.16 891.70 988.62 10 199.90 299.66 399.19 498.41 597.25 695.64 793.50 890.75 987.32 20 199.89 299.62 399.10 498.24 596.96 695.17 792.80 889.75 985.96 30 199.88 299.58 399.01 498.06 596.65 691.68 79-.'. 06 888.71 984.52 40 199.86 299.54 398.91 497.87 596.32 694.16 791.29 887.61 983.02 50 199.85 299.50 398.81 497.67 595.98 693.62 790.48 886.47 981.45 4 00 199.84 299.45 398.70 497.47 505.62 603.06 789.64 885.27 979.82 10 199.8-,' 299.41 398.59 497.SJ5 595.25 692.47 788.77 884.02 978.1 1 20 199.81 299.36 398.48 497.03 594.87 691 .85 787.85 882.73 976.34 30 199.79 299.31 308.36 496 . 79 594.40 691.22 786.91 881 .38 974.50 40 199.78 299.25 398.23 496.55 594.05 690.56 785.92 879.98 972.59 50 199.76 299.20 398.11 496.30 593.62 689.87 784.90 878.54 970.61 5 00 199.75 299.14 397.97 496.04 593.17 689.17 783.85 877.05 968.57 10 199.73 299.09 397.84 495.78 59:.'. 71 688.44 782.77 875.50 906.46 20 199.71 299.03 397.69 495.50 592.23 687.68 781.64 873.91 964.29 30 199.69 298.96 397.55 495.21 591.74 686.90 780.49 872.27 962.05 40 199.67 298.M 397.40 494.92 591.24 686.10 7 9.30 870.58 959.74 50 199.65 298.84 397.24 494.62 590.72 685.28 7 6.07 868.84 957.37 '. 00 199.63 298 77 397.08 494.31 590.18 684.43 7 6.8! 867.06 954.93 10 199.61 298 .'70 396.92 493.99 589.63 683.56 7 5.52 865.22 952.43 20 199.59 298.63 396.75 493.6(5 589.06 682.67 7 4.19 863.34 949.86 30 199.57 298.55 396.58 493. 8 J 588.48 681.75 7 2.88 801.41 947.25 40 199.55 298.48 396.40 492.98 587.89 680.81 7 1.43 859.44 944.54 50 199.53 298.40 396.22 492.62 587.28 679.85 7 0.00 857.41 941.78 7 00 199.50 298.32 396.03 492.26 586.66 678.86 768.54 855.34 938.96 10 199.48 298.24 395.84 491. 89 5S6 . 02 677.85 767.04 853.22 936.07 20 199.45 98.16 395.65 491.51 585.36 676.82 765.51 851.06 93.1.13 30 199.43 298.08 395.45 491 . 12 584.70 675.77 763.94 848.85 930.12 40 199.40 297.99 395.24 490.73 584.02 674.69 762.35 846.59 927.05 50 199.38 297.9:) 395.03 490.32 583.32 673.59 760.72 844.29 923.92 8 00 199.35 297.81 394.82 489.91 582.61 672.47 759.05 841.94 920.73 10 199.32 297.72 394.60 489.49 581.88 671.32 757.36 839.55 917.47 20 199.30 :-97.>3 394.38 489.05 581.14 670.16 755. (13 837.11 914.16 30 199.27 297.53 394.16 488.62 580.39 668.97 753 87 834.62 910.79 40 199.24 297.43 393.93 4S8.17 579.62 6U7.76 752.07 832.09 907.36 50 199.21 297.33 393.69 487.71 578.84 666.52 750.23 829.52 903.87 9 00 199.18 297.23 393.45 487.25 578.04 665.27 748.39 826.90 900.32 10 199.15 297.13 393.21 486.77 577.23 003. 99 746.50 824.24 896.71 20 -.99.12 297.02 392.96 486.29 576.40 66.'. 69 744.58 821.54 893.04 30 199.08 296.92 39->.7I 485.80 575.56 661.37 742. 0:j 818.79 889.32 40 199.05 296.81 392.45 485.31 574.71 660.02 740.64 810.00 885.55 50 199.02 296.70 392.19 484.80 573.84 6f,8.6o 788.68 813.16 881.71 10 00 198.99 296.58 391.93 484.28 572.96 657.27 736.58 810.28 877.82 TABLE IV. LONG CHORDS. | Degree 2 or' (. urve Sta. 3 Sta. 4 Sia. Sta. 6 Sta. St'a. 8 Sta. 9 Sta. 10 Sta. 10 10' 19R.95 296.47 391.66 4S3.76 1 572.06 655.83 734.50 807.37 873.88 20 198.92 296. 35 j 391.38 483.23 57J.15 654.43 732.39 804.40 869.88 30 198.88 296.24: 391.10 482.69 570.23 652.98 730.25 801 .40 865.82 40 198.85 296.12; 390.82 482.14 569.29 651.51 728.08 798.36 861.72 50 11 00 10 20 198.81 198.77 198.74 198.70 295.99 295.87 295.75 295.62 390.53 390.24 389.95 389.65 481 .59 481.02 480.45 479.87 568.34 567.37 566.39 565.40 650.01 725.88 648.50 723.65 646.96 721.39 645.4l| 719.10 795.27 792.15 788.98 785.77 857. 5C 853.34 849.08 844.76 30 198.66 295.49 389.34 479.28| 564.39 643.83! 716.78 782.53 840.40 40 198.62 295.36 389.04 478.68 563.37J 642.23 714.44 779.24 835.98 50 198.58 295.22 388.T 2 478.08 562. 34| 640.61 712.06 775.92 831.51 12 00 198.54 295.09 388.40 477.4* 561.29 638.92 709.65 772.55 826.99 10 198.50 294.95 388.08 476.84 560.23 637.31 707.22 769.15 822.43 20 198.46 294.81 387.76 476. 21 1 559. 16 i 635.63 704 75 705.7 817.81 30 198. 42 294.67 387.43 475.58 558.07 633. 9o 702.26 762.24 813.15 40 198.37 294.53 387.09 474.93; 556.97 632.21 699.74 758.72 808.44 50 198.33 294.39 386.76 474.28 555.86 630.47 697.19 755.17 803.69 13 00 198.29 294.24 386.41 473.621 554.73 628.7 694.61 751.58 798.89 10 198.24 294.09 386.07 472.951 553.59 626.93 692.01 747.95 794.04 20 198.20 293.94 385.71 472.27 552.44 625.13 689.37 744.29 789.15 30 198.15 293.79 385.36 471.58 551.27 623 31 686.71 740.60 784.21 40 198.11 293.64 3^5.00 470.89 550.09 621.47 684.03 736.87 779 2? 50 198.06 ^93.49 3S4.64 470.19) 548.90 619.62 681.32 733.10 774^21 14 00 198.02 293.33 384.27 469.48 547.69 617.74 678.58 729.30 769.15 10 1 97.97 293.17 833.90 468.76! 546.47 615.84 675.81 725.46 764.04 20 197.92 29:5.01 383.52 468.04! 545.24 613.93 673.02 721.60 758.90 30 197.87 292.85 383.14 467.30! 544.00 611.99 670.20 717.69 753.71 40 197.82 292.68 3S2.75 466.56 542.74 610.04 667.36 713.76 748.48 50 197.77 292.52 382.36 465.82 541.47 608.07 664.49 709.79 743.22 15 00 197.72 292.35 381.97 465.06 540.19 606.08 661.59 705.79 737.91 10 197.67 292.18 381.57 464.30 538.90 604.07 658.68 701.76 732.57 20 197.62 292. Oil 381.17 463.52! 537.59 602.04: 655.73 697.70 727.19 SO 197.57 291.83 380.77 462.74 536.27 600.00 652.76 693.61 721.78 40 197.52 291.66! 380.36 461.96 534.94 597.93 649.77 689.48 716.32 50 137.40 291. 48 | 379.94 461.16 533.59 595.85 646.75 685.33 710.84 16 00 197.41 291.30| 379.53 460.36 532.24 593.75 643.71 681.14 705.32 10 197.36 291.121 379.10 459.55 530.87 591.64 640.65 676.93 699.76 20 197.30 290.94 378.68 458.74 529.49 589.50 637.56 672.69 694.17 30 197.25 290.76 378.25 457.91 528.10 587. 35 i 634.45 668.42 688.55 40 197.19 290.57 377.81 457.08 526.69 585.18 631.32 664.12 682.91 50 197.14 290.38 377.38 456.24 625.28 583.00 628.16 659.80 677.22 17 00 197.08 290.19 376.93 455.39 523.85 580.80 624.98 655.44 671.50 10 197.02 290.00 376.49 454.54 522.41 578 . 58 621.78 651 .06 665.76 20 196.96 289.81 376.04 453.68 520.96 576.34 618.56 646.66 659.99 30 196.90 289.61 375.58 452.81 519.49 573.99 615.32 642.23 654.19 40 196.85 289.42 375.12 451.93 518.02 571.82 612.06 637.77 648 36 50 196.79 289.22 374.66 451.05 516.53 569. 53j 608.77 633.29 642.50 18 00 196.73 289.02 374.20 450.16 515.04 567.23 605.46 628.78 636.42 10 196.67 288.82 373.73 449.26 513.53 564.91 602.13 624.25 20 196.60 288.61 373.25 448.35 512.01 562.58 598.78 619.70 30 196.54 288.41 372.77 447.44 510.48 560.23 595.42 615.12 40 196.48 288.20 372.29 446.52 508.93 557.87 592.03 610.52 50 196.42 287.99 371.80 445.60 507.38 555.49 588.62 605.90 19 00 196.36 287.78 371.33 444.66 505.83 553.09 585.32 601.25 10 196.29 287.57 370.82 443.72 504.24 550.68 581.75 596.59 20 196.23 287.35 370.32 442.77 502.65 548.26 578.29 591.90 30 196.16 287.14 369.82 441.82 501.05 545.81 574.81 587.20 40 196.10 286.92 369.31 440.86 499.44 543.36 571.31 582.47 50 196.03 286.70 368.80 439.89 497.83 540.89 567.79 577.72 20 00 195.96 286.48 368.29 438.91 496.20 538.40 564.25 592.96 816 TABLE V. MIDDLE ORDINATES. Deg. of 2 3 4 5 6 8 9 10 Curve. Sta. Sta. Sta. Sta. Sta. Sta. Sta. Sta. Sta. 10' .145 .327 .582 .909 1 .309 1.782 2.327 2.945 3.636 20 .291 .654 1.164 1.818 2.618 3.5G3 4.654 5.889 7.272 30 .436 .982 1.745 2.727 3.927 5.345 6.981 8.KJ5 10.91 40 .582 1.309 2.327 3.636 5.235 7.1-26 9.307 11 .7* 14.54 50 1.636 2.909 4.545 6.544 8.907 11.68 11.72 18.17 1 00 .873 1.963 3.490 5.453 7.852 10.69 13.96 17.C6 21.80 10 .018 2.291 4.072 6.362 9.160 12.47 16.28 20.60 ,T>. 43 ','0 .164 2.618 4.653 7.270 10.47 14.25 18.60 23.54 29.06 30 .309 2.945 5.235 8.178 11.77 16.02 20.92 26.48 32.68 40 .454 3.272 5.816 9.086 13.08 17.80 23.24 29.41 36.30 50 .600 3.599 6.397 9.994 14.39 19.58 25 56 32.34 39.91 2 00 .745 3.926 6.978 10.90 15.69 21.35 27.88 35.27 43.52 10 .891 4.253 7.560 11. SI 17.00 23.13 30.19 38.20 47.13 20 2.036 4.580 8.140 12.72 18 30 24.90 32.51 41.12 50.73 30 2.181 4.907 8.721 13.62 19.61 26.67 34 82 44.04 54.33 40 2.329 5.234 9.302 14.53 20.91 28.44 37.13 46.95 57.92 50 2.472 5.561 9.882 15.43 22.21 30.21 39.43 49.86 61.50 8 00 2.617 5.887 10.46 16.34 23.51 31.98 41.73 52 77 65.08 10 2.763 6.214 11.04 17.24 24.81 33.75 44.04 filhW 68.65 20 2.908 6.541 11.62 18.15 26.11 35.51 46.33 58.57 72.21 30 3.053 6.868 12.20 19.05 27.41 37.27 48.63 61.46 75.77 40 3.199 7.194 12.78 19.96 28.71 39.06 50.92 64.35 79.31 50 3.344 7.520 13.36 20.86 30.01 40.79 53.20 67.23 82.85 4 00 3.489 7.847 13.94 21.76 31.30 42.55 55.49 70.11 80.38 10 3.635 8.173 14.52 22.60 32.60 44.30 57.77 72.5*8 89.91 20 3.780 8 499 15.10 23.56 33.89 46.05 60.04 75.84 93.42 30 3.925 8.826 15.68 24.46 35.18 47.80 62.32 78.70 96.92 40 4.070 9.152 16.25 25.36 36.47 49.55 64 . 58 81 .55 100.41 50 4.215 9.478 16.83 26.26 37.76 51.29 06.85 84.39 103.89 6 00 4.351 9.803 17.41 27.16 39.05 53.04 69.11 87.23 107.36 10 4.506 10.13 17.99 28.06 40.33 54.78 71.36 90.06 110.82 20 4.651 10.46 18.56 28.96 41.62 56.51 73.61 92.88 114.27 30 4.796 10.78 19.14 29.85 42.90 58.21 75.86 95.69 117.71 40 4 941 11.11 19.72 30.75 44.18 59.97 78.10 98.50 121.13 50 5.086 11.43 20 29 31.61 45.46 61.70 80.33 101.29 124.54 6 00 5.231 11.76 20.87 32.54 46.74 63.43 82.56 104.08 127.94 10 5.376 12.08 21.44 33.43 48.01 65.15 84.78 106.86 131.32 20 5.521 12.41 22.02 34.32 49.29 66.86 87.00 109.63 134.69 30 5.666 12.73 22.59 35.22 50.56 68.58 89.21 112.39 138.05 40 5.811 13.06 23.17 36.11 51.83 70.29 91.42 115.14 141.39 50 5.956 13:38 23.74 36.99 53.10 71.99 93.62 117.88 144.71 7 00 6.101 13.71 24.31 37.88 54.37 73.70 95.81 120.62 148.03 10 6.246 14.03 24.80 38.77 55.63 75.40 97.99 123.34 151.32 20 6.391 14.35 25.46 39.66 56.89 77.09 100.18 12i;. 05 154.60 30 6.536 14.68 26 03 40.54 58.15 78.78 102.35 128.75 157.87 40 6.680 15.00 26.60 41.43 59 41 80.47 104.52 131.44 161.11 50 6.825 15.33 27.17 42.31 60.67 82.15 106.67 134.11 164.34 8 00 6.970 15.65 27.74 43.19 61.92 83.83 108.83 136.78 167.56 10 .115 15.97 28.31 44.07 63.17 85.51 110.97 139.44 170.75 20 .259 16.30 28.88 44.95 64.42 87.18 113.11 142.08 173.93 50 .404 16.62 29.45 45.83 65.60 88;. 84 115.24 144.71 177.09 40 .549 16.94 30.02 46.71 66.91 90.50 117.36 147.33 180.23 50 .693 17.27 30.59 47.58 68.15 92.16 119.48 149.94 183.36 9 00 .888 17.59 31.16 48.46 69.39 93.81 21.58 152.53 186.46 10 .982 17.91 31.73 49.33 70.62 95.46 23.68 155.11 189.55 20 8.127 18.23 :52.29 50.21 71.86 97.10 25 . 77 157.68 19'.'. 61 30 8.271 18.56 32.86 51.08 73 09 98.74 27' . Sf) 100.23 195.06 40 8.416 18.88 33.42 51.95 74.81 100.37 89 JOB 162.77 19S.OH 50 8.560 19.20 33.99 52.81 75.54 102.00 131.99 165.30 201.08 10 00 8.705 19.52 34.55 53.68 76.76 103.62 134.05 167.82 204.67 TABLE V. MIDDLE ORDINATES. 317 Deg. of Curve. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. St'a. 8 Sta. 9 Sta. 10 Sta. 10 10' 8.849 19.84 35.12 54.55 77.98 105.23 136.09 170.31 207.63 20 8.993 20.17 35.68 55.41 79.20 106.85 138.13 172.80 210.57 30 9.137 20.49 36.24 56.27 80.41 108.45 140.16 175 A' 7 213. 4H 10 23 265.75 21 33 m 55.90 4 03 14 09 40.57 48.85 9 32 317.6* 1 18 21 25 59.08 4.38 13 01 42 37 62.95 6H 8 48 37-1.75 15 17 21 62.20 4.74 12 03 44.11 67. 6f 8 10 -i 37.06 13 06 81 65.25 5.09 11 14 45.78 C2.4C 7^ 7 38 50J.CS 11 21 08 6.S-.2 5.43 10 31 47.38 67. 3l 8 7 09 577.85 9 54 55 71.12 5.78 9 54 ! 4S.92 72.24 8^j 6 44 650.79 8 43 24 73.98 6.12 9 21 50.40 1 * . iC 9 22 741.67 7 43 50 76.75 6.45 8 52 61.60 82.42 9^ 6 02 (582.75 6 52 48 19.47 6.78 8 26 53.20 87. 6 10 10/4 5 43 930.4'J 1034 5C C 09 29 5 32 19 82.12 84 r '\ 7.11 7 4A 8 03 54.52 '.i.'i <>: li 5 12 1145^90 5 00 GO O-t . I O 87. '.7 t .4> 7.75 7 23 57.01 104.17 1H* 4 59 1264.40 4 31 53 88.76 8.07 7 06 58. IS ilt".l.K 12 4 46 1390.90 4 07 10 92. 20 S.37 6 50 59.30 ! 115.9(i Switch-rail 8 Feet, thrown 5 inches. No. n. Angle Radius, ("enter. Degree of Curve. Lead 11 F. No. n''. Angle JT ' ' * Dist. H'F. LF. 3 18 55 28 78.08 72 27 1 1 30.17 2.19 '< 25 45 21.91 26.41 3J4 16 15 32 108.73 52 41 44 33.31 2.80 22 15 23.65 31.10 4 14 15 00 143.56 39 54 26 36.32 2.89 1 19 38 i 25.27 35.92 41^ 12 40 48 183.98 31 03 4S 39.20 3.23 17 35 26.78 40.90 5 11 25 16 230.31 24 52 32 41.96 3.57 15 58 28.18 46.07 5}12 10 23 20 283.05 20 14 16 -1-1.60 3.89 14 38 89.49 51 .47 6 931 39 342.70 16 43 ',".) 47.15 4.21 13 32 30.70 57.12 6J^ 8 47 51 409.97 13 58 28 | 49.59 , 4.52 12 37 31.84 63 07 TABLE V1T.-TANCKNTS AND EXTERNALS OF A V CURVE. 319 Angle Tan^. T. Ex. E. Angle V. Tang. T. Ex. E. A ngle Tang. Ex. E. 90 10' 34477(5 143105 100 10' 410907 191913 110 10' 492484 256826 20 345781 142817 20 412123 19^-01 20 494013 258081 30 346788 144531 30 413324 193845 30 495549 259342 40 347798 145249 40 414569 194781 40 497091 260610 50 348811 145971 50 415798 JUB734 498767 261884 91 00 349828 146695 101 00 417032 196685 111 00 500195 263165 10 350847 147422 10 418270 191641 10 501757 264453 20 351809 148153 20 419512 198601 20 503326 265748 30 352895 14S887 30 420759 199566 30 504901 261049 40 353!i23 149624 40 422010 200536 40 506483 268351 50 354955 150365 50 423206 201510 50 508071 269673 92 00 355989 151108 102 CO 424526 202489 112 00 508667 270995 10 357027 151855 10 425191 203472 10 511269 272324 20 3580138 152606 20 427060 ','044 CO 20 512879 213660 30 359112 153359 30 428334 205453 30 514495 275C03 40 360159 154116 40 429613 206451 40 516119 276354 50 361209 154877 50 430896 207453 50 517749 217712 98 00 362203 155640 103 00 432184 208461 113 CO 519387 279077 10 363320 156408 10 433477 209473 10 521032 280449 20 364380 157178 20 434774 210490 20 522684 281829 30 365443 157952 30 436016 211512 30 524344 283216 40 366510 158730 40 437383 212539 40 526010 284611 50 3675b'0 159511 50 438695 213571 50 527685 286013 94 00 368653 160295 104 CO 44C012 214608 114 00 529367 287423 10 369730 161083 10 441333 215650 10 531056 288840 20 370810 161874 20 442660 216697 20 532753 290265 30 371898 162670 30 443991 217749 30 534457 291698 40 372i'80 163468 40 445328 218807 40 526170 293139 50 3740: 164270 50 446069 219869 50 537890 294588 95 00 375164 165076 105 00 448016 220937 115 00 539618 296045 10 316261 165886 10 449368 222010 10 541354 297509 20 SI 736-2 166699 20 450725 223088 20 548098 299051 30 37K4r,6 167516 30 452087 224172 30 544850 300463 40 379574 168336 40 453454 225261 40 546610 301952 50 380686 169161 50 454827 226355 50 548378 303450 96 00 381800 169989 '106 00 456204 221455 116 00 550155 304956 10 382919 170820 '10 457588 228560 10 551939 306412 20 384041 171656 20 458976 228671 20 553732 307992 30 385167 172495 30- 460310 230788 30 555534 309522 40 386297 173338 40 461770 232909 40 557344 311063 50 387430 174186 50 463174 233037 50 559162 312612 97 00 388567 175036 107 00 464585 234170 117 00 560989 314169 10 389707 175891 10 466001 235309 10 562825 315735 20 390852 176750 20 467422 236453 20 564670 317310 30 392000 177613 SO 46C850 237604 30 566523 318F94 40 393152 178480 40 410282 238760 40 568386 820487 50 394308 179350 50 471721 239922 50 570257 322089 98 00 395468 180225 108 00 473165 241090 118 00 572137 323700 10 396631 181104 10 474615 242264 10 574027 325320 20 397799 181987 20 476071 243443 20 515925 326950 30 398970 182873 30 477533 244629 30 511833 328589 40 400146 183764 40 479001 245821 40 579151 330237 50 401325 184660 50 480475 247019 50 581677 331895 99 00 402508 185559 109 CO 481954 248223 119 00 583614 3335C3 10 403696 186462 10 483440 249433 10 585559 335240 20 40-1887 187370 20 484932 250649 20 587515 336927 30 406083 188282 30 486430 251872 30 589480 338624 40 407283 189198 40 481934 253101 40 591455 340331 50 408487 190119 50 489444 254336 50 593440 342048 100 00 409695 191044 110 00 490961 255578 120 00 595435 343775 320 TABLE VII. TANGENTS AND EXTERNALS OF A 1' CURVE. Ansle Taiig. Ext. ' Angle Tang. Ext. Angle Tang. Ext. //; T. j E. Pi T. E. /'. T. E. 60 10' 199146 53516.2 70 10' 241460 76325.0 80 10' 289314 105540 20 199814 53851.6 20 242207 76755.0 20 290169 106091 30 200483 54188.4 30 242956 77186.8 30 291027 106645 40 201154 fi45-26.7 40 243706 77620.4 40 291886 107200 50 201826 54866.4 50 244458 78055.8 50 292748 107759 61 00 202499 55207.5 71 00 245212 78493.0 81 00 293611 108319 10 203173 55550.0 10 245967 78932.0 10 294477 108S82 20 203818 55894.0 20 246724 79372.8 20 295345 109447 30 204524 56239.4 30 247482 79815.4 30 296215 110014 40 20520 1 56586.3 40 248242 80259.8 40 297088 1105S4 50 205881 56934.6 50 249004 80706.1 50 297962 111157 62 00 206561 57284.3 72 00 249767 81154.2 82 00 298839 111731 10 20724-2 57635.6 10 250532 81604.1 10 299718 11230S 20 207924 57988.3 20 251298 82056.0 20 300599 11288S 30 208(508 58342.4 30 252066 82509.6 30 301482 113470 40 209-292 58698.1 40 252836 82965.2 40 302368 114054 50 209978 59055.2 50 253607 83422.6 50 303-256 114641 63 00 210065 59413.8 73 00 254380 83877.5 83 00 304146 115931 10 211354 59773.9 10 255154 84343.1 10 305039 115823 20 212043 60135.5 20 255931 84806.2 20 305933 116417 30 212734 60498.6 30 256709 85271.3 30 306831 11701J 40 213426 60863.2 40 257488 85738.2 40 307730 117613 .50 214120 61229.3 50 258370 86207.0 50 308632 118215 64 00 214814 61597.0 74 00 259053 86677.9 84 00 309536 118820 10 215510 61966.2 10 259838 87150.6 10 310443 119427 20 216207 62336.9 20 260624 87625.3 20 311352 120087 30 216900 62709.1 30 261412 88102.0 30 312263 120649 40 217605 63082.9 40 262202 88580.6 40 313177 121264 50 218306 63458.3 50 262994 89061.2 50 314093 121881 65 00 219009 63835.2 75 00 263788 89543.8 85 00 315011 122501 10 219712 61213.7 10 264583 90028.4 10 315933 123124 20 220417 64593.7 20 265380 90515.0 20 316856 123749 30 22 11 A3 64975 .3 30 266179 91003.6 30 317782 124378 40 221831 65358.5 40 266979 91494.3 40 318711 125008 50 222540 65743.3 50 267782 91987.0 50 3196)2 125642 60 00 223250 66129.6 76 00 208586 92481.7 86 00 320575 126278 10 2)3961 66517. 10 269392 92978.4 10 321511 12fi9i7 20 224674 66907.2 20 270200 93477.2 20 322450 127558 30 225389 67298.3 30 271010 9397'8.1 30 323391 128203 40 226104 67691.2 40 271822 94481.1 40 324335 128850 50 226821 68085.6 50 272635 94986.1 50 3-25281 129500 67 00 227540 68481.6 77 00 273451 95493.3 87 00 326230 130153 10 228259 68879.3 10 274268 96002.5 10 327182 130808 20 228980 69278.6 20 275087 96513.9 20 32H136 131466 30 229703 69679.6 30 275908 97027.4 30 :;290!)3 1321-J8 40 230427 70082.3 40 276731 97543.0 40 330052 132792 50 231152 70486.5 50 277556 98060.8 50 831014 133159 68 00 231879 70892.5 78 00 278383 98580.7 88 00 331979 134128 10 232607 71300.1 10 279212 99102.8 10 332947 134801 20 233337 71709.5 20 280043 99627.0 20 333917 135477 30 234068 721?0.5 30 280876 100153 ! 30 334890 136155 40 234800 72533.2 40 281710 100682 40 335866 136K37 50 235531 729*7.6 50 282547 101213 50 336845 137521 69 00 236270 73363.7 79 00 283386 101746 89 00 337826 138-208 10 887007 73781.5 10 284227 102281 10 338811 13S8H9 20 237745 74201.1 20 285070 102819 20 339798 139592 30 238485 74622.4 30 285914 103358 30 34078,8 140289 40 239226 75045.4 40 286761 103901 40 341780 140988 50 239969 75470.2 50 287610 104445 50 342776 141G91 70 00 240714 75896.7 80 00 288461 104991 90 00 343775 14039') TABLE VII. TANGENTS AND EXTERNALS OF A 1' CURVE. 321 Angle K Tang. T. Ex. ff. Angle V. Tang. T. Ex. E. Angle Tang. T. Ex. E. 10' 500.0 .304 10 10' 30580.3 1357.4 20 10' 61132.4 5393.2 20 1000.0 1.454 20 31084.3 1402.5 20 61648.4 5483.9 30 1500.0 3.273 30 31588.5 1448.2 30 62164.6 5575.4 40 2000.0 5.818 40 82092.7 1494.7 40 62681.1 5667.6 50 2500.0 9.090 50 32597.2 1542.0 50 63197.8 5760.7 1 00 3000.1 13.090 11 00 33101.7 1590.0 21 00 63714.9 5854.6 10 3500.1 17.818 10 33606.5 1638.7 10 64232.2 5949 2 20 4000.2 23.272 20 34111.3 1688.2 20 64749.8 6044 7 80 4500.3 29.455 30 34616.3 1738.4 30 65267.7 6140.9 40 5000.4 36.364 40 35121.4 1789.4 40 I 65785.8 6237.9 50 5500.5 44.002 50 35626.7 1841.2 50 I 66304 . 3 6335.7 2 00 6000.6 52.367 12 00 36132.2 1893.6 22 00 668-J8.0 6434.3 10 6500. S 61.459 10 36637.8 1946.8 10 67342.1 6533.7 20 7001.0 71.280 20 37143.5 2000.8 20 67861.4 6633.it 30 7501.2 81.829 30 37649.4 2055.5 30 68381.1 673r>. 40 8001.4 93.105 40 38155.5 2110.9 40 68901.0 6836.8 50 8501.7 105.11 50 38661.8 2167.2 50 69421.2 6939. 4 8 00 9002.1 117.84 13 00 39168.2 2224.1 23 00 699H.8 7042.8 10 9502.4 131.31 10 39674.8 2281.8 10 70462.6 7M7.0 20 10002.8 145.50 20 40181.5 2340.3 20 70983 . 8 7252.0 30 10.103.3 160.41 30 40688.4 2399 5 30 71505.2 7357.8 40 11003.8 176.06 40 41195.5 2459.5 40 72027.3 7464.4 50 11504.3 192.44 50 41702.8 2520.2 50 72549.1 7571.9 4 00 12004.9 209.55 14 00 42210.2 2581.7 24 00 73071.6 7680.1 10 12505.5 227.38 10 42717.9 SS43.9 10 73594.3 7789.2 20 13006.2 245.95 20 43225.7 2706.9 20 74117.4 789U.O 30 13506.9 265.24 30 43733.7 2770.7 30 74640.8 8009.7 40 14007.7 285.27 40 44241 8 2835.1 40 75164.6 8I21.2 50 14508.6 30(5.02 50 44750.2 2900.4 50 75688,6 8233 . 5 5 00 15009.5 327.51 15 00 45258.8 296G.4 25 00 76213.0 8346.7 10 15510.5 349.72 10 45767.6 3033.2 1 10 76737.8 8460.6 so 16011 6 372.67 20 46276.5 3101.7 20 77262.81 8575.4 30 16512.7 396.35 30 46785.7 3169.0 30 77788.3 8691.0 40 17013.9 420.76 40 47295.0 3238 1 40 78314.0 8807.4 50 17515.1 445.90 50 47804.6 3307.9 50 78840.2 8924.6 6 00 18016.5 471.78 16 00 48314.4 3378.5 26 00 79366.6 9042.7 10 18517.9 498.38 10 48824.4 3449.8 10 79893.5 9161.6 20 19019.4 525.72 20 49334.6 3521.9 20 80420.7 9281.3 30 19520.9 553.79 30 49845.0 *594.8 30 80948.2! 9401.8 40 20022.6 582.60 40 50355.6 3608.4 i 40 81476.1 9523.2 50 20524.3 612.13 50 50806. 4 , 3742.8 50 82004.4 9645.4 7 00 21026.2 642.41 17 00 51377. 5' 8818.0 ! 27 00 82533.0 9768.4 10 21528.1 673.41 10 51888.8! 3893.9 : 10 83062.0! 9892.3 20 22030.1 705.15 20 52400.3 3970.7 20 83591.4:10017.0 30 22532.2 737.62 30 52912 4048.1 30 84121.1 110142. 5 40 23034.4 770.83 40 53424.0 4126.4 40 84651. 31 10268.9 50 23536.7 804.78 50 53936.2: 4205.4 50 85181.8 10396.1 8 00 24039.1 839.46 18 00 54448.5 4285.2 28 00 85712.7 10524.2 10 24541.6 874.88 10 54961.2 4365.8 i 10 86243.9 10653.1 20 25044.2 911.03 20 55474.1: 4447.1 ; 20 86775.6 10782.8 30 25546.9 947.92 30 55987.3 4529.2 30 87307.6 10913.4 -JO 26049.'; 985.55 40 56500.6 4612.1 i 40 87840.1 11044.9 50 26552.6 1028.91 50 57014.3 4695.8 j 50 88372.9 11177.1 9 00 27055.7 1063.91 19 00 57528. 2' 4780.2 29 00 88906.2 11310.3 10 27558.8 1102.86 10 58042 3 4865.4 10 89439.8 11444.3 20 28062.1 1143.43 20 58556.7 4951.5 20 89973.9 11579.1 30 2S5I55.5 1184.75 30 59071.3 5038.2 30 90508.3 11714.8 40 29069.0 1226.83 40 50,^6.2 5125.8 ' 40 91043.2 11851.4 50 2'.)572.fi 1269.63 50 60101.3 5214.1 '.,0 91578.5 11988.8 10 00 30076.4 1H1S.15 20 00 60616.8' 5303.3 30 00 92114.2 12127.1 322 TABLE VII.-TANGENTS AND EXTERNALS OF A 1' CURVE. Angle Tang. Ext. T. E. Angle Tang. Ext. E. Angle ]/ , Tang. Ext. ! E. i 30 10' 92650.3; 12->66.2 40 10' 125690 22256.8 : 60 10' 160914 35796.5 20 93186.8 12406.2 20 126257 22452.0 20 101524 36055.5 30 93723.7 12547.1 30 126825 22648.1 30 102135 36315.7 40 94261.1 12688.8 40 127394 22845.2 40 102746 36577 50 94798.9 12831.4 50 127963 23043.3 50 163359 36839.4 31 00 95337.1 12974.8 41 00 128532 23242.4 51 00 1031)72 37103.1 10 95880.5 13119.2 10 129102 23442.5 10 164586 37307.9 20 96414.9 13264.3 20 129673 23643.6 20 105201 37633.9 30 96954.5 13410.4 30 130245 23845.6 30 165817 37901 . 1 40 97494.5 13557.4 40 130817 24048.7 40 100434 38169.4 50 98034.9 13705.2 50 131389 SM352.8 50 167054 384:59.0 32 00 98575.8 13853.9 42 00 131963 24457.9 52 00 167670 38709.7 10 99117.2 14003.5 10 132537 240(53.9 10 168290 38981.0 20 99658.9 14151.0 20 133111 24871 20 168910 39254 7 30 100201 14305.4 30 133687 25070.2 30 169531 39529.1 40 100744 14457.6 40 134203 25288 3 40 170158 3H804.1! 50 101287 14610.7 50 134839 25498.5 50 170770 40081.3 33 00 101831 14764.7 43 00 135416 25709.6 63 00 171400 40359.3 10 102375 14919.7 10 135994 25921 . S 10 172024 4063S.4 20 102919 15075.5 20 136573 26135.0 20 1 72050 40918 8 30 103464 15232.2 30 137152 26349.3 30 173277 41200.4 40 104010 15383. 8 40 137732 215564 6 40 173904 41483.3 50 104556 15548.3 50 138313 26780.9 50 174533 41707.4 34 00 105102 15707.7 44 CO 138894 26998.2 64 00 175162 42052.7 10 105649 15868.0 10 139476 27216.0 10 175792 42339.2 20 106197 16029.2 2u 140059 27436.1 20 176423 42627.0 30 106745 16191.3 30 140642 27656.6 30 177056 429 Hi. (1 40 107293 10354.3 40 141226 27878.1 40 177689 43200.3 50 107842 16518.2 50 141811 28100.7 50 178323 4349;. 9 35 00 108392 16683.1 45 00 142396 28324.4 55 00 178958 43790.7 10 108942 16848.8 10 142982 28549.0 10 179594 14084 S 20 109492 17015.5 20 143569 28774.8 20 180231 44380.1 30 110043 '17183.1 30. 144157 29001.6 30 180809 44676.7 40 ! 10595 17351.6 40 144745 29229.5 40 181508 41974 6 50 111147 175-21.1 50 145334 29158.5 50 182147 45273.8 36 00 111699 17691 .4 46 00 145924 29688.5 56 CO 182788 45574.2 10 112252 17802.7 10 146514 29919.6 10 1 S3 430 45S76.0 20 112806 18034.9 20 147105 30151.8 20 1S4073 46179 30 1)3360 18208.1 30 147697 30385.8 MO 184717 IK 183. 4 40 113915 18382.1 40 148290 30619.4 40 1S53itt 46189.0 50 114470 18557.1 50 148883 30854.8 50 \8MQ8 41 09.-). 9 37 00 115025 18733.1 47 00 149478 31091.4 57 00 180054 47404.2 10 115582 18909.9 10 150072 31329.0 10 1S7>,02 47713.8 20 116138 19087.8 20 150668 31567.7 20 JS7951 48024.0 30 116096 19266.5 30 151264 31807.5 30 188601 48336.9 40 117254 19446.3 40 151862 32048.4 40 189252 48650.4 50 117812 19626 9 50 152460 32290.4 50 U9905 48965.3 38 00 118371 19808.5 48 00 153058 32533. G 58 00 190557 49281.5 10 118931 19991.1 10 153658 32777.8 10 191212 49599.1 20 119491 20174.6 20 154258 33023.1 20 191867 49918.0 30 120051 20359.1 30 154859 33269.6 30 192523 50238.:' 40 120613 20544.5 40 155461 33517.2 40 193180 50559. N 50 121175 20730 9 50 156064 33765.9 50 193839 50882.8 39 00 121737 20918.2 49 00 156667 34015.8 59 00 194498 51207.1 10 122300 21106.6 10 15727! 34266.7 10 195159 51532.9 20 122864 21295.8 20 157876 34518.9 20 195821 '51859.9 30 123428 21480.1 30 158482 34772.1 30 1%IS: ;5'.'188 4 40 I1^9!W 21677.3 40 150089 35026.5 40 197147 5 25 18.3 50 124558 21869.5 50 159690 85282.01 50 197812 52849.5 40 00 125124 22062.7 60 00 160304 35538.71 1 60 00 198478 53182.1 TABLE VIII. LENGTHS OF ARC S CORRESPONDING TO ANY 333 NUMBER OF DEGREES, MINUTES, OR SECONDS FOR RADIUS = 1. fo Arc for Arc for Arc for w Arc for Arc for Arc for Degrees. Minutes. Seconds. c * Degrees. Minutes. Seconds. 11 ll 1 .0174533 .0002909 .0000048 31 .5410521 .0090175 .0001503 2 .0349086 .0005818 .0000097 32 .5585054 .0093084 .0001551 3 .0523599 .0008727 .0000145 33 .5759587 .0095993 .000! 600 4 .0698 132 .0011636 .0000194 34 .5934119 .0098902 .0001648 5 .OS72M55 .0014544 .0000242 35 .6108652 .0101811 .0001697 6 .1047198 .0017453 .0000291 36 .6283185 .0104720 .0001745 7 .1221730 .0020362 .0000339 37 .6457718 .0107629 .0001794 8 .1396263 .0023271 .0000388 38 .6632251 .0110538 .0001842 9 .1570798 .0026180 .0000436 39 .6806784 .0113416 .0001891 10 .1745329 .0029089 .0000485 40 .6981317 .0116355 .0001939 11 .1919862 .0031998 .0000533 41 .7155850 .0119264 .0001988 12 .0034907 .0000582 42 .7330383 .0122173 .0002036 13 ! 2268928 .0037815' .00()i'630 43 .7504916 .0125082 .0002085 14 .2443461 .0040724 .0000679 44 .7679449 .0127991 .0002133 15 .2617994 .0043633 .0000727 45 .7853982 .0130900 .0002182 16 .2792527 .0046542 .f000776 46 .8028515 .0133809 . .0002230 17 .2<)670i;0 .0049151 .0000824 47 .S203047 .0136717 .0002279 18 1 .3141593 .005.360 .0000873 48 .8377580 .0139626 .0002327- 19 .3316126 .00552t'9 .0000921 49 .8552118 .0142535 .0002376 20 .3490659 .0058178 .0000970 50 .8726646 .0145444 .0002424 21 .3665191 .0061087 .0001018 51 .8901179 .0148353 .0002473 22 .3839724 .0063995 .0001067 52 .9075712 .0151262 .0002521 23 .4014257 .0066904 .0001115 53 .9250245 .0154171 .0002570 24 .4188790 ! .0069K18 .0001164 54 .9424778 .0157080 .0002618 25 . 4363323 .0072722 .0001212 55 .9599311 .0159989 .0002666 26 .4537856 .0075631 .0001261 56 .9773844 .0162897 .0002715 27 .4712389 .0078540 .0001301 57 .9948377 .0165806 .0002763 28 .4886922 .0081449 .0001357 58 1.0122910 .0168715 .0002812 29 .5061455 .0084358 .0001406 59 1.0297443 .0171624 .0002860 30 .5235988 .0087266 .0001454 60 1.0471976 .0174533 .0002909 TABLE IX. ACRES FOR VARIOUS LENGTHS AND WIDTHS. 93 Widths. - I 100 90 80 70 60 50 40 30 20 lOol .229568 .206612 .183055 .160698 .137741 .114784 .091827 .068871 .045914 200 .459137 .413223 .367309 .321396 .275482 .229568 .183655 .137741 .091827 wo .688705 .619835 .550964 .482094 .413223 .344353 .275482 .206612 .137741 4(10 .918274 .8264J6 .734619 .642792 .550964 .459137 .367309 .275482 .183655 500 1.147842 1.033058 .918274 .803489 .688705 .573921 .459137 .344353 .229568 600 1.377410 1.239669 1.101928 .9641H7 .826446 .688705 .550964 .413223 .275482 700 1.606979 1.446281 1.285583 1.124885 .964187 .803489 .642792 .482094 .321396 800 1.836547 1.652893 1.469238 1.285583 1.101928 .918274 .734619 .550984 .367309 906 2.066116 1.859504 1.652892 1.446281 1.239669 1.033058 .826446 .619835 .413223 324 TABLE X. TOTAL GRADES. Grade per Station. J 00 0.1 0.2 03 0.4 0.5 0.0 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 3 0.3 0.6 0.9 1 . * 1.5 1.8 2.1 2.4 2.7 4 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3 k j . * 3.6 5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 6 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 y 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 8 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 9 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 11 1.1 2/2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 12 1.2 2.4 3.6 4.8 6.0 j 7.2 8.4 9.6 10.8 13 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 14 .4 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 15 .5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 16 .6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 17 t 1 3.4 5.1 6.8 8.5 10.2 11.9 13.8 15.3 18 !8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 19 .9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 20 2.0 4.0 6.0 8.0 10.0 1-J.O 14.0 16.0 18.0 21 2.1 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 22 2.2 4.4 6.6 8.8 11.0 13.2 15.4 17.6 19.8 23 2.3 4.6 6.9 9.2 11.5 13.8 16.1 18.4 20.7 24 2.4- 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.6 25 2^5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 26 2.6 5.2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 27 2.7 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 28 2.8 5.6 8.4 11.2 14.0 IB. 8 19.6 22.4 25 . 