m TEXT BOOK ON RAILROAD SURVEYING BY GEORGE WELLINGTON PICKELS, C.E. INSTRUCTOR IN CIVIL ENGINEERING UNIVERSITY OF ILLINOIS AND CARROLL CARSON WILEY, C.E. ASSOCIATE IN CIVIL ENGINEERING UNIVERSITY OF ILLINOIS FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON : CHAPMAN & HALL, LIMITED 1914 '/ /r& Copyright, 1913, by G. W. PICKELS AND C. C. WILEY PUBLISHERS PRINTING COMPANY 207-217 West Twenty-fifth Street, New York PREFACE THIS book was written primarily for the use of students in Railroad Surveying, and is practically a summary of the notes used by the authors in their classes for a number of years. The aim throughout has been to present the essentials of the subject in simple, concise form and to give modern practical methods only. To this end detailed mathematical solutions are omitted, obsolete methods abandoned, and only the more common problems outlined; consequently the subject-matter occupies considerably less space than the ordinary hand-book, but at the same time covers all phases of the subject thoroughly. From the standpoint of the instructor, some of the essential characteristics of the book are: (1) A co-ordination of the entire subject, indicating the general order and the solution of the various steps in the work. (2) The explanation of reasons for or conditions under which the various problems arise. (3) The assumption of a knowledge of elementary surveying and plane trigonometry on the part of the reader. (4) The consequent omission of detailed mathematics. Out- lined solutions only are given, and hence the student must follow the solution of a problem through step by step instead of sub- stituting blindly in some formula which has the appearance of being suitable. The student is therefore more thoroughly trained in the plan of attack, and consequently can more readily handle the multitude of problems which he can not find "all worked out" in this or any other book. Although designed as a book for student use, it is believed that it will appeal to the practicing engineer as well, and the latter's attention is respectfully called to the following features: (1) The simplified curve nomenclature as recommended by the American Railway Engineering Association has been adopted. (2) The " Spiral" has been co-ordinated with the circular curve, where it properly belongs under modern railroad practice. It will be noted that the spiral here given is of a general form and is iii 287914 IV PREFACE equally applicable with chords of any desired length as a so- called " Six-chord" or " Ten-chord" spiral. (3) " Turnouts" are presented in practical form as actually used on steam roads in America, i.e., with straight frogs and switches. Acknowledgment is hereby made of the courtesy of Prof. J. C. Nagle in permitting the use of Tables 5, 6, 8, and 9, which are taken from his " Field Manual for Railroad Engineers"; to Professor Carhart, for Table 7, which is taken from his "Field Book for Civil Engineers"; and to Prof. Walter Loring Webb, for Tables 9 and 10 from his work on "Railroad Construction," which appear as Tables 10 and 11 in this volume. GEORGE WELLINGTON PICKELS, CARROLL CARSON WILEY. University of Illinois, Urbana, 111., August, 1913. CONTENTS CHAPTER I RAILROAD SURVEYS PARAGRAPH PAGE 1. Terminals and Intermediate Points Connected by a Railroad 1 2. The Reconnoissance Survey 1 3. Reconnoissance in Prairie Country ...... 2 4. Reconnoissance in Rolling, Hilly Country .... 2 5. Reconnoissance in Mountainous Country .... 2 6. Preliminary Survey in Prairie Country 3 7. Preliminary Survey in Rolling, Hilly Country ... 3 8. Preliminary Survey in Mountainous Country ... 4 9. The Location Survey 4 10. Reference Stakes 4 11. Slope Stakes 5 12. Distribution Stakes 6 13. Borrow-Pit Stakes 6 14. The Location of Culverts and Trestles 6 15. The Time to Set Stakes 6 16. Monthly Estimate Surveys 6 17. Finishing Stakes 6 18. Center Stakes 6 19. Grade Stakes 6 20. Right of Way Stakes 6 21. Special Surveys 7 22. Maintenance Surveys 7 23. Monumenting 7 24. Suryej^s for Additional Tracks 7 25. The Transit Party 7 26. The Level Party 9 27. The Topography Party 9 28. The Land-line Party 9 29. Drafting 9 v VI CONTENTS PARAGRAPH PAGE 30. The Transit and Tape Method of Making Preliminary Surveys 10 31. The Transit and Stadia Method of Making Preliminary Surveys 10 32. The Plane Table Method of Making Preliminary Surveys 10 33. Remarks 11 34. Bench Marks 11 CHAPTER II MAPS 35. The Reconnoissance Map 12 36. The Reconnoissance Profile 12 37. The First Preliminary Maps . 12 38. The Preliminary Profiles 12 39. The Preliminary Map (proper) ....'... 12 40. The Projected Location 13 41. Location Notes 13 42. The Location Map 13 43. The Location Profile 14 44. Additional Profiles 14 45. Legal Right of Way Maps 15 46. Maintenance of Way Right of Way Maps .... 15 47. Station Maps 15 48. Progress Profiles 15 49. Progress Photographs 15 CHAPTER III DISTANCE, CURVATURE, AND GRADES 50. Additional Distance 16 51. Effect of Curvature 17 52. Choice of Curvature 17 53. Minor Grades 18 54. Ruling Grades 18 55. Pusher Grades . 19 56. Choice of Grades 19 57. Vertical Curves 21 CONTENTS Vll CHAPTER IV CURVES Part 1. Circular Curves PARAGRAPH PAGE 58. Functions of a Simple Curve 24 59. Relation between R and D 26 60. Field Work . 27 61. Fundamental Principles 32 62. Offsets from Chords Produced >33 63. Chord Offset Method . 34 64. Example 35 65. To Find New Radius for a Given Change in E . . 36 66. To Change T.C. to Make C.T. Fall on ParaUel Tangent 37 67. To Change R and T.C. to Make C.T. FaU on ParaUel Tangent, on same Radial Line, Approximately . 38 68. To Change R to Make T.C. Fall on Parallel Tangent 38 69. To Shift C.T. so that Forward Tangent will Pass Through Controlling Point 39 70. Compound Curves 41 71. Given the Central Angles and Radii to Find the Tangents 43 72. Given both Tangents and one Radius to Find the Other Radius 43 73. Field Work 45 74. To Change Location of C.C. so that C.T. will FaU on ParaUel Tangent 45 75. To Change Second Radius and the Location of C.C. so that C.T. wiU FaU on Parallel Tangent, on Same Radial line, Approximately 46 76. Reversed Curves 49 Part 2. Spirals 77. Definition of Spiral '49 78. Functions of Spiral 50 79. Formulas 52 80. Length of Spiral 52 81. Field Work 55 82. Deflection-Angle Method 56 83. Offset Method 60 84. To Insert Spiral between the Branches of Compound Curve ..,-.- 62 85. Deflection Angles from Point on Spiral 62 86. To Spiral Compound Curve at Each End and between Branches ... 63 Vlll CONTENTS PARAGRAPH PAGE 87. Application of Spiral to Existing Simple Curves . . 64 88. To Spiral Existing Simple Curve by Shifting Track outward near Center 64 89. To Spiral Existing Simple Curve by Sharpening Curve at Ends 66 90. To Spiral Existing Simple Curve by Compounding Curve near Center 67 91. Application of Spiral to Existing Compound Curves . 68 CHAPTER V EARTHWORK 92. Railroad Cross-Sections 73 93. Width of Roadbed 73 94. Ditches 74 95. Side Slopes 74 96. Borrow Pits 74 97. Shrinkage 75 98. Growth ' . . 75 99. Settlement 75 100. Slope Stakes 76 101. Cross-Sectioning 77 102. Field Work 77 103. Sections at Grade Points 80' 104. Side-Hill Sections 82 105. Uncompleted Sections 82 106. End Area Method of Computing Volumes ... 83 107. Prismoidal Method of Computing Volumes ... 84 108. Earthwork on Curves 85 109. Computation of Haul 88 110. Problem in the Distribution of Earth 87 111. Problem, to Find Limit of Profitable Haul ... 89 112. Effect of Trestle?, etc., on the Distribution of Earth 90 CHAPTER VI TURNOUTS, CONNECTIONS, AND CROSSINGS 113. Outline of Subjects Treated 91 114. Definitions 91 115. Switches 92 116. Frogs 94 117. Frog Angles 96 118. Frog Numbers 96 CONTENTS IX PARAGRAPH PAGE 119. Location of Turnouts 96 120. Turnouts from Straight Track 96 121. Turnouts from Curved Track ., 97 122. Double Turnouts 98 123. Practical Leads 98 124. Diverging Tracks, from Straight Tracks .... 99 125. Diverging Tracks, from Curved Tracks .... 100 126. Sidings from Straight Tracks 104 127. Sidings from Curved Tracks 106 128. Crossovers between Straight Tracks 109 129. Crossovers between Curved Tracks Ill 130. Connection between Intersecting Straight Tracks 111 131. Connection between One Straight and One Curved Intersecting Track . .113 132. Connection between Intersecting Curved Tracks . 115 133. Crossings 116 134. Crossing Data 117 135. Straight Crossings 118 136. Single-Curve Crossings . 119 137. Double-Curve Crossings 120 APPENDIX. THEORY OF SPIRAL 138. Derivation of 3 and A 121 139. Derivation of the Coordinates, x and y 121 140. Derivation of a, A, b, B ; . . 122 141. Derivation of 123 142. The Osculating Circle 124 143. Superelevation 124 TABLES 1. Spiral Table 70 2. Turnout Table 99 3. Siding Table 105 4. Crossover Table. 110 5. Table of Radii 128 6. Functions of a 1 Curve 131 7. Logarithms of Numbers 158 8. Logarithmic Sines and Cosines 176 9. Logarithmic Tangents and Cotangents 191 10. Natural Sines, Cosines, Tangents and Cotangents . . 206 11. Natural Versed Lines and External Secants . . . . 229 12. Cubic Yards per 100 ft. in Terms of Center Height . 252 13. Cubic Yards per 100 ft. in Terms of Sectional Area . 258 14. Trigonometric Formulas 262 TEXT BOOK ON RAILROAD SURVEYING CHAPTER I RAILROAD SURVEYS THE practice in railroad location surveys in one part of the United States is entirely different from that in another part, so that rules which will apply in one locality will not apply at all in others. All country can be broadly classed in three divisions: (1) level prairie country offering no obstacles in the way of hills, valleys, etc., and allowing the locating engineer much latitude in the placing of his line, and consequently re- ducing distance and curvature to a minimum; (2) rolling, hilly country through which several lines are possible, none of them departing to any great extent from the direct line between controlling points; and (3) mountainous country, which taxes the skill and ingenuity of 'the locating engineer to the utmost. Reconnoissance Surveys 1. The terminals and intermediate points connected by a railroad are determined by its promoters with a view to the amount of traffic that can be expected from them. The nature and the amount of the traffic and the direction of heaviest haul determine the maximum gradients and curvature advisable. The first thing to be determined by the locating engineer is the directions of the lines joining controlling points. This informa- tion can usually be obtained with sufficient accuracy from a map of the territory which it is generally possible to obtain. The topographic maps published by the government are the best. 2. A reconnoissance is then made of the strip of country through which the road is to pass, as the result of which some of 1 2 TEXT BOOK ON RAILROAD SURVEYING the routes are eliminated as impractical and one or more chosen for a more detailed survey. 