2 29 2.9 5.8 8.7 11.6 14.5 17.4 20.3 23 2 26.1 30 3.0 6.0 9.0 12.0 15.0 18.0 21 .0 24.0 27.0 31 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 32 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.0 28.8 33 3.3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 34 3.4 6.8 10.2 13.6 17.0 20.4 23 8 27.2 30.6 35 3.5 7.0 10.5 14.0 17.5 21.0 24.5 28.0 31.5 36 3.6 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32.4 37 3.7 7.4 11.1 14.8 18.5 22.2 25.9 29.6 33.3 38 3.8 7.6 11.4 15.2 19.0 22.8 26.6 30.4 34.2 3D 3.9 7.8 11.7 15.6 19 .5 23.4 27.3 31.2 35.1 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 41 4.1 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 4^* 4.2 8.4 12.6 16.8 21.0 25.2 29.4 33.6 37.8 43 4.3 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 44 4.4 8.8 13.2 17.6 22.0 26.4 30 8 35.-' 39.6 45 4.5 9.0 13.5 18.0 22.5 27.0 31 5 36.0 40.5 46 4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 47 4.7 9.4 14.1 18.8 23.5 28.2 32 9 37.6 4-3.3 48 4.8 9.6 14.4 19.2 24.0 28.8 33.6 38.4 43.2 49 49 9.8 14.7 19.6 24.5 29.4 34.3 39.2 44.1 50 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 51 5.1 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 52 5.2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 4-4.4 47.7 54 5.4 10.8 16 2 21.6 27.0 8.'. 4 37.8 43.2 4S.6 55 5.5 11.0 16.5 22.0 27.5 83.0 38.5 44.0 49.5 56 5.6 11.2 16.8 oo 4 28.0 33.6 39.2 44.8 50.4 57 5.7 11.4 17.1 22.8 28 . 5 34.2 39.9 45.6 51 3 58 5.8 11.6 17.4 23.2 29.0 34,8 40 (> 46.4 53.2 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.'.' 53.1 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 TABLE XI. CORRECTION FOR THE EARTH'S CURVATURE 325 AND FOR REFRACTION. Dist. D. Cor. E. Dist. D. Cor. E. Dist. D. Cor. E. Dist. D. Cor. E. 300 .00-2 1800 .066 3300 .223 4SOO .472 400 .003 1900 .074 3400 .237 4900 .492 500 .005 2000 .082 3500 .251 5000 .512 GOO .007 2100 .090 3600 .266 5100 .533 700 .010 2200 .099 3700 .281 5200 .554 800 .013 2300 .108 3800 .296 1 mile .571 900 .017 2100 .118 3900 .312 2 n iles 2.285 1000 .020 2500 .128 4000 .328 3 5.142 1100 .028 2600 .139 4100 .345 4 9.H2 1200 .030 2700 .149 4200 .362 5 14.284 1300 .035 2800 .161 4300 .379 6 20.568 1400 .040 2900 .172 4400 .397 7 27.996 1500 .046 3000 .184 4500 .415 8 36.566 1600 .05-2 3100 .197 4600 .434 9 46.279 1700 .059 3200 .210 4700 .453 10 " 57.135 TABLE XII.-ELEVATION OF OUTER RAIL ON CURVES. Degree Velocity in Miles per Hour. D. 10 15 20 25 30 35 40 45 50 60 1 .006 .013 .023 .036 .051 .070 .091 .115 .143 .205 2 .011 .026 .046 .071 .103 .140 .182 .231 .285 .410 3 .017 .038 .068 .107 .154 .209 .274 .346 .428 .616 4 .023 .051 .091 .142 .205 .279 .365 .462 .570 .821 5 .029 .064 .114 .178 .257 .349 .456 .577 .712 6 .034 .077 .137 .214 .308 .419 .547 .693 .855 7 .040 .090 .160 .249 .359 .489 .638 .808 .997 8 .046 .103 .182 .285 .410 .559 .730 .923 9 .051 .115 .205 .321 .402 .628 .821 10 .057 .128 .228 .356 .513 .698 .912 12 .068 .153 .274 .428 .616 .838 14 .080 .180 .319 .499 .718 .978 16 .091 .05 .365 .570 .821 18 .103 .231 .410 .641 .923 20 .114 .257 .456 .712 25 * .143 .321 .570 .891 30 .171 .385 .684 35 .200 .449 .798 40 .228 .513 .912 50 .285 .641 326 TABLE XIII. COEFFICIENTS FOR REDUCING INCLINED STADIA MEASUREMENTS TO THE HORIZONTAL. Inclination. 0' 10' 20' 30' 40' 60' 1.00000 .99999 .99997 .99992 .99986 .99979 1 .99970 .99959 .99946 .99931 .99915 .99898 2 .99878 .99857 .99834 .99810 .99784 . 99756 3 .99726 .99695 . 99062 .99627 .99:>91 .99553 4 .99513 .99472 .99429 .99384 .99338 . 99290 5 .99240 .99189 .99136 .99081 .99025 .98967 6 .98907 .98846 .98783 .98718 .98052 .98384 p j- .98515 .98444 .98371 .98296 .98220 .98142 8 .98063 .97982 .97899 .97815 .97729 97642 9 .97553 .97162 .97370 .97276 .97180 .97083 10 .96985 .96884 .96782 .96679 .96574 .96467 11 .96359 .96249 .96138 .96025 .95911 .95795 12 .95677 .95558 .95438 .95315 .95192 .95066 13 .94940 .94881 .94682 .94550 .94417 .94283 14 .94147 .94010 .93871 .93731 .93589 .93446 15 .93301 .93155 .93007 .92858 .92708 .9. '556 16 .92402 .92247 .92091 .91933 .91774 .91614 17 .91452 .91288 .91124 .90957 90790 .90621 18 .90451 .90279 .90106 .89932 .89756 .89579 19 .89400 .89221 .89040 .88857 .88673 .88488 20 .88302 .88114 .87926 .87735 .87544 .87:,51 21 .87157 .86962 .86765 .86568 .86309 .86168 22 .85967 .85764 .85560 .85355 .85149 .84941 23 .84733 .84523 .84312 .84100 .83886 .83672 24 .83456 .83240 .83022 .82803 .82583 .82301 25 .82139 .81916 .81691 .81466 .81239 .81011 26 .80783 .80553 .80322 .80091 .79858 .79624 27 .79389 .79153 .78916 .78679 .78440 .78200 28 .77959 .77718 .77475 .77232 .76987 .76742 29 .76496 .76249 .76001 .75752 .75502 .75251 30 .75000 .74747 .74494 .74240 .73985 .73730 81 .73473 .73216 .72958 .72699 .72440 .72179 32 .71918 .71656 .71394 .71131 .70867 .70602 33 .70336 .70070 .69804 .69536 .69268 .68999 34 .68730 .68460 .68189 .67918 .67646 .67374 35 .67101 .66827 .66553 .66278 .66003 .65727 36 .65450 .65174 .64896 .64618 .64340 .6-10(51 37 .63781 .63502 .63221 .62941 .62659 .62378 38 .62096 .61813 .61530 .61247 .60963 .60679 39 .60395 .60110 .59825 .59540 .59254 .58968 40 .58682 .58395 .58108 .57821 .57534 '.57246 TABLE XIV.-COEFFICIENTS FOR REDUCING URADIENTER 327 MEASUREMENTS TO THE HORIZONTAL. Inclination. 0' 10' 20' 30' 40' 60' 100.00 99.99 99.99 99.98 99.97 99.96 1 99.95 99.94 99.92 99.91 99.89 99.87 2 99.84 99.82 99.79 99.77 99.74 99.71 3 99.67 99.64 99.60 99.57 99.53 99.49 4 99.45 99.40 99.35 99.31 99.26 99.21 5 99.15 99.10 99.04 98.98 98.93 98 87 6 98.80 98.74 98.67 98.61 98.54 98.47 7 98.39 98.32 98.24 98.17 98.09 98.01 8 97.93 97.84 97.76 97.66 97.58 97.49 9 97.40 97.31 97.21 97.11 97.02 96.91 10 96.81 96.71 96.60 96.50 96.39 96.28 11 96.17 96.06 95.94 95.83 95.71 95.59 12 95.47 95.35 95.23 95.10 94.97 94.85 13 94.72 94.59 94.46 94.3-2 94.18 94.05 14 93.91 93.78 93.63 93.49 93.31 93.20 15 93.05 92.90 92.75 92.60 92.45 92.29 16 92.14 91.98 91.82 9J.66 91.50 91.34 17 91.18 91.01 90.84 90.67 90.50 90.33 18 90.16 89.98 89.81 89.63 89.45 89 27 19 89.09 88.91 88.73 88.54 88.36 88.17 20 87.98 87.79 87.60 87.41 87.21 87.01 21 86.82 86.63 86.43 86.23 86.02 85.82 23 85.62 85.42 85.21 85.00 84.79 84.58 23 84.38 84.16 83.95 83.73 83.52 83.30 24 83.08 82.87 82.65 82.43 82.20 81.98 25 81.76 81.53 81.31 81.08 80.85 80.62 26 80. 30 80.16 79.93 79.69 79.45 79.22 27 78.99 78.75 78.51 78.27 78.03 77.79 28 77.54 77.30 77.06 76.81 76.57 76.32 29 76.07 75.82 75.57 75.32 75.07 74.82 30 74.57 74.31 74.06 73.80 73.55 73.29 328 TABLE XV. OFFSETS FOR TRANSITION CURVES. s o AK=t = 8iH=. Fi>?. 130. ' Deg. of Offset Rad of Offset Lengths of Transition Curves. Curve Curve 100 150 200 240 250 300 320 360 400 10' 34377 .01 .03 .05 .07 .08 .11 .12 .16 .19 20 17189 .02 .05 .10 .14 .15 .81 .:>/, 30 11459 .04 .08 .15 .21 22 .'33 .37 .47 .5s 40 8595 4 .05 .11 .19 .28 iao .44 .50 .63 .78 50 6875.5 .06 .14 .24 .35 .38 .55 .62 .79 .97 1 00 5729.6 .07 .16 .29 .42 .46 .65 .74 .94 1.1(5 10 4911 1 .08 ,19 .34 .49 .53 .76 .8? 1.10 1.36 20 4297.2 .10 22 .39 .56 .61 .87 .!)'.) 1 . 26 1.55 30 3819.7 .11 !25 .44 .63 .68 .98 1.12 1.41 1 . 75 40 3437.8 .12 .27 .48 .70 .76 1.09 1.24 1.57 1.94 50 3135.2 .13 .30 .53 . 77 .83 1.20 1.37 1 .73 2.13 2 00 2S64.8 .15 .33 .58 .84 .91 1.31 1.49 1.83 2.33 10 2644.4 .16 .35 .63 .91 .98 1.42 1.61 2.01 2.52 20 2455.5 .17 .38 .68 .98 .06 1.53 f,78 2.^0 ij- 30 2291.8 .18 .41 .73 . 05 .14 1.64 1.86 2.36 2^91 40 2148.6 .19 .44 .78 .12 .21 1.74 1.99 2.51 3.10 50 2022.2 .21 .46 .82 .19 .29 1.85 2.11 X.' 67 3.30 3 00 1909.9 .22 .49 .87 .26 .36 1.96 2.23 2. S3 3.49 10 1809.3 .23 .5-2 .92 .33 .44 2.07 2.86 2 . 98 3.68 20 1718.9 .24 .55 .97 .40 .51 2.18 2. 48 3.14 3.88 30 1637.0 .25 .57 1.02 .47 .59 2.29 2.61 3.30 4.07 40 1562.6 .27 .60 1.07 .54 .67 2.40 2.73 3.45 4.26 50 1494.7 .28 .63 1.11 .61 .74 2.51 2.85 8. (51 4.46 4 00 1432.4 .29 .65 .16 1.68 .82 2.62 2.98 3.77 4.65 10 1375.1 .30 .68 .21 1.74 .89 2.73 3.10 3.92 4.S4 20 13-^.2 .32 .71 .26 1.81 .97 2.83 3.23 4.08 5.04 30 1273.2 .33 .74 .31 1.88 .04 2.94 3.35 4.24 5.23 40 1227.8 .34 .76 .36 1.95 .12 3.05 3.47 4.39 5.48 50 1185.4 .35 .79 .41 2.02 .20 3.16 3. GO 4.55 5.62 5 00 1145.9 .36 .82 .45 2.09 2.27 3.27 3.72 4.71 5.81 10 1109.0 .38 .85 .50 2.16 2.35 3.38 3.84 4.86 6.00 20 1074.3 .39 .87 .55 2.23 2.42 3.49 3.97 5.02 6.20! 30 1041.7 .40 .90 .60 2.30 2.49 3.60 4.09 5.18 6.39! 40 1011.1 .41 .93 .65 2.37 2.57 3.71 4.22 5.33 6.58 50 982.2 .42 .95 .70 2.44 2.65 3.81 4.34 5 49 6.78 6 00 954.9 .44 .98 .74 2.51 2.73 3.92 4.46 5.65 6.97 10 929.1 .45 1.01 79 2.58 ' 2.80 4.03 4.59 5.80 7.16 20 904.7 .46 1.04 .84 2.65 < 2.88 4.14 4.71 fi . 96 7.36 30 881.5 .47 1.06 .89 2.72 2.95 4.25 4.83 6.12 7.55 40 859.4 .48 1.09 .94 2.79 3.03 4.36 4.96 6.27 7.74 50 838.5 .50 1.12 .99 2.86 3.10 4.47 5.08 6.43 7.93 7 00 818.5 .51 .14 2.03 2.93 3.18 4.1$ 5.20 G . 58 8.12 10 799.5 .52 .17 2.08 3.00 3.25 4.68 5.33 6.74 8.32 20 781.3 .53 .20 2.13 3.07 3.33 4.79 5.45 6.SK) 8.51 30 763.9 .55 1.23 2.18 3.14 3.41 4.90 5.57 7.05 8.70 40 747.3 .56 .25 2.23 3.21 3.48 5.01 5.70 7.21 8.89 50 731.4 .57 .28 2.28 3.28 3.56 5.12 5.82 7.36 9.09 8 00 716.2 .58 .31 2.33 3.35 3.63 5 23 5.95 7 . 5 '2 9 28 10 701.6 .59 .34 2.37 3.42 3.71 5.33 6.07 7.68 9.47 20 687.5 .61 .36 2.42 3 49 3.78 5.44 6.19 7.83 9.66 30 674.1 .62 1.39 2.47 3.56 3.86 5.55 6.31 7.99 9.85 40 661.1 .63 .42 2.52 3.63 3.93 5.66 6.44 8.14 10.05 50 648.6 .64 .44 2.57 8.69 ; 4.01 5 . 77' 0.56 8. 80 10.24 9 00 636.6 .65 .47 2.62 3.76 j 4.08 5.88 6.68 8.45 10.43 10 625.0 .67 .50 2.66 383 4.16 5.99 6. SI 8.61 10.62 20 613.9 .68 .53 2.71 3.90 4.23 6.09 (5.93 8.76 10.81 30 603.1 .69 .55 2.76 3.97 I 4.31 6.20 7.05 8.92 11. i'(i 40 592.7 .70 .58 2.81 4.04 4.39 6.31 7.18 9.08 11.19 50 582.7 .71 .61 2.86 4.11 4.46 6.42 7.30 9.23 11.39 10 00 573.0 .73 1.64 2.91 4.18 4.54 6.53 7.42 9.39 11.58 TABLE XV. -OFFSETS FOR TRANSITION CURVES. 329 Deg. of Offset Curve. Had. ol Offsei Curve. Lengths of Transition Curves. 100 150 200 240 250 300 320 360 400 10 10' 563.0 .74 .66 2.95 4.25 4.61 6.63 7.55 9.54 11.77 20 554.5 .75 .69 3.00 4.32 4.69 6.74 7.67 9.70 11.96 30 546.7 .16 .72 i 3.05 4.39 4.76 6.85 7.79 9.85 12.15 40 537.1 .78 .74 3.10 4.46 4.84 6.96 7.91 10. 0: 12.34 50 528.9 .79 .77 3.15 4.53 4.91 7.07 8.03 10.16 12.53 11 00 520.9 .80 .80 3.19 4.60 4.99 7.17 8.16 10.3', 12.7 10 513.1 .81 .83 3 24 4.67 5.06 7.28 8.28 10.47 12.91 20 505.6 .82 .86 3.29 4.74 5.14 7.39 8.40 10.6:: 13.10 30 498.2 .84 .88 3.34 4.81 5.21 7.50 8.53 10.78 13.29 40 491.1 .85 .91 3.39 4.87 5.29 .61 8.65 10.93 13.48 50 484.2 .86 .93 3.44 4.94 5.36 .71 8.77 11.09 13.67 12 00 477.5 .87 .96 3.48 5.01 5.44 .82 8.89 11.24 13.86 10 470.9 .88 .99 3.53 5. OS 5.51 .93 9.02 11.40 14.05 20 464.6 .90 2.02 3.58 5.15 5.59 8.04 9.14 11.66 14.24 30 458.4 .91 2.04 3.63 5.22 5.66 8.14 9.26 11.71 14.43 40 45-2.3 .92 2.07 3.68 5.29 5.74 8.25 9.38 11.86 14.62 50 446.5 .93 2.10 3.73 5.36 5.81 8.36 9.51 12.01 14.81 13 00 440.7 .94 2.12 3.77 5.43 5.89 8.47 9.63 12.17 15.00 10 435.2 .96 2.15 3.82 5.50 5.96 8.58 9.75 12.32 15.19 20 429.7 .97 2.18 3.87 5,57 6.04 8.68 9.87 12.47 15.37 30 424.4 .98 2.21 3 92 5.61 6J1 8.79 9.99 12.63 15.56 40 419.2 .99 2.23 3.97 5.71 6.19 8.90 10.12 12.78 15.75 50 414.2 1.01 2.26 4.01 5.77 6.26 9.00 10.23 12.94 15.94 14 00 409.3 .02 2 29 4.06 5.84 6.34 9.11 10.36 13.09 16.13 10 404.4 .03 2^31 4.11 5.91 6.41 9.22 10.48 13.24 16.32 20 399.7 .04 2.34 4.16 5.98 6.49 9.33 10.60 13.40 16.50 30 395.1 .05 2.37 4.21 6.05 6.56 9.43 10.72 13.55 16.69 40 390.7 .07 2.40 4.25 6.12 6.64 9.54 10.85 13.70 16.88 50 388.3 .08 2.42 4.30 6.19 6.71 9.65 10.97 13.85 17.07 15 00 382.0 .09 2.45 4.35 6.26 6.79 9.75 11.09 14 01 17.25 10 377.8 .10 2.48 4.40 6.33 6.86 9.86 11.21 14.16 17.44 20 373.7 .11 2.50 4.45 6.40 6.94 9.97 11.33 14.31 17.63 :30 369.7 .13 2.53 4.50 6.46 ".01 10.08 11.45 14.46 17.82 40 365.7 .14 2.56 4.54 6.53 /.09 10.18 11.57 14.62 18.00 50 361.9 .15 2.59 4.59 6.60 7.16 10.29 11.69 14.77 18.19 16 00 358.1 .16 2.61 4.64 6.67 ^.24 10.40 11.82 14.92 18.38 10 354.4 .17 2.64 4.69 6.74 ".31 10.50 11.94 15.07 18.56 20 350.8 .19 2.67 4.74 6 81 '.38 10.61 12.06 15.23 18.75 30 347.2 .20 2.69 4.78 6.88 ".48 10.72 12.18 15.38 18.93 40 343.8 .21 2.72 4.83 6.95 ".53 10.82 12.30 15.53 19.12 50 340.4 .22 2.75 4.88 .01 ".61 10.93 12.42 15.68 19.31 17 00 337.0 .24 2.78 4.93 .08 ^.68 11.03 12.54 15.83 19.49 10 333.8 .25 2.80 4.98 .15 ".76 11.14 12.66 15.98 1 19.68 20 330.6 .26 2.83 5.02 .22 ".83 11.25 12.78 16.14 19.86 30 327.4 27 2.86 5.07 .29 ".91 11.35 12.90 16.29! 20.05 40 324.3 ^28 2.88 5.18 .36 ".98 11.46 13.02 16.44 1 20.23 50 321.3 .30 2.91 5.17 .43 8.05 11.57 13.14 16.59 20.42 18 00 318.3 .31 2.94 5.21 .50 8.13 11.67 13.26 16.74 20.60 10 315.4 .32 2.97 5.26 .51) 8.20 11.78 13.38 16.89 20.79 20 312.5 .33 2.99 5.31 .63 8.28 11.88 13.50 17.04 20.97 30 309.7 .34 3.02 5.36 .70 8.35 11.99 13.62 17.19 21.15 40 306.9 .36 3.05 5.41 77 8.43 12 10 13.74 17.34 21.34 50 304.2 .37 3.07 5.45 '.84 8.50 12.20 13.86 17.49 21.52 19 00 301.6 .38 3.10 5.50 .91 8.57 12.31 13.98 17.64 21.70 10 298.9 .39 3.13 5.55 .97 8.65 12.41 14 10 17.79 21.89 20 296.4 .40 3.16 5.60 8.04 8.72 12.52 14.22 17.941 22.07 30 293.8 .42 3.18 5.65 8.11 8.80 12.62 14.34 18.09 22.25 40 291.3 .43 3.21 5 69 8.18 8.87 12.73 14.46 18.24 22.44 50 288.9 .44 3 . 24 5.74 8.25 8.94 12.84 14.58 18 39 22.62 20 00 286.5 1.45 3.26 5.79 8.32 9.02 12.94 14.70 18.54 22.80 330 TABLE XVI. TRANSITION CURVES. Table gives d = ^ vers ^. Then AO = f- - d. Degree Lengths of Transition Curves. of Offset Curve. 100 150 200 240 250 300 320 360 400 10' 20 .001 .001 30 .001 .001 .001 .002 40 .001 .001 .001 .002 .002 .003 50 .001 .001 .001 .002 .003 .004 .005 1 00 .001 .002 .002 .003 .004 .006 .008 10 .001 .001 .002 .003 .005 .005 .008 .01 20 .001 .002 .003 .003 .006 .007 .01 .01 30 .001 .002 .004 .004 .007 .009 .01 .02 40 .001 .003 .005 .005 .009 .01 .02 .02 50 .001 .003 .006 006 .01 .01 .02 .03 2 00 .002 .004 .007 .007 .01 .02 .02 .03 10 .001 .002 .005 .008 .009 .02 .02 .03 .04 20 .001 .002 .005 .009 .01 .02 .02 .03 .04 30 .001 .002 .006 .01 .01 .02 .02 .03 .05 40 .001 .003 .007 .01 .01 .02 .03 .04 .05 50 .001 .003 .008 .01 .01 .03 .03 .04 .06 3 00 .001 1 .004 .009 .01 .02 .03 .04 .05 .07 10 .001 .004 .01 .02 .02 .03 .04 .06 .08 20 .001 .004 .01 .02 .02 .04 .04 .06 .08 30 .001 .005 .01 .02 .02 .04 .05 .07 .09 40 .002 .005 .01 .02 .03 .04 .05 .07 .10 50 .002 .006 .01 .02 .03 .05 .06 .08 .11 4 00 .002 .006 .02 .03 .03 .05 .06 .09 .12 10 .002 .007 .02 .03 .03 .06 .07 .10 .13 20 .002 .008 .02 .03 .03 .06 .07 .10 .14 30 .002 ! .008 .02 .03 .04 .07 .08 .11 .15 40 .003 .009 .02 .04 .04 .07 .08 .12 .17 50 .003 .009 .02 .04 .04 .08 .09 .13 .18 5 00 .003 .01 .02 .04 .05 .08 .10 .14 .19 10 .003 .01 .03 .04 .05 .09 .10 .15 .20 20 .003 .01 .03 .05 .05 .09 .11 .16 .22 30 .004 .01 .03 .05 .06 .10 .12 .17 .23 40 .004 .01 .03 .05 .06 .10 .13 .18 .24 50 .004 .01 .03 .06 .06 .11 .13 .19 .26 6 00 .004 .01 .03 .06 .07 .12 .14 .20 .27 10 .005 .02 .01 .06 .07 .12 .15 .21 .29 20 .005 .02 .04 .07 .07 .13 .16 .22 .31 30 .005 .02 .04 .07 .08 .14 .16 .23 .32 40 .005 .02 .04 .07 .08 .14 .17 .25 .34 50 .006 .02 .04 .08 .09 .15 .18 .26 .36 7 00 .006 .02 .05 .08 .09 .16 .19 .27 .37 10 .006 .02 .05 .08 .09 .16 .20 .28 .39 20 .006 .02 .05 .09 .10 .17 .21 .30 .41 30 .007 .02 .05 .09 .10 .18 .22 .31 .43 40 .007 .02 .06 .10 .11 .19 .23 .33 .45 50 .007 .02 .06 .10 .11 .20 .24 .34 .47 8 00 .008 ! .03 .06 .11 .12 .21 .25 .86 .49 10 .008 i .03 .06 .11 .12 .21 .26 .37 .51 20 .008 .03 .07 .11 .13 .22 .27 .39 .53 30 .009 1 .03 .07 .12 .13 .23 .28 .40 .55 40 .009 .03 .07 .12 .14 .24 .29 .42 .57 50 .009 .03 .07 .13 .15 .25 .30 .43 .59 9 00 .01 .03 .08 .13 .15 .26 .32 .45 .62 10 .01 .03 .08 .14 .16 .27 .33 .47 .64 20 .01 .03 .08 .14 .16 .28 .34 .48 .66 30 .01 .04 .09. .15 .17 .29 .35 .50 .69 40 .01 .04 .09 .15 .17 .30 .36 .52 .71 50 .01 .04 .09 .16 .18 .31 .38 .54 .74 10 00 .01 .04 .10 .16 .19 .32 .39 .55 .76 TABLE xvi. TRANSITION CURVES. 331 Degree Lengths of Transition Curves. of Offset Curve. 100 150 200 240 250 300 320 3(50 400 10 10' .01 .04 .10 .17 .19 .33 .40 .57 .79 20 .01 .04 .10 .18 .20 .34 .42 .59 .81 30 .01 .04 .10 .18 .20 .35 '.43 .61 .84 40 .01 .05 .11 .19 .21 .37 .44 .63 .87 50 .01 .05 .11 .19 .22 .38 .46 .65 .89 11 00 .01 .05 .12 .20 .23 .39 .47 .67 .92 10 .01 .05 .12 .21 .23 .40 .49 .69 .95 20 .02 .05 .12 .21 .24 .41 .50 .71 .98 30 .02 .05 .13 .22 .25 .42 .52 .73 .01 40 .02 .05 .13 .22 .25 .44 .53 .76 .04 50 .02 .06 .13 j .23 .26 .45 .55 .78 .07 12 00 .02 .06 .14 .24 .27 .46 .56 .FO .10 10 .02 .06 .14 .24 .27 .48 .58 .82 .13 20 .02 .06 .14 25 .28 .49 .59' .84 .16 30 .02 .06 .15 ^26 .29 .50 .61 .87 .19 40 .02 .06 .15 .26 .30 .52 .63 .89 .22 50 .02 .07 .16 .27 .31 .53 .64 .91 .25 13 00 .02 .07 .16 .28 .32 .54 .66 .94 .29 10 .02 .07 .16 .-.'9 .32 .55 .68 .96 .32 30 .02 * .07 .17 .29 .33 .57 .69 .99 .35 30 .02 .07 .17 .30 .34 .59 .71 .01 .39 40 .02 .07 .18 .31 .35 .60 .73 .04 .42 50 .02 .08 .18 .31 .36 .61 .75 .06 .46 U 00 .02 .08 .19 .32 .36 .62 .76 .09 .49 10 .02 .08 .19 .33 .37 i .64 .78 .11 .53 20 .02 .08 .20 .34 .38 .66 .80 .14 .56 ::o .03 .08 .20 .35 .39 .68 .82 .17 .60 40 .03 .09 .20 .35 .40 .69 .84 .19 .64 50 .03 .09 .21 .36 .41 .71 .86 .22 .67 15 00 .03 .09 .21 .37 .42 .72 .88 .25 .71 10 .03 .09 .22 38 '.43 .74 .90 .28 .75 20 .03 .09 .22 .39 .44 .75 .92 .30 .79 30 .03 .10 .23 .39 .45 77 .94 .33 .83 40 .03 .10 .23 .40 .46 '.79 .96 .36 .87 50 .03 .10 .24 .41 .47 .80 .98 .39 .91 16 00 .03 .10 .24 .42 .48 .82 1.00 .42 1.95 10 .03 .10 .25 .43 .49 .84 1.02 .45 1.99 20 .03 .11 .2') .44 .50 .86 .04 .48 2.03 80 .03 .11 .26 .45 .51 .87 .06 .51 2.07 40 .03 .11 .26 .46 .52 .89 .08 .54 2.11 50 .03 .11 .27 .47 .53 .91 .10 .57 2.15 17 00 .03 .12 .27 .48 .54 .93 .13 .60 2.20 10 .04 .12 .28 .48 .55 .95 .15 .63 2.24 20 .04 .12 .29 .49 .56 .96 .17 .66 2.28 30 .04 .12 .29 .50 .57 .98 .19 .70 2.33 40 .04 J3 .30 .51 .58 1.00 .22 .73 2.37 50 .04 .13 .30 .52 .59 1.02 .24 .76 -2.42 18 00 .04 .13 .31 .53 .60 1.04 .26 .80 2.46 10 .04 .13 .*! .54 .61 1.06 .29 .83 2.51 20 .04 .13 .32 .55 .62 .08 !ai .86 2.55 30 .04 .14 .33 .56 .64 .10 .33 .90 2.60 40 .04 .14 .33 .57 .65 .12 .36 .93 2.65 50 .04 .14 .34 .58 .66 .14 .38 .96 2.70 19 00 .04 .14 .34 .59 .67 .16 .41 2.00 2.74 10 .04 .15 .35 .60 .68 .18 .43 2.04 2.79 20 .04 .15 .36 .61 .69 .20 .46 2.07 2.84 30 .05 .15 .36 .63 .71 .22 .48 2.11 2.89 40 .05 .16 .37 .64 72 1.24 .51 2.14 2.94 50 .05 .16 .37 .65 78 .26 .53 2.18 2.99 20 00 .05 .16 .38 .66 .74 1.28 1.56 2.22 3.04 B32 TABLE XVII.-DEFLECTION ANGLES FOR TRANSITION CURVES, TO TENTHS OU % A MINUTE, FOR CHORDS OF 1 TO 5 CHAINS, IN TERMS OF ANGLE FOR CHORD OF 1 CHAIN. 1 2 g 4 i 1 a I 4 5 O'.l 0'.4 0'.9 i'.a 2'. 5 6'.1 24'. 4 54'. 9 1 38' 2 32' .2 .8 1 .8 3 .2 5 .0 G .2 24 .8 55 .8 39 35 .8 1 .2 2 .7 4 .8 7 .5 G .3 25 .2 56 .7 41 37 .4 1 .0 3 .G G .4 10 .0 6 .4 25 .6 57 .6 42 40 .5 2 .0 4 .5 8 .0 12 .5 6 .5 26 .0 58 .5 44 42 .0 2 .4 5 .4 9 .(i 15 6 .'i 26 .4 59 .4 46 45 .7 2 .8 G .3 11 .2 17 .5 G .7 26 .8 1 47 47 .8 3 .2 7 .2 1.' .8 20 .0 6 .8 27 .2 1 49 50 .9 3 .6 8 .1 14 .4 22 .5 6 .9 v'7 .6 B 50 52 1 .0 4 .0 9 .0 10 .0 25 .0 .0 Jd .0 3 52 55 1 .1 4 .4 9 .9 17 .G 27 .5 .1 28 .4 4 54 57 i .2 4 .8 10 .8 19 .2 30 .0 28 .8 5 55 3 1 .3 5 .2 11 .7 20 .8 3-J .5 !a 29 .2 G 57 2 1 .4 5 .6 12 .G 2-2 .4 35 .0 .4 29 .6 7 58 5 1 .5 6 .0 13 .5 24 .0 37 .5 .5 30 .0 8 -' 7 1 .6 G .4 14 .4 25 .G 40 .0 .6 30 .4 8 2 10 1 .7 .8 15 .3 27 .2 42 .5 r- 30 .8 5) 3 12 1 .8 7 2 1G .2 28 .8 45 .0 7 !B 31 .2 10 5 15 1 .9 7 !e 17 .1 30 .4 47 .5 7 .9 31 .6 Ik G 17 2 .0 8 .0 18 .0 32 .0 50 .0 8 .0 32 .0 12 8 20 2 .1 8 .4 18 .9 33 .G 52 .5 8 .1 3-2 .4 13 10 22 8 .8 19 .8 35 .2 55 .0 8 .2 32 .8 14 11 25 2 !a 9 .2 i.0 .7 36 .8 57 .5 8 .3 38 .2 15 13 27 2 .4 9 .G 21 .6 36 .4 1 0' 8 .4 33 .6 1G 14 30 2 .5 10 .0 S2 .5 40 .0 2 8 .5 34 .0 17 16 32 2 .6 10 .4 *3 .4 41 .6 5 8 .6 34 .4 17 18 35 2 7 10 .8 21 .3 4.J .4 7 8 .7 34 .8 18 19 37 2 is 11 .2 25 .2 44 .8 10 8 .8 35 .2 19 21 40 2 .9 11 .6 26 .1 46 .4 12 8 .9 35 .6 20 22 42 3 .0 12 .0 27 .0 48 .0 15 9 .0 26 .0 21 24 45 3 .1 12 .4 27 .9 49 .6 17 9 .1 3(5 .4 22 26 47 3 .2 12 .8 28 .8 51 .2 20 9 .2 815 .8 23 27 50 3 .3 13 .2 29 .7 52 .8 22 9 .8 3; .2 24 29 52 3 .4 13 .G 80 .6 54 .4 25 9 .4 37 .6 25 30 55 3 .5 14 .0 3! .5 56 .0 er 9 .5 38 .0 26 32 57 3 .6 14 .4 32 .4 57 .6 so 9 .6 38 .4 26 34 4 3 .7 14 .8 33 .3 59 .2 32 9 .7 38 .8 27 35 2 3 .8 15 .2 34 2 1 1' 35 9 .8 39 .2 28 37 5 3 .9 15 .G 35 .1 2 37 9 .9 39 .6 n 38 7 4 .0 1G .0 36 .0 4 40 10 .0 40 .0 30 40 10 4 .1 16 .4 36 .9 6 42 10 .1 40 .4 31 42 12 4 . 16 .8 37 .8 45 10 .2 40 .8 32 43 15 4 .3 17 .2 38 .7 9 47 10 .3 41 .2 88 45 17 4 .4 17 .G 3D .6 10 50 10 4 41 .6 34 46 20 4 .5 18 .0 40 .5 12 52 10 .5 42 .0 35 48 22 4 .0 18 .4 41 .4 14 55 10 .6 42 .4 35 50 25 4 -.7 18 .8 4-2 .3 15 57 10 .7 42 .8 * 51 27 4 .8 19 .2 43 .2 17 2 10 .8 i:] .2 37 53 30 4 .9 19 .G 44 .1 18 2 10 .9 43 .6 38 54 32 5 .0 20 .0 45 .0 20 5 11 .0 44 .0 39 50 35 5 .1 20 .4 45 .9 22 7 11 .1 44 .4 40 58 37 5 .2 20 .8 46 .8 23 10 11 .2 44 .8 41 59 40 5 .3 21 .2 47 .7 25 12 11 .3 45 .-J 42 3 1 42 5 .4 21 .G 48 .G 26 15 11 .4 45 .6 43 2 45 5 .5 22 .0 49 .5 28 17 11 .5 46 .0 44 4 47 5 .0 22 .4 50 .4 30 20 11 .6 40 .4 44 6 50 5 .7 22 .8 51 .3 31 22 11 .7 46 .8 45 7 52 5 .8 23 .2 52 .2 33 25 11 .8 47 .2 40 y 55 5 .9 23 .6 53 .1 34 27 11 .9 47 .0 47 10 5T 6 .0 24 .0 54 .0 36 30 12 .0 48 .0 48 12 _5 j TABLE XVIII. VOLUME FOR LENGTH 100 FEET. BASK = 8. SIDE ST.OPES 2 TO 1. D .1 .2 .:; 4 i . 5 <; .7 | .S .9 { I K u 17 20 24 2e 33 1 3" 4; 4( 51 56 61 66 72 r? 83 2 8 9E 101 107 114 120 127 154 141 148 3 156 m 171 178 186 194 203 211 220 228 4 23? 246 255 264 274 283 293 303 313 323 5 333 344 354 365 376 387 398 410 421 433 6 414 456 468 481 493 506 518 531 544 557 570 584 597 611 625 639 653 667 682 696 8 711 726 741 756 WWfl 787 803 818 834 850 9 8(i7 883 900 916 933 950 967 984 1002 1019 to 1037 1055 1073 1091 1109 1128 1146 1165 1184 1203 11 1222 1242 1261 1281 1300 1320 13401 1361 1381 1402 12 1422 1443 1 1(54 1485 1506 1528 1549 1571 1593 1615 18 16:57 16591 1(582 1704 1 727 1750 1773 1796 1820 1843 14 1867 1890) 1914 1938 1963 1987 2012 2036 2061 2086 15 2111 2136 2162 2187 2213 2239 2265 2291 2317 2344 16 2370 2397 2424 2151 2478 2506 2533 2561 2588 2616 17 2614! 2673 2701 2730 2758 2787 2816 2845 2874 2904 18 2933 : 2963 ','993 302: 3053 3083 3114 3144 3175 3206 19 3237 3268 3300 3331 3363 3394 3426 3458 3491 3523 20 3556 3588 3621 3654 3687 3720 3754 3787 3821 3855 21 3889 3923 3957 3992 4026 4061 4096 4131 4166 4202 2-2 4237 4273 4308 4344 4380 4417 4453 4490 4526 4563 23 4 GOO 4637 4(i74 4712 4749 4787 4^2;V 4863 4901 4939 24 4978 ! 501(1 5055 5094 5133 5172 5212 5251 5291 5380 OK 53?0: 5410 5451 5491 5532 5572 5613 5654 5695 5736 26 t 5778 5819 5861 5<)03 5915 5987 6029 6072 6114 6157 Off 6200 6243 6286 6330 (5373 6417 6460 6504 6548 6593 28 6637 6682 6726 6771 6816 6861 6906 6952 6997 7043 29 7089 7135 7181 7227 7274 7320 7367 7414 7461 7508 BASE = 8. SIDE SLOPES 3 TO 1. 3 6 10 14 18 22 26 31 36 1 41 46 52 57 63 69 76 82 89 96 2 104 111 119 127 135 144 152 161 170 179 3 1S9 199 209 219 229 240 251 262 273 285 4 296 308 320 333 345 358 371 385 398 412 5 426 440 455 469 484 499 514 530 546 562 6 578 594 611 628 645 6()2 680 697 715 733 752 770 789 808 828 847 867 887 907 928 8 948 969 990 1011 1033 1055 1077 1099 1121 1144 1167 1190 1213 1237 1260 1284 1308 1333 1357 1382 10 1407 1433 1458 1484 1510 1536 1563 1589 1616 1643 1! 1(570 1698 1726 1754 1782 1810 1839 1868 1897 1926 12 1956 1985 2015 2045 2070 2106 2137 2168 2200 2231 13 2263 2295 2327 2360 2392 2425 2458 2491 2525 2559 14 2593 2627 2661 2U96 2731 2766 2801 2837 2872 2908 15 2914 2981 3017 3054 3091 3129 3166 3204 3242 3280 16 3319 3357 3396 3435 3474 3514 3554 3594 3634 3674 17 3715 3756 3797 3838 38SO 3921 3963 4005 4048 4090 18 4133 4176 4220 4263 4307 4351 4395 4440 4484 4529 19 4574 4619 4665 4711 4757 4803 4849 4896 4943 4990 20 5087 5085 5132 5180 5228 5277 5325 5374 5423 5473 21 5522 5572 5622 5072 5723 5773 5824 5875 5926 5978 22 6030 6082 6134 6186 6239 6292 6345 6398 6452 6505 23 6559 6613 66C8 6722 6777 6832 6888 (5943 6999 70" 5 24 7111 7168 7224 7281 7338 7395 7453 7511 7569 7627 25 7085 7744 7803 7862 ? ( .'21 7981 8040 HI 00 8160 8221 26 8281 8342 SI 03 8485 8f>2(; 8588 8650 87' 12 8775 8837 27 8900 8963 9026 U090 9154 9218 9282 9346 9411 9476 28 9r>41 9C06 96W MK HSO? 9869 9986 10002 10069 10136 29 10204 10271 10339 10407 10475 105441 106121 10681 10750 10819 TABLE xvni. VOLUME FOR LENGTH 100 FEET. BASE =14. SIDE SLOPES f TO 1. D .0 .1 .2 .3 .4 .5 .6 .7 .s .9 5 11 16 22 27 33 39 45 51 1 57 64 70 Mjff 83 90 97 104 111 119 2 126 133 141 149 156 1G4 172 181 IS!) 197 3 206 214 223 232 241 250 259 2(58 277 287 4 296 306 316 826 336 346 356 306 377 887 5 398 409 420 431 442 453 4(55 476 488 499 6 511 523 535 547 559 572 584 597 609 G22 y 635 648 661 675 688 701 715 729 742 75G 8 770 785 799 813 828 842 857 887 902 9 917 932 947 903 978 994 1010 1096 1042 1058 10 1074 1090 1107 1123 1140 1157 1174 1191 1208 1225 11 1243 12GO 1278 1295 1313 1331 1349 1367 1385 1404 19 1422 1441 1459 1478 1497 1536 1535 1555 1574 1593 13 1613 1633 1652 1672 1692 1713 1733 1753 1771 1794 14 1815 1836 1857 1878 1 Si>9 1920 1941 1963 1984 2006 15 2028 2050 2072 2094 2116 2138 2161 2183 2206 8229 16 2252 2275 2298 2321 2345 2368 2392 2415 2439 2468 17 2487 2511 2535 2560 2584 2C09 2633 2(558 2683 2708 18 2733 2759 2784 2809 2835 2861 2886 2912 2938 29G5 19 2991 3017 3044 3070 3097 3124 3151 3178 3205 3232 20 3259 3287 3314 3342 3370 3398 3426 3454 3482 3510 21 3589 3567 3596 3625 3654 3683 3712 3741 3771 8800 22 8830 3859 38*9 3919 3949 3979 4009 4040 4070 4101 23 4131 4162 4193 4224 4255 4287 4318 4349 4381 4413 24 4444 4476 4508 4541 4573 4(505 4638 4670 4703 4736 25 4769 4802 4835 4868 4901 4935 4968 5002 5036 5070 26 Oft 5104 5450 5138 5485 5172 5521 5206 5556 5241 5592 5275 5627 5310 5663 5345 5699 5380 5735 5415 5771 28 5807 5844 5880 5917 5953 5990 6027 6064 6101 6139 29 6176 6213 6251 6289 6326 63G4 6402 6441 6479 6517 BASE = 10. SIDE SLOPES 1 TO 1. 4 ! 8 ' 11 15 19 24 28 32 I 3G 1 41 45 50 54 59 64 69 74 79 H4 2 89 91 99 105 110 116 121 127 133 139 3 144 150 156 163 169 175 181 188 194 ! 201 4 207 214 221 228 235 242 249 256 263 , 270 5 278 285 293 300 308 316 324 331 339 347 G 356 364 372 380 389 397 406 414 423 432 7 441 450 ! 459 468 477 486 495 505 514 524 8 533 543 553 563 572 582 592 603 613 623 9 633 644 654 665 675 686 697 708 719 730 10 741 752 763 774 786 797 809 820 832 844 11 856 867 879 891 904 916 928 940 953 965 12 978 990 1003 1016 1029 1042 1055 1068 1081 1094 13 1107 1121 1134 1148 1161 1175 1189 1203 1216 1230 14 1244 1259 1273 1287 1301 1316 1330 1345 1359 1374 15 1389 1404 1419 1484 1449 1464 1479 1494 1510 1525 10 1541 1556 1572 1588 1604 1619 1635 1651 1668 1684 17 1700 1716 1733 1749 1766 1782 1799 181G 1833 1S50 18 1867 1884 1901 1918 1935 1953 1970 1988 2005 2023 19 2041 2059 2076 2094 2112 2131 2149 2107 2185 2204 20 222 2241 2259 2278 2297 2316 2335 2354 2373 2302 24U 2430 2450 24ti9 2489 2508 2528 2548 2568 I 2587 22 2607 2027 2648 2668 2688 2708 2729 2749 2770 ! 2790 23 2811 2832 2853 2874 2895 2916 2937 2958 2979 3001 24 3022 3044 3065 3087 3109 3131 3152 3174 3196 3219 25 3241 3263 3285 3308 3330 3353 3375 3398 3421 8444 26 3467 3490 3513 3536 3559 3582 3606 3629 3G53 3676 27 3700 3724 37 IS 3771 3795 3819 3844 3868 3892 3916 28 3941 3965 3990 4014 4039 4064 4089 4114 4139 4164 29 4189 4214 4239 4265 4290 4316 4341 4367 4393 4419 TAfcLE XVlll. VOLUME FOR LENGTH 100 FEET. BASE = 16. SIDE SLOPES f TO 1. 335 D .0 .1 .2 .3 .4 .6 .6 . 7 .8 .9 6 12 18 25 31 38 44 51 58 1 65 r "*2 79 86 94 101 109 117 125 133 2 141 149 157 166 174 183 192 201 209 219 3 228 237 247 256 266 275 285 295 305 316 4 326 336 347 358 368 379 390 401 412 424 5 435 447 458 470 482 494 506 518 531 543 556 568 581 594 607 620 633 646 660 673 ij* 687 701 715 729 743 757 771 786 800 815 8 830 845 859 875 890 905 921 936 952 967 9 983 999 1015 1032 1048 1064 1081 1098 1114 1131 10 1148 1165 1182 1200 1217 1235 1252 1270 1288 1306 11 1324 1342 1361 1379 1398 1416 1435 1454 1473 1492 12 1511 1530 1550 1569 1589 1609 1629 1649 1669 1689 13 1709 1730 1750 1771 1792 1813 1833 1855 1876 1897 14 1919 1940 1962 1983 2005 2027 2049 2072 2094 2116 15 2139 2162 2184 2207 2230 2253 2276 2300 23-23 2347 16 2370 2394 2418 2442 2466 2490 2515 2539 2564 2588 17 2613 2638 2663 2688 2713 2738 2764 2789 2815 2841 18 2867 2893 2919 2945 2971 2998 3024 3051 3078 3105 19 3131 3159 3186 3213 3241 3268 3296 3323 3351 3379 20 3407 3436 3464 3492 3521 3550 3578 3607 3636 3665 21 3694 3724 3753 3783 3812 3842 3872 3902 JJ932 3962 22 3993 4023 4054 4084 4115 4146 4177 4208 4239 4270 23 4302 4333 4365 4397 4429 4461 4493 4525 4557 4590 24 1622 4655 4688 4721 4753 4787 4820 4853 4887 4920 25 49:>4 4987 5021 5055 5089 5124 5158 5192 5227 5262 26 5296 5331 5366 5401 5436 5472 5507 5543 5578 5614 27 5650 5686 5722 5758 5795 5831 5868 5904 5941 5978 28 6015 605-2 6089 6126 6164 6201 6239 6277 6315 6353 29 6391 6429 6467 6506 6544 6583 E622 6661 6699 6739 BASE = 18. SIDE SLOPES 1 TO 1. r- 13 20 27 34 41 48 56 63 1 70 78 85 93 101 108 116 124 132 140 2 148 156 165 173 181 190 198 207 216 224 3 233 242 251 260 269 279 288 297 307 316 4 326 336 345 355 365 375 385 395 405 416 5 426 436 447 457 468 479 489 500 511 522 6 533 544 556 567 578 590 601 613 625 636 648 660 672 684 696 708 721 733 745 758 8 770 783 796 808 821 834 847 860 873 8S7 9 900 913 927 940 954 968 981 995 1009 1023 10 1037 1051 1065 1080 1094 1108 1123 1137 1152 1167 11 1181 1196 1211 1226 1241 1256 1272 1287 1302 1318 12 1333 1349 1365 1380 1396 1412 1428 1444 1460 1476 13 1493 1509 1525 1542 1558 1575 1592 1608 1625 1642 14 1659 1676 1693 1711 1728 1745 1763 1780 1798 1816 15 1833 1851 1869 1887 1905 1923 1941 1960 1978 1996 16 2015 2033 2052 2071 2089 2108 2127 2146 2165 2184 17 2204 2-223 2242 2262 2-281 2301 2321 2340 2360 2380 8 2400 24-20 2440 2460 2481 2501 2521 2542 2562 2583 19 2604 2624 2645 2666 2687 2708 2729 2751 2772 2793 20 2815 2836 2858 2880 2901 2923 2945 2967 2989 3011 21 3033 3056 3078 3100 3123 3145 3168 3191 3213 3236 22 3259 3282 3305 3328 3352 3375 3398 3422 3445 3469 23 3493 3516 3540 3564 3588 3612 3636 3660 3685 3709 24 3733 3758 3782 3807 383-2 3856 3881 3906 3931 3956 25 3981 4007 4032 4057 4083 4108 4134 4160 4185 42'1 26 4237 42(53 4-2S9 4315 4341 43f)8 4394 4420 4447 4473 27 4500 -15-27 4553 4580 4607 4634 4661 4688 471(5 4743 28 4770 4798 48-25 -1853 4F8I 4!K)8 4S36 4964 4992 5020 29 5048 5076 5105 5133 5161 5190 5218 5247 5276 5304 3B6 TABLE xvin VOLUME FOR LENGTH MO FEET, BASE = 20. SIDE SLOPES 1 TO 1. I) t a ;j 4. t >>')> 2272 2292 2312 2332 2353 2373 23'.)4 2414 2435 17 2456 2476 2497 251H 2539 2560 2581 2603 2624 2015 18 2667 688 2710 2731 2753 2775 2707 2819 2841 2863 19 2885 2907 2930 2H52 2975 2997 3020 3043 3065 8088 20 3111 3134 3157 3180 3204 3227 3250 3274 :;-ji)7 3321 21 3344 3368 3392 3416 3440 3404 3488 3512 3530 3501 22 3585 3610 3634 3659 3684 8708 3733 3758 3783 3808 23 3833 3859 3884 3909 3935 3960 3986 4011 4037 4063 24 4089 4115 4141 4107 1193 4219 4240 4-.'72 4299 4325 25 4352 4319 4405 4132 4459 4488 4513 4540 4568 4595 *26 4622 4650 1677 4705 4732 4760 4788 4^16 1844 4872 27 4900 4928 4956 4985 5013 5012 5070 5099 5128 5156 28 5185 5214 5243 5272 r.30i 5331 5360 5389 5419 5448 29 5478 5507 5537 5567 5597 5627 5657 5687 5717 5747 TABLE XIX. VOLUME FOR LENGTH 100 FEET. BASE = 0. SIDE SLOPES 1 TO 1. 