3. If the country is of the first class, no reconnoissance is necessary and the survey is in the nature of a preliminary. 4. For country of the second class, the reconnoissance should be made across-country on horseback, following the direction of the line determined from the map. If in following the direct line between controlling points obstacles are met with which can not be surmounted, such as high hills, the engineer should explore on both sides of the obstacle and decide if possible which route offers the least resistance. Before deflecting from a straight line, the engineer must be sure that his reasons for so doing are justified from an economic standpoint. In country of the second class it is seldom that grades cause much trouble, and the main lookout of the engineer is to keep the amount of curvature as low as possible. The result of the reconnoissance through country of the second class is that one or more routes are selected for a more detailed survey. 5. It is in mountainous country third class that all the skill of the engineer is brought into play. The drainage of the country should be carefully studied, as it plays a very important part in the location of a railroad. If the controlling points are in the same valley, the main problem is solved; and the con- forming of the alinement to the topography is merely a matter of detail. But when the controlling points are in different valleys, the ridges between them have to be crossed, and the principal object of the reconnoissance is to discover the most favorable crossing places in the valleys and on the ridges. The saddles in the ridges and the most favorable river crossings become secondary controlling points. The location of all such points is platted on the map, and their elevations and distances apart are recorded. The most important and useful " instruments " used on reconnoissance are the judgment and experience of the locating engineer, as upon these depend the amount and cost of more detailed surveys and the cost of construction and of operation. RAILROAD SURVEYS 3 Preliminary Surveys 6. In country of the first class, a preliminary line is run for direction. The magnetic bearing of the direct line between controlling points is scaled off the map, and a line having this direction is initiated from the first controlling point and pro- duced to the second controlling point. Since the direction of the line as obtained from the map and this direction as laid off in the field are subject to considerable error, the first line run will probably pass to one side of the second controlling point. The distance by which the transit line misses its mark is noted and the correction that must be applied to the first line is com- puted. The next line run will be the location. In running the preliminary line stakes should be placed only at transit points. A straight line is the most difficult one to run, and particular care must be taken to avoid errors. It must be remembered that the preliminary is a reference line, and if it is not straight the purpose for which it is run is defeated. The distances between hubs can be determined with sufficient accuracy by means of the stadia. 7. In country of the second class, preliminary lines are run over each of the routes chosen by the locating engineer on the reconnoissance survey. The data taken are such that the several routes can be compared with respect to distance, grades, and curvature. Usually lines run with the transit and stadia will give sufficient data as regards distance and curvature; and the elevations of enough commanding points can be taken with the stadia to indicate the grades that will be required by the several routes. From these data one of the routes will usually appear superior to the others, and what is generally known as a preliminary survey is then made over the selected route. A preliminary survey as generally understood is a topographic survey of a narrow strip of country within which the road must pass. The purpose of this survey is to secure data from which a topographic map can be platted upon which the paper location is projected. The transit and stadia line already run over the chosen route is used as the base line from which the topography is taken. Although this line is usually measured with the tape, time used for this purpose is wasted, as it is impossible to plat the traverse distances to a consistent degree TEXT BOOK ON RAILROAD SURVEYING of accuracy. Spirit levels are then run over the line to determine the elevations of the transit points, and bench marks are es- tablished at half-mile intervals. 8. In country of the third class, several preliminary lines are required, each following one of the routes chosen by the locating engineer. The approximate grades of the lines between secondary controlling points are obtained from the data taken on the re- connoissance survey, and the preliminary line should be chosen so that the grade line so determined will conform as nearly as possible to the surface of the ground. If this is done, topography will not have to be taken as far on each side of the line as would otherwise be necessary. In mountainous country topography can be taken more accurately with the hand level than with the stadia, and line stakes are placed every 100 feet and their eleva- tions determined by spirit leveling so that this can be done. In other respects the methods of surveying in mountainous coun- try are the same as those used in country of the second class. Location Survey 9. On location the line is run