337 D .0 .1 2 .3 .4 .5 .6 . 7 .8 .9 1 1 1 2 3 1 4 4 5 6 8 9 11 12 13 8 15 16 18 20 21 23 25 27 29 31 3 33 36 38 40 43 45 48 51 53 56 4 59 62 65 68 72 75 78 82 85 89 98 96 100 104 108 112 116 120 1 25 129 6 133 138 142 147 152 156 161 166 171 176 181 187 198 197 203 208 214 220 2-25 231 8 237 243 249 255 2(51 268 274 280 287 SP8 g 800 307 313 sao 327 334 341 348 | 350 363 10 370 378 385 393 401 408 416 424 1 432 440 11 448 456 466 473 481 490 498 507 i 516 524 12 533 542 551 560 509 579 588 597 ; C07 616 13 6-26 636 645 655 665 675 685 695 705 716 14 7-26 736 747 757 768 779 789 800' 811 822 15 833 S44 856 867 878 1 890 901 913 925 936 16 948 960 972 984 996 1008 1021 1033 1045 1058 17 1070 1083 1096 1108 1121 1134 1147 1160 1173 1187 18 1200 1213 1227 1:240 1254 1268 1-281 15-95 1309 1323 19 1337 1351 1305 1380 1394 1408 1423 1437 1452 1467 20 1481 1496 1511 1556 1541 1556 1572 1587 1602 1618 21 1633 1649 1665 16SO 1696 1712 1728 1744 1760 1776 22 1793 1809 1825 1842 1858 1875 1892 1908 1925 1942 23 1959 1976 1993 2011 20-28 2045 2063 2080 2098 2116 24 2133 2151 2169 2187 2205 2223 2241 2260 2278 'J296 25 2314 2333 2352 2371 2389 2408 2427 244(J 2465 2484 26 2504 2523 2542 25(52 2581 2601 2021 2640 2060 2680 27 2700 2720 2740 2700 2781 2801 2821 2842 2S63 2883 28 2904 2924 2945 2966 2987 3008 3029 3051 3072 3093 29 3115 3136 3158 3180 3201 3223 324 b 3267 3289 3311 BASE = 0. SIDE SLOPES f TO 1. 1 1 1 2 3 4 5 1 6 y 8 9 11 13 14 16 18 20 a 22 25 27 29 32 35 38 41 44 47 3 50 53 57 61 64 68 72 76 80 85 4 89 93 98 103 108 113 118 123 128 133 5 139 145 150 156 162 168 174 181 187 193 6 200 207 214 221 228 235 242 249 257 265 7 272 280 288 293 304 313 321 3:29 338 347 8 836 365 374 383 392 401 411 421 430 440 9 450 460 470 481 491 501 512 523 534 545 10 556 567 578 589 601 613 624 636 648 6CO 11 072 685 697 709 722 735 748 761 774 787 12 800 813 827 841 854 868 882 896 910 925 13 939 953 968 983 998 1013 1028 1043 1058 1073 14 1089 1105 1120 1136 1152 1168 1184 1201 1217 1233 15 1250 1267 1284 1301 1318 1335 1352 1309 1387 1405 16 1422 1440 1458 1476 1494 1513 1531 1549 1508 1587 17 1606 1625 1644 1663 1682 1701 1721 1741 1760 1780 18 1800 1820 1840 1861 1881 1901 1922 1943 1964 1985 19 2006 2027 2048 2069 2091 2113 2134 2156 2178 2200 20 2222 2245 2267 2289 2312 2335 2358 2381 2404 2427 21 2450 2473 2497 2521 2544 2568 2592 2616 2640 2665 22 2689 2713 2738 2763 2788 2813 2838 2863 2888 2913 23 2939 2965 2990 3016 3042 3068 3094 3121 3147 3173 24 3200 3227 3254 3281 3308 3335 33fc2 3389 3417 3445 25 3472 3500 3528 3556 3584 3613 3641 3069 3098 3727 26 3756 3785 3814 3843 3872 3901 3931 3961 3990 4020 27 4050 4080 4110 4141 4171 4201 4232 4263 4x'94 4325 28 4356 4387 4418 4449 4481 4513 4544 4576 4608 4640 29 4672 4705 4737 4769 4802 4835 4868 4901 4934 4967 TABLE XX. -SINES AND COSINES. 1 2 30 40 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine | Cosin \ .00000 One. .01745 .99985 ".03490 99939 .05234 91(863 .06976.99756:60 1 ; 1-0029 One. .01774 .99984 .03519 .99938 .05263 99801 .070051.99754159 2 .00058 One. .01803 .99984 .03548 .99937 .0-3292 99860 .07034 .99752 58 3 .00087 One. .01832 .99983 .03577 .99936 .05321 .99858; .070631.99750 57 4 .00116 One. .01862 .99983 .03606 .99935 .05350 .99857 .07092 .99748 56 5 .00145 One. .01891 .99982 .03635 .99934 .05379 .99855; .07121 .99746 55 6 .00175 One. .01920 .99982 .03664 .99933 .05408 .99854 .07150 .99744 54 7 .00204 One. .01949 .99981 .03093 .99932 .05437 .99852 .07179 .99742 53 8 .00243 One. .01978 .99980 .03723 .99931 .05466 .99851 .07208 .99740 52 9 .00262 One. .02007 .99980 .03752 .99930 .05495 .99849 .07237 .99738 51 iO .00291 One. .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 50 11 .00320 .99999 .02065 .99979 .03810 .99927 .05553 .99846 .07295 .99734 49 12 .00349 .99999! .02094 .99978 .03839 .99926 .05582 .99844 .07324 .99731 48 13 .00378 .99999; .02123 .99977 1 .03868 .99925 .05611 .99842 .07353 .99729 47 14 .00407 .99999' .02152|. 99977 1.03897 .99924 .05640 .99841 .07382 .99727 46 15 .00436 .99999 ..021811.99976 .03926 .99923 .05669 .99839; .07411 .99725 45 10 .00465 '.99999 .02211 .99976 .03955 .99922 .05698 .99838: .07440 .99723 44 17 .00495 .99999 .02240 .99975 .03984 .99921 .05727 .99836; .07469 .99721 43 18 .00524 .99999 .02269 .99974 .04013 .99919 .05756 .99834 .07498 .99719 42 19 .00553 .99998 .02298 .99974 .04042 .99918 .05785 .99833| .07527 .99716 41 20 .00582 .99998 .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356 .99972 .04100 .99916 .05844 .99829 .07585 .99712 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .05873 .99827! .07614 .99710 as 23 .00669 .99998 .02414 .99971 .04159 .99913 .05902 .99826 .(7643 .99708 37 24 .006981.1)9998 .02443 .99970 .04188 .99912 .05931 .99824 .07672 .99705 36 25 .00727 .99997 .02472 .99969 .04217 .99911 .05960 .99822 .07701 .99703 35 26 .00756 .99997 .02501 .99969 .04246 .99910 .05989 .99821 .07730 .99701 | 34 27 .00785 .99997 .02530 .99968 .04275 .99909 .06018 .99819 .07759 .99699 33 28 .00814 .99997 .02560 .99967 .04304 .99907 .06047 .99817 .07788 .99696 32 29 .00844 .99996 .025891.99966 .04333 .99906 .06076 .99815 .07817 .99694 31 30 .00873 .99996 .02618 .99966 .04362 .99905 .06105 .99813 .07846 .99692 30 31 .00902 .99996 .02647 .99965 .04391 .99904 .06134 .9981 2 ' .07875 .99689 29 32 .00931 .99996 .02676 .99964 .04420 .99902 .06163 .99810 .07904 .99687 28 33 .00960 .99995 .027051.99963 .04449 .99901 .06192 .99808 .07933 .99685 27 34 .00989 .99995 .02734 .99963 .04478 .99900 .06221 .99806 .07962 .99683 26 35 .01018 .99995 .02763 .99962 .04507 .99898 .06250 .99804 .07991 .99680 25 36 .01047 .99995 .02792 .99961 ! .04536 .99897 .06279 .99803 .08020 .99678 2-1 37 .01076 .99994 .02821 .99960 .04565 .99896 .06308 .99801 .08049 . 99676 j 23 38 .01105 .99994 .02850 .99959 .04594 .99894 .06337 .99799 .08078 .99673! 22 39 .01134 .99994 .02879 .99959 .04623 .99893 .06366 .99797 .08107 .99671 i 21 40 .011G4 .99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 20 41 .01193 .99993 .02938 .99957 .04682 .99890 .06424 .99793 .08165 .99666 19 42 .01222 .99993 .02967 .99956 .04711 .99889 .06453 .99792 .08194 .99664 18 43 .01251 .99992 .02996 .99955 .04740 .99888 .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04769 .99886 .06511 .99788 .08252 .99659 16 45 .01309 .99991 .0305* .99953 .04798 .99885 .06540 .99786 .08281 .99857 15 46 .01338 .99991 .03083 .99952 .04827 .99883 .06569 .99781 .08310 .99654 14 47 .01367 .99991 .03112 .99952 .04856 .99882 .06598 .99782 .08339 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .99990, .03170 .99950 . 04914 .99879 .06656 .99778 .08397 .99647 11 50 .01454 .99989 .03199 .99949 .04943 .99878 .06685 .99776 : .08426 .99644 10 51 .01483 .99989 .03228 .99948 1.04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989' .03-.T.7 .99947 ! .05001 .99875 .067431.99772 .08484 .99639 8 53 -01542 .99988 .03286 .99946 I. 05030 .99873 .06773 .99770 .08513 .99637 7 54 .01571 .99988 .03316 .99945 .05059 .99872 .06802 .99768 .08542 .99635 6 55 .01600 .99987 .0-3345 .99944 1 .05088 .99870 .06831 .99766 .08571 .99632 5 56 .01629 .99987 .03374 .99943 .05117 .99869 .06860 .99764 .08600 .99630 4 57 .01658 .99986 .03403 .99942 1.05146 .99867 .06889 .99762 .08629 .99627 8 58 .01687 .99986; .03432 -.99941 .05175 .99866 .06918 .99760 .08658 .99625 2 i 59 .01716 .99985 .03461 .99940 .05205 .99864 .06947 .99758 .08687 .99622 1 60 .01745 . 99985 1 .03490 .99939 .05234 .99863 .06976!. 99756 .08716 .99619 / Cosin Sine I Cosin Sine i Cosin Bine Cosin Sine Cosin Sine / 89 88' H 87 86 85 TABLE XX. -SINES AND COSINES. I 5 6 7 |! 8* 9 Sine Cosin Sine Cosin Sine ! Cosin Sine Cosin Sine | Cosin g Tl .0871(5 .99619 .10453 .99452 .12187 .99255 .13917 .99027 .15648 .98769 (50 1 .08745 .99617 .10482 .99449 .12216 .99251 .13946 .99023 .15672 .98764 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .13975L99019 .15701 .98760 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004;. 99015 .15730 .98755 57 4 .08831 .99609 .10569 .99440 .18808 .99240 .14033 .99011 i .15758 .98751 r>6 5 .08860 .99607 .10597 .99437 .12331 .99237 .14061 .99006 .15787 .98740 55 6 .08889 .99604 .10626 .99434 .12360 .99233 .14090 .99002 .15816 .98741 54 7 .08918 99602 .10655 .99431 .12:389 .99230 .14119 .98998 .15845 .98737 53 8 .08947 .99599 .10684 .99428 .12418 .99226 . 14148 98994 15873 98732 52 9 .08976 .99596 .10713 .99424 .12447 .99222 .14177 .98990 .15902 .98728 51 10 .09005 .99594 .10742 . 99421 j .12475 .99219 I .14205 .98906 .15931 .98723 50 11 .09034 .99591 .10771 .99418 .12504 .99215 .14234 .98982 i. 15959 .98718 40 12 .09063 .99588 .10800 .99415 .135331.99211 .5)8978 .15988 .98714 48 13 .09092 .99586 .10829 .99412 .12562!. 99208 ! 14292 .98973 .16017 .9870S 47 14 .09121 .99583 .10858 .99409 . 12591 |. 99204 .14320 .98969 .16046 .98704 46 15 .09150 .99580 1.10887 .99406 .12620 |. 99200 .14346 .98965 .16074 .98700 45 16 .09179 .9957'8 .10916 .99402 .12649 .99197 .14378 .98961 .16103 .98695 44 17 .09208 .99575 .10945 .99399 .12678 .99193 .14407 .98957 .16132 .98690 43 18 ! .09237 .99572 .10973 .99396 .127061.99189 .14436 .98953 .16160 .98686 48 19 .09^66 .99570 .11002 .99:39.3 .12735 .99186 .14464 .98948 .16189 .98681 41 20 .09295 .99567 .11031 .99390, .12764 .99182 .14493 .98944: .16218 .98676 40 21 .09324 .99564 .11060 .99386 .12793 .99178 .14522 .98940 .16246 .98671 30 22 .09353 .99562 .110891.99383 .12822 .99175 .14551 .98936 .16275 .98667 38 23 .09382 .99559 .11118 .99380 .12851 .99171 .14580 .98931! .16304 .98662 37 24 .09411 .99556 .11147 .99377 .12880 .99167 . 14608 i. 98927 1 .16333 .98657' 36 25 .09440 .99553 .11176 .99374 .12908 .99163 .14637 >.98923i .16361 .98652 39 26 .09469 .99551 .11205 .99370 .12937 .99160 .14666 . 98919 ' .16390 .98648 84 27 .09498 .99548 .11234 .99367 .12966 .99156 .14695!. 98914 .16419 .98643 38 28 .09527 .99545 .11263 .99364 .12995 .99152 .14723 . 98910 j .16447 .98638 38 29 .09556 .99542 .11291 .99360 .13024 .99148; .147521.98906 .16476 .98633 31 30 .09585 .99540 .11320 .99357; .13053 . 99144 j .147811.98902 .16505 .98629 .'50 31 .09614 .99537 .11349 . 99354 ! .13081 .99141 .14810 .98897 i .16533 .98624 29 32 .09642 .99534 .11378 .99351 .13110 .99137 .14838 .98893 : .16562 .98619 28 33 .09671 .99531 .11407 .99:347 .13139 .99133 .14867 .98889 : .16591 .98614 27 34 .09700 .99528 .11436 .99344 .13168 .99129 .148961.98884 i .16620 .98609 26 35 .09729 .99526 .11465 .99341 .13197 .99125 .14925 .98880 .16648 .98604 25 36 .09758 .99523 .11494 .99337 .13226 .99122 .14954 .98876 .16677 .98600 24 37 .09787 .99520 .11523 .99334 .13254 .99118 .14982 .98871 !.16706[.98595 23 38 .09816 .99517 .11552 .99331 .13283 .99114 .15011 .98867 .16734 .98590 22 39 .09845 .99514 .11580 .99327 .13312 .99110 .15040 .98863 .16763 .98585 21 40 .09874 .99511 .11609 .99324 .13341 .99106, .15069 .98858 .16792 .98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102: .15097 .98854 .16820 .98575 19 42 1 .09932 .99506 .11667 .99317 .13399 .99098 .15126 .98849 .16849 .98570 18 43 ! .09961 .99503 .11696 .99314; .13427 .99094 .15155 .98845 .16878 .98565 17 44 ! .09990 .99500 .11725 .99310 .13456 .99091 .15184 .98841 .16906 .98561 16 45 .10019 .99497 .11754 99307 .13485 .99087! .15212;. 98836 ! .16935 .98556 15 46 .10048 .99494 .11783 .99303 .13514 .99083 .15241 i .98832 i .16964 .98551 14 47 .10077 .99491 .11812 .99300 .13543 .99079 .152701.98827 .16992 .98546 18 48 .10106 .99488 .11840 .99297 .18572 .99075 .152991.98823 .17021 .98541] 12 49 50 .10135 .10164 .99485 .99482 .11869 .11898 .99293 .99290i .13600 .13629 .99071 .99067 .15327 .15356 .98818 .98814 .17050 .98536 .17078 .98531 11 10 51 .10192 .99479 .11927 .99286* .13658 '.99063 .15385 .98809 .17107 .98526 9 52 .10221 .99476 1.11956 .992831 .13687 .99059 .15414 .98805 .17136 .98521 8 53 .10250 .99473 !ll985 .99279 .13716 .99055 . 15442 j. 98800 .17164 .98516 7 54 .10279 .99470 .12014 .99276 .13744 .99051 .154711.98796 .17193 .98511 6 55 .10308 .99467 .12043 .99272 .13773 .99047 .15500 .98791 .17222 .98506 5 56 .10337 .99464 .12071J.99269 .138021.99043 .155291.98787 .17250 .98501 4 57 10366 .99461 ! .12100 .99265 .13831 '.99039 .15557 .98782 .17279 .98496 3 58 .10395 .99458 i. 12129 1.99262 .138601.99035 M55861. 98778 1.17308 .98491 2 59 .10424 .99455 ! . 12158 1 . 99258 . 13889 i . 99031 : .15615 .98773 .17336 .98486 1 6(' .10453 .99452 ! .12187 .99255 .13917 .99027 .15643 .98769 .17365 .98481 t Sine Cosin 1 Sine Cosin \ Sine Cosin Sine Cosin Sine r 84* 83 82' 81 80 TABLE XX. SINES AND COSINES, 4 10 11 12 13 I 14 Sine Cosin Sine Cosin ! Sine Cosin Sine Cosin Sine Cosin ; "o 717365 .'98481 Tl908l .98163 1.20791 .97815 .22495 '! 97437 .24198 797030 i fiO 1 .17393 .98476 .1910* .98157 .20820 .97801) .22523 .974:30 .24220 .1)7023 59 2 .17422 .98471 .19138 .98152 .20848 .97803 .22552 .97424 .24241) .97015 , 58 3 .17451 .98466 .19167 .98146 .20877 .97797 .22580 .97417 .24277 .97008 ; 57 4 .17479 .98461 . 19195 .98140 .20905 .97791 .22608 .97411 i .24305 .97001 56 5 .17508 .98455 .19224 .98135 .209:331.97784 .22637 .97404 ! .24:333 .96994 55 6 .17537 .98450 .19252 .98129 .20962 .97778 .22665 4)735 IS .24362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .22693 .97391 .24390 .96980 53 8 .17591 .98440 .19:309 .98118 .21019 .97766 .22722 .973S4 .24418 .96973 52 9 .17623 .98435 .19338 .98112 .21047 .97760 .22750 .97378 .24446 .96966 51 10 .17651;. 98430 .19366 .98107 .21076 .97754 .22778 .97371 .24474 .96959 50 11 .17680 .98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 12 .17708 .98120 .19423 .98096 .21132 .97742 .22835 .97358 .24531 .96945 48 13 .17737 .98414 .19452 .98090 .21161 .97735 .2286:3 .97:351 .24559 .1)61)37 47 14 .17766 .98409 .19481 .98084 .21189 .97729 .22892 .97345 .24587 961)30 46 15 .17794 .98404 .19509 .98079 .21218 .97723 .22920 .97338 .24615 .96923 45 16 |. 17823 .98399 .19538 .98073 .21246 .97717 .22948 .97331 .24644 .96916 44 17 .17852 .98394 .19566 .98067 .21275 .97711 .22977 .97325 .24672 .96909 43 18 .17880 .983891 .19595 .98061J .21303 .97705 .23005 .97318 .24700 .%<)(>> 42 19 .17909 .98:383 .19623 93056 .21331 .97698 .23033 .97311 .347281.96894 41 20 .17937 .98378 .19652 98050 .21360 .97692 .23062 .97'304 .24750 .96887 40 21 .17966 .98373 .19680 98044 .21388 .97686 .23090 .97298 .24784 .96880 39 22 1.1791)5 .98368 23 .180231.98362 .19709 .19737 98039 93033 .21417 .21445 .97680 .97673 .23118 .23146 .97291 .97284 ; .24813 i .24841 .96873 .1)6866 38 37 24 .13052 .98:357 .19766 93027 .21474 .97667 .23175 .97278 .24869 .1)6858 36 25 .18081 .98352 .19794 98021 .21502 .97661 .23203 .97271 .24897 .96851 35 26 .18109 .98347 .19823 98016 .21530 .97655 .23231 .97'264 .24925 .96844 34 27 .18138 .98341 .19851 93010 ' .21559 .97648 .2,3260 .97257! .24954 .1)6837 33 28 .18166 .98a36 .19880 98004 ! .21587 .97642 .23288 .97251! .24982 .96829 32 29 .18195 .98331 .19908 97913 i .21616 .97636 .23310 .97244; .25010 .96822 31 30 .18224 .98325 .19937 97932 .21644 .97630 .23345 .97237 .25038 .96815 30 31 .18252 .98320 .19965 97987 .21672 97623 .23373 .97230! .25066 .96807 29 32 .18281 .98315 .19994 97981 .21701 97617 .2-3401 .97223 .25094 .96800 28 33 .18309 .98310 .20022 97975 ! .21729 97311 .23429 .97217 .25122 .96793 27 34 .18338 .98304 .20051 97969 i .21758 97604 .23458 .97210! .25151 .96786 26 35 .18367 .98299 .20079 97963 | .21786 97598 .23486 .97203 .25179 .96778 25 36 .18395 . 93294 | .20108 97958 .21814 97592 .23514 .97196 .25207 .96771 24 37 .18424 .98288 .20136 97952 .21843 97585 .23542 .97189 .25235 .96764 23 38 .184521.98283 .20165 97946 .21871 97570 .23571 .97182 .25263 .96756 22 39 .18481 . 98277 ii .20193 97940 .21839 97573 .23599 .97176 .25291 .96749 21 40 .18509 .98272! .20222 97934 .21928 97566, .23627 .97169 .25320 .96742 20 41 .18538 .98267 .20250 97928 .21956 97560 .23656 .97162 .25348 .96734 19 42 .18567;.98261 .20279 97922 .21985 97553 .23684 .97155 .25376 .96727 18 43 .18595|.98256 .20307 97916 .22013 97547 .23712 .97148 .25404!. 9671 9 17 44 .18624;.98250 .20336 97910! .22041 97541 .23740 .97141' .25432 .96712 16 45 .18652 .98245 .20364 97905 .22070 97534 .23769 .97134 . 25460 i. 96705 15 46 .18681 .98240 .20393 97899 .22098 97528 .23797 .97127 ; .254881.96697 14 47 .18710 .98234 .20421 97893 .22126 97521 .23825 .97120 .25516 .96690 13 48 .18738 .98229 .20450 97887 .22155 .97515 .23853 .97113 .25545 .96682 12 49 .18767 .98223 .20478 97881 .22183 .97508 .23882 .97106 .255731.96675 11 50 .18795 .98218 ! .20507 .97875 .22212 .97502 .23910 . 97100 j .25601 .96667 10 51 .18824 .98212 .20535 .978691 .22240 .97496 .23938 .97093 .25629 .96660 9 52 .18852 .98207 .20563 .97863 .222681.97489 .23966 .97086 .25657 .96653 8 53 .18881 .98201 .20592 .97857 .222971.97483 .23995 .97079 .25685 .96645 7 54 .18910 .98196 .20620 .97851 .223251.97476 .24023 .97072 .25713 .96638 6 55 .18938 .98190 .20649 .97845 .223531.97470 .24051 .97065 .25741 .96630 5 56 . 18967 j. 98185 1 1.20677 .97839 .22382i.97463 .24079 .97058 .25769 .96623 4 57 .18995 .98179; 1.20706 .97833 .224101.97457 .24108 .97051 .25798 .96615 3 58 .19024 .981741 .20734 .22438 .97450 .241361.97044 .25826 .96608 2 59 . 19052 ; . 98168 j .20763 .97821 .22467 .97444 .241641.97037 .25854 .96600 1 60 .19081;. 98163 .20791 .97815 .22495 .97437. .24192 .97030 .25882 .96593 Cosin Sine i Cosin Sine Cosin Sine Cosin | Sine | Cosin Sine 79 II 78 77 76 ! 75 TABLE XX.-S1NES AND COSINES. 34 15 16 17 ; 18 19 Sine Cosin i Sine Cosin Sine Cosin Sine Cosin I Sine Cosin Ol. 25883 .90593 .275(54 .96126 .29237 .95630 .30902 .95106 .32557 .94552 BC 1 .25910 .96585 .27592 .90118 .29205 .95022 .30929 .95CJ7 .32584 .94542 59 2 j. 25938 .96578 .27020 .90110 .29293 .95013 .30957 .95088 ' .32612 .94533; 58 3 .25906:. 96570 : .27648 .90102 .29321 .95605 .30985 .95079 .32639 .94523 57 4 .25994 .96562 .27676 .90094 .29348 .95596 .31012 .95070 .32667 .94514 58 5 .26022;. 96555 .27704 .96086 .29376 .95588 .31040 .95061 .32694 .94504 55 6 .20050 .96547 .27731 .96078 .29404 .95579 .310081.95052 .32722 .94495 51 7 .26079 .96540 .27759 90070 .29432 .95571 .31095 .95043 .32749 .94485 K 8 .26107 .96532: .27787 .96002 .29460 .95502 . 31123 1.95033 .32777 : 94470 52 9 '.26135 .96524 .27815 .9605-4' .29487 .95554' .31151 .95024 .32804 .94460 51 10 .26163 . 96517 j .27843 .90040^.29515 .95545 .31178 .95015 .32832 .94457 H 11 .26191 .96509 .27871 .96037 .29543 .95536! .31206 .95006 ' .32859 .91447 49 12 .26219 .96502 .27899 .96029 .29571 .95528 .31233 .94997 1 .32887 .5X438 4:^ 13 .26247 .96494 .27927 .96021 .29599 .9551 9 S .31201 .94988 i .32914 .94428 47 14 .26275 .96486 .27955 .96013; .29626 .95511! .31289 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .955021 .31316 .94970 .32969 .94409 16 . 26331 !. 96471 .28011 .95997 .29682 . 95493 ( .31344 .94961 .32997 .94399 44 17 .26359 .96463 .28039 .95989 .29710 .95485! .31372 .94952 .33024 .94390 43 18 .26387 .96456 .28067 .95981 .29737 .95476 .31399 .94943 i .33051 .94380 42 19 .26415 .96448 .28095 .95972 .29765 .95467! .31427 .94933 1 .33079 .94370 41 20 .26443 .96440 .28123 .95964 .29793 .95459; .31454 .94924 '.33106 .94361 K 21 .26471 . 96433 ' .28150 .95956 .29821 .95450 .31482 .94915 .33134 .94351 3! 22 .26500 .96425 .28178 .95948 .29849 .95441! .31510 .94900 i .33161 .94342 88 23 .26528 . 96417 > .28206 .95940 .29876 .95433 .31537 .94897 .33189 .94332 37 24 1 .26556 .96410 .28234 .95931 .29904 .95424 .31565 .94888 .33216 .94:WC 30 25 '.26584 .90402 .28262 .95923 .29932 .95415 .31593 .94878 .33244 .94313 35 26 [ .26612 .96394 .28290 .95915 .29960 .95407 .31620 .94869 .33271 .943031 34 27 ; .26640 .96386 .28318 .95907 .29987 .95398 .31648 .94860 .33298 .94293 33 28 .26668 .96379 .28340 .95898 .30015 .95389 .31675 .94851 .33326 .<>42Ht 32 29 .26696 .96371 .28374 .95890 [80043 .95380 .31703 .94842 .33353 .94274 31 30 .26724 .96363 .28402 .95882 .30071 .95372! .31730 .94832 .33381 .94264 30 31 .26752 .96355 .28429 .95874 .30098 .95363 .31758 .94823 .33408 .94254 21 32 .20780 .90347 .28457 .95865 .30126 .95354 .31786 .94814 .33436 .94245 28 83 .26808 .96340 .28485 .95857 .30154 .95345 .31813 .94805 :. 33463 .94235 27 34 .26836 .96332 .28513 .95849 .30182 .95337 .31841 .94795 .33490 .94225 >< 35 .26864 .96324 .28541 .95841 .30209 .95328J .31868 .94780 i. 33518 .94215 X 36 .26892 .96316 .28509 .95832 .30237 .95319 .31896 .94777 1 .33545 .94200 24 37 .26920 .90308 .28597 .95824 .30205 .953101 .31923 .94768 .33573 .94190 58 38 .26948 .90301 .28625 .95816 .30292 .95301 .31951 .94758 .33600 .94186 '.'.. 30 .26976 .96293 .28652 .95H07 .303201.95293 .31979 .94749 .33627 .94176 2] 40 .27004 .96285, .28680 .95799 .30348 .95284 .32000 .94740 i .33655 .94167 3C 41 .27032 .96277 .28708 .95791 .30376 .95275 .32034 .94730 .33682 .94157 fl 42 .27060 .96269 .28736 .95782 .30403 .95266! .32061 .94721 .a3710 .94147 18 43 .27088 .96201 .28764 .95774 .30431 .95257 .32089 .94712 .33737 .94137 K 44 .27116 .96253 .28792 .95766 .30459 .95248 .32116 .94702 .33764 .94127 ]( 45 .27144 .96246 .95757 .304861. 95240 i .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 18 48 .27228 .96222 .28903 .95732 .30570 .95213 .32227 .94665 ;. 33874 1. 94088 12 49 .27256 .96214 .28931 .95724 .30597 .95204 .32254 .94656 33901 .94078 11 50 .27284 .96206; .28959 .95715 .30625 .95195 .82282 .94640 ,.33929 .94068 10 51 .27312 .96198 .28987 .95707 .30653 .95186 .32309 .94637 1 .33956 .94058 9 52 .27340 .96190 .29015 .95698 : .30680 .95177 .32337 .94627 .33983 .94049 8 53 .27368 .96182 .29042 .95690 ] .30708 .95168 .32364 .94618 .34011 .94039 7 54 .27396 .96174 .29070 .95681 .30736 .95159 .32392 .94609 .34038 .94029 B 55 .27424 .96166 .29098 .95073 .30763 .95150 .32419 .94599 .34065 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142' .32447 .94590 .34093 .94009 4 57 .27480 .96150 .29154 .95656 .30819 .95133 .32474 .93580 .34120 .93999 8 58 .27508 .96142 .29182 .95647 .30846 .95124 .32502 .94571 .34147 .93989 2 59 .27536 .96134 .29209 .95639 .30874 .95115 .32529 .94561 .34175 .939791 1 60 .27564 .96126: .29237 .95630 .30902 .95106; .32557 .94552 .34202 .93969 / Cosin Sine ' Cosin "Sine Cosin Sine Cosin Sine Cosin Sine / 74 i 73 72 71 to 34: TABLE XX. SINES AND COSINES. 20 21 22 || 23 24 Sine Cosin Sine Cosin , Siue Cosin Sine Cosin Sine Cosin / .84302 .93969 .35837 ~.9#;5rt : .374(11 ~ 927 18 739073 792050 .40674 T 91 355 60 1 .34229 .93959 .35804 ,9ms .374SS .92707 .39100 .92039! .40700 .91343 59 2 ! .34257 .93949 .35891 .93837 .37515 .92097 .39127 .92028 .40727 .91331 5;? 3 .34284 .93939 .35918 .93337:!. 37543 .92080 .39153 .92010 .40753 .91319 57 4 .34311 .93929 .35945 .9:3310 .37509 .92675 .39180 .93005: .40780 .91307 56 5 .34339 .93919 .35973 .93306 .37595 .92604 .39207 .91994! .40800 .9121)5 55 6 .34366 .93909 .36000 .'.>:.!,*. C> .37022 .92053 .39234 .91982 .40833 .91283 54 y .34393 .9:3899 .36027 .'.i:.5->5 .37649 .92042 .39200 .91971 .40800 .91272 53 8 .34421 .93889 .36054 .93274 .37076 .92031 .39287 .91959 .40; Hi .91260 52 9 .34448 .93879 .36081 .93204 .37703 .92620 .39314 .91948 .4001.3 .91248 51 10 .34475 .93809 .36108 .9325:3 .37730 .92009 .39341 .91936 .40939 .91230 50 11 . 34503 .93859 .36135 .93243 .37757 .92598 .39367 .91925 .40900 .!M-::i 40 12 .34530 .9:3849 .30102 .93232 .37784 .92587 .39394 .91914 .40992 .91212 48 13 .34557 .9:3839 ! 36190 .93222 .37811 .92570 .39421 .91902 .41019 .01000 14 .34584 .93829 .30217 .93211 .37838 .92505 .39448 .91891 .41045 .911FH HO 15 .34012 .93819 .36244 .93201 .37865 .92554 .39474 .91879 .41072 .91176 46 16 .31639 .93809 .36271 .93190 .37892 .92543 .39501 .91868 .41098 .91104 44 17 .34000 .93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .0115.' 4-3 18 .34694 19 .34721 .93789 .93779 .30325 .93169 .363521.93159 .37940 .92521 .37973 .92510 .39555 .39581 .91845 .918:33 .41151 .41178 .91140 91128 42 41 20 .34748 .93709 ,3637'9 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93750 .36406 .93137 .38026 .92488 .39635 .91810 .41231 .91104 39 22 .34803 .93748 .36434 .93127 .38053 .92477 .39661 .91799 .41257 .91092 38 23 .34830 .937:38 .36461 .93116 .38080 .92466 .39688 .91787 .41284 37 24 .34857 .93728 .36488 .93106 .38107 .92455 .39715 .91775 .41310 .91068 25 .34884 .93718 .36515 .93095 .38134 .92444 .39741 .91764 .41337 .91056 35 26 .34912 .93708 .36542 .93084 .38101 .92432 .39768 .91752 .41363 .91044 34 27 .34939 .93698 . 36569 !.9307'4 .38188 .92421 .39795 .91741 .41390 .91032 33 28 .34966 .93688 .36596 .93063 .38215 .92410 .39822 .01721' .41416 32 29 .34993 .93677 .36623 .93052 .8241 .92399 .39848 .91718 1 1 ! i H IN 31 30 .35021 .93667 .36650 .93042 .38208 .92388 .39875 .1,1706 .41469 ! 90998 30 31 .35048 .93657 .36677 .93031 .38295 .92377 .39902 01004 .41496 .90984 29 32 .35075 .93647 .30704 .93020 .38322 .92366 .399.28 .H08 .41522 .90! 72 38 33 .35102 .93637 .36731 .93010 . 38349 : . 92355 .89955 .91671 .41549 34 .35130 .93626 .30758 .92999 .38376 .92343 : .39982 .91660 .415751.90948 26 35 .35157 .93616 .36785 .92988 .384031.92332 i .40008 .91648 .41602 .90930 25 36 .35184 .93606 .36812 .92978 .38430 .92321 .40035 .91630 .41628 .90924 24 37 .35211 . 93596 .36839 .92967 .38456 .92310 .40062 ! 91(125 .41655 .90911 23 38 .35239 .93585 .36867 .92956 .38483 .92299 .40088 .9161=3 .41681 .90899 22 39 .35266 .93575 .36894 .92945 .38510 .92287 .40115 .91601 .417071. 90887 21 40 .35293 .93565 .369211.92935 .38537 .92276 .40141 .91590 .41734 .90875 20 41 .35320 .93555 .36948 .9292-1 .38564 .92265 .40168 .91578 .41760 .90863 10 42 .35347 .93544 .36975 .92913 .38591 .92254 .40195 .91566 .41787 .00851 18 43 .35375 .93534 .37002 .92902 .38617 .92243 .40221 .91555 .41813 .90S39 17 44 .35402 .93524 .370291.92892 .38044 .92231 .40248 .91543 .41840 ;;i !>:_>i; 16 45 .35429 .93514 . 371)56 : . 92881 .38071 .92220 .4027'5 .91531 .41806 /90814 15 46 .35456 .93503 .37083 .92870 .38098' 92209! .40:301 91519 41892 14 47 .35484 .93493 .371101.92859 .387251.92198! .40328 .91508 .41919 !il(/7'90 13 48 .35511 .93483 .37137 .92849 .88752 .92186 .40355 .91490 .41945 . 90778 12 49 .35538 .93472 .37164 .92838 .38778 .92175 .40381 .91484 .41972 11 50 .35565 .93462 .37191 .92827 .38805 .92164' .40408 .91472 .41998 ! 90758 10 1 51 .35592 .93452 .37218 .92816 .38832 .92152 .40434 .91461 . 42024 !. 90741 9 52 .35619 .93441 .37245 .92805 .388.7,) .92141 .40461 .91449 .42051 .90729 8 53 .35647 .93431 .37272 .92794 .38886!. 92130 .40488 .91437 .42077 .90717 7 54 .35674 .93420 .37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 6 55 .35701 .93410 .37326 .92773 .38939 .92107 . 40541 '. 91414 .42130 .90082 5 56 57 58 .35728 .35755 .35782 .93400 .93389 .93379 .37353 .37380 .37407 .92762 .92751 .92740 .389661.92096 .40567 .91402 .38993 .92085 ! .40594 .91390 .39020 .92073 .40621 .91378 .42156 .42183 .42209 .90680 4 .90668 3 .90655 2 59 .35810 .93368 .37434 .92729 .39046 .92062 .40647 .91366 .42235 .90643 1 60 .35837 .93358 .37461 .92718 .31)073 .92050 .40074 .91355 .42262 .90631 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine 69 68 67 il 66 6 II 65 TA11LE XX. -SINES AND COSINES. 343 25 26 27 28 29 / Sine jCosin Sine Cosin Sine Cosin Sine I Cosin Sine Cosin 742262 790631 .43837 .89879 .45399 789101 74694 ?! 788295 748481 .87462 60 1 .42288!. 9061 8 .43863 .89867 .45425 .89087 .469731.88281 .48506 .87448 59 2 .42315 .90606 i .43889 .89854 .45451 .89074 .46999|. 88267 .48532 .87434 53 3 .42341 .90594 .43916 .89841 .45477 .83061 .470241.88254 .48557 .87420 57 4 .42367 .90582 .43942 .89828 .45503 .89048 .47050 .K32-10 .48583 .87406 56 5 .42394 .90569 .43968 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213 .48634 .87377 54 .42446 '.90545 .44020 .89790 .45580 .89008 .47127 .88199 .48659 .87363 53 g .424731.90532 .44046 .89777 .45600 .83995 .47153 .88185 .48684 .87349 52 9 .42499 .90520 .44072 .89764 .45632 .83981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578 .90483 .441511.89726 .45710 .88942 .47255 .83130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 9 22 .42841 .90358 .44411 .89597: .45968 .88808 .47511 .87993 .49040 .87150 *'O 23 1 .42867 .90346 .44437 .89584 .45994 .88795 .47537 .87979 .49065 87136 37 24 .42894 .90334 .44464 .89571 ! .46020 .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90321 .44490 .89558: .46046 .88768 .47588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47014 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .46097 .88741 .47039 .87923 .49166 .87079 33 28 .42999 .90284 .44568 .89519 .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .9^271 .44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .446721.89467 .46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .40252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90203; .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 2G 35 .43182 .90196: .44750 .89428 .46304 .8863-1 .47844 .87312 .49369 .86964 25 36 .43209 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88607 .478951.87784 .49419 .86935 23 38 .43261 .90153 .44828 .89389 .46381 .88593 .47920 .87770 .49445 .86921 23 39 .43287 .90146 .44854 .80376 .46407 .88580 .47946 .87756 .49470 .80906 21 40 .43313 .90133 .44880 .89363 .46433 .88566 .47971 .87743 .43495 .86892 20 41 .4&340 .90120 .44906 .89350 .46458 .88553 .47997 .87729 .49521 .86878 10 42 .43386 .90103 .449321.89337 .46484 .88539 .48022 .87715 .49546 .86863 lu i 43 .43392 .90035 .44958 .89364 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90-082 .44334 .89311 .46536 .88512 .480731.87687 .49590 .86834 16 45 .4:3445 .90070 .45010 .89298 .46561 .83499 .480991.87673 .49622 .86820 15 46 .43471 .90057 .45036 .89285 .46587 .88485 .481241.87659 .49647 .86805 14 j 47 .43497 .90045 .45082 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 i 48 .43523 .900-52 .45088 .89259 .46639 .88458 .48175 .87631 49697 .86777 12 i 49 .43549 .90019 .45114 .89245 .46664 .88445 .48201 .87617 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 r) 52 .43628 .801)31 .45192 .89203 .46742 .88404 .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390 .48303 .87561 .49884 .86704 7 54 .43680 .89956 .45-243 .89180 .46793 .88377 .48328 .87546 .49849 .86690 C 55 .43706 .891)43 . 45269 .89167 .46819 .88363 .48354 .87532 .49874 .86675 5 56 .43733 .89930 .45295 .89153 .46844 .88349 .48379 .87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 .46870 .88336 .48405 .87504:1.49924 .86646 3 58 .437&5 .89905 .45347 .89127 .46896 .88322 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .453731.89114 .46921 .88308 .48456 .87476 .49975 .86617 1 60 .48837 .89879 .45399 .89101 .46947 .88295 .48481 .67462 .50000 .86603 _0 / Cosin | Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine / 64 ll GD a i C2 6i u II , 60 TABLE XX. SIXES AND COSINES. 30* 31 |1 32 qoo OO 34' Sine Cosin Sine Cosin Sine Cosin Sine 1 Cosin Sine j Cosin f 750000 .86003 .51504 .85717 752992 784K05 7544(54 ~. 83807 .55919 '.K2901 60 1 .50025 .86588 .51529 .85702 .53017 .84789 .54488 .83851 .55943 .82887 69 2 .50050 .86573 .51554 .85687 .530411.84774 .64513 .83835 .55908 .82871 58 8 .50076 .86559 .51579 .85672 .53066 1.84759 ; .54537 .83819 .55992 L 82855 57 4 .50101 .86544 .51604 .85657 .530911.84743 .51561 .83804 .56016 .82839 56 5 .50126 .86530 .51628 .85642 .53115 .84728 .54586 .83788 .56040 .82822 55 6 .50151 .86515 .51653 .85627 .53140 .84712 ! .546101.83772 .56064 .S2S06 54 ? .50176 .86501 .51678 .85612 ,53164 .84697 .546351.83756 .50088 .S2',yo 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54659 .83740 .56112 .8-2773 53 9 .80827 .86471 .51728 .85582 .63214 .84666 i .546831.83724 .56136 .82757 51 10 .50252 .86457 .51753 .85567 .53238 .84650 .54708 .837u8 .50160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635" .54732 .83692 .561 SI .82724 49 12 .50302 .86427 .51803 .85536 .532881.84619 .54756 .83'.j;y .50208 .82708 48 13 .50327 .86413 .51823 .85521 .53312 .84604 .54781 .83600 .56232 .82092 47 14 .50352 .86398 .51852 .85506 .53337 .84588 .54805 .83645 .56256 .82675 46 15 .50377 .86384 .51877 .85491 .53361 .84573 .54829 .83629 .56280 .82059 45 16 .50403 .86369 .51902 .85476 .53386 .84557 .54854 .83613 .56305 .82643 44 17 .50428 .86:354 .51927 .85461 .53411 .84542 54878 83597 .56329 82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .56353 .82610 42 19 .50478 .86325 .51977 .85431 .53460!.84311 ..'.4927 .83365 .56377 .82593 41 20 .50503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 22 ! .50553 .86281 .52051 .85385 .533341.84464 .54990 .83517 .56449 .82344 33 23 .50578 .86266 .52076 .85370 .53558 .84448 .5502-4 .83501 .56473 .82528 37 24 .50603 .86251 .52101 .85355 .53583 .84433 .55048 .83485 .56497 .8251 1 3G 25 .50628 .86237 .52126 .85340 .53607 .84417 .55072 .83469 .56521 .82495 85 26 .50654 .86222 .52151 .85325 .53632 .84402 .55097 .83453 .56545 .82478 34 Of" .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437 .56569 .82462 83 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 .83421 .56593 .82446 32 29 .50729 .83178 .52225 .85279 .53705 .84355 .55169 .83405 .56617 .82S29 81 30 .50754 .86163 .52250 .85264 .53730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .56665 .82396 ! 29 32 .50804 .86133 .5.3299 .85234 .53779 .84308 .55243:. 8=3356 .56689 .82380 28 33 .50829 .86119 .53324 .85218 .53804 .84292 .P5266 .83340 i .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277 .55291 .83321 ! .56736 .82347 26 35 .50879 .86089 .52374 .85188 .53853 \. 84261 .55315 .83308 i .56760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 .55339 .83292 ! .56784 .8231-1 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .55363 .83276 ! .56808 .82297 23 38 .50954 .86045 .52448 .85142] .53926 .84214 .55388 .83200 .56832 .82281 22 39 .50979 .860301 .52473 .85127 .53951 .84198 .55412 .83244 .56856 .82204 21 40 .51004 86015 1 .52498 .85112 .53975 . 84182 .55436 .83228 .50880 .82248 20 41 .51029 .860001 .52522 .85096 .54000 .84167! .55460 .83212 h .56904 .82231 19 42 .51054 . 859::15 .52547 .85081 .54024 .84151 .55484 .83195: .56928 .82214 18 43 .510791.85370 .52572 .85066 .54049 .84135 .55509 .83179 .56952 82198 17 44 .51104 .85356 ,52597 .85051 .54073 .84120 .55533 j .83163 '. ' .50976 i .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .55557 .83147: .57000 .82165 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .53671 .85005 .54146 .84072 .55605 .83115; .57047 .82132 13 48 .51204 .85896 .52696 .84989 .54171 .84057 .55630 .83098 .57071 .82115 12 49 .51229|.85881 .52?20 . 84974 i .54195 .84041 .55654 . 83082 ii .57095 .82098 11 50 .51254 .85866 .52745 .84959 .54220 .8-1025 .55078 .83006 .57119 .82082 10 51 .512791.85851 .52770 .84943 .54244 .84009! .55702 .83050 .57143 .82065 9 52 . 51304 .85836 .53794 .84928 .542691.83994 .55726 .83034 .571(57 .82048 8 53 .51329 .85821 .52819 .84913 .54293 |.83978 .55750 .83017,;. 57191 .82032 7 54 .51354 .85806 .52844 .84897 .54317.88962 .55775 .83001! .57015 .82015 ; 6 55 .51379 .85792 .52869 .84882 . 54342 1.83946 .55799 .82985' .57238 .81999 5 56 .51404 .85777 .52893 .84866 .543661.83930 .55823 .82969 .57262 .81982 4 57 .51429 .85762 .52918 .84851 .54391 |. 83915 .55847 .82953- .57286 .81905 3 58 .51454 .85747 .52943 .84836! ,64415.88899 .55871 .82936, .57310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .53895 .82920 ,57334 .si >:;_' 1 60 .51504 .85717 .52992 .84805 .54464 . 8:3867 Ij .55919 .82904 .57358 .81915 / Cosin Sine Cosin Sine Cosin | Sine Cosin Sine Cosin Sine i 59 58 57 56 55 TABLE XX. SINES AND COSINES. 345 35 ! | 36 37 H 38 || 39 / Sine i Cosin Sine ICosin Sine Cosin Sine Cosin !| Sine Cosin ~o . 57358 LsiiUo .58779 .80902 .60182 79864 .61566 .78801 .62932 .77715 60 1 .57381 .81899 .58802 .80885 .60205 .79846 .61589 .78783 .62955 .77696! 59 2 .57405 :.M8AJ .58826 .80867 .60228 .79829 .61612 .78765 .62977 .77678) 58 3 .57429 .81 865 ! .58849 .80850! .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453 .81848 .58873 .80833 .60274 .79793 .61658 .78729! .63022 .77641 56 5 .57477 . 81832 i .58896 .80816 .60298 .79776 .61681 .78711 .63045 .77623 56 C .57501 .57524 .81815| .81798 .58920 .58943 .80799' .80782 .60321 .79758 . 60344 S. 79741 .61704 .61726 .78694 .78676J .63068 .63090 .77605 54 .77586 53 8 .57548 .81782 .58967 .80765 .60367 .79723 .61749 .78658J .63113 .77568 52 9 .57572 .81765 .58990 .80748! .60390 .79706 .61772 .78640! .63135 .77550' 51 10 .57596 .8ir48| .59014 .80730; .60414 .79688. .61795 .78622 .63158 .77531 50 11 .57619 .81731 .59037 .80713 .60437 . 79671 ! .61818 .7'8604 .63180 .77513 40 12 .57643 .81714 .59061 .80696 .60400 .79653 .61841 .78586 .63203 .77494 18 13 .57667 .81698 .59084 .80679 .60483 .79635 .01864 .7'8568 .63225 .77476 47 14 .57691 .81681 .591081.80662 .60506 .79618 .61887 .78550 .63248 .77458 46 15 .57715 .81664 .59131 .80644 .60529 .79600 i .61909 .78532, .63271 .77439 45 16 .57738 .81647 .59154 .80627i .60553 .79583 .61932 .78514 .63293 .77421 44 17 .57762 .81631 .59178 .80610! .60576 .79565 .61955!. 78496 .63316 .77402 43 18 .57786 .81614 .59201 .80593' .60309 .79547 .61978 .78478 .63338 .77384' 42 10 .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .633611.77366 41 20 .57833 .81580 .59248 .80558; .60645 .79512 .62024 . 78442 | .63383 .77347 40 21 .57857 . 81563 ! .59272 . 80541 .60668 . 79494 p .62046 .78424 .63406 .77329J 39 oo .57881 .81546 .59295 .80524 .60691 .79477 . .62069 .78405 .63428 .77310 38 23 .57904 .81530 .59318 .80507 .60714 .79459 .62092 .78387' .63451 .77292 37 24 .57928 .81513 .59342 .80489 .60738 .79441 .62115 .788691 . 63473 ! . 77273! 36 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 .77218! 33 28 .58023 .81445; .59436 .80420 .60830 .79371! .62206 .78297 .63563 .77199 32 29 .58047 .81428 .59459 .80403 60853 .79353 .62229 .78279! .63585 .77181 ! 31 30 .58070 .81412 .59482 .80386; .60876 .79335 .62251 .78261 .63608 .77162 30 31 .580941.81395 .59506 .80368 .60899 .79318 .62274 .78243; .63630 .77144 29 32 .58118!. 81378 i .593291.80351 .60922 .793001! .62297 .78225 .63653 .77125 28 33 .581411.81361 .595521.80334 .60945 .79282 .62320 .78206 .63675 .77107 27 34 1.581 65 i. 813441 .59576 .80316 .60968 .79264 ! .62342 .78188 .63698 .77088 26 35 .58189; . 81327 .59599 .80299! .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310; .59622 .802821 .61015 .79229 i .62388 .78152 .63742 .77051 24 37 .58236.81293 .59646 .80264 .61038 .79211!! .63411 .78134; .63765 .77033 23 38 .58260 .81276 .59069 .80247 .61061 .79193 .62433 .78116! .63787 .77014 22 39 .58283 .81259 .59693 .80230 .61084 .79176 .62456 .78098 .63810 .76996! 21 40 . 58307 i. 81242 .59716 .80212: .61107 .79158 .62479 .78079 .63832 .76977 20 41 .58-330 '.81225 .59739 .80195 I .61130 .79140 .62502 .78061 .63854 .76959 19 42 .58334 .81208 .59763 .80178 .61153 .79122 .625241.780431 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .62547 .78025 .63899 .76921 17 44 .58401 >.81 174! .59809 .80143 .61199 .79087 j .62570 .78007 .639221.76903! 16 45 .58425 .81157 .59832 .80125 .61222 .79069 '.62592 .77988! .63944 .7'6884 15 46 .58449 .811401 .59856 .80108 .61245 .79051 .62615 .77970 .63966 .76866 14 47 .58472,. 81 123 S .59879 .80091 .61268 .79033 .62638 . 77952 ! .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62660 . 77934 ' .64011 .76828 12 49 .58519 .81089 .59926 . 80056 ! .61314 .78998 .62683 .77916 .64033 .768101 11 50 .58543 .81072 .59949 .80038 .61337 . 78980 i .62706 .77897 .64056 .76791 10 51 .58567 .81055 .59972 .80021! .61360 .78982 .62728 .77879 .64078 .76772 9 52 .58590 .81038' .59995 .80003 .61383 . 78944 il .62751 .77861 ! .64100 .76754 8 53 .58614 .81021i .60019 .79986 .61406 .78926 .62774 .778431 .64123 .76735 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .70717 6 55 .58661 .80987 .60065 .79951 .61451 .78891 .62819 .77806 .64167 .76698 5 56 .58684 . 80970 .60089 .79934 .61474 .78873 .62842 .77788 .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61497 .78855 .62864 .77769 .642121.76661 3 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 . 77731 .64234 .76642 2 59 .58755 .80919! .60158 . 79881 i .61543 .78819 .62909 .77733 .64256 .76623 60 .58779 .80902 1 .60182 .79864 .61566 .78801 .62932 .77715 .64279 .76604 / Cosin 1 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine / 54 II 53 52 51 50 TABLE XX. SINES AND COSINES. 40 41 1! 42 43 44 Sine ! Cosin | Sine Cosin Sine j Cosin Sine Cosin Sine Cosin ' .64279 .76604 .65000 .75471 .66913 .74314 .68200 .73135 .69406 .71934 60 .61301 .76586 .65628 .75452 .66935 .74295 .68221 .73116 .69487 .71911 59 .64333 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 : 53 .64346 . 76548 l .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873! 57 .64368 .76530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 r>r> .64390 .76511 .65716 .75375 .67021 .74217 .68306 .73036 .69570 .71833 55 .64412 .76492 .65738 .75356 .67043 .74198 .68327 .73016 .69591 .71813 M .64435 .76473 .65759 .75337 .67064 .7-1178 .68349 .72996 .69612 .71792 53 .61457 .76455 .65781 .75318 .67086 .74159 .68370 .72976 .69633 .71772 i 52 .61479 .64501 .76436 .76417 .65803 .65825 .75233, .75280 .07107 .67129 .74139 .74120 .68391 .68412 .72957 .72937 .69654 .69675 .717521 51 .71732, 50 .64524 .76398' .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 1*9 . 64546 !. 763301 .653o3 .75211 .67172 .7'4080 .68455 .72897 .69717 .71691 1 48 .6 1563'. 76331 | .65391 .75:i, A 2 .67194 .74061 .68476 .72877 .69737 .71671 ! 47 .64590 .76343 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71650 ; 46 .64612 .76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 .71030 45 .64635 .763041 .65953 75165 .67258 .74002 .68539 .72817 .69800 .71610 44 .64657 .76236 .65378 75146 .67280 .73983 .68561 .72797 .69821 .71590 43 .64679 .76267 .63000 75128 .67301 .73963 .68582 . 72777 .69842 .71569 42 ,64701 .76248 .63022 75107 i .67323 .73944 .68603 .72757 .698621.71549 41 .64723 .76229 ; .66044 75088 .67344 .73924 .68624 .72737 .69883 .71529 140 .64746 .76210 '1.66066 75039 .67366 .73904 .68645 .72717 .69904 .71508 '39 .64768 .76192 .66033 7535J .67387 .73885 .68666 .72697 .69925 .71488 38 .64790 .76173 .63103 75030 .67409 .73865 .68688 .72677 .69946 .71468 37 .64812|.76154 .66131 75011 .67430 .73846 .68709 .72657 .69966 .71447 36 .64834 j.76135 .66153 74332 .67452 .73820 .63730 .73637 .699871.71427 35 .648561.76116 .66175 74373 ! .67473 .73808 .63751 .72617 .70008 .71407 34 . 64878 !. 76097: .66197 74953 ! .67495 .73787 .68772 .72597 .700291.71386 33 .649011.76078 .63318 74331 .67516 .73767 .68793 .72577 .700491.71366 32 .64923 ! 76059 .60240 71915 .67538 .73747 .68814 .72557 .700701.71345 31 .64945;. 76041 .66262 74896 .67559 .73728 .68835 .72537 .70091 .71325J 30 .64967 .76022 .66284 74876' .67580 .73708 .68857 .72517 .70113 .71305! 29 .64939 .76003 i .68306 74857 .67602 .73683 .68878 .7.2497 701 32 1.71284 28 .650111.75984 j .66327 74833 .67623 .73669 .6:5899 .72477 .70153:. 71204 27 . 65033 j. 75965 .66349 74818 .67645 .73649 .68920 .72457 .70174 .71243 26 .65055 .75946 .66371 74799 ! .67666 .73629 .68941 . 72437 ; .70195 .71223 35 . 65077 j. 75927 .6(3393 74780 .67688 .73610 .68962 .72417 .70215 .71203 24 . 65100 . 75903 .66414 74760 .67709 73590 .68983 .72397' .702361.71182 88 .65122 .75383 .66436 74741 .67730 73570 .69004 .72377 .70257 .71162 22 .65144 .75870 .60 158 74722 .67752 73551 .69025 .72357 . 70277 i. 71 141 21 .65166 .75851 .68480 74703 j .67773 73531 .69046 .72337 .70298 .71121 20 .65188 . 75832 ! .66501 74683; .67795 73511 .69067 72317 .70319 .71100 19 .65^10 .75813 .66523 74664 ! .67816 73191 .69088 72297 .703391.71030 18 .65232 .75794 .66545 74644; .67837 73172 .69109 72277 .70300 .71(559 17 .65254 .75775 .68566 74625 .67859 .73452 .69130 72257 .70381 .71039 16 .65276 .75756 .66588 74606 .67880 .73432 .69151 72236 .70401 .7101!) 15 .652981.75738 .66610 .74588 i .67901 .73413 .69172 72216 i .70422 .70W!-i 14 .65320 .75719 .66632 .74567 .67923 .73393 .69193 72196 .70443;. 70978 13 .65342 .75700 .666531.74548 j .67944 .73373 .69214 72176 .70463 .70957 12 .65364 .75680 .66675 .74528 .67965 .73353 .69235 72156 .704H4 .roirrr 11 .65386 .75661 .66697 .74509 .67987 .73333 .69256 72136 i .70505 .70J10 10 .65408 .75642 .66718 '.74489 .68008 .7.3314 .69277 .72116 .70525 .70896 9 .654301.75623 .66740!. 74470 .68029 .73294 .69298 72095 .70546 .70875 8 . 65452 i. 75604 .66762 .74451 ! .68051 .73274 .69319 .72075 .70567 '.70355 7 .65474 .75585 .66783 .74431 ! .68072 .73254 .69340 72055 .70587 .70834 6 .65496 .75566 .66805 .74412 ! .6,8093 .73234 .69361 .72035 .70608 .70813 5 .655181.75547 .66827 .74392 j .68115 .73215 .69382 .72015 .70628.. 70793 4 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995! .706491.70772 3 .65562 .75509 .66870 .74353 .68157 .73175 . 69424 . 71974 1 1 . 70670 . 70752 2 .65584 .75490 .66891 .74334 .68179 .73155 . 69445 i . 71954 . 70690 . 70731 1 .65606 .75471 .66913 .74314 .68200 .73135 . 69466 j. 71 934 .70711 .70711 Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin , Sine / 49 48 47 46 !l 45 1 TABLE XXI.-TANGENTS AND COTANGENTS. 1 ! 2 3 Tang Cotang Tang Cotang Tang Cotang Tang 1 Cotan.Gr .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 1 .00029 3437.75 .01775 58.3506 .03521 28.3994 .05270 18.97'55 2 .00058 1718.87 .01804 55.4115 .03550 28.1C64 .05299 18.8711 3 .0003? 1145.92 .01833 54.5G13 .03579 27.9372 .05328 18.7678 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 6 .00145 687.549 .01091 52.C321 .03638 27.4G99 .05387 18.5645 .0017'5 572.957 .01920 52.C307 : .03667 27.2715 .05416 18.4645 7 .00204 431.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 8 .00333 429.713 .01978 50.5185 .03725 26.8150 .0547'4 18.2677 ! .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 11 .00320 312.521 .02066 48.4121 .03812 2G.2296 .05562 17.9802 12 .00343 JioO.47'8 .02005 47.7395 .03J42 26.0307 .05501 17.8863 13! .00313 2G4.441 .02124 47.0853 .03871 25.8348 .056CO 17.7934 14 1 .0040? 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 18 .C043G 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 1J, .00-105 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 1?J .00^3 5^2.213 .02240 44.6386 .03987 25.0798 .05737 17.4314 18 .00524 130.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 19 .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 go .00583 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 . i .00o-10 150.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 23 .OOOJ9 1-19.465 .02415 41.4106 .04162 24.0263 .06012 16.9150 24 1 .00698 113.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 85 .00737 137.507 .02473 40.4358 .01220 23.6945 .05970 16.7496 ;.-; .00756 132.219 ,02502 89.9655 .04250 23.5321 .05999 16.6681 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 88 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 ;.".' .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 1 .00902 110.892 .02648 37.7680 .04395 22.7519 .06145 16.2722 : : : .00^31 107.426 .02677 37.3579 .04124 22.6020 .C6175 16.1952 83 .000 JO 104.171 .02706 86.9500 .04454 22.4541 .06204 16.1190 84 1J1.1J7 .02735 3i).r,6-.7 .04483 22.3081 .06233 16.0435 86 .01018 9.12179 [02764 3(5.1776 .04512 22.1640 .06262 15.9687 86 .01047 95.4SU5 .02793 35.8006 .01511 22.0217 .06291 15.8945 87 .01076 02.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 38 .01105 90.4633 .02351 35.0695 .04599 21.7426 .06350 15.7483 39l .01185 8^.1436 .02881 34.7'im .016-28 21.6056 .06379 15.6762 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 , (< .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 12 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 43 .01251 79.9434 .02997 a3. 3662 .04745 21.0747 .06496 15.3943 n .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 45 .OKJOO 76.3900 .03055 32.7303 .01803 20.8188 .06554 15.2571 46 .C1338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.189'J 47 .01367 73.1390 .03114 3->.1181 .04862 20.5691 .06613 15.1222 48 .01396 71.6151 .03113 31.8205 .04891 20.4465 .06642 15.0557 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 BO .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 531 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 55 .0160C (52.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 56 .01629 61.3829 .03376 29.6245 .05124 19.5156 .06876 14.5438 57! .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 58 .01687 59.2659 .03434 29.1220 ; .05182 19.2959 .06934 14.4212 r.i .01716 58.2612 .03463 28.8771 j .05212 19.1879 .06963 14.3607 60 .01746 57.2900 .0341)2 28.6363 .05241 19.0811 .06993 14.3007 / Cotang Tang Cotang Tang Cotang Tang Cotang i Tang 89 88 87<> 86 348 TABLE XXI.-TANfiF.NT3 AND COTANGENTS. Tang Cotang Tang Cotang Tang Cotang Tang Cotang ' Ti .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 60 i .07023 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12481 59 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .123-J8 8.10536 r,s 8 .07U80 14. 12; .08837 11.3163 .10599 9.413515 ! .12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 i .12397 8.06674 56 6 .07139 14.0079 .08895 11.2417 .10657 9.38307 .12426 8.04756 55 .07168 13.9507 .08925 11.2048 i .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 ' .08954 11.1081 .10710 9.33155 .12485 8.00948 58 s .07227 13.8378 .08983 11.1316 ! .10746 9.30599 i .12515 7.99058 59 9 .07256 13.7821 .09013 11.0954 . 10775 9.28058 i .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 | .10805 9.25530 .12574 .95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 i .12603 .93438 4'.) 18 .07344 13.6174 .09101 10.9882 .10863 9.20516 .12633 .91582 48 13 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12662 .89734 47 11 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12692 .87895 413 ir> .07431 13.4566 .09189 10.8829 | .10952 9.13093 .12722 .86064 4~> 16 .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 .84242 44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 .82428 48 i;, .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 .80622 48 l:i .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 .78825 11 90 .07578 13.1969 .09335 10.7119 1 .11099 9.00983 .12869 .77035 to 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 .75254 39 x^ .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 .73480 3ti 28 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 .71715 87 -i .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 .69957 36 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 .68208 85 26 .07753 12.8981 .09511 10.5136' .11276 8.86862 .13047 .66466 84 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 .64732 88 28 .07812 12.8014 .09570 10.4491 ! .11335 8.82252 .13106 .63005 38 29 .07041 12.7536 .09600 10.4172 .11364 8.79964 .13136 .61287 81 30 .07870 12.7062 .09629 10.3854 j .11394 8.77689 .13165 .59575 30 31 .07899 12.6591 .09658 10.3538 ! .11423 8.75425 .13195 .57872 eo 32 .07'929 12.6124 .09688 10.3224 ! .11452 8.73172 .13224 .56176 28 88 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 .54487 87 34 .07987 12.5199 .09746 10.2602 ! .11511 8.68701 .13284 .52806 26 85 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 .51132 25 38 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 .49465 24 87 .08075 12.3838 .09834 10.1683 | .11600 8.62078 .13372 .47806 23 88 .03104 12.3390 .09864 10.1381 1 .11629 8.59893 .13402 .46154 22 :.!) .08134 12.2946 .09893 10.1080 .11659 8.57718 .1:3432 .44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 .42871 20 41 .08192 12.2067 .09952 10.0483 .11718 F. 63402 .13491 .41240 10 .; : j .08221 12.1632 .09981 10.0187 i .11747 8.51269 .13521 .39616 IS 43 .08251 12.1201 .10011 9.98931 i .11777 8.49128 .13550 .37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 .36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 .34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 .33190 14 ..'- .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 .31600 18 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 .30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 .28442 11 50 .08456 11.8262 .10216 9.T8817 .11983 8.34496 .13758 .26873 10 51 .08485 11.7853 .10246 9.76009 .12013 8.32446 .13787 .25310 '. 53 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 .23754 8 58 .Oa544 11.7045 .10305 9.70441 .12072 8.28376 .13846 .22204 7 64 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 .20661 8 65 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 .19125 6 66 .08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 .17594 4 67 .086(51 11.5461 .10422 9.59490 .12190 8.20352 .13965 .16071 3 68 .08690 11 5072 .10452 9.56791 .12219 8.18370 .13995 .14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 .13042 1 60 .08749 11.4301 . 10510 9.51436 .12278 8.14435 .14054 7.11537 f Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 8 5 I 8 4 8 3 8 2 TABLE XXI. TANGENTS AND COTANGENTS. 8 i 9 10 !, IP Tang | Cotang |j Tang Cotang Tang Cotang Tang Cotang .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 60 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .11113 7.08.546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 3; .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.6321)5 .19559 5.11279 56 5 .14202 7.04105 .15988 6.25186 .17783 5.62344 .19589 5.10490 55 (j .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 54 .14262 1 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 53 g .14291 i 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 52 8 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 G. 95385 .16167 6.18559 .17963 5.56706 .39770 5.05809 49 12 .14410 6.93952 .16196 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5. 042:, 7 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14199 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .10982 5.00451 42 19 .14618 6.840S2 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 90 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 . 14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 i> > .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 23 .14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 | .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 36 25 .14796 6.75838 ' .16585 6.02962 .18384 5.43966 .20194 4.95201 35 96 .14826 6.74483 1 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.0079? .18444 5.42192 .20254 4.93721 33 28 i .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 & 29 .11915 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 .14975 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 29 3;.' .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 83 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 3J .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 25 86 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 J7 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.5S627 ' .16974 5.89151 .18775 5.32631 .20588 4.85727 22 3D .15213 6.57339 .17004 5.88114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 C. 54777 .17063 5 86051 .18865 5.30080 .20679 4.83590 19 4;! .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.840C1 .18925 5.28393 .20739 4.8217'5 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81171 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.807'69 15 46 . 15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .1907'6 5.24218 .20891 4.78673 12 19 .15511 6,44720 .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15640 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 I" 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.7590G 8 .',-.! .15630 6.39804 .17423 5.73960 .1922? 5.20107 .21043 4.75219 7 51 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 f: :':> .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6.36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 i .19347 5.16863 .21164 4.72190 3 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.32566 .17603 5.68094 ' .19408 5.15256 .21225 4.71137 1 ;o .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 / Cotang Tang Cotang Tang Cotang j Tang Cotang Tang 81 I 1 80 il 79 ' ' 78 TABLE XXI. TANGENTS AND COTANGENTS. 4 1 5 6 7 Tang Cotang Tang Cotang Tang Cotang i Tang Cotang .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 :60 1 .07022 14.2411 .08778 11.3919 ! .10540 9.48781 ; .12308 8.12481 si 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .Itfi'JS 8.10530 56 3; .07080 14.1285 .08837 11.3163 .10599 9.43515 .12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10028 9.40904 ; .12397 8.06674 56 5 .07139 14.0079 .08895 11.2417 .10657 9.38307 .12426 8.04756 55 G .07168 13.9507 .08925 11.2048 1 .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 .08954 11.1081 .10710 9.33155 .12485 8.00948 58 8 .07237 13.8378 .08983 11.1316 i .10746 9.30599 .12515 7.99058 58 9 .07250 13.7821 .09013 11.0954 . 1077 5 9.28058 .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 1 .10834 9.23016 .12603 7.93438 49 121 .07344 13.6174 .09101 10.9882 i .10863 9.20516 .12633 7.91582 48 13 .07373 13.5634 .09130 10.9529 i! .10893 9.18028 .12662 7.89734 47 14 .07402 13.5098 : .09159 10.9178 1 .10922 9.15554 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12728 7.86064 45 10 .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 7.84242 ?44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 1 43 18 .07519 13.2996 1 .09277 10.7797 .11040 9.05789 .12810 7.80622 48 19 j .07548 13.2480 .09306 10.7457 I .11070 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 II .11099 9.00983 .12869 7.77035 10 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 38 23 | .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 87 24 .07085 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 36 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 7.68208 35 26 .07753 12.8981 .09511 10.5136' .11276 8.86862 .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 7.64732 38 2s .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.63005 32 29 .07841 12.7536 .09600 10.4172 .11364 8.79964 .13136 7.61287 81 80 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.59575 30 81 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 & 32 .07929 12.6124 .09688 10.3224 i .11452 8.73172 .13224 7.56176 28 88 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 26 85 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 36 .08046 12.4288 .09805 10.1988 I .11570 8.64275 .13343 7.49465 24 37 ! 68075 12.3838 .09834 10.1683 ! .11600 8.62078 .13372 7.47806 23 38 .03104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 89 .08134 12.2946 .09893 10.1080 .11659 8.57718 .13432 7.44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 .11718 F. 63402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 48 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 IT -1! .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 7.36389 16 46 .08309 12.0346 .10069 D. 93101 .11836 8.44896 .13609 7.34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 ,-- .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 18 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 7.30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 60 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 9.76009 .12013 8.32446 .13787 7.25310 9 58 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .0&544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .OS573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 56 .08602 11.62-18 .10363 9.64935 .12131 8.24345 .13906 7.19125 5 58 .08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08681 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 3 58 .08690 11 5072 i .10452 9.56791 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 60 .08749 11.4301 .10510 9.51436 .12278 8.14435 ! .14054 7.11537 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 85 i 84 83 82 TABLE XXI TANGENTS AND COTANGENTS. I, 8 9 10 11 Tang | Cotang Tang Cotang Tang Cotang Tang Cotang ~0 .14054 7.11537 .15838 G. 31375 .17633 5.67128 .19438 5.14455 1 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 2 .11113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 41 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 5 . 14302 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 6 .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 3 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 9 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 JO .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.0G584 11 .14381 G. 95385 .16167 6.18559 .17963 5 . 56706 .39770 5.05809 12 .14410 6.93952 .16196 6.17419 .17993 5.55777 .19801 5.05037 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.0421.7 11 .14470 6.91104 .16256 G. 15151 .18053 5.53927 .19861 5.03499 15 .14199 G. 89688 .16286 6.14023 .18033 5.53007 .19891 5.02734 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 18 .14588 G. 85475 .16376 6.10664 .18173 5.50264 .1^982 5.00451 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 20 .14648 6.82694 1 .16435 6.08444 I .18233 5.48451 .20042 4.98940 21 . 