very carefully, stakes are driven every 100 feet, the plusses and angles of all property lines are taken, and curves are run in where indicated on the map. The distances from intersections with property lines to the nearest government section monument, or other legal monument, are carefully chained, so that accurate descriptions of the right of way can be drawn up. All buildings near the line which will be damaged by it must be located and an estimate of the damage be made. A level party follows the transit party taking profile levels and establishing bench marks unless previously established. From these data alinement maps, right of way maps, and pro- files are made; and after the grade line is established on the profile, construction can begin. Construction Surveys 10. Reference stakes. After the location has been made and accepted, and just before construction begins, the beginning and end of all curves and intermediate points on long tangents are " tied-in " by reference stakes so that after the construction RAILROAD SURVEYS 5 woik is completed these points can be re-located in their correct positions. There are several methods of referencing a point; the one shown in Fig. 1 is very satisfactory. A is the point to be referenced. First choose permanent points at C and E at least 300 feet from A, making the angle CAE as nearly a right angle as possible. A distant windmill or house-chimney makes an ideal point. If these are not to be had, a nail FIG. i. driven into the trunk of a tree about five feet from the ground is good. If hubs are used they should be made as permanent as possible, and located so that they will not be disturbed. Af- ter C and E have been determined, set a hub at point B on the line AC, and another one at D on the line AE. B and D should be placed far enough from the center line of the road so that they will not be disturbed during construction. Point A is re- located after construction by the intersection of the lines CB and ED, the transit being set-up at B and D. Care and judg- ment should be used in locating reference points, as they should be used after the roadbed has settled for locating per- manent monuments. 11. Slope stakes are then set at each station on each side of the center line at the points where the side slopes of the cut or fill will intersect the ground surface. These stakes are for the guidance of the contractor, and have marked on them the vertical distance from the ground at the stake to the level of the roadbed. (For method of setting slope stakes, see paragraph 100.) 6 TEXT BOOK ON RAILROAD SURVEYING 12. Distribution stakes are set to show the contractor the desired movement of the earth from the cuts into the fills. (See paragraph 108.) 13. Borrow-pit stakes are set to indicate to the contractor the limits within which he may borrow earth for making the fills. 14. The location of culverts and trestles must also be staked out. If possible, these should be constructed in advance of the grading so that openings will not have to be left in the fill to be completed later. 15. The time to set stakes is just before they are needed. If they are set too long a time ahead, some of them will invariably be knocked out and will require re-setting. 16. Monthly estimate surveys. During the last few days of every month surveys are made to determine the amount of earth moved by the contractor during the month, so that he can be paid. 17. Finishing stakes. From the slope stakes the contractor can construct the roadbed to within a few inches of the correct grade. At this stage of the work it is customary to give finishing stakes, which are stakes driven to grade at the edges of the roadbed at each station. From these the contractor is able to finish the roadbed to the correct grade and width. 18. Center stakes. After the earthwork is completed, the important points on the center line are re-located from the reference stakes, and center stakes (usually untacked) are driven from which the track is laid. 19. Grade stakes. After the track is laid, grade stakes are driven at every station and at those points where the grade changes with their tops to the grade of the final top or base or rail. Grade stakes are placed on the inside of curves. 20. Right of way stakes. The right of way fences are usually built as soon as the materials for construction can be hauled over the line. For the guidance of the fence foreman, stakes are placed on each right of way line (1) opposite the beginning and RAILROAD SURVEYS 7 end of all curves, (2) opposite each station on curves, (3) from 300 to 500 feet apart on tangents, and (4) at all jogs in the right of way. These stakes should be long enough to be seen above the weeds, wheat, oats, etc. Laths are excellent for this purpose. 21. Special surveys. After the track is laid, special structures, such as station buildings, water tanks, cattle pens, etc., will require staking out. In addition, the parts of the right of way leased to coal, lumber, and grain companies must be staked. Maintenance Surveys 22. Due to the fact that it takes two or three years for the roadbed to settle and for the track to become thoroughly em- bedded in the ballast, it will be necessary to re-set center and grade stakes frequently during this period. All center stakes which are set after the track is laid should be tacked. 23. Monumenting. After the track and roadbed have settled thoroughly, permanent monuments should be placed at the beginning and end of all spirals and circular curves, between the branches of compound curves, and at intermediate points on long tangents. 