14G78 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 22 . 14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 2; J > .14737 6.?'8564 .16525 6.05143 .18323 5.45751 .20133 4.96690 24 .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 25 .14796 6.75838 ' .16585 6.02962 .18384 5.43966 .20194 4.95201 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 27 .14856 6.73133 .16645 6.0079^ .18444 5.42192 .20254 4.93721 2S .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 .20315 4.92249 30 . 14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 81 .14975 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 32 . 15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 34 .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 87 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 38 .15183 6.58627 ' .16974 5.89151 .18775 5.32631 .20588 4.85727 89 .15213 6.57339 .17004 5.88114 .18805 5.31778 .20618 4.85013 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 41 .15272 6.54777 .17063 5 86051 .18865 5.30080 .20679 4.83590 48. .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 43 .15332 6.52234 .17123 5.840C1 .18925 5.28393 .20739 4.8217'5 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.814T1 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.807'69 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 'i!< .15511 6.44720 .17303 5.77936 .19106 6.23391 .20921 4.77978 60 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 M .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 5-2 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 63 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 54 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 55 .15689 6.37374 .17483 5.71992 .192S7 5.18480 .21104 4.73851 56 .15719 6.36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 5? .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72190 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 59 .15809 6.32566 .17603 5.68094 .19408 5.15256 .21225 4.71137 60 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 / Cotang Tang Cotang Tang Cotang j Tang Cotang Tang 81 l j 80 il 79 78 TABLE XXL TANliKNTS AND COTANUKNTS. 12 l| 13 ! 14 15 Tang Cotang Tang Cotang Tang Cotang Tang | Cotang f .21256 4.70463 .23087 4.33148 .24933 4.01078 .20795 3.73205 60 1 .21286 4.69791 .23117 4.32573 .24904 4.00583 .2(5820 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.00080 i .20857 3.72338 58 a .21347 4.68452 .23179 4.31430 .25026 3.995U2 .20888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98007 .26951 3.71046 55 6 .21438 4.66458 .23271 4.29724 .25118 3.98117 .20982 3.70016 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97027 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.09761 52 9 .215S9 4.64480 .23363 4.28032 .25211 3.96051 .27070 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.90105 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95080 .27138 3.68485 49 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27109 3.68001 48 13 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.07638 47 14 .21682 4.61219 .23516 4.25239 .25366 3.94232 .27232 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.00796 45 16 .21743 4.59927 .23578 4.24132 .25128 3.93271 .27294 3.60370 44 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27326 3.65957 43 a8 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.05538 42 19 .21834 4.58001 .23670 4.22481 .25521 3.91839 .27388 3.05121 41 90 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.04705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 83 .21925 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 38 83 .21956 4.55453 .23793 4.20298 .25645 3.89945 .27513 3.03401 37 21 .21986 4.54826 .23823 4.19756 .25676 3.8947'4 .27545 3.03048 36 25 .J2017 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 86 .22047 4.53568 .23885 4.18675 .25738 3.88536 .2?'G07 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.88068 .27038 3.61814 33 88 ..22108 4.52316 .23946 4.17600 .25800 3.87601 .27070 3.61405 32 2:) .22139 4.51693 .23977 4.17064 .25831 3.87136 .277'01 3.60990 31 30 .22109 4.51071 .24008 4.16530 .25862 3.86671 .27732 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 88 .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 4.14405 .25986 3.84824 .27853 3.58966 26 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 36 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58160 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 4.11778 .26141 3.82537 .28015 3.50957 "1 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 19 42 .22536 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.55761 18 43 .22567 4.43134 .24408 4.09699 .26266 3.80726 .28140 3.55304 17 44 .22597 4.42534 .24439 4.09182 ,20297 3.80276 .28172 3.54908 16 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 46 1 .22658 4.41340 .24501 4.08152 .20359 3.79378 .28234 3.54179 14 47 i .22689 4.40745 .24532 4.07639 .28390 3.78931 .28260 3.53785 13 48! .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 | .22750 4.395GO .24593 4.06616 .20152 3.78040 .28329 3.53001 11 50 .22781 4.38969 .24624 4.06107 .20483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 58 .22842 4.37793 .24686 4.05092 .20546 3.76709 .28423 3.51829 8 53 .22872 4.37207 .24717 4.04586 .26577 3.76268 .28454 3.51441 7 51 .22903 4.3'J023 .24747 4.04081 .26008 3.75828 .28486 3.51053 6 55 .22934 4.3G040 .24778 4.03578 .26039 8.75388 .28517 3.50666 5 50 .22964 4.35459 .24809 4.03076 .26670 3.74950 .28549 3.50279 4 57 .22995 4.34879 .24840 4.02574 .26701 3.74512 .28580 3.49894 3 53 .23026 4.34300 .24871 4.02074 .26733 3.74075 .28012 3.49509 2 53 .23056 4.33723 .24902 4.01576 .26764 3.73640 .28043 3.49125 1 GO .23087 4.33148 .24933 4.01078 .26795 3.73205 .28675 3.48741 / Cotang Tang Cotang Tang Cotang Tang Cotang | Tang / 77 76 75 74 TABLE XXI. TANGENTS AND COTANGENTS. 1 16 1 17 18 19 / Tang Cotang Tang Cotang Tang ; Cotang Tang j Cotang ~o .28675 ! 3.48741 .30573 3.27085 .32492 3.07768 .34433 1 2.90421 60 i .28706 ! 3.48:359 .30605 3.26745 .32524 3.07464 .34465 2.90147 59 g .28738 1 3.47977 .30637 3.26406 .32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.26067 .32588 3.06857 .34530 2.89600 57 4 "28800 a.ffllG .30700 3.25729 .32621 3.06554 .34563 2.89327 56 5 .'28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 55 G .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 54 7 .28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.88511 53 8 .28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 9 .28958 3.43337 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10. .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2 87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 12 i .29053 3.44202 .30955 3.23048 .32878 3.04152 .34824 2.87161 48 13 .29084 3.43829 .30987 3.22715 .32911 3.03854 .34856 2.86892 47 14 .29116 3.43456 .31019 3.22384 .-32943 3.03556 .34889 2.86624 46 15 .211147 3.43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 -29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 H .29242 3.41973 1 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 i: . 29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 2.85289 1 41 go .29305 3.41236 | .31210 3.20406 .33136 3.01783 .35085 2.85023 40 01 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 S. 84758 39 22 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 28 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 37 -.'4 .29432 3.39771 .31338 3.19100 .83360 3.00611 .35216 2.83965 i36 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 26 i .29495 3.39012 .31402 3.18451 .33330 3.00028 .35281 2.83439 34 27 ! .29526 3! 33079 .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 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29(585 3.36875 .31594 3.16517 .33524 2.98292 .35477 2.81870 28 33 .20716 3.3G516 .31626 3.16197 .33557 2.S8004 .35510 2.81610 27 34 .29748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 26 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29311 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37! .29843 3.3508? .31754 3.14922 .33686 2.96858 .35641 2.80574 23 38 29' '75 3.34732 , .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .99906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 3.33670 .31882 3.13656 .33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .35805 2.79289 18 43 j .30033 3.32965 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .80065 3.32814 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 48 .30193 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .30224 3.30S68 .32139 3.11153 .34075 2.93468 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10843 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 8 53 .30351 3.29483 .32267 3.09914 -.34205 2.92354 .36167 2.76498 7 54: .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 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 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 3 58 .30509 3.27767 .32428 3.08379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32460 3.0R073 .34400 2.90696 .36364 2.74997 1 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 / Cotanr' Tang Cotang Tang Cotang Tang Cotang Tang / V3 il 72 71 70 352 TABLE XXT.-TANOENTS AND COTANGENTS. ! 20 || 21 22 23 I Tang | Cotuiig Tang Cotang Tang Cotang Tang | Cotang .36397 2.74748 ! .38386 2.60509 . 40403 2.47509 .424ir" 2.35585 60 1 .38430 2.74499 .38420 2.60283 .40436 2.47302 .42483 2.35395 59 2 .36463 2.74251 .38453 2.60057 .40470 a. 47095 .42516 2.35205 r,s 3 .36496 2.74004 .38487 2.59831 .40504 2.46888 .42551 2.35015 5 1 ! 4 .86539 2.73756 .38520 2.59G06 .40538 2.46682 .4&ZS5 2.34825 56 5| .36363 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42054 2.34447 54 7 .36623 2.73017 .38620 2.58932 .40640 2.46065 .43688 2.34258 53 8 .36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 .42757 2.33881 5] 10, .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 50 11 .36760 2. 72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 10 12 .36793 2 71792 .38787 2.57815 .40809 2.45043 .42800 2.33317 48 13 .36825 Si 71548 .38821 2.57593 .40843 2.44839 .42894 g! 33130 47 1! .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.82943 46 15 .36893 2.71083 .38888 2.57150 .40911 2.44433 .42963 2.32756 K 16 .38925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 H 17 .36953 2.70577 .38055 2.56707 .40979 2.44027 .43032 2.32883 43 18 .36991 2.70335 .38988 2.56487 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39022 2.56266 .41047 2.43623 .43101 2.32013 11 20 .37057 2.69853 .39055 2.56046 .41081 2.4:3422 .43136 2.31826 W 211 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.69371 .39122 2.55608 .41149 2.43019 .43205 2.31456 88 23 .37157 2.69101 .39156 2.55389 .41183 2.42819 .43230 2.31271 24 .37190 2.68892 .39190 2.5517'0 .41217 2.42618 .43274 2.31086 86 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.30902 35 26 1 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 84 27 .37289 2.68175 .39290 2.54316 .41319 2.42019 .43378 2.80534 83 ^ .37322 2 67937 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 ! 39357 2.54082 .41387 3.41620 .43447 2.30167 31 80 .37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 J30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 !29 82 .37455 2.66989 .39453 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37483 2.66752 .39493 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39528 2.53001 .41553 2.40629 .43620 2.29254 26 35 .37554 2.66281 ! .39559 2.52786 .41592 2.40432 .43654 2.29073 25 86 .37588 2.66046 . .39593 2.52571 .41626 2.40235 .43689 2.28891 21 37 .37621 2.65811 i .39826 2,52357 .41660 2.40038 .43724 2.28710 23 88 .3765i 2.65576 .39660 2.52142 .41694 2.39841. .43758 ,'.'.2S52S 22 39 .37687 2.65342 .39894 2.51929 [41728 2.39645 .43793 y. 28348 21 10 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.279S7 19 42 37787 2.64643 .39r93 2.51289 .41831 2.39058 .43897 2.27806 18 48 .37823 2.64410 .39829 2.51076 .41865 2.38863 .43933 2.27626 17 44 .37853 2.64177 .39362 2.50864 41899 2.38668 .43966 2.27'447 16 45 37887 2.63945 .39896 2.50653 .41933 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .41968 2 38279 .44036 2.27088 114 47 .37953 2.63483 .39963 2.50229 .42002 2.'38084 .44071 2.26909 18 18 .37985 2.63253 .39997 2.50018 .42036 2.37891 .44105 2.26730 (2 48 .38020 2.63021 .40031 2.49807 .42070 2.37697 .44140 2.26553 11 50 .33053 2.62791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .38088 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 a .38120 2.62333 .40132 2.49177 .42173 2.37118 .44244 2.26018 H 53 .38153 2.62103 .40166 2.48967 .42207 2.3G925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25603 (i 55 .38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 B 56 .38253 2.61418 .40267 I 2 48340 .42310 2.36349 .44384 2.25309 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 3 58 .38320 2.60963 .40335 ! 2.47924 .42379 2.359fi7 .44453 2.24956 % 59 ! 38353 2.(i()73ri .40369 ! 2.47716 .42413 2.35776 .44488 224780 1 60 .38386 2.60509 .40403 2.47509 \ .42147 2.35585 .44523 2.24604 U / Cotang | Tang Cotang ; Tang Cotang Tang Cotang Tang > 69 !| 68* II 67 H 66 TABLE xxi. TAX<;F.XTS AND COTANGENTS. } \ 24 ' 25 ' 26 i 27 Tang i Cotang i Tang Cotang Tang Cotang Tang Cotang t. .44523 2.24604 .46631 2.14451 .48773 2.05030 "7511953" 1.96261 (50 1 .44558 2.244.28 .46666 2.14288 .48809 2.04879 .50989 1.96120 ! 59 2 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 1.95838 !57 1 .44662 2.23903 .46772 2.13801 .48917 2.04426 .51099 1.95698 '56 5 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.95557 55 G .44732 2.23553 .46843 2.13477 .48989 2.04125 .51173 1.95417 ,54 7 .44767 a!23378 i .46879 2.13316 .49026 2.0397'5 .51209 1.95277 53 6 .44802 2.2-5204 .46914 2.13154 .49062 2.03825 .51240 1.95137 52 9 .44837 2.23030 .46950 2.12993 .49098 2.03S75 .51283 1.94997 151 10 .44873 2.22857 i .46985 2.12832 .49134 2.0352G .51319 1.94858 50 1! .44907 2.22683 ! .47021 2.12671 .49170 2.03376 .51:356 1.94718 49 12 .44912 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 '48 13 .44977 2.22337 1 .47002 2.12350 .49242 2.03078 .51430 1.94440 47' 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 '45 in .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 1.94023 44 ir .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 1.93885 43 is .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 1.93746 !42 19 .45187 2.21304 .47305 2.11392 .49459 2.02187 .51G51 1.93608 41 90 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 81 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 1.93195 38 23 .45327 2.20619 .47443 2.10758 .49604 2.01596 .51798 1.93057 |37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 ! .51835 1.92920 |36 26 .45397 2.20378 .47519 2.10442 .49677 2.01302 i .51873 1.92782 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 1 .51909 1.92645 34 27 .45467 2.19938 .47590 2.10126 .497'49 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47626 2.09969 .497'86 2.00862 .51983 1.92371 32 29 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 i. 91 962 29 32 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 1.91626 28 33 .4567'8 2.18923 .47805 2.09184 .49967 2.00131 .521G8 1.91G90 ^7 84 .45713 2! 18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 30 .45784 2.18419 .47912 2.08716 .5007'6 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 3.08560 .50113 1.99550 .52316 1.91147 23 58 .45854 2.18084 .47984 2.08405 .50149 .99406 .52353 1.91012 22 31) .45889 2.17916 .48019 2.08250 .50185 .99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 .99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 .98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 i .98828 .52501 1.90472 18 43! .46030 2.17249 .48163 2.07630 .50331 ! .98684 1 .52538 1.90337 17 44 j .46065 2.17083 .48198 2.07476 .50368 | .98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 .98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 .98253 .52650 1.89935 i!4 47 .46171 2.16585 .48306 2.07014 .50477 .98110 .52687 1.8S801 13 4S .46206 2.16420 .48342 2.06860 .50514 1.97966 .52724 1.89667 12 4!) .46242 2.1 6255 .48378 2.06706 .50550 1.97823 .52761 1.89533 11 50 .46277 2.16090 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.8926b 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46383 2.15596 .48521 2.06094 .50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 5 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 .53022 1.88602 4 57 1 .46525 2.14940 .48665 2.05485 .50843 1.96685 .53059 1.88469 1 3 58 i .46560 2.14777 .48701 2.05333 .50879 1.96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 1.96261 .53171 1.88073 J Cotang Taug Cotang Tang Cotang Tang Cotang Tang / 65 64 63 62 TABLE XXI. TAN(JKXTS AND r< >TAN< JKNTS. 28 29 30 31 Tang Cotang Tang Cotang Tang | Cotang Tang Cotang Of .53171 1.88073 .55431 1.80405 .57735 1.73205 .ooost; i.(;0428 60 j .53208 1.87941 .55469 1.80281 .57774 1.73089 .60126 1.66318 59 .53246 1.87809 . .55507 1.80158 .57813 1.72973 .60105 1.06209 58 8 .53283 1.87677 .55545 1.80034 .57851 1.7'2857 .60205 ! 1. (56099 |57 4 .53320 1.87546 .55583 1.79911 .57890 1.72741 j .60245 1.65990 :Z6 5 .53358 1.87415 .55621 1.79788 , .57929 1.72625 .60284 1.65881 155 (1 .53395 1.87283 .55659 1.79665 .57908 1.72509 -6032J 1.6S772 T4 7' .53432 1.87152 .55697 1.79542 .58007 1.72393 j! .60364 ; 1.05663 5o 8 .53470 1.87021 .55736 1. 71)41 9 .58046 1.72278 .60403 1.65554 JM .53507 1.86891 .55774 1.711296 ! .58085 1.72163 .60443 1.05445 51 10 .53545 1.86760 .55812 1.79174 .58124 1.72047 i .00483 1.05337 BO 11 .53582 1.86630 .55850 1.79051 .58162 1.71932 j .60522 1.65228 49 1-2 .53620 1.86499 .55888 1. 78929 .58201 1.71817 i! .60562 1.05100 4S 13 .53657 1.86369 .55926 1.78807 .58240 1.71702 .(1000:2 1.05011 :47 14 .53694 1.86239 i .55964 1.78685 .58279 1.71588 i .60642 1.645)03 40 15 .53732 1.86109 1 .56003 1.78563 .58318 1.71473 i .60681 1.04795 45 16 .53769 1.85979 .56041 1.78441 .58357 1.71358 1 .60721 1.040H7 44 17 .53807 1.85850 .56079 1.78319 .58396 1.71244 .60761 1.01571) 4:i 18 .53844 1.85720 .56117 1.78198 .58435 1.71129 .60801 1.64471 4.'2 19: .53882 1.85591 .56156 1.78077 .58474 1.71015 .OOS41 1.C480S 41 20 .:>:fc>o 1.85462 .56194 1.77955 : .58513 1.70901 .60881 1.04250 40 21 .53957 1.85333 .56232 1.77834 .58552 1.70287 1 .60921 1.64148 80 22 .53995 1.85204 .56270 1.77713 .58591 1.70673 .60960 1. MO 11 88 28 .54032 1.85075 .56309 1.77592 .58631 1.70560 : .61000 1.CS934 :j< 24 .54070 1.84946 .56347 1.77471 .58670 1.70446 .01040 1.63820 ^30 26 .54107 1.84818 .56385 1.77a51 .58709 1.70332 .61080 1.03719 85 26 .54145 1.84689 .56424 1.77230 ! .587'48 1.70219 .61120 1.03612 |34 27 .54183 1.84561 .56462 1.77110 .58787 1.70106 .61160 1.03505 83 28 .54220 1.84433 .56501 1.76990 .58826 1.69992 .01200 1.63S98 82 29 .54258 1.84305 .56539 1.76869 .58865 1.69879 .61240 1.63292 ,-51 80 .54296 1.84177 .56577 1.78749 .58905 1.69766 .01280 1.03185 80 31 .54333 1.84049 .56616 1.76629 1 .58944 1.69653 1 .61320 1.G3079 89 82 .54371 1.83922 .50G54 1.76510 ! .58983 1.C9541 .61300 1.0^972 28 S3 .54409 1.83794 .56693 1.76390 .50022 1.69428 .61400 1.62866 27 ;n .54446 1.83667 .56731 1.76271 .59061 1.69316 .61440 1.62700 20 35 .54484 1.83540 .56769 1.76151 .59101 1.69203 .61480 1.62654 25 36 .54522 1.83413 .56808 1.76032 .59140 1.69091 .61520 1.62548 x>4 37 .54560 1.83286 .56846 1.75913 .59179 1.68979 .61561 1.02442 2;> 38: .54597 1.83159 .56885 1.75794- 1 .59218 1.68866 .61601 1.02336 22 39 .54635 1.83033 .56923 1.75675 .C9258 1.68754 .61641 1.62230 83 40 .54673 1.82906 .56962 1.75556 .59297 1.68G43 .61681 1.62125 2.) 41 .54711 1.82780 .57000 1.75437 .59336 1.68681 .61721 1.02019 19 4.2 .54748 1.82654 .57039 1.75319 .59376 1.08419 .61761 1.01914 18 43 .54786 1.82528 1 .57078 l! 75200 1 .59415 1.68308 .61801 i Gi SOB 17 44 .54824 1.82402 .57116 1.75082 .59454 1.68196 .61842 1.01703 10 45 .54862 1.82276 .57155 1.74964 .59494 1.68085 .61882 1.01598 15 46 .54900 1.82150 .57193 1.74846 i .59533 1.67974 .61922 1.01493 14 47 .54938 1.82025 .57232 1.74728 .59573 1.67863 : 61962 1.61388 18 48 .54975 1.81899 .57271 1.74610 .59612 1 . 67752 I .62003 1.01283 12 4!) .55013 1.81774 .57309 1.74492 .59651 1.C7641 .62043 1.01179 11 50 .55051 1.81649 .57348 1.7437'5 .59691 1.67530 .02083 j 1.01074 10 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 1.00865 8 53 .55165 1.81274 .57464 1.74022 : .59809 1.67198 .62201 1.60761 7 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 .62245 1.00057 6 55 .55241 1.81025 .57541 1.73788 .59888 1.66978 .02285 1.00553 5 66 .55279 1.80901 .57580 1.73671 .59928 1.66867 .02325 1.00449 4 57 .55317 1.80777 .57619 ! 1.73555 .59967 1.66757 : .62800 1.00345 3 58! .55355 1.80653 .57657 1.73438 .60007 1.66647 i .62406 1.00241 2 59j .55393 1.80529 .57696 1.73321 .60046 1.66538 .62446 1.60137 i 60 .55431 1.80405 .57735 1.73205 | .60086 1.06428 .02487 1.00033 ^ j Cotang Tang Cotang Tang ! Cotang Tang Cotang Tang 1 61 60 59 ll 68 TABLE XXI. TANGENTS AND COTANGENTS. 355 32 33 34 35 Tang Cotang i Tang Cotang Tang Cotang Tang Cotang .62487 1.600->! .f: 11 1.53986 .67451 1.48256 .70021 1.42815 60 1 .62527 1.51 '.; JO .04982 1.53888 .67493 1.48163 .70004 1.42720 59 2 .62568 1.59820 ll .65024 1.53791 .67536 1.48070 . 70107 1.42638 58 3 .62608 1.59723 .05005 1.53093 .67578 1.47977 ! .70151 1.42550 57 4 .62649 1.59020 .65106 1.53595 .67620 1.47885 i .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 6 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7 .62770 1.59311 .65231 1.53302 .67748 1.47607 ,70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70308 1.42110 52 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 '.65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 .629S3 1.58900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 12 .62973 1.5871)7 .65438 1.52816 .67900 1.47146 .70543 1.41759 48 13 .63014 1.58606 .65480 1.52719 .08002 1.47053 .705v3 1.41672 47 14 .63055 1.58593 .65521 1.52022 .68045 1.46962 !rOG29 1.41584 46 15 .63095 1.58490 .65563 1.52525 .68088 1.46870 .70673 1.41497 45 6 .63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1.41409 44 .63177 1.58286 .65646 1.52332 ! .68173 1.46686 .70760 1.41322 43 8 .63217 1.58184 .65688 1.52235 i .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.52139 ! .68258 1.46503 .70848 1.41148 41 20 .63299 1.57981 .65771 1.52043 ; .68301 1 .4647-1 .70891 1.41061 40 21 .6.3340 1.57879 .65813 1.51946 .68343 1.46320 .70935 1.40974 39 22 .63380 1.57773 .65854 x. 51850 .08386 1.46229 .70979 1.40887 38 23 .63421 1.57676 .65896 1.51754 .68429 1.46137 .71023 1.40800 37 24 .63462 1.57575 .65938 1.51058 .68471 1.46046 .71066 1.40711 36 25 .63503 1.57474 .65980 1.51562 .68514 1.45955 .71110 1.40027 35 26 .63544 1.57372 .66021 1.51466 i .68557 1.45864 .71154 1.40.540 ; .4 27 .63584 1.678*1 .66063 1.51370 | .68600 1.45773 .71198 1.40454 33 28 .63625 1.57170 .66105 1.51275 .68642 1.45682 .71242 1.40367 32 2U .63666 1.57069 .66147 1.51179 .68685 1.45592 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 ! .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.45410 .71373 1.40109 29 32 .63789 1.56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 33 .63830 1.56007 .66314 1.50797 .68857 1.45229 .71461 1.89936 27 34 .63871 1.56566 .66356 1.50702 .68900 1.45139 .71505 1.39850 26 35 .63912 1.56466 .66398 1.50607 .68942 1.45049 .71549 1.39764 25 36 .63953 1.56366 .66140 1.50512 .68985 1.44958 .71593 1.39679 24 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .00524 1.50322 .09071 1.44778 .71681 1.39507 22 39 .64076 1.5GOG5 .66506 1.50228 .69114 1.44688 .71725 1.39421 21 40 .64117 1.55966 .60608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 18 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .06776 1.49755 .69329 1.44239 .71946 1.38994 16 45 .64322 1.55467 .66818 1.49661 .6937'2 1.44149 .71990 1.38909 15 46 .64363 1.55368 .86860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .09502 1.43881 .72122 1.38653 12 49 .64487 1.55071 .66986 1.49284 .69545 1.43792 .72167 1.38568 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.88399 9 52 .64610 1.54774 .67113 1.49003 .69675 1.43525 .7'2299 1.38314 8 53 .64652 1.54675 ! .67155 1.48909 .69718 1.43436 .72344 1.38229 54 .64693 1.54576 i .67197 1.48816 .69761 1.43347 .72388 1.38145 6 55 .64734 1.54478 .67239 1.48722 .69804 1.43258 .72432 1.38060 P 56 .64775 1.54379 ! .67282 1.48629 .69847 1.43169 .72477 1 37976 i 5? .64817 1.54281 i .67324 1.4R536 .69891 1.43080 .72521 1.37891 t 58 .64858 1.54183 i .67366 1.48442 .69934 1.42992 .72565 1 37807 5 .64899 1.54085 1 .67409 1.48849 .69977 1.42903 .72610 1.37722 6C .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 ( / Cotang Tang Cotang Tang i Cotang j Tang Cotang Tang i 57 56 55 II 64 TABLE XXI. TANGENTS AND COTANGENTS. 36 37 i 88 || 39 Tang | Cotang |j Tang j Cotang Tang i Cotang il Tang Cotang .7^654 1.37638 ! .75355" l.r->704 .78129 I .27994 .80978 1.23490 60 1 .72099 1.37554 .75401 l.JfcttWl .78175 1.27917 .81027 1.23416 59 2 .72743 1.37470 .75447 1.32544 .78228 1.27841 .81075 1.23343 ;58 3 1 .72788 1.37386 .75493 1.32464 .78209 1.27704 .81123 1.23270 57 4 72832 1.37302 .75538 1.32384 .78316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 .78303 1.27611 .81220 1.23123 55 6 .72921 1.37134 .75029 1.32224 .78410 1.27535 .81268 1.23050 54 7 .72906 1.37050 .75075 1.32144 .78457 1.27458 .81316 1.22977 53 8 .73010 1.30967 .75721 1.32064 .7'8504 1.27383 .81364 1.22904 52 9 .73055 1.30883 .75767 1.31984 .78551 1.27306 .81413 1.22831 51 10 .73100 1.36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 50 11 .73144 1.36716 .75858 1.31825 .78645 1.27153 .81510 1.22685 49 12 .73189 1.36633 .75904 1.31745 .78698 1.27077 .81558 1.22012 48 13 .73234 1.36549 .75950 1.31666 .78739 1.27001 .81606 1.22539 47 14 .73278 1. 30466 .75996 1.31586 .78786 1.20925 .81655 1. 22407 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 .81703 1.22394 45 16 .73368 1.30300 .76088 1.31427 .78881 1.26774 .81752 1.22321 44 17 .73413 1.30217 i .76134 1.31348 .78928 1.26698 .81800 1/22249 43 18 .73457 1.30134 i .76180 1.31209 .78975 1.26622 .81849 1.22178 JO 19' .73503 1.30051 .76226 1.31190 i .79022 1.26546 .81898 1.22104 41 20 .73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 1.21959 39 22 .73637 1.33803 ! .76304 1.30952 .79104 1.26319 .82044 1.21886 38 23 .73681 1.35719 ! .76410 1.30873 .79212 1.26244 .82092 1.21814 37 24 ,73726 1.35037 .76456 1.30795 .79259 1.26169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 35 26 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 27 .73861 1.35389 .76594 1.30558 .79401 1.25943 .82287 1.21526 33 28 .73906 1.35307 i .76640 1.30480 .79449 1.25867 .82336 1.21454 32 29 .73951 1.352^4 .76686 1.30401 .79496 1.25792 .82385 1.21382 31 30 .73996 1.35142 i .76733 i.30323 .79544 1.25717 .82434 1.21310 30 31 .74041 1.35060 ' .76779 1.30244 .79591 1.25642 .824&3 1.21238 29 32 .74086 1.34978 .76825 1.30166 .79639 1.25567 .82531 1.21166 28 33 .74131 1.34806 .70871 1.30087 .79686 1.25492 .82580 1.21094 27 34 .74176 1.34814 .70918 1.30009 .79734 1.25417 .82629 1.21023 26 35 .74221 1.34732 .70964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .74207 1.34650 .77010 1.29853 1 .79829 1.25268 .82727 1.20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .71402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20605 21 40 .7414!? 1.34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 .74492 1.34242 .77243 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1.34160 .77289 1.29385 .80115 3.24820 .83022 1.20451 18 43 .74583 1.34079 .77335 1.29307 .80163 1.24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.21672 .83120 1.20308 16 45 .74674 1.33916 .77428 1.39152 .80258 1.24597 .83169 1.20237 15 46 .74719 1.33835 .77475 1.29074 .80306 1.24523 .83218 1.2016G 14 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 .74855 1.33592 .77615 1.28842 .80450 1.24301 .83366 1.19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 1.19740 8 53 .75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 1.19609 7 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 .83613 1 . 19599 6 55 .75128 1.33107 ,77895 1.28379 .80738 1.23858 .83662 1.19528 5 56 .75173 1.33026 .77941 1.28302 .80786 1.23784 ! 83712 1.19457 4 57 .75219 1.32946 .77938 1.28225 [80884 1.23710 .83761 1.19387 3 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .88811 1 . 19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.23563 .83860 1.19246 1 60 .75355 1.32704 .78129 1.27994 .80978 1.23490 .831/10 ; 1.19175 I Cotang Tang Cotang Tang Cotang Tang ',' Cotang Tang / 53 52 51 60 TABLE XXI.-TANGENTS AND COTANGENTS. 4 4 1 4 2 4 3 Taiig Cotang Tang Cotang Tang Cotang Tang Cotang .83910 1.19175 .86929 1.15037 .90040 1.11061 .93252 1.07237 60 1 .83960 1.19105 .86980 1.14969 .90093 1.10996 .93306 1.07174 59 9 .84009 1.19035 .87031 1.14902 .90146 1.10931 .93360 1.07112 58 a .84059 1.18964 .87082 1.14834 .90199 1.10867 .93415 1.07049 57 4 .84108 1 . 18894 .87133 1.14767 .90251 1.10802 .93469 1.06987 56 6 .84158 1.18824 .87184 1.14699 .90304 1.10737 .93524 1.06925 55 C .84208 1.18754 .87236 1.14632 .90357 1.10672 .93578 1.06862 54 7 .84258 1.18684 .87287 1.14565 .90410 1.10607 .93633 1.06800 53 8 .84307 1.18614 .87338 1.14498 .90463 1.10543 .93688 1.06738 52 i> .84357 1 . 18544 .87389 1.14430 .90516 1.10478 .93742 1.06676 51 10 .84407 1.18474 .87441 1.14363 .90569 1.10414 .93797 1.06613 50 11 .84457 1.18404 .87492 1.14296 .90621 1.10349 .93852 1.06551 41) 12 .84507 1.18334 .87543 1.14229 .90674 1.10285 .93906 1.06489 48 13 .84556 1 . 18264 I .87595 1.14162 .90727 1.10220 .93961 1.06427 4. 14 .84606 1.18194 .87646 1.14095 .90781 1.10156 .94016 1.06365 16 15 .84656 1 . 18125 ! .87698 1.14028 .90834 1.10091 .94071 1.06303 15 16 .84706 1.18055 .87749 1.13961 .90887 1.10027 .94125 1.06241 44 17 .84756 1.17986 .87801 1.13894 ' .90940 1.09963 .94180 1.06179 13 18 .84806 1.17916 .87852 1.13828 : .90993 1.09899 I .94235 1.06117 I'.! 19 .84856 1.17846 ! .87904 1.13761 .91046 1.09834 .94290 1.06056 11 20 .84906 1.17777 ; .87955 1.13694 .91099 1.09770 .94345 1.05994 10 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 1.05932 39 22 .85006 1.17638 .88059 1.13561 .91206 1.09642 .94455 1.05870 88 23 .85057 1.17569 i .88110 1.13494 .91259 1.09578 .94510 1.05809 87 2-1 .85107 1 . 17500 .88162 1.13428 .91313 1.09514 .94565 1.05747 80 25 .85157 1.17430 .88214 1.13361 .91366 1.09450 .94620 1.05685 35 26 .85207 1.17361 .88265 1.13295 .91419 1.09386 .94676 1.05624 34 27 .85257 1.17292 .88317 1.13228 .91473 1.09322 .94731 1.05562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 1.05501 32 29 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 1.05439 81 30 .85408 1.17085 .88473 1.13029 .91633 1.09131 .94896 1.05378 30 31 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 1.05317 29 32, .85509 1.16947 .88576 1.12897 .91740 1.09003 .95007 1.05255 28 33 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 1.05194 27 34 .85609 1.16809 '.88680 1.12765 .91847 1.08876 .95118 1.05133 26 35 .85660 1.16741 .88732 1.12699 .91901 1.08813 .95173 1.05072 25 36 .85710 1.16672 .88784 1.12633 I 91955 1.08749 I .95229 1.05010 24 37 .85761 1.16603 .88836 1.12567 .92008 1.08686 i .95284 1.04949 23 38 .85811 1.16535 .88888 1.12501 : .92062 1.08622 .95340 1.04888 23 39 .85862 1.16466 .88940 1.12435 l .92116 1.08559 .95395 1.04827 -,'1 40 .85913 1.16398 .88992 1.12369 .92170 1.08496 .95451 1.04766 20 41 .85963 1.16329 .89045 1.12303 .92224 1.08432 .95506 1.04705 1!) 42 .86014 1.16261 .89097 1.12238 .92277 1.08369 .95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92331 1.08306 .95618 1.04583 17 44 .86115 .16124 .89201 1.12106 .92385 1.08243 .95673 1.04522 16 45 .86166 .16056 .89253 1.12041 .92439 1.08179 .95729 1.04461 15 46 .86216 .15987 .89306 1.11975 .92493 1.08116 .95785 1.04401 14 47 .86267 .15919 .89358 1.11909 .92547 1.08053 .95841 1.04340 13 48 .86318 .15851 .89410 1.11844 .92601 1-07990 .95897 1.04279 13 49 .86368 .15783 .89463 1 11778 .92655 1.07927 .95952 1.04218 11 50 .86419 .15715 .89515 1.11713 .92709 1.07864 : .96008 1.04158 10 51 .86470 .15647 .89567 1.11648 .92763 1.07801 ! .96064 1.04097 9 52 .86521 . 15579 .89620 1.11582 .92817 1.07738 i .96120 1.04036 8 53 .86572 .15511 .89672 1.11517 .92872 1.07676 ! .96176 1.03976 7 54 .86623 .15443 .89725 1.11452 .92926 1.07613 i .96232 1.03915 6 55 .86674 .15375 .89777 1.11387 .92980 1.07550 .96288 1.03865 5 56 .86725 .15308 .89830 1.11321 .93034 1.07487 i .96344 1.03794 4 57 .86776 .15240 .89883 1.11256 .93088 1.07425 .96400 1.03734 3 58 .86827 .15172 .89935 1.11191 .93143 1.07362 .96457 1.0:3674 L' 59 .86878 1.15104 .89988 1.11126 .93197 1.07299 .96513 1.03613 1 00 .86929 1.15037 .90040 1.11061 .93252 1.07237 .96509 1.03553 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang i 4 9 4 8 i 4 7 4 6 TABLE XXI. TANGENTS AND COTANGENTS. 4 4 4 4 , II 4 4 Tang Cotang Tang Cotang Tang Cotang .96569 .03553 60 20 .97700 1.02355 -40 i .98843 1.01170 20 1 .96625 .03493 59 21 .97756 1.02295 39 - 1 .5)8901 1.01112 19 2 .96681 .03433 58 22 .97813 1.02236 ' 38 - 2 : .'.)*. )5S 1.01053 18 3 .96738 .03372 57 23 .97870 1.02116 37 ' 3 .won; 1.00994 17 4 .96794 .03312 56 24 .97927 1.02117 ,36 H I .1KK173 1.00935 16 6 .96850 .03252 55 25 .97984 1.02057 jay 1 4 5 .99131 1.00876 15 i 6 .96907 .03192 54 26 .98041 1.01998 34 -J .-; .ii;ii8;) 1.', OS IS 14 7 .96963 .03132 53 27 .98098 1.011)31) 33 4 7 .91)24? 1.00759 13 8 .97020 .03072 53 28 .98155 1.01879 :w 8 .99304 1.00701 12 .97076 .03012 51 29 .98213 1.01820 31:14 9 .99362 1.00642 11 10 .97133 .02952 50 30 .98270 1.01761 30 .99-420 1.00583 10 11 .97189 .02892 49 31 .98327 .01702 29! 'f 1 .99478 1.00525 9 .97246 .02832 48 32 .98384 : .01642 2H f 1.00467 8 13 .97302 .02772 47 83 .98441 .01583 27 F 3 .99594 1.00108 7 14 .97359 .02713 46 34 .98499 .01524 26 T 4 .9965 J 1 ~>0')50 6 15 .97416 .02653 45 85 .98556 .01465 25 T 5 .W710 1. W291 5 16 .97472 .02593 44 36 .98613 .01406 24 T 8 .99768 1 . 10233 4 17 .97529 .02533 43 37 .98671 : .01347 23 f 7 .99826 1. )0 175 3 18 .97586 .02474 42 38 .98728 .012S8 22 f 8" .99884 1.10116 19 .97643 .02414 41 39 .98786 .01229 21 ifi !) . !Ht:M. 1 'Oii.i,*. T 20 .97700 .02355 40 40 .98843 .01170 20: ( | l.OOOUO 1 . HKH'M) Cotang Tang Cotang Tang j Cotang Tang 4 5 I 4 5 1 1 4 5 J TABLE XXII.-VKRSIXES AND KXS ' 1 2 3 ' Vers. ! Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec, .00000 .00000 .00015 .00015 .00061 .00061 .00137 .00137 1 .00000 ; .00000 .00016 .00016 .00062 .00062 .00139 00139 1 2 1 .00000 .00000 .00016 .00016 .00063 .00063 .00140 .00140 2 3 .00000 .00000 .00017 .00017 .00064 .00064 .00142 .00142 3 4 .00000 .00000 .00017 .00017 .00065 .00005 .00143 .00143 4 5 .00000 .0)000 (1 .00018 .00018 .00066 .00066 .00145 .00145 5 6 .00000 .00000 1 .00018 .00018 .00067 .00067 .00146 .00147 6 7 .00000 .00000 .00019 .00019 .00068 .00068 .00148 .00148 7 8 .00000 .00000 .00020 .00020 .00069 .00069 .00150 .00150 8 9 .00000 .00000 .00020 .00020 .00070 .00070 .00151 .00151 9 to .00000 .00000 .00021 .00021 .00071 .00072 .00153 .00153 10 11 .00001 .00001 .00021 .00021 .00073 .00073 .00154 .00155 11 .00001 .00001 .00022 .00022 .00074 .00074 .00156 .00156 12 13 .00001 .00001 .00023 .00023 .00075 .00075 .00158 .00158 13 14 .00001 ! .00001 .00023 .00023 .00076 .00076 .00159 .00159 14 15 .00001 .00001 .00024 .00024 .00077 .00077 .00161 .00161 15 16 .00001 .00001 .00024 .00024 .00078 .00078 .00162 .00163 16 17 .00001 .00001 .00025 .00025 .00079 .00079 .00164 .00164 17 18 .00001 .00001 .00026 .00026 .00081 .00081 .00166 .00166 18 19 .00002 .00002 .00026 .00026 .00082 .00082 .00168 .00168 19 20 .00002 .00002 .00027 .00027 .00083 .00083 .00169 .00169 20 21 .00002 .00002 .00028 .00028 .00084 .00084 .00171 .00171 21 22 .00002 .00002 .00028 .00028 .00085 .00085 .00173 .00173 22 3 .00002 .00002 .00029 .00029 .00087 .00087 .00174 .00175 23 24 .00002 .00002 .00030 .00030 .00088 .00088 .00176 .00176 24 85 .00003 .00003 .00031 .00031 .00089 .00089 .00178 .00178 25 26 .00003 .00003 .00031 .00031 .00090 .00090 .00179 .00180 27 .00003 .00003 .00032 .00032 .00091 .00091 .00181 .00182 H~ 28 .00003 .00003 .00033 .00033 .00093 .00093 .00183 .00183 28 29 .00001 .00004 .00031 .00034 .00091 .00094 .00185 .00185 29 30 .00004 .00004 .00034 .00034 .00095 .00095 .00187 .00187 30 31 .00001 .00004 .00035 .00035 .00096 .00097 .00188 .00189 31 32 .00004 .00004 .00036 .00036 .00098 .00098 .00190 .00190 32 83 .00005 .00005 .00037 .00037 .00099 .00099 .00192 .00192 33 34 .00005 .00005 .00037 .00037 .00100 .00100 .00191 .00194 34 35 .00005 .00005 .00038 .00038 .00102 .00102 .00196 .00196 35 36 .00005 .00005 .00039 .00039 .00103 .00103 .00197 .00198 36 37 .00006 .00006 .00040 .00040 .00104 .00101 .00199 .00200 37 38 .00006 .00006 .00041 .00041 .00106 .00106 .00201 .00201 38 39 .00006 .00006 .00041 .00041 .00107 .00107 .00203 .00203 39 40 .00007 .00007 .00042 .00042 .00108 .00108 .00205 .00205 40 41 .00007 .00007 .00043 .00043 .00110 .00110 .00207 .00207 41 42 .00007 .00007 \ .00014 .00044 .00111 .00111 .00208 .00200 42 43 .00008 .00008 .00045 .00045 .00112 .00113 .00210 .00211 43 44 .00008 .00008 .00016 .00046 .00114 .00114 .00212 .00213 44 46 .00009 .00009 .00047 .00047 .00115 .00115 .00214 .00215 45 46 .00009 .00009 .00018 .00048 .00117 .00117 .00216 .00216 46 47 .00009 .00009 .00048 .00048 .00118 .00118 .00218 .00218 47 48 .00010 .00010 .00019 .00049 .00119 .00120 .00220 .00220 48 49 .00010 .00010 .00050 .00050 .00121 .00121 .00222 .00222 49 50 .00011 .00011 .00051 .00051 .00122 .00122 .00224 .00224 50 51 .00011 .00011 .00052 .00052 .00124 .00124 .00226 .00226 51 52 .00011 .00011 .00053 .00053 .00125 .00125 .00228 .00228 5.0 53 .00012 .00012 .000.54 .00054 .00127 .00127 .00230 .00230 53 54 .00012 .00012 .00055 .00055 .00128 .00128 .00232 .00232 54 55 .00013 .00013 .00056 .00056 | .00130 .00130 .00234 .00231 55 56 .00013 .00013 ,00057 .00057 I .00131 .00131 .00236 .00236 57 .00014 .00014 .00058 .00058 I! .00133 .00133 .00238 .00338 57 58 .00014 .00014 .00059 .00059 .00134 .00134 .00240 .00240 58 59 I .00015 .00015 .00060 .00060 .00136 .00133 .00212 .00242 59 60 1 .00015 .00015 .00061 .00061 i .00137 .00137 1 .00244 .00244 60 TABLE XXII. VEK.SINES AND KXSKCANTS, 12 1 13 14 15 Vers. Exsec. Vers. Exsec. . Vers. Exsec. Yers. Ex?ec. .(1-2185 .02191 .02197 . ea j .02210 .02216 . .-. ; .02340 .02246 .02253 .02271 .02277 .02314 j - ! . .02364 .02370 .02377 . $429 .02434 .02440 .02447 . J453 .02472 .03479 .1 3493 .02504 .02511 .08517 .06530 .03695 . 3548 .02550 .02834 .02340 .02247 .02253 I ! 02258 .02266 .02272 .02279 .0-2285 ! .02291 .03398 , .02304 | .02:311 .02317 ! .02323- .03330 \ .02343 .02349 . -.. 56 .02:369 .1 ,-. " . -:: 3 .02415 .02421 .02441 .02454 .02461 .02474 .03509 .02515 . 2521 . 3548 .02555 .03560 .02570 .02576 .02616 .02623 .02642 .02649 .02655 .1 . H .02! .02716 .03722 .03729 . .; . .02743 .' B56 .02770 .02777 !l 3788 [02790 02569 i .03907 02576 .02914 .02921 .03589 .03596 .096 1 .02928 .02935 .02943 .02630 .02637 .02644 .02651 ' . : 165 . 8673 .02679 .C26S6 .< J834 . 2870 . . . : .02987 .02994 .02970 .03046 ': 8970 .02977 . :,.- 09 19 .02700 .02707 '] .02728 . :' .( 3749 .02749 .02756 . ->-, .03770 .02777 .02784 .- .; -1 . ,- 6 .03027 .1 90M .03041 .03048 :; . 070 .090 1 .03106 .03113 .03120 .0312! .03142 . H4S .03163 .03171 .03178 .03185 .03193 . .; / 814 .1 3244 . ./I 08258 .( 879 .0 . ) .03076 ! .03084 | .03108 .03114 .03121 .03129 .03137 .03144 .03152 .03159 .03167 .03175 !08190 .1 .' 18 .a aa .03244 i .08251 ! . :: .03275 .03321 . 8337 .03345 .033.33 .03061 |, .03-107 .08416 .08424 .03432 .03439 .03447 .03340 .03455 ! .03463 i .03471 .03479 .03487 .03495 03503 .08512 .03920 .03370 .03400 .03407 .03415 .03433 .034:30 .03438 .03445 .03453 .03476 .03483 .03491 .( M98 .03506 . 3514 .03529 .03537 .03560 .03567 . /.. * .03598 .08006 .03614 .03621 .03637 .03645 .03676 .03707 .03715 .03723 .03731 .03739 .03762 .03770 [03778 [03788 .0791 .08810 [03818 .06842 .08868 !03B74 ANTS. 363 16 17 18 Vers. Exsec. Vers. ; Exsec. ; Vers. Exsec. to n i-J 18 14 15 16 17 18 .' U 34 35 :', : 88 89 40 41 -I-.! 43 44 45 46 47 48 49 60 M .088T4 i .04030 .03914 i .03953 .03071 .04011 I .04019 ! . 1088 .04036 | .04044 i . : ' i .04085 .04 03 .04110 .04118 .04126 .04135 .04143 .04151 .04159 [04168 .04176 0*193 .04501 .04334 .1 1868 .04303 .04810 [04819 .04337 j .0483 : .< 1844 .043(51 .04047 .04056 .04065 .04073 .04083 .04091 .04100 I .04108 .04117 .04136 .04135 .04144 .04161 .04170 .04179 .04188 .04306 .04314 i .04333 .04333 ! .04341 .04350 .04359 41968 .04377 .04395 .04304 .04313 .04388 [04331 .04:340 .04349 04S58 .04867 .04376 .04385 .04394* .04403 .04413 .04433 .04431 .04440 .04449 .04458 .04468 .04477 j .04486 j .04496 .04504 .04514 .04583 .04589 .04541 .04551 .04370 .04378 .04387 .04395 .04404 .04413 .04431 .04420 .04438 .04446 .04455 .04464 .04473 .04481 .04493 .04507 .04515 .04534 .04541 .04550 .04559 44567 .04576 [04585 .04593 44602 .04611 .04569 .04578 .04588 .04597 .04606 .< 1616 .04635 .04635 .04644 .04653 .04663 .04683- j .04894 i .04903 ; .04913 .04931 !04980 .04948 .04957 .05146 .05156 .05166 .05176 .05186 .05196 [05806 .05816 19' Vers. Exsec. .05448 .05458 .05467 .05477 [05486 .04637 .04646 .04655 .04668 [04673 .04681 [04600 .04700 .04710 .04719 .04739 .04738 .04748 , .04757 .04767 I .04776 .04786 .04795 . 1805 .04815 .04834 .04843 .04853 ; .04976 .04985 .04994 .05003 i .05013 [05021 .05346 .06856 .065 5 .05515 .05534 .05534 .05543 .05553 [06562 .06572 .66768 .05773 .06783 [05794 .05806 [05815 .' : 88 .05048 .05067 .05067 j .05076 ! .05065 . . : .05103 .05113 .05133 .05131 .05140 .05149 .05886 .0589! .05307 .05317 .0632! !05837 .05347 .05357 !05367 .05878 .1 5691 .05610 .05630 .<",: 9 .05847 8 --- .05869 10 .05879 ! 11 .05890 13 .05901 13 .05911 14 .05933 16 .05944 17 .05955 ! 18 .05965 19 .05976 20 .05649 .05987 . 0541 - .05418 .05489 .05678 .05687 .04879 .04703 .04716 .04725 .04734 .04743 .04753 .04760 [04769 .04778 44387 .04796 .04806 .04814 .04841 .04860 44858 .( 1891 [04901 .04911 .05168 : .05177 ! [05186 .05195 i [osaos [05814 .04940 .0495J .04959 .04979 .05848 45851 .05870 .05807 .05! 18 .05088 .05047 .05057 .050 7 .06077 45087 .05697 .05107 .05116 .06844 .05354 .06873 .04885 ! .05136 .04894 .05146 .05401 .05410 45480 .05489 !05448 .05449 .05460 .05470 .05480 .05490 [06601 [05511 .06581 .05543 .05573 [06684 .06604 .05604 [05615 05636 [05646 .05657 .05667 [05678 .05699 .05709 [05780 .05730 .05741 .05751 [05762 .05716 .05726 .05736 .05746 .05755 .05765 .05775 .06785 [06794 [06804 [05814 06020 .06030 .06041 .06052 .06074 .05843 .05863 [05873 .05912 .06951 .06971 .05991 .06001 [06011 [06681 30 31 .06107 .06118 [06189 .06140 .06151 .06173 38 .06184 39 .06195 40 .06206 .06817 41 48 .06339 44 .06860 4.-' [06861 -tf .06879 6 46889 H GO .06306 .06317 .06373 .06407 .06418 3G4 TABLE XXII.-VERSINES AND EXSECANTS. / 2( > 2] L 22 28 Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. ~~0~ .06031 .06418 .06642 .07115 .07282 .07853 .07950 .08636 1 .06041 .06429 .(XH553 .07126 .07293 .07866 .07961 .08649 1 2 .06051 .06440 .06663 .07138 .07303 .07879 .07972 .08663 2 3 .06061 .06452 .06673 .07150 .07314 .07892 .07984 .08676 3 4 .06071 .06463 .06684 .07162 .07325 .07904 .07995 .08690 4 5 .06081 .06474 .06694 .07174 .07336 .07917 .08006 .08703 5 6 .06091 .06486 .06705 .07186 .07347 ,07930 .08018 .08717 6 7 .06101 .06497 .06715 .07199 .07358 070-43 .08029 .08730 7 8 .06111 .06508 .06726 .07211 .07369 .07955 .08041 .08744 8 9 .06121 .06520 .06736 .07223 .07380 .07968 .08052 .08757 9 10 .06131 .06531 .06747 .07235 .07391 .07981 .08064 .08771 10 11 .06141 .06542 .06757 .07247 .07402 .07994 .08075 .08784 11 12 .06151 .06554 .06768 .07259 .07413 .08006 .08086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 .06171 .06577 .06789 .07283 .07435 .08032 .08109 .08825 14 15 .06181 .06588 .06799 .07295 .07446 .08045 .08121 .08839 15 16 .06191 .06600 .06810 .07307 .07457 .08058 .08132 .08852 16 IT .06201 .06611 .06820 .07320 ! .07468 .08071 .08144 .08866 17 18 .06211 .06622 .06831 .07332 .07479 .08084 .08155 .08880 18 19 .06221 .06634 .06841 .07344 .07490 .08097 .08167 .08893 19 20 .06231 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .06252 .06668 .06873 .07380 .07523 .08135 .08201 , 08934 22 23 .06262 .06680 .06884 .07393 .07534 .08148 .08213 .08948 23 24 .06272 .06691 .06894 .07405 .07545 .08161 .08225 .08962 24 25 .06282 .06703 .06905 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 .07568 .08187 .08248 .08989 26 27 .06302 .06726 .06926 .07442 .07579 .08200 .08259 .09003 27 28 .06312 .06738 .06937 .07454 .07590 .08213 .08271 .09017 28 29 .06323 .06749 .06948 .07466 .07601 .08226 .08282 .09030 29 30 .06333 .06761 .06958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .06773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06353 .06784 .06980 .07503 .07634 .08265 .08317 .09072 32 33 .06363 .06796 .06990 .07516 .07645 .08278 .08329 .09086 33 34 .06374 .06807 .07001 .07528 .07657 .08291 .08340 .09099 34 35 .06384 .06819 .07012 .07540 .07668 .08305 .08352 .09113 35 36 .06394 .06831 .07022 .07553 .07679 .08318 .08364 .09127 36 37 .06404 .06843 .07033 .07565 .07690 .08331 .08375 .09141 37 38 .06415 .06854 .07044 .07578 .07701 .08344 .08387 .09155 38 39 .06425 .06866 .07055 .07590 .07713 .08357 .08399 .09169 39 40 .06435 .06878 .07065 .07602 .07724 .08370 .08410 .09183 40 41 .06445 .06889 .07076 .07615 .07735 ..08383 .08422 .09197 41 42 .06456 .06901 .07087 .07627 .07746 .08397 .08434 .09211 42 43 .06466 .06913 .07098 .07640 .07757 .08410 .08445 .09224 43 44 .06476 .06925 .07108 .07652 .07769 .08423 .08457 .09238 44 45 .06486 .06936 .07119 .07665 .07780 .08436 .08469 .09252 45 46 .06497 .06948 .071:30 .07677 .07791 .08449 .08481 .09266 46 47 .06507 .06960 .07141 .07690 .07802 .08463 .08492 .09280 47 48 .06517 .06972 .07151 .07702 .07814 .08476 .08504 .09294 48 40 .06528 .06984 .07162 .07715 ! .07825 .08489 .08516 .09308 49 50 .06538 .06995 .07173 .07727 .07836 .08503 .08528 .09323 50 51 .06548 .07007 .07184 .07740 .07848 .08516 .08539 .09337 51 52 .06559 .07019 .07195 .07752 07859 .08529 .08551 .09351 52 53 .06569 .07031 .07206 .07765 .07870 .08542 .08563 .09365 53 54 .06580 .07043 .07216 .07778 .07881 .08556 .08575 .09379 54 55 .06590 .07055 .07227 .07790 .07893 .08569 .08586 .09393 55 56 .06600 .07067 .07238 .07803 .07904 .08582 .08598 .09407 56 57 .06611 .07079 .07249 .07816 .07915 .08596 .08610 .09421 57 58 .06621 .07091 .07260 .07828 .07927 .08609 .08622 .09435 58 59 .00632 .07103 .07271 .07841 .07938 .08623 .08634 .09449 59 60 .06642 .07115 .07282 .07853 .07950 .08636 .08645 .09464 60 TABLE XXH.-VERSINES AND EXSEOANTS. ' 24 25 26 27 / Vers. Exsec. Vcrs. Exsec. Vers. Exsec. Vers. Exsec. .08645 .094(54 .09369 .10338 .10121 .11200 .10899 .12233 1 .08657 .09478 .09382 .10353 .10133 .11276 .10913 .12249 1 2 .08669 .09492 .09394 .10368 .10146 .11292 .10926 .12266 2 3 i .08681 .09506 .09406 .10383 .10159 .11308 .10939 .12283 3 4 .08693 .09520 .09418 .10398 .10172 .11323 .10952 .12299 4 5 .08705 .09535 .09431 .10413 .10184 .11339 .10965 .12316 5 6 .08717 .09549 .09443 .10428 .10197 .11$55 .1097'9 .12333 6 7 .08728 .09563 .09455 .10443 .10210 .11371 .10992 .12349 7 8 .08740 .09577 .09468 .10458 .10223 .11387 .11005 .12366 8 9 .08752 .09592 .09480 .10473 .10236 .11403 .11019 .12383 9 10 .08764 .09606 .09493 .10488 .10248 .11419 .11032 .12400 10 11 .08776 .09620 .09505 .10503 .10261 .11435 .11045 .12416 11 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .12433 12 13 .08800 .09649 .09530 .10533 .10287 .11467 .11072 .12450 13 14 .08812 .09663 .09542 .10549 .10300 .11483 .11085 ..12467 14 15 .08824 .09678 .09554 .10564 .10313 .11499 .11098 .12484 r> 16 .08*36 .09692 .09567 .10579 .10326 .11515 .11112 .12501 16 17 .08848 I .09707 .09579 .10594 .10338 .11531 .11125 .12518 17 18 .08860 .09721 .09592 .10609 .10351 .11547 .11138 .12534 18 19 .08873 .09735 .09604 .10625 .10364 .11563 .11152 .12551 19 20 .08884 .09750 .09617 .10640 .10377 .11579 .11165 .12568 20 21 .08896 .09764 .09629 .10655 .10390 .11595 .11178 .12585 21 22 .08908 .09779 .09642 .10670 .10403 .11611 .11192 .12602 22 23 .08920 .09793 .09654 .10686 .10416 .11627 .11205 .12619 23 24 .08932 .09808 .09666 .10701 .10429 .11643 .11218 .12636 24 25 .08944 .09822 .09679 .10716 .10442 .11659 .11232 .12653 25 26 .08956 .09837 .09691 .10731 .10455 .11675 .11245 .12670 26 27 .08968 .09851 .09704 .10747 .10468 .11691 .11259 .12687 27 28 .08980 .09866 .09716 .10762 .10481 .11708 .11272 .12704 28 29 .08992 .09880 .09729 .10777 .10494 .11724 .11285 .12721 29 30 .09004 .09895 .09741 .10793 .10507 .11740 .11299 .12738 30 31 .09016 .09909 .09754 .10808 .10520 .11756 .11312 .12755 31 32 .09028 .09924 .09767 .10824 .10533. .11772 .11326 .12772 32 33 .09040 .09939 .09779 . 10839 .10546 .11789 .11339 .12789 33 31 .09052 .09953 .09792 .10854 .10559 .11805 .11353 .12807 34 35 .09064 .09968 .09804 .10870 .10572 .11821 .11366 .12824 35 36 .09076 .09982 .09817 .10885 .10585 .11838 .11380 .12841 36 37 .09089 .09997 .09829 .10901 .10598 .11854 .11393 .12858 37 38 .09101 .10012 .09842 .10916 .10611 .11870 .11407 .12875 38 39 .09113 .10026 .09854 .10932 .10024 .11886 .11420 .12892 39 40 .09125 .10041 .09867 .10947 .10637 .11903 .11434 .12910 40 41 .09137 .10055 .09880 .10963 .10650 .11919 .11447 .12927 41 42 .09149 .10071 .09892 .10978 .10663 .11936 .11461 .12944 42 43 .09161 .10085 .09905 .10994 .10676 .11952 .11474 .12961 43 44 .09174 .10100 .09918 .11009 .10689 .11968 .11488 .12979 44 45 .09186 .10115 .09930 .11025 . 10702 .11985 .11501 .12996 45 46 .09198 .10130 .09943 .11041 .10715 .12001 .11515 .13013 46 47 .09210 .10144 .09955 .11056 .10728 .12018 .11528 .13031 47 48 .09222 .10159 .09963 .11072 .10741 .12034 .11542 .13048 48 49 .09234 .10174 09981 . 11087 .10755 .12051 .11555 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11569 .13083 50 51 .09259 .10204 .10006 .11119 .10781 .12084 .11583 .13100 51 52 .09271 .10218 .10019 .11134 .10794 .12100 .11596 .13117 52 53 .09283 .10233 ! .10032 .11150 .10807 .12117 .11610 .13135 53 54 .09296 .10248 .10044 .11166 .10820 .12133 .11623 .13152 54 55 .09308 .10263 .10057 .11181 .10833 .12150 .11637 .13170 55 56 .09320 .10278 .10070 .11197 .10847 .12166 .11651 .13187 56 57 .09&32 .10293 .10082 .11213 .10860 .12183 .11664 .13205 57 58 .09345 .10308 .10095 .11229 .10873 .12199 .11678 .13222 58 59 .09357 .10323 .10108 .11244 .10886 .12216 .11692 .13240 59 60 .09369 .10338 .10121 .11260 .10899 .12233 .11705 .13257 60 vKi;siM-:s AND r.\sKr.\NTs. / 2 8 2 9 3( ) 3 L t Vers. Exsec. Vers. Exsec. Vers. ( Exsec. Vers. Exsec. ~ .11705 .13257 .12538 .14335 .13397 .15470 .14283 .10603 1 .11719 .13275 .12552 .14354 .13412 .15489 .14298 .10084 1 2 .11733 .13202 .12566 . 14372 .13427 .15509 .14313 .10704 2 8 .11746 .13310 .12580 .14391 .13441 .15528 .14328 .10725 3 4 .11760 .13327 .12595 .14409 .13456 .15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .13470 . 15507 .14358 .16706 5 6 .11787 .13362 .12623 .14446 .13485 .15587 .14373 .16786 6 7 .11801 .13380 .12637 .14405 .13499 .15606 .14388 .10806 7 8 .11815 .13398 .12651 .14483 .13514 .15020 .14403 .10827 8 9 .11828 .13415 .12665 .14502 .13529 .15645 .14418 .16848 9 10 .11*12 .13433 .12079 .14521 .13543 .15005 .14433 .16868 10 11 .11856 .13451 .12694 .14539 .13558 .15684 .14449 .10889 11 12 .11870 .13468 .12708 .14558 .13573 .15704 .14404 .10909 12 13 .11883 .13486 .12722 .14576 .13587 .15724 .14-179 .10930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .14-194 .10950 14 15 .11911 .13521 .12750 .14614 .13616 .15763 .14509 .16971 15 16 .11925 .13539 .12705 .14632 .13031 .15782 .14524 .16992 16 17 .11938 .13557 .12779 .14651 .13646 .15802 .14539 .17012 17 18 .11952 .13575 .12793 .14670 .13660 .15822 .14554 .17033 18 19 .11006 .13593 .12807 .14689 .13675 .15841 .14569 .17C54 19 20 .11980 .13610 .12822 .14707 .13690 .15861 .14584 .17075 20 21 .11994 .13628 .12836 .14726 .13705 .15881 .14599 .17095 21 22 .12007 .13646 .12850 .14745 .13719 .15901 .14615 .17116 22 23 .12021 .13664 .12864 .14764 .13734 .15920 .14630 .17137 23 24 .12035 .13682 .12879 .14782 .13749 .15940 .14645 . 7158 24 25 .12049 .13700 .12893 .14801 .137(33 .15960 .14660 . 7178 25 26 .12063 .13718 .12907 .14820 .13778 .15980 .14075 . 7199 26 27 .12077 .13735 .12921 .14839 .13793 .16000 .14090 . 7220 27 28 .12091 .13753 .12936 .14858 .13GC8 .16019 .14700 . 7241 28 29 .12104 .13771 .12950 .14877 .13822 .16039 .14721 . 7362 29 30 .12118 .13789 .12964 .14896 .13837 .16059 .14736 . 7283 30 31 .12132 .13807 .12979 .14914 .13852 .16079 .14751 .17304 31 32 .12146 .13825 .12993 ..14933 .13867 .16099 .14766 .17325 32 33 .12160 .13843 .13007 .1495 .13881 .16119 .14782 .17346 33 34 .12174 .13861 .13022 .14971 .13896 .16139 .14797 .17367 34 35 .12188 .13879 .13036 .14990 .13911 .16159 .14812 .17388 35 36 .12202 .13897 .13051 .15009 .13926 .16179 .14827 .17409 36 37 .12216 .13916 .13065 .15028 .13941 .16199 .14843 .17430 37 38 .12230 .13934 .13079 .15047 .13955 .16219 .14858 .17451 38 39 .12244 .13952 .13094 .15006 .13970 .16239 .14873 .1747'2 39 40 .12257 .13970 .13108 .15085 .13985 .16259 .14888 .17493 40 41 .12271 .13988 .13122 .15105 .14000 .16279 .14904 .17514 41 42 .12285 .14006 .13137 .15124 .14015 .16299 .14919 .17535 42 43 .12299 .14024 .13151 .15143 .14030 .16319 .14934 .17556 43 44 .12313 .14042 .13106 .15102 .14044 .16339 .14949 . 7577 44 45 .12327 .14061 .13180 .15181 .14059 .16359 .14965 .17598 45 46 .12341 .14079 .13195 .15200 .14074 .16380 .14980 .17620 46 47 .12355 .14097 .13209 .15219 .14089 .16400 .14995 .17641 47 48 .12369 .14115 .13223 .15239 .14104 .16420 .15011 .17662 48 49 .12383 .14134 .13238 .15258 .14119 .16440 .15026 .17083 49 50 .12397 .14152 .13252 .15277 .14134 .16460 .15041 .17704 50 51 .12411 .14170 .13267 .15296 .14149 .16481 .15057 .17726 51 52 .12425 .14188 .13281 .15315 .14164 .16501 .15072 .17747 52 53 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 53 54 .12454 .14225 .13310 .15354 .14194 .16541 .15103 .17790 54 55 .12468 .14243 .13325 .15373 .14208 .16562 .15118 .17811 55 56 .12482 .14262 .1.3339 .15393 .14223 .16582 .15134 .17832 56 57 .12496 .14280 .13354 .15412 .14238 .16602 .15149 .17854 57 58 .12510 .14299 .1,3368 .15431 .14253 .16623 .15164 .17875 58 59 . 12524 .14317 .13383 .15451 .14268 .16643 .15180 .17896 59 60 .12538 .14335 .13397 .15470 .14283 .16663 .15195 .17918 60 TAKLK XXII. -VKKS1NKS AND EXSECANTS. ' 32 33 C 34 35 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .15195 .17918 .161:8 .192MJ .17096 .20622 .18085 .22077 1 .15211 .17939 .10149 .1-9259 .17113 .20645 .18101 .22102 1 2 .15226 .17961 .10105 .19281 .17129 .20669 .18118 .22127 2 3 . 15241 .17982 .16181 .19304 .17145 .20693 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19349 .17178 .20740 .18168 .22202 5 6 .15288 .18047 i .16228 .19372 .17194 .20764 .18185 .22227 6 7 .15303 .18068 .16244 .19394 .17210 .20788 1 .18202 .22252 7 8 .15319 .18090 .16260 .19417 .17227 .20812 .18218 .22277 8 9 .15334 .18111 .16276 .19440 .17243 .20836 .18235 .22302 9 10 .15350 .18133 .16292 .19463 .17259 .20859 .18252 .22327 10 11 .153G5 .18155 .16308 .19485 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17292 .20907 .18286 .22377 12 13 .15396 .18198 .16340 .19531 .17308 .20931 .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 .20955 !l88*9 .22428 14 15 .15427 ' .18241 .16371 .19576 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18353 .22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .18369 .22503 17 18 .15474 .18307 .16419 .19645 .17390 .21051 .18386 .22528 18 19 .15489 .18328 .16435 .19668 .17407 .21075 .18403 .22554 19 20 .15505 .18350 .16451 .19691 .17423 .21099 .18420 .22579 20 21 .15520 .18372 .1G467 .19713 .17439 .21123 .18437 .22604 21 22 .15536 .18394 .16483 .19736 .17456 .21147 .18454 .22629 22 23 .15552 .18416 .16499 .19?'59 .17472 .21171 .18470 .22655 23 24 .15567 .18437 .16515 .19782 .17489 .21195 .18487 .22680 24 25 .15583 .18459 .16531 .19805 1 .17505 .21220 .18504 .22706 25 26 .15598 .18481 .16547 .19828 .17522 .21244 .18521 .22731 26 7 .15614 .18503 .16563 .19851 .17538 .21268 .18538 .22756 27 28 .15630 .18525 .16579 .19874 : .17554 .21292 .18555 1 .22782 28 29 .15645 .18547 .16595 .19897 .17571 .21316 .18572 ; .22807 29 30 .15661 .18569 .16611 .19920 .17587 .21341 .18588 .22833 30 31 .15676 .18591 .16627 .19944 .17604 .21365 .18605 .22858 31 32 .15692 .18613 .16644 .19967 .17620 .21389 .18022 .22884 32 33 .15708 .18635 .16660 ! .19990 .17637 .21414 .18639 .22909 33 34 .15723 .18657 .16676 i .20013 .17653 .21438 .18656 .229a5 34 35 .15739 .18679 .16692 .20036 .17670 .21462 .18673 .22960 35 36 .15755 .18701 .16708 .20059 .17686 i .21487 .18690 .22986 36 37 .15170 .18723 .16724 .20083 .17703 .21511 .18707 .23012 37 38 .15786 .18745 .16740 .20106 .17719 .21535 .18724 .23037 38 39 . 15S02 .18767 .16756 .20129 : .17736 .21560 .18741 .23063 39 40 .15818 .18790 .16772 .20152 .17752 .S1584 .18758 .23089 40 41 .15833 .18812 .16788 .20176 .17769 .21609 .18775 .23114 41 42 .15849 .18834 .16805 .20199 .17786 .21633 .18792 .23140 42 43 .15865 .18856 .16821 .20222 .17802 .21658 .18809 .23166 43 44 .15880 .18878 .16837 .20246 j .17819 .21682 .18826 .23192 44 45 .15896 : 18901 .16853 1 .20269 .17835 .21707 .18843 .23217 45 46 .15912 .18923 .16869 .20292 .17852 .21731 .18860 .23243 46 47 .15923 .18945 .16885 .20316 .17868 .21756 .18877 .23269 47 48 .15943 .18967 .16902 .20339 .17885 .21781 .18894 .23295 48 49 .15959 .18990 .16918 .20363 .17902 .21805 .18911 .23321 49 50 .15975 .19012 .16934 .20386 ; .17918 .21830 .18928 .23347 50 51 .15991 .19034 .16950 .20410 .17935 .2ia55 .18945 .23373 51 52 .16006 .19057 .16966 .20433 .17952 .21879 .18962 .23399 52 53 .16022 .19079 .16983 .20457 .17968 .21904 .18979 .23424 53 54 .10038 .19102 .16999 .20480 i .17985 .21929 .18996 .23450 54 55 .16054 .19124 .17015 .20504 1 .18001 .21953 .19013 .23476 55 56 .10070 .19146 .17031 .20527 !' .18018 .21978 .19030 .23502 56 57 ! 16085 .19169 .17047 .20551 .18035 .22003 .19047 .23529 57 58 .16101 .19191 .17064 .205;-) .18051 .22028 .19064 .23555 58 59 .16117 .19214 .17080 .20598 .18008 .22053 .19081 .23581 59 60 .16133 .19236 .17096 .20022 .18085 .22077 U .19098 .23607 60 36S AULE xxii. VEfcsiNEs AND EXSECANTS. / 36 37 38 30 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. ~T .19098 .23007 .201S6 .25214 .21199 .26902 .22285 .28676 i .19115 .23633 .20154 .25241 .21217 .26931 .22304 .287'06 1 2 .19133 .23659 .20171 .25269 .21235 .26960 .22322 .28737 2 3 .19150 .23685 .20189 .25296 .21253 .26988 .22340 .28767 3 4 .19107 .23711 .20207 .25324 .21271 .27017 .22359 .28797 4 5 .1918! .23738 .20224 .25351 .21289 .27046 .22377 .28828 5 6 .19201 .23764 .20242 .25379 .21307 .27075 .22395 .28858 6 7 .19218 .23790 .20259 .25406 .21324 .27104 .22414 .28889 7 8 .19235 .23816 .20277 .25434 .21342 .27133 .22432 .28919 8 9 .19252 .23843 .20294 .25462 .21360 .27162 .22450 .28950 9 10 .19270 .23869 .20313 .25489 .21378 .27191 .22469 .28980 10 11 .19287 .23895 .20329 .25517 .21396 .27221 .22487 .29011 11 12 .19304 .23922 .20347 .25545 .21414 .27250 .22500 .29042 12 13 .19321 .23948 .20365 .25572 .21432 .27279 .22524 .29072 13 u .19338 .23975 .20382 .25600 .21450 .27308 .22542 .29103 14 15 .19356 .24001 .20400 .25628 .21468 .27337 .22561 .29133 15 16 .19373 .24028 .20417 .25656 .21486 .27366 .22579 .29164 16 17 .19390 .24054 .20435 .25683 .21504 .27396 .22598 .29195 ! 17 18 .19407 .24081 .20453 .25711 .21522 .27425 .22616 .29226 18 19 .19424 .24107 .20470 .25739 .21540 .27454 .22634 .29256 19. 20 .19442 .24134 .20488 .25767 .21558 .27483 .22653 .29287 20 21 .19459 .24160 .20506 .25795 i .21576 .27513 .22671 .29318 21 22 .19476 .24187 .20523 .25823 i .21595 .27542 .22690 .29349 22 23 .19493 .24213 .20541 .25851 i .21613 .27572 .22708 .29380 23 24 .19511 .24240 .20559 .25879 j .21631 .27601 .22727 .29411 24 25 .19528 .24267 .20576 .25907 i .21649 .27630 .22745 .29442 25 26 .19545 .24293 .20594 .25935 .21667 .27660 .22764 .29473 26 27 .19502 .24320 .20612 .25963 .21685 .27689 .22782 .29504 27 28 .19580 .24347 .20629 .25991 .21703 .27719 .22801 .29535 28 29 .19597 .24373 .20647 .26019 .21721 .27748 .22819 .29566 29 30 .19614 .24400 .20665 .26047 .21739 .27778 .22838 .29597 30 31 .19632 .24427 .20682 .26075 .21757 .27807 .22856 .29628 31 82 .19649 .24454 .20700 .26104 .21775 .27837 .22875 .29659 32 33 .19666 .24481 .20718 .26132 .21794 .27867 .22893 .29690 33 34 .1^684 .24508 .20736 .26160 .21812 .27896 .22912 .29721 34 35 .19701 .24534 .20753 .26188 .21830 .27926 .22930 .29752 35 36 .19718 .24561 .20771 .26216 .21848 .27956 .22949 .29784 36 37 .19736 .24588 .20789 .26245 .21866 .27985 .22967 .29815 37 38 .19753 .24615 .20807 .26273 .21884 .28015 .22986 .29846 38 39 .19770 .21642 .20824 .26301 .21902 .28045 .23004 .29877 39 40 .19788 .24669 .20842 .26330 .21921 .28075 .23023 .29909 40 41 .19805 .24696 .20860 .26358 .21939 .28105 .23041 .29940 41 42 .19822 .24723 .20878 .26387 .21957 .28134 .23060 .29971 42 43 .19840 .24750 .20895 .26415 .21975 .28164 .23079 .30003 43 44 .19857 .24777 .20913 .26443 .21993 .28194 .23097 .30034 44 45 .19875 .24804 .20931 .26472 .22012 .28224 .23116 .30066 45 46 .19892 .24832 .20949 .26500 .22030 .28254 .23134 .30097 48 47 .19909 .24859 .20967 .26529 .22048 .28284 .23153 .30129 47 48 .19927 .24886 .20985 .26557 .22066 .28314 .23172 .30160 43 49 .19944 .24913 .21002 .26586 .22084 .28344 .23190 .30192 49 50 .19962 .24940 .21020 .26615 .22103 .28374 .23209 .30223 50 51 .19979 .24967 .21038 .26643 .22121 .28404 .23228 .30255 61 r Q .19997 .24995 .21056 .26072 .22139 .28434 .23246 .30287 52 53 .20014 .25022 .21074 .26701 .22157 .28464 .23265 .30318 53 54 .20032 .25049 .21092 .26729 .22176 .28495 .23283 .30350 54 55 .20049 .25077 .21109 .26758 .22194 .28525 .23302 .30382 55 56 .20066 .25104 .21127 .26787 .22212 .28555 .23321 .30413 56 57 .900*4 .25131 .21145 .26815 .22231 .28585 .23339 .30445 57 58 .20101 .25159 .21163 .26844 .22249 .28615 .23358 .30477 53 59 .20119 .25186 .21181 .26873 .22267 .28646 i .23377 .30509 59 60 .20136 .25214 .21199 .26902 1 .22285 .28676 .23396 .30541 60 TABLE XXII. VERSINES AND EXSECANTS. 