24. Additional tracks, such as side tracks, business tracks, branch-line tracks, yard tracks of various kinds, cross-overs, etc., are required to be staked out from time to time. There is no end to surveys of this kind, and all the large railroads employ maintenance parties who do nothing else. Organization of Parties The field corps is usually divided into (1) a transit party, (2) a level party, (3) a topography party, and sometimes (4) a land-line party. 25. TRANSIT PARTY. The members of the transit party and their duties are as follows: The Locating Engineer is the chief of the entire surveying corps, and receives his instructions from and reports to the Chief 8 TEXT BOOK ON RAILROAD SURVEYING Engineer of the railroad company. His duties are: to direct all the surveys from the reconnoissance to the location, to pro- vide accommodations for his party, to pay all general expenses, and in case a camp is necessary to purchase all supplies and to manage the camp. The Transitman is next in rank to the locating engineer, and in his absence is in charge of the party. His duties are: to do the transit work, which consists of lining in the chainmen, measuring the angles between successive tangents, noting the bearings of the tangents, measuring the angles which the line makes with all railways, highways, streams, and property lines, and record- ing the plusses at which they cross the line; and to keep the notes of the transit party. On construction the transitman usually becomes Resident Engineer and has charge of from 8 to 15 miles of construction. The Head Chainman ranks next to the transitman in the transit party, and is directly in charge of the rear chainman, stakeman, and axmen. His duties are: to see that the distances are chained correctly; to see that the stakes are driven on line, that they are driven straight, and that they are marked cor- rectly; to direct the axmen where to cut in opening up the line; to set new transit points ; and to direct the taking of plusses. The head chainman has a very important position, as he regulates the speed of the entire party. In open country frequently the locating engineer takes this position. The head chainman carries the zero end of the tape. The Rear Chainman's duties are : to hold his end of the tape on the last stake driven while the head chainman gets the distance; and to take and record all plusses which he turns over to the transitman at frequent intervals. The Rear Flagman's duties are: to give the transitman a sight on the back sight station whenever he signals for it; and to carry excess baggage. The rear flagman should be a wide- awake man with good eyesight. The Stakeman's duties are: to carry the stakes; to mark the station numbers on the stakes; and to drive the stakes as directed by the head chainman. The Axmen do all the necessary clearing in order that the transit and level parties may have a clear path. They are sometimes required to make the stakes. RAILROAD SURVEYS 26. THE LEVEL PARTY. The members of the level party and their duties are as follows: The Levelman is chief of the level party and ranks next to the transitman in the surveying corps. His duties are: to run profile levels over the line and to establish bench marks; and to keep the level notes. The Rodman's duties are: to hold the rod vertical upon the ground at each station, and at those intermediate points where the longitudinal slope of the ground changes; and to keep " peg notes " as a check on the levelman's computations. 27. THE TOPOGRAPHY PARTY. The members of the topography party and their duties are as follows: The Topographer usually holds equal rank with the levelman in the surveying corps. This position is a very important one and should be filled by an experienced man. The duties of the topographer are: to take all data necessary for making an accurate contour map of a strip of country sufficiently wide to enable the engineer to make an intelligent projected location; and to record these data in such a way that they will be readily understood by the draftsman. The topographer is assisted in his work by a rodman and a tapeman. 28. THE LAND-LINE PARTY. The duties of this party are : to measure the angles which the line makes with all railways, highways, streams, and property lines; to tie-in the line to the nearest government monuments so that legal descriptions may be prepared of the required right of way ; and to secure the names of the property owners. 29. Drafting. In addition to the above field parties there is the field draftsman, who does his work in camp. His duties are to plat the notes taken by all the parties the previous day, which necessitates the use of two sets of field note-books or else loose-leaf note-books. In some cases the draftsman, with the help of tha locating engineer and the transitman, plats the notes each night ; and the levelman plats the profile of the line over which he ran levels that day. Thus the map is kept up to date, and the locating engineer can project his location as the line advances. 10 TEXT BOOK ON RAILROAD SURVEYING Methods of Making Preliminary Surveys The preliminary survey is the most expensive survey, and is of primary importance since the location depends directly upon it. Hence the method of making it should be given considerable thought in order that it may be done with accuracy and economy. There are at present three general methods used: (1) the transit and tape method, (2) the transit and stadia method, and (3) the plane table method. 30. The transit and tape method is by far the most common. The transit party runs the line with transit and tape, the level party follows taking profile levels, and the topography party follows the level party; the land-line party may come in any- where after the transit party. This kind of a survey will require from six to fifteen men, depending on whether the several parties have a separate personnel and on the number of axmen required to open the line. 31. The transit and stadia method consists in running the line with the transit and stadia. Stakes are placed only at transit stations, and the elevations of these points are determined by transit and trigonometric leveling. The location and eleva- tions of important intermediate points along the line are de- termined in a similar manner. While the transit is at each station, the topography around that point is taken with the stadia. Thus all the needed data are taken as the line advances. These data may be recorded in the note-book and worked up later by the party draftsman; but it is much better if the draftsman plats the notes as the transitman takes them and draws in the contours while the landscape is before him. The draftsman holds a very important position in this party and should be an expert in that line of work. A survey by this method is very accurate as re- gards the contours, which are the most important item on a preliminary map; and, if good men are employed, it is more efficient than the first method. This method requires a transit- man, a draftsman, two rodmen, and as many axmen as the nature of the country may require. 32. The plane table method is very similar to the one just described and differs from it mainly in the use of instruments. RAILROAD SURVEYS 11 The plane table takes the place of the transit, and the plane tableman does the drafting. Owing to the difficulty in handling and setting up the plane table, it is doubtful whether this method is as efficient as the transit and stadia method; and although the services of the draftsman are dispensed with, yet the progress is possibly not as rapid. 33. Remarks. For long lines that justify the employment of a large number of men the transit and tape method is probably the most efficient. In other cases, however, the transit and stadia method will prove the more economical. The latter method has not been used to any considerable extent, due to the fact that few engineers fully appreciate the advantages of the stadia method; but in the few cases in which it has been tried it has fully demonstrated its superiority, particularly for open country. 34. Bench marks. When spirit-levels are run over the line, bench marks should be established at half-mile intervals, ap- proximately, and should be placed far enough from the center line so that they will not be disturbed during construction. After construction, permanent bench marks should be established on all permanent structures, such as concrete bridges, and at every station building along the line. The common practice of using spikes driven in telephone poles and mile posts is a very dangerous one, as these are frequently moved and re-set. If the road has few concrete or steel structures, then bench marks may be established on trestles. When these are renewed, the elevations of the several parts will rarely be changed more than an inch; while a bench mark on a telephone pole may be changed several feet. Note. For further information on the subject of Railway Surveys and the Economics of Railway Location, see A. M. Wellington's "Economic Theory of the Location of Railways; " F. Lavis' " Railway Location Surveys and Estimates," or W. L. Webb's "Economics of Railroad Construction." CHAPTER II MAPS Reconnoissance Maps 35. The reconnoissance map consists of a general sketch of the country which the locating engineer has investigated, and shows the several routes that are possible. Only controlling points, such as towns to be passed through, available stream crossings, saddles in the ridges, etc., are shown. If an existing map of the country is available, it is best to draw in the routes directly upon it. The notes taken on the reconnoissance survey regarding the geological formations, the cultural features of the country, the width, depth, and current of streams, etc., are considered part of the reconnoissance map, and frequently have great weight in the choice of routes. 36. The reconnoissance profile is made from the elevations of the controlling points and the distances between them. Pro- files are made for each of the several routes and are frequently placed on the same sheet, so that a more intimate comparison can be made. The purpose of the reconnoissance maps is to eliminate the impractical routes and to determine which ones will bear a more detailed investigation. Preliminary Maps 37. The first preliminary maps show the transit and stadia lines run, and all railroads, highways* and streams that cross the lines. 38. The preliminary profiles corresponding to the first pre- liminary maps are made from the elevations of the traverse stations. From a study of these maps one route occasionally two will appear superior to the others, and the preliminary survey proper is then made over this route. 