369 ' 40 41 42 43 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .23396 .30541 .24529- .32501 .25686 .34563 .26865 .36733 1 .23414 .30573 .24548 .32535 .25705 .34599 .26884 .36770 1 2 .23433 .30605 .24567 .32568 .25724 .34634 .26904 .36807 2 3 .23452 .30636 .24586 .32602 .25744 .34669 .26924 .36844 3 4 .23470 .30668 .24605 .32636 .25763 .34704 .26944 .36881 4 5 .23489 .30700 .24625 .32669 .25783 .34740 .26964 .36919 5 6 .23508 .30732 .24644 .32703 .25802 .34775 .26984 .36956 6 7 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 .32770 .25841 .34846 .27024 .37030 8 9 .23564 .30829 .24701 .32804 .25861 .34882 .27043 .37068 9 10 .23583 .30861 .24720 .32838 .25880 .34917 .27063 .37105 10 11 .23603 .30893 .24739 .32872 .25900 .34953 .27083 .37143 11 12 .23620 .30925 .24759 .32905 .25930 .34988 .27103 .37180 12 13 .23639 .30957 .24778 .32939 .25939 .35024 .27123 .37218 13 14 .23658 .30989 .24797 .32973 .25959 .35060 .27143 .37255 14 15 .23677 .31022 .24816 .33007 .25978 .35095 .27163 .37293 15 16 .23696 .31054 .24835 .33041 .25998 .35131 .27183 .37330 16 17 .23714 .31086 .24854 .33075 .26017 .35167 .27203 .37368 17 18 .23733 .31119 .24874 .33109 .26037 .35203 .27223 .37406 18 19 .23752 .31151 .24893 .33143 .26056 .35238 .27243 .37443 19 20 .23771 .31183 .24912 .33177 .26076 .35274 .27263 .37481 20 21 .23790 .31216 .24931 .33211 .26096 .35310 .27283 .37519 21 22 .23808 .31248 .24950 .33245 .26115 .35346 .27303 .37556 22 23 .23827 .31281 .24970 .33279 .26135 .35382 .27323 .37594 23 24 .23846 .31313 .24989 .33314 .26154 .35418 .27343 .37632 24 25 .23865 .31346 .25008 .33348 .26174 .35454 .27363 .37670 25 26 .23884 .31378 .25027 .33382 .26194 .35490 .27383 .37708 26 27 .23903 .31411 .25047 .33J16 .26213 .35526 .27403 .37746 27 28 .23922 .31443 .25066 .33451 .26233 .35562 .27423 .37784 28 29 .23941 .31476 .25085 .33485 .26253 .35598 .27443 .37822 29 30 .23959 .31509 .25104 .33519 .26272 .35634 .27463 .37860 30 31 .23978 .31541 .25124 .33554 .26292 .35670 .27483 .37898 31 32 .23997 .31574 .25143 .33588 .26312 .35707 .27503 .37936 32 33 .24016 .31607 .25162 .33622 .26331 .35743 .27523 .37974 33 34 .24035 .31640 .25182 .33657 .26351 .35779 .27543 .38012 34 35 .24054 .31672 .25201 .33691 .26371 .35815 .27563 .38051 35 36 .24073 .31705 .25220 .33726 .26390 .35852 .27583 .38089 36 37 .24092 .31738 .25240 .33760 .26410 .35888 .27603 .38127 37 38 .24111 .31771 .25259 .33795 .26430 .35924 .27623 .38165 33 39 .24130 .31804 .25278 .33830 .26449 .35961 .27643 .38204 39 40 .24149 .31837 .25297 .33864 .26469 .35997 .27663 .38242 40 41 .24168 .31870 .25317 .33899 .26489 .36034 .27683 .38280 41 42 .24187 .31903 .25336 .33934 .26509 .36070 .27703 .38319 4:? 43 .24206 .31936 .25356 .33968 .26528 .36107 .27723 .38357 43 44 .24225 .31969 .25375 .34003 .26548 .36143 .27743 .38396 4i 45 .24244 .32002 .25394 .34038 .26568 .36180 .27764 .38434 43 46 .24262 .32035 .25414 .34073 .26588 .36217 .27784 .38473 46 47 .24281 .32068 .25433 .34108 .26607 .36253 .27804 .38512 47 48 .24300 .32101 .25452 .34142 .26627 .36290 .27824 .38550 43 49 .24320 .32134 .25472 .34177 .26647 .36327 .27844 .38589 43 50 .24339 .32168 .25491 .34212 .26667 .36363 .27864 .38628 50 51 .24358 .32201 .25511 .34247 .26686 .36400 .27884 .38666 51 52 .24377 .32234 .25530 .34282 .26706 .36437 .27905 .38705 5:3 53 .24396 .32267 .25549 .34317 .26726 .36474 .27925 .38744 53 54 .24415 .32301 .25569 .34352 .26746 .36511 .27945 .38783 54 55 .24434 .32334 .25588 .34387 .26766 .36548 .27965 .38822 55 56 .24453 .32368 .25608 .34423 . .26785 .36585 .27985 .38860 56 57 .24472 .32401 .25627 .34458 .26805 .36622 .28005 .38899 57 58 .24491 .32434 .25647 .34493 .26825 .36659 .28026 .38938 58 59 .24510 .32468 .25666 .34528 .26845 .36696 .28046 .38977 59 60 .24529 .32501 .25686 .34563 .26865 .ijG733 | .28066 .39016 60 TAHLK. \\n. VKKSINKS ANP KXSI-VANTS. 44 45 46 47 Vers. BXSBO. V(M-S. Kxstv. \Vrss. Kxscv. Vrrs. Kxser. ' .88066 TsTwiT .29289 .41121 .30584 .*;'.).'.(> .31SOO .411628 1 .88088 .89065 .29310 .41468 .80555 .1: !'.);)<) .:us-ji .40674 1 9 .28108 .89095 .29830 .41604 .80576 .44042 .31848 .46719 2 3 38127 .89184 .29851 .41545 .80597 .44086 .81864 .40705 8 4 [28147 .39173 [29372 .41586 .80818 . Hi2!) .48811 4 5 .38167 .39912 .29398 .m>27 .3o t i;) .44178 .:M'.H>7 .40857 5 G .28187 .39251 [29418 .41669 .80660 .11217 .:!28 .46903 6 7 .28808 .89991 .29433 .41710 .30681 .41260 .:51!H!) .46949 7 8 . 2S22S .39330 .29454 .41759 .80708 .44804 .31971 .46995 8 . 2S2 1 S [41793 .80728 .44347 .81992 .17011 9 10 .38988 .3940S) .99495 .41835 .30744 .44391 .32013 .47067 10 It ,28289 .89448 .90516 .41876 .30765 .44485 .89085 .47184 11 u .28309 39487 .ir.u -; .44479 .82056 .47180 12 13 .28329 .39527 .41959 [30807 .44523 .32077 [47226 13 14 [39568 .29578 .42001 [80828 .44567 .32099 .-17272 14 15 .89606 .29599 .10012 .80849 .44610 .32120 .47819 15 13 .28890 .89646 .29619 .42084 .30870 .44654 .32141 .47865 10 17 [98410 .39685 .29640 .48126 .80891 .44898 .88163 .47411 17 18 .26431 .89601 .421G8 .80912 .44742 .82184 .47458 18 19 .28461 .89764 .29081 [42210 .44787 .82205 .47604 19 80 [98471 [89804 .49951 .30951 .44881 .82287 .41551 20 81 .28499 .89844 .99793 .30075 .44875 .32948 .47598 21 .28518 .89884 .42335 .44919 .83270 .47644 22 83 .28582 .39924 .29764 .42377 .31017 .44963 .47691 83 XT t .28553 .39963 .29785 .42419 .81038 .45007 .32812 .47788 N 85 .28573 [40008 .48461 .81059 .45052 .32384 .47784 95 88 .40043 .29826 .42508 .81080 .45096 .4783] 86 97 .'.'SOI ( .40083 .29847 .49545 .81101 .45141 .89377 .47878 27 88 .28634 .40198 .49587 .81122 .45185 .82398 .47925 2S 99 .28655 .40168 .29888 .42680 .81143 .47972 99 GO [98875 .40908 .90909 .42072 .31105 ,45874 .32111 .48019 30 si .98695 .IOC 13 .29030 .1271 I .31180 .45819 .89482 .48086 31 82 .88716 .40283 .99951 . 12756 .81207 .45363 .32484 .48113 8S 88 [28788 .40324 .99971 .49799 .81228 .45408 .48160 33 34 .88757 .40364 .29992 .42841 .31219 .45452 .82527 .48207 84 85 [28777 ,40404 .30013 .49888 .81270 .45497 .32548 . 18254 ;t r > 36 .28797 .40 III .30034 .42926 .81291 .45542 .82570 .48801 30 87 .28818 .40485 .88054 .49968 .81812 .45587 .82591 .48849 37 38 .40595 .30075 .43011 .81884 .45681 .82813 .48398 88 39 .88859 .40565 .30096 .48058 .81855 .45678 .32884 .48448 39 -10 .40000 .30117 .48098 .31370 ,45721 .89668 .48491 40 41 .28900 .40646 .30138 .43139 .31307 .45766 .89677 .48588 41 12 .30158 .48181 [31418 .45811 .32899 .48586 42 43 .88941 .40797 .80179 .43224 .81489 .45856 .32720 .48633 43 44 .28901 .40708 .80200 .48867 .81461 .45901 .327:2 .48681 44 45 .28981 .40808 .80221 .43810 .81482 .45946 .32763 .48728 45 46 .29008 .40849 .80942 .43352 .81508 .45992 .82785 .48776 46 47 .29028 .40890 .30203 .48895 .81584 .46037 .82806 .48824 47 48 [89048 .40930 .30283 .43488 .81545 .46062 .38828 .48871 48 49 .89068 .40971 .30304 .43481 .81587 .-11 H27 .82849 .48919 49 50 .29084 .41012 .80395 .43524 .31588 .40173 .88871 .4SU67 50 51 .29104 .41053 .30346 .43567 .31009 .46218 .32S93 .49015 51 52 .29125 .41093 .30307 .43010 .81030 ..(('-.Hi;? .."2;M 1 .49063 58 53 .80145 .41134 .30388 .43053 .31051 .48309 .82986 .49111 53 54 .89166 .41175 .30409 .48696 .31073 .46354 [82957 .49159 64 55 .89187 .41216 .30430 .48789 .31694 .46400 .321)79 .49207 55 56 .89907 .41257 .80451 .43783 .31715 .46445 .88001 56 57 89928 .41298 .30471 .48826 .31730 .46491 .83022 [49808 57 58 .99948 .41839 [30492 .43869 .81758 .88044 58 59 .99989 .41380 .80518 .43912 .31779 .46582 .49399 59 60 .90989 .41491 .30534 .48956 .31800 .46628 .33087 .49443 60 TAKLK XXII. VERSINES AND KXSErANTS t 48 49 50 51 f Vers. Exsec. Vers. E::sec. Vers. Exsec. Vers. Kxstx-. o .33087 .49448 .34304 .52425 .35721 .55572 .37068 .58902 1 .33109 .49496 .34410 .52471) .35744 .55626 .37091 .58968 1 2 .33130 .49544 : .84438 .58687 .35766 .55080 .37113 .59016 2 3 .33152 .34460 .52579 .35788 .55734 .37130 .59073 3 4 .33173 .49641 .34482 .59680 .35810 .55789 1 .37158 .59130 4 5 .33195 .49090 .34504 .52681 .35833 .55843 .37181 .59188 5 6 .33217 .49738 .84536 .52732 .35855 .55897 .37204 .59245 6 7 .3323$ .49787 .34548 .52784 .35877 .55951 .372215 .59300 7 8 .33260 .49835 .34570 .35900 .56005 .37249 .59360 8 9 .33282 .49884 .34592 .52886 .85922 .56060 .37272 .59418 9 10 .33303 .49933 .34614 .52938 .35944 .56114 .87894 .59475 10 11 .33325 .49981 .34636 .52989 .35967 .56169 .37317 .59533 11 12 .33347 .50030 .34658 [53041 .35989 .56833 .37340 .59590 12 13 .33303 .50079 .346:X> .53092 .36011 .37362 .59648 13 14 .33390 .501:28 .34702 .53144 .36034 .37385 .59706 14 13 .33412 .50177 .31721 .53196 .36056 .56387 .37408 .59764 15 HI .33434 [50236 .34746 .53247 .36078 .56442 .37430 .69823 10 17 .33455 .50275 .34768 .53299 .36101 .56497 . 37453 .59880 17 18 .33477 .50324 .34790 .53351 .36123 .56551 i .37476 .59938 13 19 .33199 .50373 .31812 .5:1403 .36146 .56606 .37498 .59996 19 20 .33520 .50422 .34834 .53455 .36168 .56661 .37521 .60054 30 21 .33542 .50471 .34856 .53507 .36190 .56716 .37544 .60112 21 22 .33504 .50521 | .34878 .53559 .36213 .50771 .37567 .60171 22 23 .33580 .5(1570 : .34900 .53611 .36235 .37589 23 .33S03 .51065 .35122 .54134 .36460 .57380 .37817 .60815 33 34 .33S25 .51115 .35144 .54187 .36482 .57436 .37840 .60874 34 35 .33847 .51165 .35106 .54240 .36501 .57491 .37862 .60933 35 36 .33809 .51215 '.35188 .54292 .36527 .57547 [87885 .60992 36 37 .33891 .51265 .35210 .54345 .36549 .57603 .37908 .61051 37 38 .33912 .51314 .35232 .54398 .3657'2 .57659 .37931 .61111 38 39 .33934 .51364 .35254 .54451 .36594 .57715 .37951 .61170 39 40 .33956 .51415 .35277 .54504 .36617 .57771 .37976 .61229 40 41 I .33978 .51465 .35299 .54557 .36639 .57827 i .379P9 .61283 41 42 .34000 .51515 .35321 .54610 .36602 .57883 1 .38022 .61348 42 43 ! .34022 .51565 .35343 .54663 .36684 .57939 || .38045 .61407 43 44 .34044 .51615 .35365 .54716 .36707 .57995 .38068 .61467 44 45 .34065 .51665 .85388 .54769 .36729 .58051 ! .38091 .61526 45 46 .34087 .51716 .35410 .54822 .36752 .58108 .38113 .61586 46 47 .34109 .51766 .85433 .54876 .36775 .58164 .38136 .61646 47 48 .34131 .51817 .35454 .54929 .36797 .5-001 .38159 .61705 48 49 .34153 .51867 .5476 .54982 .36820 [58277 [88188 .61765 49 50 .34175 .51918 .35499 .55036 .36842 .58333 .38205 .61825 50 51 .34197 .51968 .35521 .55089 .36865 .58390 .38228 .61885 51 52 .34219 .52019 .35543 .55143 .86887 .58447 .38251 .61945 52 53 .34241 [69069 .35565 .55196 .36910 .58503 .38274 .62005 53 54 .34262 .52120 .35588 .55250 .36932 .58560 [88896 .62065 54 55 ! .31284 .52171 .35610 .55303 .36955 .58617 .38319 .62125 55 56 .34306 .522-22 .35632 .55357 .36978 .58674 .38342 .62185 56 57 .34328 .52273 .35654 .55411 .37000 .58731 .38365 .62246 57 53 .34350 .52323 .35877 .5541.5 .37023 .58788 .88888 .62306 58 59 .34372 .52374 .35699 .37045 .58845 .38411 .62366 59 60 .34394 .52425 .35721 .55572 .37008 .58903 .38434 .62427 6Q TABLE XXII. YERSINES AND EXSEL'ANTS. 52 53 54 55 YITS. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .38434 .62427 .39819 .06164 | .41221 .70130 .42642 .74345 1 .38457 .62487 .39842 .66228 .41245 .70198 .42666 .74417 1 o .38480 .62548 ! .39865 .66292 .41269 .702(57 .42690 .74490 2 3 .38503 .62609 .39888 .66357 .41292 .70335 .42714 .74568 3 4 .38526 .62669 .39911 .66421 .41316 .70403 .42738 .74635 4 5 .38549 .02730 .39935 .66486 .41339 .70472 .42702 .74708 5 6 .38571 .(52791 .39958 .66550 .41363 .70540 ll .42785 .74781 6 7 .38594 .62852 : .39981 .66615 .41386 .70809 .42809 .74854 7 8 .38617 .62913 ! .40005 .66679 .41410 .70677 i .42833 .74927 8 9 .38640 .62974 ' .40028 .66744 .41433 .70746 i .42857 .75000 9 10 .38663 .63035 .40051 .66809 .41457 .70815 ; .42881 .75073 10 11 .38686 .63096 .40074 .66873 .41481 .70884 .42905 .75146 11 12 .38709 .63157 .40098 .66938 .41504 .70953 .42929 .75219 12 13 .38732 .63218 .40121 .67003 .41528 .71022 ! .42953 .75293 ! 13 14 .38755 .63279 j .40144 .67068 .41551 .71091 .42976 .75366 ! 14 15 .38778 .63341 i .40168 .67133 .41575 .71160 .43000 .75440 15 16 .38801 .63402 .40191 .67199 .41599 .71229 .43024 .75513 16 17 .38824 .63464 .40214 .67264 .41622 .71298 .43048 .75587 ! 17 18 .38847 .63525 .40237 .67329 .41646 .71368 i .43072 .75661 18 19 .38870 .63587 ! .40261 .67394 .41670 .71437 .43096 .75734 19 20 .38893 .63&4S ; .40284 .67460 .41693 .71506 : .43120 .75808 20 21 .38916 .63710 .40307 .67525 .41717 .71576 .43144 .75882 21 22 .38939 .63772 .40331 .67591 .41740 .71646 .43168 .75956 22 23 .38902 .63834 .40354 .67656 .41764 .71715 .43192 .76031 23 24 .38985 .63895 .40378 .67722 .41788 .71785 .43216 .76105 24 25 .39009 .63957 .40401 .67788 .41811 .71855 .43240 .76179 25 26 .39032 .64019 .40424 .67853 .41835 .71925 .43264 .76253 26 27 .39055 .64081 .40448 .67919 .41859 .71995 I .43287 .76328 27 28 .39078 .64144 .40471 .67985 .41882 .72065 .43311 .76402 28 29 .39101 .64206 .40494 .68051 .41906 .72135 .43335 .76477 29 30 .39124 .64268 .40518 .68117 .41930 .72205 .43359 . 76552 30 31 .39147 .64330 .40541 .68183 .41953 .72275 .43383 .76626 31 32 .39170 .64393 .40565 .68250 .41977 .72346 .43407 .76701 32 33 .39193 .64455 .40588 .68316 .42001 .72416 .43431 .76776 33 34 .39216 .04518 .40611 .68382 .42024 .72487 .43455 .76851 34 35 .39239 .64580 .40635 .68449 .42048 .72557 .43479 .76926 35 36 .39262 .64643 .40658 .68515 .42072 .72628 .43503 .77001 36 37 .39286 .64705 .40682 .68582 .42096 .72698 .43527 .77077 37 38 .39309 .64768 .40705 .68648 .42119 .72769 ! .43551 .77152 38 39 .39332 .64831 .40728 .68715 .42143 .72840 .43575 .77227 39 40 .39355 .64894 .40752 .68782 .42167 .72911 .43599 .77303 40 41 .39378 .64957 .40775 .68848 .42191 .72982 ' .43623 .77378 41 42 .39401 .65020 .40799 .68915 .42214 .73053 .43647 .77454 42 43 .39424 .65083 .40822 .68982 .42238 .73124 .43671 .77530 43 44 .39447 .65146 .40846 .69049 .42262 .73195 .43695 .77606 44 45 .39471 .65209 .40869 .69116 .42285 .73267 .43720 .77681 45 46 .39494 .65272 .40893 .69183 .42309 .73338 j .43744 .77757 46 47 .39517 .65336 .40916 .69250 .42333 .73409 .43768 .77833 47 48 .39540 .65399 .40939 .69318 .42357 .73481 .43792 .77910 48 49 .39563 .65462 .40963 .69385 .42381 .73552 .43816 .77986 49 50 .39586 .65526 .40986 .69452 .42404 .73624 .43840 .78062 50 51 .39610 .65589 .41010 .69520 .42428 .73696 .43864 .78138 51 52 .39633 .65653 .41033 .69587 .42452 .73768 .43888 .78215 62 53 .39656 .65717 .41057 .69655 .42476 .73840 .43912 .78291 53 54 .39679 .65780 .41080 .69723 .42499 .73911 .43936 .78368 54 55 .39702 .65844 .41104 .69790 .42523 .73983 I .43960 .78445 55 56 .39726 .65908 .41127 .69858 .42547 74056 i .43984 .78521 56 57 .39749 .65972 .41151 .69926 .42571 .74128 I .44008 .78598 57 58 .39772 .66036 .41174 .69994 .42595 .74200 .44032 .78675 58 59 .39795 .66100 .41198 .70062 .42619 .74272 .44057 .78752 59 60 .39819 .66164 .41221 .70130 .42642 .74345 ,1 .44081 .78829 60 TABLE XXII. VEKS1NES AND EXSECANTS. 373 56 57 58 59 Vers. Exsec. Vers. Exsec. | Vers. Exsec. Vers. Exsec. o .44081 .78829 .45536 .83608 .47008 .88708 .48496 .94100 1 .44105 .78906 .45500 .83690 .47033 .88796 : .48521 .94254 1 2 .44129 .78984 .4.5585 .83773 1 .47057 .88884 .48546 .94349 2 3 .44153 .79001 .45009 .83855 .4J082 .88972 .48571 .94443 3 4 .44177 .79138 .45634 .83938 .47107 .89060 .48596 .94537 4 5 .44201 .79216 .45058 .84020 .47131 .89148 .48621 .94632 5 6 .44225 .79293 .45083 .84103 1 .47156 .89237 .48646 .94726 6 7 .44250 .79371 .45707 .84186 j .47181 .89325 .48671 .94821 7 8 .44274 .79449 .45731 .84209 .47206 .89414 .48696 .94916 8 9 .44298 .79527 .45756 .84352 .47230 .89503 .48721 .95011 9 10 .44322 .79604 .45780 .84435 .47255 .89591 .48746 .95106 10 11 .44346 .79682 .45805 .84518 .47280 .89680 .48771 .95201 11 12 .44370 .79761 .45829 .84601 .47304 .89769 .48796 .95296 12 13 .44395 .79839 .45854 .84685 .47329 .89858 .48821 .95392 13 14 .44419 .79917 .45878 .84708 .47354 .89948 .48846 .95487 14 15 .44443 .79995 .45903 .84852 .47379 .90037 .48871 .95583 15 16 .44407 .80074 .45927 .84935 .47403 .90126 .48896 .95678 16 17 .44491 .80152 .45951 .85019 .47428 .90216 .48921 .95774 17 18 .44516 .80231 .45976 ..85103 .47453 .90305 .48946 .95870 18 19 .44540 .80309 .46000 .85187 .47478 .90395 .48971 .95966 19 20 .44504 .80388 .40025 .85271 .47502 .90485 .48998 .96062 20 21 .44588 .80467 .46049 .85355 .47527 .90575 .49021 .96158 21 oo .44612 .80546 .46074 .85439 j! .47552 .90605 .49046 .96255 22 23 .44637 .80625 .46098 .85523 .47577 .90755 .49071 .96351 23 24 .41601 .80704 .40123 .85608 .47601 .90845 .49096 .96448 24 25 .44685 .80783 .40147 .85692 .47626 .90935 .49121 .96544 25 26 .44709 .80862 .46172 .85777 .47051 .91026 .49146 .96641 26 27 .44734 .80942 .46196 .85861 .47676 .91116 .49171 .96738 27 28 .44758 .81021 .46221 .85946 .47701 .91207 .49196 .96835 28 29 .44782 .81101 .46246 .86031 .47725 .91297 .49221 .96932 29 30 .44806 .81180 .46270 .86116 .47750 .91388 .49246 .97029 30 31 .44831 .81260 .46295 .86201 .47775 .91479 .49271 .97127 31 32 .44855 .81340 .46319 .86286 .47800 .91570 .49296 .97224 32 33 .44879 .81419 .40344 .86371 .47825 .91061 .49321 .97322 33 34 .44903 .81499 .46368 .86457 .47849 .91758 .49346 .97420 34 35 .44928 .81579 .46393 .86542 .47874 .91844 .49372 .97517 35 36 .44952 .81659 .46417 .86627 .47899 .91935 .49397 .97615 36 37 .44976 .81740 .46442 .86713 .47924 .92027 .49422 .97713 37 38 .45001 .81820 .46466 .86799 .47949 .92118 .49447 .97811 38 39 .45025 .81900 .46491 .86885 .47974 .92210 .49472 .97910 39 40 .45049 .81981 .46516 .80990 .47998 .92302 .49497 .98008 40 41 .45073 .82061 .46540 .87056 .48023 .92394 .49522 .98107 41 42 .45098 .82142 .46565 .87142 .48048 .92486 .49547 .98205 42 43 .45122 .82222 .46589 .87229 .48073 .92578 .4957'2 .98304 43 44 .45146 .82303 .46614 .87315 .48098 .92670 .49597 .98403 44 45 .45171 .82384 .46639 .87401 .48123 .92762 .49623 .98502 45 46 .45195 .82465 .46663 .87488 .48148 .92855 .49648 .98601 46 47 .45219 .82546 .46688 .87574 .48172 .92947 .49673 .98700 47 48 .45244 .82627 .46712 .87661 .48197 .93040 .49698 .98799 48 49 .45268 .82709 .46737 .87748 .48222 .93133 .49723 .98899 49 50 .45292 .82790 .46762 .878:34 .48247 .93226 .49748 .98998 50 51 .45317 .82871 .46786 .87921 .48272 .93319 .49773 .99098 51 52 .45341 .82953 .46811 .88008 .48237 .93412 .49799 .99198 52 53 .45305 .83034 .408:36 .88095 1 .48322 .93505 .49824 .99298 53 54 .45390 .83116 .46800 .88183 .48347 .93598 .49849 .99398 54 55 .45414 .a3198 .46885 .88270 .48372 .93692 .49874 .99498 55 56 .45439 .83280 .46909 .88357 .48396 .93785 .49899 .99598 56 57 .45403 .83362 .46934 .88445 .48421 .93879 .49924 .99698 57 58 .45487 .83444 .46959 .88532 .48446 .93973 .49950 .99799 58 59 .45512 .83526 .46983 .88620 .48471 .94066 .49975 .99899 59 60 .45536 .83608 .47008 .88708 i .48496 .94160 .50000 .00000 60 374 TABLE XXII.-VERS1XES AND EXSECANTS. / i c 6 i ! 2 e 3 Vers. Exsec. Vers. Exsec. Yers. Exsec. Vers. Exsec. .50000 1.00000 .51519 1.06267 .53053 .13005 .54GOl" .20269 1 .50025 1.00101 .51544 1.06375 .53079 .13122 .54627 .20395 1 8 .50050 1.00202 .51570 1.06483 .53104 .13239 .54653 .20521 2 I .50076 1.00303 .51595 .06592 .53130 .13356 .54679 .20647 3 4 .50101 1.00404 .51621 .06701 .53156 .13473 .54705 .20773 4 5 .50126 1.00505 .51646 .06809 .53181 .13590 .54731 .20900 B 6 .50151 1.00607 .51672 .06918 .53207 .13707 .54757 .21026 6 7 .50176 1.00708 .51697 .07027 .53233 .13825 .54782 .21153 7 B .50202 1.00810 .51723 .07137 .53258 .13942 .54808 .21280 B '. .50227 1.00912 .51748 .07246 .53284 .14060 .54834 .21407 10 .50252 1.01014 .51774 .07356 .53310 .14178 .54860 .21535 10 11 .50277 1.01116 .51799 .07465 .53336 .14296 .54886 .21662 11 19 .50303 1.01218 .51825 .07575 .5:3361 .14414 .54912 .21790 12 18 .50328 1.01320 .51850 .07685 .53387 .14533 .54938 .21918 13 14 .50353 1.01422 .51876 .07795 .53413 .14651 .54964 .22045 14 15 .50378 1.01525 .51901 .07905 .5.3439 .14770 .54900 .22174 15 16 .50404 1.01628 .51927 .08015 .53464 .14889 .55016 .22302 16 17 .50429 1.01730 .51952 .08126 .53490 .15008 .55042 .22430 17 18 .50454 1.01833 .51978 .08236 .53516 .15127 .55068 .22559 IS 19 .50479 1.01936 .52003 .08347 .53342 .15246 .55094 .22688 & 90 .50505 1.02039 .52029 .08458 .53567 .15306 .55120 .22817 20 21 .50530 1.02143 .52054 .08569 .53593 .15485 .55146 .22946 21 32 .50555 1 02246 .52080 .08680 .53619 .15605 .55172 .23075 22 28 .50581 1.02349 .52105 .08791 .53645 .15725 .55198 .23205 23 34 .50606 1.02453 .52131 .08903 .53670 .15845 .55224 .23334 24 26 .50631 1.02557 .52156 .09014 .53696 .15965 .55250 .23464 25 {6 .50656 1.02661 .52182 .09126 .53722 .16085 .55276 .23594 26 87 .50682 1.02765 .52207 .09238 .53748 .16206 .55302 .23724 27 28 .50707 1.02869 .52233 .09350 .53774 .16326 .55328 .23855 28 89 .50732 1.02973 .52239 .09462 .53799 .16447 .55354 .23985 29 to .50758 1.03077 .52284 .09574 .53825 .16568 .55380 .24116 SO 31 .50783 1.03182 .52310 .09686 .53851 .16689 .55406 .24247 31 82 .50808 1.03286 .52335 .09799 .53877 .16810 .55432 .24378 32 88 .50834 1.03391 .52361 .09911 .53903 .16932 .55458 .24509 3:5 84 .50859 1.03496 .52386 .10024 .53928 .17053 .55184 .24640 34 88 .50884 1.03601 .52412 .10137 .53954 .17175 .55510 .24772 35 86 .50910 1.03706 .52438 .10250 .53980 .17297 .55536 .24903 86 37 .50935 1.03811 .52463 .10363 .54006 .17419 .55563 .25035 37 88 .50960 1.03916 .52489 .10477 .54032 .17541 .55589 25167 38 89 .50986 1.04022 .52514 .10590 .54058 .17663 .55615 .25300 89 40 .51011 1.04128 .52540 .10704 .54083 .17786 .55641 .25432 40 41 .51036 1.04233 .52566 .10817 .54109 .17909 .55667 .25565 41 42 .51062 1.04339 .52591 .10931 .54135 .18031 .55693 .25697 42 43 .51087 1.04445 .52617 .11045 .54161 .18154 .55719 .25830 43 44 .51113 1.04551 .52642 .11159 .54187 : .18277 .55745 .25963 44 45 .51138 1.04658 .52668 .11274 .54213 .18401 .55771 .26097 45 46 .51163 1.04764 .52694 .11388 .54238 .18524 .55797 .26230 46 47 .51189 1.04870 .52719 .11503 .54264 .18648 .55823 .26364 47 48 .51214 1.04977 .52745 .11617 .54290 .18772 .55849 .26498 4S 4!) .51239 1.05084 .52771 .11732 .54316 .18895 .55876 .26632 4!) 5U .51265 1.05191 .52796 .11847 .54342 .19019 .55902 .26766 50 51 .54290 1.05298 .52822 .11963 .54368 .19144 .55928 .26900 Bl 52 .51316 1.05405 .52848 .12078 .54394 .19268 .55954 .27035 52 58 .51341 1.05512 .52873 .12193 .54420 .19393 .55980 .27169 58 54 .51366 1.05619 .52899 .12309 .54446 .19517 .56006 .27304 51 55 .51392 1.05727 .52924 .12425 .54471 .19642 .56032 .27439 66 56 .51417 1.05835 .52950 .12540 .54497 19767 .56058 .27574 56 57 .51443 1.05942 .52976 .12657 i .54523 19892 .5(5084 .27710 57 58 .51468 1.06050 .53001 .12773 .54549 20018 .56111 .27845 58 59 .51494 1.06158 .53027 : .12889 .54575 20143 .5(5137 .27981 59 60 .51519 1.06267 .53053 .13005 .54601 20269 i .56163 .28117 60 TABLE XXII. VEKSINES AND EXSECANTS. 6 40 6 5 ! i 6 6 7 Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .56163 .28117 .57738 1.36620 .59326 1.45859 .60927 1.55930 1 .56189 .28253 .57765 .36768 .59353 1.46020 .60954 1.56106 2 .56215 .28390 .57791 .36916 .59379 1.46181 .60980 1.56282 3 .56241 .28526 ! .57817 .37064 .59406 1.46342 .61007 1.56458 4 .56267 .28663 i .57844 .37212 .59433 1.46504 .61034 1.56684 5 .56294 .28800 .57870 .37361 .59459 1.46665 .61061 1.56811 G .56320 .28937 .57896 .37509 .59486 1.46827 .61088 1.56988 7 .56346 .29074 .57923 .37658 i .59512 1.46989 .61114 1.57165 8 .56372 .29211 ! .57949 .37808 : .59539 1.47152 .61141 1.57342 g .56398 .29349 i .57976 .37957 : .59566 1.47314 .61168 1.57520 10 .56425 1.29487 i .58002 .3810? .59592 1.47477 .61195 1.57698 11 .56451 1.29625 .58028 .38256 .59619 1.47640 .61222 1.57876 12 .56477 1-29763 .58055 .38406 .59645 1.47804 .61248 1.58054 13 .56503 1.29901 .58081 .38556 .59672 1.47967 .61275 1.58233 14 .56529 1.30040 .58108 .38707 i .59699 1.48131 .61302 1.58412 15 .56555 1.30179 .58134 .38857 .59725 1.48295 .61329 1.58591 16 .56582 1.30318 .58160 .39008 .59752 1.48459 .61356 1.58771 17 .56608 1.30457 .58187 .39159 .59779 1.48624 .61383 1.58950 IS .56634 .30596 .58213 .39311 .59805 1.48789 .61409 1.59130 19 .56660 .30735 .58240 .39462 .59832 1.48954 .61436 1.59311 20 .56687 .30875 .58266 .39614 .59859 1.49119 i .61463 1.59491 21 ,56713 .31015 .58293 .39766 .59885 1.49284 .61490 1.59672 22 .56739 .31155 .58319 .39918 .59912 1.49450 .61517 1.59853 23 .56765 1.31295 .58345 .40070 .59938 1.49616 .61544 1.60035 24 .56791 1.31436 .58372 .40222 | .59965 1.49782 .61570 1.60217 25 .56818 1.31576 .58398 .4037'5 | .59992 1.49948 .61597 1.60399 26 .56844 1.31717 .58425 .40528 .60018 1.50115 .61624 1.60581 27 .56870 1.31858 .58451 .40681 .60045 1.50282 .61651 1.60763 38 .56896 1.31999 .58478 .40835 .60072 1.50449 i .61678 1.60946 29 .56923 1.32140 .58504 .40988 .60098 1.50617 .61705 1.61129 30 .56949 1.32282 .58531 .41142 .60125 1.50784 .61732 1.61313 31 .56975 1.32424 .58557 .41296 .60152 1.50952 .61759 1.61496 32 .57001 1.32566 .58584 .41450 .60178 1.51120 .61785 1.61680 33 .57028 1.32708 .58610 .41605 .60205 1.51289 , .61812 1.61864 34 .57054 1.32850 .58637 .41760 .60232 1.51457 .61839 1.62049 35 .57080 1.32993 .58663 .41914 .60259 1.51626 .61866 1.62234 30 .57106 1.33135 .58690 .42070 .60285 1.51795 .61893 1.62419 37 .57133 1.33278 .58716 .42225 .60312 1.51965 : .61920 1.62604 38 .57159 1.. 33422 .58743 .42380 .60339 1.52134 .61947 1.62790 39 .57185 1.33565 .58769 .42536 .60365 1.52304 .61974 1.62976 40 .57212 1.33708 .58796 .42692 .60392 1.52474 .62001 1.63162 41 .57238 1.33852 .58822 .42848 .60419 1.52645 .62027 1.63348 42 .57264 1.33996 .58849 .43005 .60445 1.52815 j .62054 1.63535 43 .57291 1.34140 .58875 .43162 .60472 1.52986 .62081 1.63722 44 .57317 1.34284 .58902 .43318 .60499 1.53157 .62108 1.03909 45 .57343 1.34429 .58928 .43476 .60526 1.53329 .62135 1.64097 46 .57369 1.34573 .58955 .43633 .60552 1.53500 .62162 1.64285 47 .57396 1.34718 .58981 .43790 .60579 1.53672 .62189 1.64473 48 .57422 1.34863 .59008 .43948 .60606 1.53845 ] .62216 1.64662 49 .57448 1.35009 .59034 .44106 .60633 1.54017 .62243 1.64851 50 .57475 1.35154 .59061 .44264 .60659 1.54190 .62270 1.65040 51 .57501 1.35300 .59087 .44423 .60686 1.54363 .62297 1.65229 52 .57527 1.35446 .59114 .44582 .60713 1.54536 .62324 1.65419 58 .57554 1.35592 .59140 .44741 .60740 1.54709 .62351 1.65609 54 .57580 1.35738 .59167 .44900 .60766 1.54883 .62378 1.65799 55 .57606 1.35885 .59194 .45059 .60793 1.5505? .62405 1.65989 56 .57633 1.36031 .59220 .45219 .60820 1.55231 .62431 1.66180 57 .57659 J. 36173 .59247 1.45378 .60847 1.55405 .62458 1.66371 58 .57685 1.36325 .59273 1.45539 .60873 1.55580 .62485 1.66563 59 .57712 1.36473 .59300 1.45099 .60900 1.55755 .62512 1.66755 60 .57738 1.36620 .59326 1.45859 .60927 1.55930 .62539 1.66947 376 TABLE XXII.-VERSINES AND EXSECANTS. / 38' ( 59 \ ro i 1 Vers, Exsec. Vers. Exsec. 1 Vers. Exsec. Vers. Exsec. 1 .62539 1.66947 .64163 1.79043 .65798 1.92380 .67443 2.07155 .62566 1.67139 .64190 1.79254 i .65825 1.92614 .67471 2.07415 < .62593 1.67332 .64218 1.79466 .65853 1.92849 .67498 2.07675 2 ] .62620 1.67525 .64245 1.79679 .65880 1.93083 .67526 2.07936 g / .62647 1.67718 .64272 1.79891 .65907 1.93318 .67553 2.08197 i 5 .62674 1.67911 .64299 1.80104 .65935 1.93554 .67581 2.08459 i 6 .62701 1.68105 .64326 1.80318 .65962 1.93790 .67608 2.08721 ( 5 .62728 1.68299 .64353 1.80531 1 .65989 1.94026 .67636 2.08983 7 8 .62755 1.68494 .64381 1.80746 .66017 1.94263 .67663 2.09246 8 9 .62782 1.68689 .64408 1.80960 : .66044 1.94500 .67691 2.09510 9 10 .62809 1.68884 .64435 1.81175 .66071 1.94737 .67718 2.09774 10 11 .62836 1 69079 .64462 1.81390 .66099 1.94975 .67746 2.10038 11 12 .62863 1.69275 .64489 1.81605 .66126 1.95213 .67773 2.10303 12 13 .62890 1.69471 .64517 1.81821 .66154 1.95452 .67801 2.10568 13 14 .62917 1.69667 .64544 1.82037 .66181 1.95691 .67829 2.10834 14 15 .62944 1.69864 .64571 1.82254 .66208 1.95931 .67856 2.11101 15 16 .62971 1.70061 .64598 1.82471 .66236 1.96171 .67884 2.11367 16 17 .62998 1.70258 .64625 1.82688 .66263 1.96411 .67911 2.11635 17 18 .63025 1.70455 .64653 1.82906 .66290 1.96652 .67939 2.11903 18 19 .63052 1.70653 .64680 1.83124 .66318 1.96893 .G7966 2.12171 19 20 .63079 1.70851 .64707 1.83342 .66345 1.97135 .67994 2.12440 20 21 .63106 1.71050 .64734 1.83561 .66373 1.97377 .68021 2.12709 21 22 .63133 1.71249 .64761 1.83780 .66400 1.97619 ' .68049 2.12979 22 23 .63161 1.71448 .64789 1.83999 .66427 1.97862 .68077 2.13249 23 24 .63188 1.71647 .64816 1.84219 .66455 1.98106 .68104 2.13520 24 25 .63215 1.71847 .64843 1.84439 ! .66482 1.98349 .68132 2.13791 25 26 .63242 1.72047 .64870 1.84659 I .66510 1.98594 .68159 2.14063 26 27 .63269 1.72247 .64898 1.84880 .66537 1.98838 .68187 2.14335 27 28 .63296 1.72448 .64925 1.85102 .66564 1.99083 .68214 2.14608 28 29 .63323 1.72649 .64952 1.85323 .66592 1.99329 .68242 2.14881 29 30 .63350 1.72850 .64979 1.85545 .66619 1.99574 .68270 2.15155 30 31 .63377 1.73052 .65007 1.85767 .66647 1.99821 .68297 2.15429 31 32 .63404 1.73254 .65034 1.85990 .66674 2.00067 .68325 2.15704 32 33 .63431 1.73456 .65061 1.86213 .66702 2.00315 .68352 2.15979 33 34 .63458 1.73659 1 .65088 1.86437 .66729 2.00562 .68380 2.16255 34 35 .63485 1.73862 .65116 1.86661 .66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 .65143 1.86885 .66784 2.01059 .68435 2.16808 36 37 .63539 1.74269 .65170 1.87109 .66811 2.01308 .68463 2.17085 37 38 .63566 1.74473 .65197 1.87334 j .66839 2.01557 .68490 2.17363 38 39 .63594 1.74677 .65225 1.87560 .66866 2.01807 .68518 2.17641 39 40 .63621 1.74881 .65252 1.87785 .66894 2.02057 .68546 2.17920 40 41 .63648 1.75086 .65279 1.88011 .66921 2.02308 .68573 2.18199 41 42 .63675 1.75292 .65306 1.88238 .66949 2.02559 .68601 2.18479 42 43 .63702 1.75497 .65334 1.88465 .6697'6 2.02810 .68628 2.18759 43 44 .63729 1.75703 .65361 1.88692 .67003 2.03062 .68656 2.19040 44 45 .63756 1.75909 .65388 1.88920 .67031 2.03315 .68684 2.19322 45 46 .63783 1.76116 .65416 1.89148 .67058 2.03568 .68711 2.19604 46 47 .63810 1.76323 .65443 1.89376 .67086 2.03821 .68739 2.19886 47 48 .63838 1.76530 j .65470 1.89605 .67113 2.04075 .68767 2.20169 48 49 .63865 1.76737 .65497 1.89834 .67141 2.04329 .68794 2.20453 49 50 .63892 1.76945 .65525 1.90063 .67168 2.04584 .68822 2.20737 50 51 .63919 1.77154 .65552 1.90293 .67196 2.04839 .68849 2.21021 51 52 .63946 1.77362 .65579 1.90524 .67223 2.05094 .68877 2.21306 52 53 .63973 1.77571 ' .65607 1.90754 .67251 2.05350 .68905 2.21592 53 54 .64000 1.77780 , .65634 1.90986 .67278 2.05607 .68932 2.21878 54 55 .64027 1.77990 .65661 1.91217 .67306 2.05864 .68960 2.22165 55 56 .64055 1.78200 .65689 1.91449 .67333 2.06121 .68988 | 2.22452 56 57 64082 1.78410 .65716 1.91681 .67361 2.06379 .69015 2.22740 57 58 .64109 1.78621 .65743 1.91914 .67388 2.0663? .69043 2.23028 58 59 .64136 1.78832 .65771 1.92147 .67413 Xi.lKJWMi .