39. The preliminary map is generally made to a scale of from 200 to 400 feet to the inch, and is a complete topographic map of 12 MAPS 13 a strip of country from 100 to 1,000 feet wide. All highways, railways, streams, and buildings are shown. Frequently it is desirable to show the property lines, the names of land owners, and government monuments. Also all geological formations and other features that might affect the location of the line are indicated. Occasionally, special features may require an aux- iliary map to a scale of 50 or 100 feet to the inch over short stretches of the line. 40. The projected location is then made on the preliminary map. First project that line which will give the best alinement and grades. This will be the best line from the standpoint of operation, and the one to be used unless the cost of construction is prohibitive. In this case determine the line whose construction will come with- in the allowable cost and at the same time keeps within the allowable limits of curvature and grade. This can be done only by trial and requires a large amount of skill and judgment. In order to determine the grades and the amount of earthwork for the several projected lines, it will be necessary to construct a profile for each of them from the contour map. In locating grade lines on these profiles it must be remembered (1) that intersecting railways must be crossed either at grade or at a clearance distance above or below grade, (2) that highways can be raised or lowered within certain limits, and (3) that streams must be crossed a safe distance above high-water mark. After satisfactory tangents have been projected, they are connected by curves which most nearly conform to the contour of the ground, and at the same time keep within the maximum curvature. If the curves are to be spiraled, allowance must be made at this time. 41. Location notes are then made from the accepted projected location. The bearings of tangents, the plusses of the be- ginnings and ends of all curves, the central angles of all curves, and the degrees of curve are scaled off the map and recorded. These condensed notes are used in making the final location. Location Map 42. The location map is usually made to the same scale as the preliminary map, and shows all railways, highways, streams, and property lines that cross the line, together with their plusses; 14 TEXT BOOK ON RAILROAD SURVEYING the names of the property owners and the amount of right of way required from each; the government subdivision lines and the numbers of sections, etc.; the distances from the line to govern- ment monuments; the boundaries of each field through which the line passes; the location and size of all openings; and any other data that will be of use to the construction engineer or to the right of way agent. The location map can often be made by adding the necessary data to the preliminary map, and then making a new tracing, omitting the contours and such other features that are not de- sired on the final map. 43. The location profile is made from the profile levels which were run over the located line, and shows: the ground line; the grade line, including the per cent, of grade of the different por- tions and the elevations of all points where the grade changes and all points on vertical curves; the location and dimensions of all openings in the embankment ; the plusses of railways, highways, streams, and openings in the embankment ; the elevations of the top of rail of all intersecting railroads, and the high-water mark of streams; the descriptions and elevations of all bench marks; a rectified alinement map at the bottom of the profile sheet, con- taining practically all the data that are shown on the location map; the distribution diagram, which is drawn between the profile proper and the alinement; on the profile proper the econom- ical movement of the earth from the cuts into the fills; and the amount of excavation, overhaul, and borrow at all points along the line. In fact, the location profile contains practically all the information needed by the resident and construction engineers. It is used by them in staking out all the construction work, with the exception of large bridges and trestles, for which separate plans are made. 44. Additional profiles are made for any change in the eleva- tions of highways and railways. These profiles are short and are sometimes (and preferably) placed on the location profile opposite the points where the changes are made. Right of Way Maps Right of way maps are of two kinds: (1) legal maps and (2) maintenance of way maps. MAPS 15 45. Legal right of way maps are made on sheets which are the same size as the sheets on which the conveyance deeds are drawn up (usually 8J x 13). A separate map is made for each de- scription, and a blue-print is attached to the deed and becomes a part of it and is recorded along with the deed. The railroad company preserves its copies of the deeds along with the right of way maps in the form of a book, which is known as the right of way book. 46. Maintenance of way right of way maps are made from the location map and a separate 'sheet (about 8J x 10) is used for each section (i.e., government section) or for each mile of track. These sheets are bound into a book and are used by the main- tenance of way department. Construction Maps 47. Station maps are usually made to a scale of 50 or 100 feet to the inch, and show the proposed buildings and tracks at each station. Any changes in the plans which are made during con- struction are recorded on the map. Some railroads have stand- ard plans for station layouts, and these are followed as closely as conditions will permit. 