(59071 2.23317 59 60 .64163 1.79043 .65798 1.92380 I .67443 2.07155 .69098 2.23607 60 TABLE XXII. VERSINES AND EXSECANTS. ' 72 73 740 75 ,/ Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .69098 2.23607 .70763 2.42030 .72436 2.62796 .74118 2.86370 1 .69126 2.23897 .70791 2.42356 .72464 2.63164 .74146 2.86790 1 2 .69154 2.24187 .70818 2.42683 .72492 2.63533 .74174 2.87211 2 3 .69181 2.24478 .70846 2.43010 .72520 2.63903 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 .74231 2.88056 4 5 .69237 2.25062 .70902 2.436G6 .72576 2.64645 .74259 2.88479 5 C 69264 2.25355 .70930 2.43995 .72604 2.65018 .74287 2.88904 6 7 .69292 2.25648 .70958 2.44324 .72632 2.65391 .74315 2.89330 7 8 .69320 2.25942 .70985 2.44655 .72660 2.65765 .74343 2.89756 8 B .69347 2.26237 .71013 2.44986 .72688 2.66140 .74371 2.90184 9 10 .69375 2.26531 .71041 2.45317 .72716 2.66515 .74399 2.90613 10 11 .69403 2.26827 .71069 2.45650 .72744 2.66892 .74427 2.91042 11 12 .69430 2.27123 .71097 2.45983 .72772 2.67269 .74455 2.91473 12 18 .69458 2.27420 .71125 2.46316 .72800 2.67647 .74484 2.91904 13 14 .69486 2.27717 .71153 2.46651 .72828 2.68025 .74512 2.92337 14 15 .69514 2.28015 .71180 2.46986 .72856 2.68405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 .72884 2.68785 .74568 2.93204 16 17 .69569 2.28612 .71236 2.47658 .72912 2.69167 .74596 2.93640 17 18 .69597 2.28912 .71264 2.47995 .72940 2.69549 .74624 2.94076 IS 19 .69624 2.29212 .71292 2.48333 .72968 2.69931 .74652 2.94514 19 20 .69652 2.29512 .71320 2.48671 .72996 2.70315 .74680 2.94952 20 21 .69680 2.29814 .71348 2.49010 .73024 2.70700 .74709 2.95392 21 22 .69708 2.30115 .71375 2.49350 .73052 2.71085 .74737 2.95832 -.';! 23 .69735 2.30418 .71403 2.49691 .73080 2.71471 .74765 2.96274 23 24 .69763 2.30721 .71431 2.50032 .73108 2.71858 .74793 2.96716 24 25 .69791 2.31024 .71459 2.50374 .73136 2.72246 .74821 2.97160 25 26 .69818 2.31328 .71487 2.50716 .73164 2.72635 .74849 2.97604 26 27 .69846 2.31633 .71515 2.51060 .73192 2.73024 .74878 2.98050 27 28 .69874 2.31939 .71543 2.51404 .73220 2.73414 .74906 2.98497 88 29 .69902 2,32244 .71571 2.51748 .73248 2.73806 .74934 2.98944 J'.i 30 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 80 31 .69957 2.32858 .71626 2.52440 .73304 2.74591 .74990 2.99843 SI 32 .69985 2.33166 .71654 2.52787 .73332 2.74984 .75018 3.00293 :w 33 .70013 2.33474 .71682 2.53134 .73360 2.75379 .75047 3.00745 88 34 .70040 2.33783 .71710 2.53482 .73388 2.75775 .75075 3.01198 34 35 .70068 2.34092 .71738 2.53831 .73416 2.76171 .75103 3.01652 35 36 .70096 2.34403 .71766 2.54181 .73444 2.76568 .75131 3.02107 86 37 .70124 2.34713 .71794 2.54531 .73472 2.76966 .75159 3.02563 87 38 .70151 2.35025 .71822 2.54883 .73500 2.77365 .75187 3.03020 88 39 .70179 2.35336 .71850 2.55235 .73529 2.77765 .75216 3.03479 :) 40 .70207 2.35649 .71877 2.55587 .73557 2.78166 .75244 3.03938 40 41 .70235 2.35962 .71905 2.55940 .73585 2.78568 .75272 3.04398 41 42 .70263 2.36276 .71933 2.56294 .73613 2.78970 .75300 3.04860 42 43 .70290 2.36590 .71961 2.56649 .73641 2.79374 .75328 3.05322 43 44 .70318 2.36905 .71989 2.57005 .73669 2.79778 .75356 3.05786 44 45 .70346 2.37221 .72017 2.57361 .73697 2.80183 .75385 3.06251 45 46 .70374 2.37537 .72045 2.57718 .73725 2.80589 .75413 3.06717 46 47 .70401 2.37854 .72073 2.58076 .73753 2.80996 .75441 3.07184 47 48 .70429 2.38171 .72101 2.58434 .73781 2.81404 .75469 3.07652 48 49 .70457 2.38489 .72129 2.58794 .73809 2.81813 .75497 3.08121 49 50 .70485 2.38808 .72157 2.59154 .73837 2.82223 .75526 3.08591 50 51 .70513 2.39128 .72185 2.59514 .73865 2.82633 .75554 3.09063 51 52 .70540 2.39448 .72213 2.59876 .73893 2.83045 .75582 3.09535 52 53 .70568 2.39768 .72241 2.60238 .73921 2.83457 .75610 3.10009 58 54 .70596 2.40089 .72269 2.60601 .73950 2.83871 .75639 3.10484 54 55 .70624 2.40411 .72296 2.60965 .73978 2.84285 .75667 3.10960 55 56 .70652 2.40734 .72324 2.61330 .74006 2.84700 .75695 3.11437 5G 57 .70679 2.41057 .72352 2.61695 .74034 2.85116 .75723 3.11915 57 58 .70707 2.41381 .72380 2.62061 .74062 2.855J33 .75751 3.12394 58 59 .70735 2.41705 .72408 2.62428 .74090 2.85951 .75780 3.12875 59 60 .70763 2.42030 .72436 2.62796 .74118 2.86370 .75808 3.13357 60 TABLE XXtL-VERSlNES AND EXSEC A NTS. ' 76 77 78 79 / Vers. Exsec. Vers. Exsec, Vers. Exsec. Vers; Exsec, .75808 3.13357 .77505 3.44541 .79209 3.80973 .80919 4.24084 1 .75836 3.13839 .77533 3.45102 .79237, 3.81633 .80948 4.24870 1 2 .75864 3.14323 .77562 3.45664 .79266 3.82294 .80976 4.25658 2 3 .75892 3.14809 .77590 3.46228 .79294 3.82956 .81005 4.26448 3 4 .75921 3.15295 .77618 3.46793 .79323 8.83621 .81033 4.27241 4 5 .75949 3.15782 .77647 3.47360 i 79351 3.84288 ,81062 4.28036 5 6 .75977 3.16271 .77675 3.47928 ; 79380 3.84956 ,81090 4.28833 6 7 .76005 3.16761 .77703 3.48498 ,79408 3,85627 .81119 4.29634 7 8 .76034 3.17252 .77732 3.49069 .79437 3.86299 .81148 4.30436 8 9 .76062 3.17744 .77760 3.49642 .79465 3.86973 .81176 4.81241 9 10 .76090 3.18238 .77788 3.50216 .79493 3.87649 .81205 4.32049 10 11 .76118 3.18733 .77817 3.50791 .79522 3.88327 .81233 4.32859 11 12 .76147 3.19228 .77845 3.51368 .79550 3.89007 .81262 4.33671 12 13 .76175 3.19725 .77874 3.51947 .79579 3.89689 .81290 4.34486 13 14 .76203 3.20224 .77902 3.52527 .79607 3.90373 .81319 4.35304 14 15 .76231 3.20723 .77930 3.53109 .79636 3.91058 .81348 4.36124 15 16 .76260 3.21224 .77959 3.53692 .79664 3.91746 .81376 4.36947 16 1? .76288 3.21726 .77987 3.54277 .79693 3.92436 .81405 4.37772 17 18 .76316 3.22229 .78015 3.54863 .79721 3.93128 .81433 4.38600 18 19 .76344 3.22734 .78044 3.55451 .79750 3.93821 .81462 4.39430 19 20 .78373 3.23239 .78072 3.56041 .79778 3.94517 .81491 4.40263 20 81 .76401 3.23746 .78101 3.56632 .79807 3.95215 .81519 4.41099 21 22 .76429 3.24255 .78129 3.57224 .79835 3.95914 .81548 4.41937 22 23 .76458 3.24764 .78157 3.57819 .79864 3.96616 .81576 4.42778 23 24 .76486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 85 .76514 3.25787 .78214 3.59012 .79921 3.98025 .81633 4.44468 25 26 .76542 3.26300 .78242 3.59611 .79949 3.98733 .81662 4.45317 26 27 .76571 3.26814 .78271 3.60211 .79978 3.99443 .81691 4.46169 27 38 .76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 28 29 .76627 3.27847 .78328 3.61417 .80035 4.00869 .81748 4.47881 29 30 .76655 3.28366 .78356 3.62023 .80063 4.01585 .81776 4.48740 SO 31 .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 3:2 .76712 3.29406 .78413 3.63238 .80120 4.03024 .81834 4.50466 32 33 .76740 3.29929 .78441 3.63849 .80149 4.03746 .81862 4.51337 33 34 .76769 3.30452 .78470 3.64461 .80177 4.04471 .81891 4.52208 34 35 .76797 3.30977 .78498 3.65074 .80206 4.05197 .81919 4.53081 35 36 .76825 3.31503 .78526 3.65690 .80234 4.05926 .81948 4.53958 36 87 .76854 3.32031 .7'8555 3.66307 .80263 4.06657 .81977 4.54837 37 38 .76882 3.32560 .78583 3.66925 .80291 4.07390 .82005 4.55720 38 39 .76910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 40 .78938 3.33622 .78640 3.68167 .80348 4.08863 .82063 4.57493 40 41 .76967 3.34154 .78669 3.68791 .80377 4.09602 .82091 4.58383 41 42 .76995 3.34689 .78697 3.69417 .80405 4.10344 .82120 4.59277 142 43 .77023 3.35224 .78725 3.70044 .80434 4.11088 .82148 4.60174 43 44 .77052 3.35761 .78754 3.70673 .80462 4.11885 .82177 4.61073 1 44 45 .77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61976 45 46 .77108 3.36839 .78811 3.71935 .80520 4.13334 .82234 4.62881 46 47 .77137 3.37380 .78839 3.72569 .80548 4.14087 .82263 4.63790 47 48 .77165 3.37923 .78868 3.73205 .80577 4.14842 .82292 4.64701 48 49 .77193 3.38466 .78896 3.73843 .80605 4.15599 .82320 4.65616 49 50 .77222 3.39012 .78924 3.74482 .80634 4.16359 .82349 4.66533 50 51 .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 51 :,2 .77278 3.40106 .78981 3.75766 .80691 4.17886 .82406 4.68377 52 53 .77307 3.40656 .79010 3.76411 .80719 4.18652 .82435 4.69304 53 .54 .77335 3.41206 .79038 3.77057 .80748 4.19421 .82463 4.70234 54 55 .77363 3.4175P .79067 3.77705 .80776 4.20193 .82492 4.71166 55 66 .77392 S. 42315* .79095 3.78355 .80805 4.20966 .82521 4.72102 56 57 .77420 S. 42867 .79123 3.79007 .80&33 4.21742 .82549 4.73041 57 58 .77448 3.43424 .79152 3.79661 .80862 4.22521 .82578 4.73983 58 59 .77477 3.43982 .79180 3.80316 .80891 4.23301 .82607 4.74929 59 60 .77505 3.44541 .79209 3.80973 .80919 4.24084 .82635 4.75877 60 TABLE XXII. VERSINES AND EXSECANTS. ' 80 81 82 83 ' Vers. Exsec. Vers. Exsec. I Vers. Exsec. Vers. Exsec. .82635 4.75877 .84357 5.39245 1 .86083 6.18530 .87813 7.20551 1 .82664 4.76829 .84385 5.40422 .86112 6.20020 .87842 7.22500 1 8 .82692 4.777'84 .84414 5.41602 .86140 6.21517 .87871 7.24457 2 3 .82721 4.78742 .84443 5.42787 .86169 6.23019 .87900 7.26425 3 4 .82750 4.79703 .84471 5.43977 ! .86198 6.24529 .87929 7.28402 4 5 .82778 4.80667 .84500 5.45171 .86227 6.26044 .87957 7.36388 5 6 .82807 4.81635 .84529 5.46369 .86256 6.27566 .87986 7.32384 6 7 .82836 4.82606 .84558 5.47572 .86284 6.29095 .88015 7.34390 7 8 .82864 4.83581 .84586 5.48779 .86313 6.30630 .88044 7.36405 8 9 .82893 4.84558 .84615 5.49991 .86342 6.32171 .88073 7.38431 9 10 .82922 4.85539 .84644 5.51208 .86371 6.33719 .88102 7.40466 10 11 .82950 4.86524 .84673 5.52429 .86400 6.35274 .88131 7.42511 11 12 .82979 4.87511 .84701 5.53655 .86428 6.36835 .88160 7.44566 12 13 .83003 4.88502 .84730 5.54886 .86457 6.38403 .88188 7.46632 13 14 .83036 4.89497 .84759 5.56121 .86486 6.39978 .88217 7.48707 14 15 .83065 4.90495 .84788 5.57361 .86515 6.41560 .88246 7.50793 15 16 .83094 4.91496 .84816 5.58606 .86544 6.43148 .88275 7.52889 '16 ir .83122 4.92501 .84845 5.59855 .86573 6.44743 .88304 7.54996 17 18 .83151 4.93509 .84874 5.61110 .86601 6.46346 .88333 7.57113 |18 19 .83180 4.94521 .84903 5.62369 .86630 6.47955 .88362 7.59241 ! 19 20 .83208 4.95536 .84931 5.63633 .86659 6.49571 .88391 7.61379 20 21 .83237 4.96555 .84960 5.64902 .86688 6.51194 .88420 7.63528 21 38 .83266 4.97577 .84989 5.66176 .86717 6.52825 .88448 7.65688 22 88 .83294 4.98603 .85018 5.67454 .86746 6.54462 .88477 7.67859 23 24 .83323 4.99633 .85046 5.68738 .86774 6.56107 .88506 7.70041 24 86 .83352 5.00666 .85075 5.70027 .86803 6 57759 .88535 7.72234 25 2Q .83380 5.01703 .85104 5.71321 .86832 6.59418 .88564 7.74438 26 27 .83409 5.02743 .85133 5.7'2620 .86861 6.61085 .88593 7.76653 27 2* .83438 5.03787 .85162 5.73924 .86890 6.62759 .88622 7.78880 28 99 .83467 5.04834 .85190 5.75233 .86919 6.64441 .88651 7.81118 29 30 .83495 5.05886 .85219 5.76547 .86947 6.66130 .88680 7.83367 30 31 .83524 5.06941 .85248 5.77866 .86976 6.67826 .88709 7.85628 31 82 .83553 5.08000 .85277 5.79191 .87005 6.69530 .88737 7.87901 32 33 .83581 5.09062 .85305 5.80521 .87034 6.71242 .88766 7.90186 33 84 .83610 5.10129 !85334 5.81856 .87063 6.72962 .88795 7.92482 34 33 .83639 5.11199 .85363 5.83196 .87092 6.74689 .88824 7.94791 35 36 .83667 5.12273 .85392 5.84542 .87120 6.76424 .88853 7.97111 36 37 .83696 5.13350 .85420 5.85893 .87149 6.78167 .88882 7.99444 37 38 .83725 5.14432 .85449 5.87250 .87178 6.79918 .88911 8.01788 38 39 .83754 5.15517 .85478 5.88612 .87207 6.81677 .88940 8.04146 39 40 .83782 5.16607 .85507 5.89979 .87236 6.83443 .88969 8.06515 40 41 .83811 5.17700 .85536 5.91352 .87265 6.85218 .88998 8.08897 41 42 .83840 5.18797 .85564 5.92731 .87294 6.87001 .89027 8.11292 42 43 .83868 5.19898 .85593 5.94115 .87322 6.88792 .89055 8.13699 43 44 .83897 5.21004 .85622 5.95505 .87351 6.90592 .89084 8.16120 44 45 .83926 5.22113 .85651 5.96900 .87380 6.92400 .89113 8.18553 45 46 .83954 5.23226 .85680 5.98301 .87409 6.94216 .89142 8.20999 46 47 .83983 5.24343 .85708 5.99708 .87438 6.96040 .89171 8.23459 47 48 .84012 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 49 .84041 5.26590 .85766 6.02538 .87496 6.99714 .89229 8.28417 49 50 .84069 5.27719 .85795 6.03962 .87524 7.01565 .89258 8.30917 50 51 .84098 5.28853 .85823 6.05392 .87553 7.03423 .89287 ' 8.33430 fel fig .84127 5.29991 .85852 6.06828 .87582 7.05291 .89316 8.35957 :52 53 .84155 5.31133 .85881 6.08269 .87611 7.07167 .89345 8.38497 53 54 .84184 5.32279 .85910 6.09717 .87640 7.09052 .89374 8.41052 54 55 .84213 5.33429 .85939 6.11171 .87669 7.10946 .89403 8.43620 55 56 .84242 5.34584 .85967 6.12630 .87698 7.12849 .89431 8.46203 56 57 .84270 5.35743 .85996 6.14096 .87726 7.14760 .89460 8.48800 57 58 .84299 5.36906 .86025 6.15568 .87755 7.16681 .89489 8.51411 58 501 .84328 5.38073 .86054 6.17046 .87784 7.18612 .89518 8.54037 59 60! .84357 5.39245 .86083 6.18530 .87813 7,20551 .89547 8.56677 60 580 TABLE XXII. VERSINES AND EXSEOAXTS. / 84 85 86 / Vers. Exsec. Vers. Exsec. Vers. Exsec. .89547 8.56677 .91284 10.47371 .93024 13.33559 1 .89576 8.59332 .91313 10.51199 .93053 13.39547 1 2 .89605 8.62002 .91342 10.55052 .93082 13.45586 2 3 .89634 8.64687 .91371 10.58932 .93111 13.51676 3 4 .89663 8.67387 .91400 10.62837 .93140 13.57817 4 5 .89692 8.70103 .91429 10.66769 .93169 13.64011 5 6 .89721 8.72833 .91458 10.70728 .93198 13.70258 6 7 .89750 8.75579 .91487 10.74714 .93227 13.76558 7 8 .89779 8.78341 .91516 10.78727 .93257 13.82913 8 9 .89808 8.81119 .91545 10.82768 .93286 13.89323 9 10 .89836 8.83912 .91574 10.86837 .93315 13.95788 10 11 .89865 8.86722 .91603 10.90934 .93344 14.02310 11 12 .89894 8.89547 .91632 10.95060 .93373 14.08890 12 13 .89923 8.92389 .91661 10.99214 .93402 14.15527 13 14 .89952 8.95248 .91690 11.0339? .93431 14.22223 14 15 .89981 8.98123 .91719 11.07610 .93460 14.28979 15 16 .90010 9.01015 .91748 11.11852 .93489 14.35795 16 17 .90039 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90068 9.06849 .91806 11.20427 .93547 14.49611 18 19 .90097 9.09792 .91835 11.24761 .93576 14.56614 19 20 .9C126 9.12752 .91864 11.29125 .93605 14.63679 20 21 .90155 9.15730 .91893 11.33521 .93634 14.70810 21 22 .90184 9.18725 .91922 11.37948 .93663 14.78005 22 23 .90213 9.21739 .91951 11.42408 .93692 14.85268 23 24 .90242 9.24770 .91980 11.46900 .93721 14.92597 24 25 .90271 9.27819 .92009 11.51424 .93750 14.99995 25 26 .90300 9.30887 .92038 11.55982 .93779 15.07462 26 27 .90329 9.33973 .92067 11.60572 .93808 15.14999 27 28 .90358 9.37077 .92096 11.65197 .93837 15.22607 28 29 .90386 9.40201 .92125 11.69856 .93866 15.30287 29 30 .90415 9.43343 .92154 11.74550 .93895 15.38041 30 31 .90444 9.46505 .92183 11.79278 .93924 15.45869 31 32 .90473 9.49685 .92212 11.84042 .93953 15.53772 32 33 .90502 9.52886 .92241 11.88841 .93982 15.61751 33 34 .90531 9.56106 .92270 11.93677 .94011 15.69808 34 35 .90560 9.59346 .92299 11.98549 .94040 15.77944 35 36 .90589 9.62605 .92328 12.03458 .94069 15.86159 36 37 .90618 9.65885 .92357 12.08040 .94098 15.94456 37 38 .90647 9.69186 .92386 12.13388 .94127 16.02835 38 39 .90676 9.72507 .92415 12.18411 .94156 16.11297 39 40 .90705 9.75849 .92444 12.23472 .94186 16.19843 40 41 .90734 9.79212 .92473 12.28572 .94215 16.28476 41 42 .90763 9.82596 .92502 12.33712 .94244 16.37196 42 43 .90792 9.86001 .92531 12.38891 .94273 16.46005 43 44 .90821 9.89428 .92560 12.44112 .94302 16.54903 44 45 .90850 9.92877 .92589 12.49373 .94331 16.63893 45 46 .90879 9.96348 .92618 12.54676 .94360 16.72975 46 47 .90908 9.99841 .92647 12.60021 .94389 16.82152 47 48 .90937 10.03356 .92676 12.65408 .94418 16.91424 48 49 .90966 10.06894 .92705 12.70838 .94447 17.00794 49 50 .90995 10.10455 .92734 12.76312 .94476 17.10262 50 51 .91024 10.14039 .92763 12.81829 .94505 17.19830 51 52 .91053 10.17646 .92792 12.87391 .94534 17.29501 52 53 .91082 10.21277 .92821 12.92999 .94563 17.39274 53 54 .91111 10.24932 .92850 12.98651 .94592 17.49153 54 55 .91140 10.28610 .92879 13.04350 .94621 17.59139 55 50 .91169 10.32313 .92908 13.10096 .94650 17.69233 56 57 .91197 10.36040 .92937 13.15889 .94679 17.79438 57 58 .91296 10.39792 .92966 13.21730 .94708 17.89755 58 59 .91255 10.43569 .92995 13.27620 .94737 18.00185 59 60 .91284 10.47371 .93024 13.33559 1 .94766 18.10732 60 TABLE XXII.-VERSlNES AND EXBEOANTS. 381 / 87 88 89 / Vers. Exsec. Vers. Exsec. Vers. Exsec. .94766 18.10732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 2 .94825 18.32182 .96568 28.13917 .98313 58.27431 2 3 .94854 18.43088 .96597 28.38812 .98342 59.31411 3 4 .94883 18.54119 .96626 28.64137 .98371 60.39105 4 5 .94912 18.65275 .96655 28.89903 .98400 61.50715 5 6 .94941 18.76560 .96684 29.16120 .98429 62.66460 6 .94970 18.87976 .96714 29.42802 .98458 63.86572 7 8 .94999 18.99524 .96743 29.69960 .98487 65.11304 8 9 .95028 19.11208 .96772 29.97607 .98517 66.40927 9 10 .95057 19.23028 .96801 30.25758 .98546 67.75736 10 11 .95086 19.34989 .96830 30.54425 .98575 69.16047 11 12 .95115 19.47093 .96859 30.83623 .98604 70.62285 12 13 .95144 19.59341 .96888 31.13366 .98633 72.14583 13 14 .95173 19.71737 .96917 31.43671 .98662 73.73586 14 15 .95202 19.84283 .96946 31.74554 .98691 75.39655 15 16 .95231 19.96982 .96975 32.06030 .98720 77.13274 16 17 .95260 20.09838 .97004 32.38118 .98749 78.94968 17 18 .95289 20.22852 .97033 32.70835 .98778 80.85315 18 19 .95318 20.36027 .97062 33.04199 .98807 82.84947 19 20 .95347 20.49368 .97092 33.38232 .98836 84.94561 20 21 .95377 20.62876 .97121 33.72952 .98866 87.14924 21 22 .95406 20.76555 .97150 34.08380 .98895 89.46886 22 23 .95435 20.90409 .97179 34.44539 .98924 91.91387 23 24 .95464 21.04440 .97208 34.81452 .98953 94.49471 24 25 .95493 21.18653 .97237 35.19141 .98982 97.22303 25 26 .95522 21.33050 .97266 35.57633 .99011 100.1119 26 27 .95551 21.47635 .97295 35.96953 .99040 103.1757 27 28 .95580 21.62413 .97324 36.37127 .99069 106.4311 28 29 .95609 21.77386 .97353 36.78185 .99098 109.8966 29 30 .95638 21.92559 .97382 37.20155 .99127 113.5930 30 31 .95667 22.07935 .97411 37.63068 .99156 117.5444 31 32 .95696 22.23520 .97440 38.06957 .99186 121.7780 32 33 .95725 22.39316 .97470 38.51855 .99215 126.3253 33 34 .95754 22.55329 .97499 38.97797 .99244 131.2223 34 35 .95783 22.71563 .97528 39.44820 .99273 136.5111 35 36 .95812 22.88022 .97557 39.92963 .99302 142.2406 36 37 .95842 23.04712 .97586 40.42266 .99331 148.4684 37 38 .95871 23.2163? .97615 40.92772 .99360 155.2623 38 39 .95900 23.38802 .97644 41.4452E .99389 162.7033 39 40 .95929 23.56212 .97673 41.97571 .99418 170.8883 40 41 .95958 23.73873 .97702 42.51961 .99447 179.9350 41 42 .95987 23.91790 .97731 43.07746 .99476 189.9868 42 43 .96016 24.09969 .97760 43.64980 .99505 201.2212 43 44 .96045 24.28414 .97789 44.23720 .99535 213.8600 44 45 .96074 24.47134 .97819 44.84026 .99564 228.1839 45 46 .96103 24.66132 .97848 45.45963 .99593 244.5540 46 47 .96132 24.85417 .97877 46.09596 .99622 263.4427 47 48 .96161 25.04994 .97906 46.74997 .99651 285.4795 48 49 .96190 25.24869 .97935 47.42241 .99680 311.5230 49 50 .96219 25.45051 .97964 48.11406 .99709 342.7752 50 51 .9G248 25.65546 .97993 48.82576 .99738 380.9723 51 52 .%277 25.86360 .98022 49.55840 .99767 428.7187 52 53 .96307 26.07503 .98051 50.31290 .99796 490.1070 53 54 .96336 26.28981 .98080 51.09027 .99825 571.9581 54 55 .96365 26.50804 .98109 51.89156 .99855 686.5496 55 56 .96394 26.72978 .98138 52.71790 .99884 858.4369 56 57 .96423 26.95513 .98168 53.57046 .99913 1144.916 57 58 .96452 27.18417 .98197 54.45053 .99942 1717.874 58 59 .96481 27.41700 .98226 55.35946 .99971 3436.747 59 60 .96510 27.65371 .98255 56.29869 1.00000 Infinite 60 382 TABLE XXIII. USEFUL NUMBERS AND FORMULAS. Ratio of circumference to diameter n 3.1415906530 Reciprocal of same .3183098862 /T 4/iT 1.7724 538509 rr2 9.8696044011 Degrees in arc equal to radius 12 57 295779513 IT Minutes in arc equal to radius - - 3437.74677078 I Seconds in arc equal to radius - 206264 .80024 ' Length of seconds pendulum at New York in feet 3.25938 Square root of same 1.8054 I Acceleration due to gravity at New York g 32.1688 Square root of same ^g 5.67175 Cubic inches in U. S. gallon 231 Cubic inches in Imperial gallon 277.274 Cubic inches in U. S. bushel 2150.42 Cubic feet in U. S. bushel 1.244456 U. S. gallons in one cubic foot 7.4805 I iirperial gallons in one cubic foot 6.2321 AUvsJmum weight of one cubic foot of water (% 100 nearly)- 62.4 :\ ii'titoer of gr.iins in one pound avoirdupois 7000 Number of grains in one ounce avoirdupois 437.5 Number of grains in one pound troy 5760 Number of grains in one ounce troy 480 Circumference of circle (radius r) 2nr Area of circle (radius r) 7r> 2 Area of sector (arc of a degrees) ~ -Trr 2 oOO Area of sector (length of arc /) Area of segment (chord = c, middle ordinate = m), nearly.. Surface of a sphere (diameter d) Volume of a sphere (diameter d) Area of a triangle each side unity .4330 Area of a pentagon each side unity 1 . 7205 Area of a hexagon each side unity' = .4330 X 6 = 2.5980 Area of an octagon each side unity 4.8284 Volume spherical segment of one base of radius r and altitude a is = ^irar 2 -f %na 3 = 1.5708ar a + .5236a 3 ; = 1.58a?-' 2 -j- .5a 3 , very nearly; = 1.6ar 3 + .5a 3 , nearly. L^t A and a represent the bases of a frustum of a pyramid, L and I the lengths of corresponding sides, h the altitude and a l the area of either base for side equal to unity. Then volume = ^ (L* + Z 2 + IL). o This formula is immensely shorter and better than that given in the books, namely, vol = (A -\~ a -\- \ Aa), since this latter requires extra and needless computation in each term, besides the extraction of the square root. ! When I = 0, we get volume of pyramid = ; . o i Let R and r represent the radii of a frustum of a cone, and h its altitude. Then volume = ^nh(R^ + r 2 + rR). If r we have, volume of cone = J^/ijTrR 3 . TABLE XXIV. 383 CONVERSION OF ENGLISH FEET INTO METRES. 6 Feet. 1 2 8 4 5 7 8 9 Met. Met. Met. Met. Met. Met. Met. Met. Met. Met. 0.000 0.3048 0.609( 5 0,9144 1.2192 1.5239 1.82 872.1335 2.4383 2.7431 10 3 0479 3.3527 3. 6575 ) 3.9623 4.2671 4.5719 4.87 6715.1815 5.4863 5.7911 20 6.0959 6.4006 6.705J > 7,0102 7.3150 7.6198 7.92468.2294 8.5342 8.8390 30 9.1438 9.4486 9.753^ [ 10.058 10.363 10.668 10.9 7211.277 11.582 11.887 40 12.192 12.496 12.80 13.106 13.411 13.716 14. C 2014.325 14.630 14.935 50 15.239 15.544 15.841 ) 16.154 16.459 16.763 17.068 17.373 17.678 17.983 60 18.287 18.592 18.89' " 19.202 19.507 19.811 20.11620.421 20.726 21.031 70 21.335 21.640 21.94. ) 22.250 22.555 22.859 23.1 6423.46t) 23.774 24.079 80 24.383 24.688 24.99- J 25.298 25.602 25.907 86.5 1226.517 26.822 27.126 90 100 27.431 30.479 27.736 30.7S4 28.04 31.08< 28.346 ) 31.394 28 . 651 28 . 955 29 . 260 29 . 565 31.6981 32.003132.30832.613 29.87(1 32.918 30.174 33.222 CONVERSION OF METRES INTO ENGLISH FEET. Met, 1 2 3 4 5 6 7 8 9 Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet Feet. Feet. 0.000 3.2809 6.5018 9.8427 13.123 16.404 19.685 22.966 26.247 29.528 10 32.809 36.090 39.371 42.651 45.932 49.213 52.4 94 55.77E 59.056 62.337 20 65.618 68.899 72.179 75.461 78.741 82.02-2 85.303 88.584 91.865 95.146 30 98.427 101.71 104.99 108.27 111.55 114.83 118 11 121.3S 124.6" 127.96 40 131.24 134.52 137.80 141.08 144.36 147.64 150 9-2 154.2C 157.48 160.76 50 164.04 167.33 170 61 173.89 177.17 180.45183.73 187.01 190.29 193.57 60 196.85 200.13 203.42 206.70 209.98 213.26 210 r,4 219.82 223.10 226.38 70 229.66 232.94 236.22 239.51 242.79 246.07249. 3r. 252.69 255.91 259.19 80 262.47 265.75 269.03 272.31 275.60 278.88 js-j. 1(5 285.44 288.72 292.00 00 295.28 298.56 391.84 305.12 308.40 311.69314. 97 318.25 321.53 324.81 ing 328.09 331.37 334.65 337.93 341.21 344.49347.78 351.06 354.34 357.62 CONVERSION OF ENGLISH STATUTE-MILES INTO KILOMETRES. Miles. 1 2 3 4 5 6 7 8 9 Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. 0.0000 1.60933.2186 t.8279 6.4372 8.0465 9.6558 11.265212.8745 14.4848 10 16. 093 117.702 19. 312 20.921 22.530 24.139 25 74!) 27.358 28.967 30.577 20 32.18633.79535.405 J7.014 38.6-23 40.232 41 .842 43.451 45.060 46.670 30 48.27949.88851.498 33.107 54.716 56.325 57 .935 1 59.544 61.153 62.763 40 64.37265.98167.591 59.200 70.8091 72.418 74 .028 75.637 <7.246 78.856 50 i 80. 465 82. 074 83. 684 35.293 86.902 88.511 90 .121 91 730 93.339 94.949 60 '96.55898.16799.777 101.39 102.99 104.60 10 (3.21 107.82 109.43 111.04 70 112.65ill4.26115.87 117.48 119.08 120.69 122.30 123. 91j 125.52 127.13 80 128.74 130.35131.96 133.57 135.17 136.78 13 8.89 140.00 141.61 143.22 90 144. 851146. 44 148.05 149.66 151.26 152.87 154.48 156.09 157.70 159.31 : 100 160.93162.53164 14 165 75 167.35 168.96 170.57 172.181 173.79 175.40 [CONVERSION OF KILOMETRES INTO ENGLISH STATUTE-MILES ! Kilom. 1 2 3 4 5 6 7 8 9 Miles. Miles. O.OOOOiO.6214 Miles. 1.2427 Miles. 1.864! Miles. 2.4855 Miles. 3.1069 Miles. 3.7282 Miles. 4.3497 Miles' 4.9711 Miles. 5.59?4 10 6.213b 6.8352 7.4565 8.0780 8.6994 9.3208 9.94211 10.562 11.185 11.805 20 12.427 13.049 13.670 14.292 14.913 15.534 16 .156 16.776 17.399 18.019 30 18.641 19.263 19.884 20.506 21.127 21.748 23 370 22.990 23.613 24.233 1 40 24.85f 25.477 26.098 26.72027.341 27.962 28.584| 29.204 29.827' 30.-J47 50 31.06< 31.690 32.311 32.033J33.554 34.175 34.7971 35.417 36.040 36 660 60 37.28:; 37.904 38.525 39.147 39.768 40.3891 41 .011 41.631 42. 254 42.874 70 43.49' 44.118 44.739 45.361 45.982 46.603 47 47.845 iS.468 49.088 80 49.711 50.332 50.953 51 575 52 196 52.8171 53.4391 54.059 54.682 55.302 90 55.924 56.545 57.166 57.78858.400 59 050 58 . o.Y, 30.272 60.895 61.515 100 62. 13* !6,.759 63.380 64.002164.623 35.244 (55.866 ! 66.486 67.109 67.729 386 INDEX. PAGE Curves, how to lay out on the ground: With transit and chain ............................................... 53 With chain only ...................................................... - Curves, slackening speed on ........................................... nj Should be flattened on grades ...................................... n Middle ordinates, Table V, and ordinates to short chords, approxi- mate values. 51 Problems on curves 65 To locate curve when vertex is inaccessible 87 To locate when the vertex and the ends are inaccessible 87 Problems applicable to passing obstacles generally 65, 74, 88 Curves, compound, formulas for 115, 178 Radii and tangents compared 119 Problems pertaining to 120, 150 Problems, special 124 General relations deduced by substitutions 145 Curves tangent to curves, problems on 151 To find tangents of a compound curve of any number of branches 176 Wye problems 165, 170 'Curves, concentric, problems concerning 170, 173 Curves, distance apart starting from same point, etc 175 To locate the second branch of a compound curve from a point on the first branch 174 Curves, reversed, problems on 184 Curves, turnout, difference between ending on tangents and con- necting with tracks 199 Curve, the true transition: Definitions and essential requisites x?3 Elementary relations To lay out the curve Special problems and laying out the curve To lay out the curve from different points on it 245 Comparison of the true curve with the compound curve com- monly used as such 151 Curves, vertical 107 Curved tracks, angle of intersection 178 Datum in leveling 90 Deflection angles 53 Degree of curve 42 Distances of frogs from switch, Table VI 187 Earthwork: Area of level section 254 Area of section not level 255 Formulas for volumes 256 "End-area volume^" and "middle-area volumes" compared 257 Special formulas and cases 259 Loaded flat cars, piles of broken stone, etc 263 Ends of embankments or "dumps" 264 Ground irregular laterally , -'64 INDEX. 387 PAGE Karthwork : Mixed work 265 Correction for curvature 267 Overhaul 268 Monthly estimates 269 Final estimates 270 Computation of volume of prismoids level laterally 270 Clearing and grubbing 284 Staking out work 286 Borrow-pits 294 Shrinkage 294 Two important principles in ''staking out" 291 Elevation a relative question 90 Elevation of outer rail 112 Simple and accurate formula for elevation 113 Proper elevation to use 113 1'ield-book, form of, for leveling .' 92 Frogs and switches, Table VI 318 Grade line -'84 Gradienter, formulas deduced for it require no computation 105 Horizon. To find height of object by the dip of the horizon. Special and exact formulas apply without computation 98 Level, to adjust 18 Use and care of 22 Leveling , 90 How to keep notes on 92 Proof of correctness 93 Trigonometric 97 Locate a level line, to 94 Locate a grade line, to.... 9; Location 5 Locating engineer, qualifications for 9 Maps to be used freely in studying the country 7 Needle, magnetic, to adjust 23 How to use 23 Numbers forming integral sides of right triangles 76 Obstacles ,in surveying 76 To erect a perpendicular 76 To let fall a perpendicular 77 Ditto from an inaccessible point 77 To prolong a line 78 Obstacles to measurement of line: When one end is inaccessible 79 When both ends of the line are inaccessible 80 A line perpendicular to an inaccessible line 81 A line determined from another line across an inaccessible space. 81 Offsets, to calculate 48 Preliminary survey 5, 82 Plane right triangles, solution of 3t Plane oblique triangles, solution of ...,,... , 35 388 INDEX. PAGE Piers, bridge 298 Reconnoissance c Pocket compass for 8 Requirements for g Resistance, train 9 Sines, tangents and secants defined 22 Slope stakes, to set 286 Stadia. Formulas deduced and simplified 101 Streams, bends in, to avoid 7 Surveys, preliminary 5, 82 "Tangent proportion" deduced 33 Tangent to a curve from fixed point, how to locate 75 Tangent of any compound curve, how to find the length 176 Tables. See Contents. Track-laying 296 Transit, to adjust 12 Use and care of 15 Trigonometry 25 The application of but one geometrical principle 27 Tunnels 298 Turnouts: Single 196, 212, 216 Double 208, 210, 214, 217, 219 Traversing 82 JOHN WILEY & SONS, 53 E. Tenth Street, New York, PUBLISH: INSPECTION OF THE MATERIALS AND WORKMANSHIP EMPLOYED IN CONSTRUCTION. A Reference Book for the Use of Inspectors, Superinten- dents and Others Engaged in the Construction of Public and Private Work, Etc. By Austin T. 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