48. Progress profiles. Each month after the monthly es- timate surveys have been made, the amount of grading done during the month is shown graphically on the location profile in colored pencil. Each month a different color is used, so that the chief engineer and other railroad officials can see at a glance the progress of the construction work and can compare the amount done each month with that of former months. 49. Progress photographs. For the same reason photographs should be taken each month or oftener of the various structures under construction, such as concrete arches, bridges, trestles, station buildings, etc. Photographs are excellent auxiliaries to a written report. CHAPTER III DISTANCE, CURVATURE, AND GRADES IN the location of a railroad there are three factors that make for success or failure from a financial standpoint, namely, dis- tance, curvature, and grades. The best location is that in which each of these is a minimum. This would be when the line be- tween terminals was perfectly straight, and on a uniform grade. This ideal line involves a maximum expenditure for construction and a minimum outlay for operation. Such a line, of course, is a financial if not a physical impossibility. It must be re- membered that the first cost is a fixed amount, while the cost of operation continues as long as the road exists; and in time the additional cost of operating a cheaply located road will more than offset the amount saved in the construction. This state- ment is borne out by the fact that all the large railroads are spending huge amounts yearly for the reduction of curvature and grades, in order that operating expenses may be reduced. The locating engineer must take all of these things into con- sideration in making his location. Of the three named items, grades is by far the most important, while distance is the least important. Distance 50. Additional distance in the length of a line affects the initial cost of construction, the cost of maintenance, and the cost of operation. The cost of construction and maintenance is proportional to the distance and may be readily determined. The cost of operation does not vary directly as the distance. The cost of operating a small additional distance will be only about one-third as much per train-mile as the average cost over tne whole line. When the additional distance becomes several miles, the cost is increased to about one-half of the average cost per train-mile. 16 DISTANCE, CURVATURE, AND GRADES 17 Curvature 51. Curvature affects the cost of construction, the cost of maintenance, and the cost of operation. The cost of construction is increased since curvature means an increase in the length of the line; and also if structures such as trestles and bridges come on curves their cost is largely in- creased thereby. The cost of maintenance is increased since curved track requires more care in alinement and surface than straight track, and also there is greater wear on both the track and the rolling stock due to the curvature. The cost of operation is increased because more power is re- quired on curves than on straight track, and frequently a sharp curve will limit the weight of train that can be operated over the line. Also, sharp curves limit the speed of passenger trains, which may be an important item where there is a competing line. It has been determined by experiment that the additional tractive force required to draw a train around a one-degree curve is the same as that required by a grade of from 0.03 to 0.05 per cent. This resistance does not vary directly as the degree of curve, but decreases as the degree of curve increases. For curves up to about 8 degrees, however, the curvature resistance is usually taken as 0.04 per cent, per degree of curve; for curves from 8 to 15 degrees, 0.03 per cent, per degree; and for curves above 15 degrees, 0.02 per cent, per degree. The resistance to a train in starting is about double that given above, and must be considered if a stopping place comes on a curve. In general, curve resistance is not serious until it becomes so great that when added to grade resistance the total resistance limits the weight of train that can be hauled over the line. Therefore all the steeper grades should be compensated for curvature. 52. Choice of curvature. The total resistance offered by a curve depends directly upon the central angle, and is practically independent of the radius. Hence the important point is to keep the amount of curvature a minimum, and after the tangents have been decided upon the problem is to make the rate of curvature as uniform as the topography will permit. Little is gained by having a few flat curves and a number of sharp ones. The proof of this statement is in the fact that in the vast amount 18 TEXT BOOK ON RAILROAD SURVEYING of re-alinement work that the railroads have been doing in the past few years there are very few, if any, cases where an increase in radius is the only benefit derived from the improvement. The object of all this work has been to reduce the total amount of central angles in curves and at the same time to reduce the maximum rate of curvature. Grades Grades may be classified as (1) minor grades, (2) ruling grades, and (3) pusher grades. 53. Minor grades are those which do not limit the weight of train which can be hauled over the line by one engine. Such grades are of three kinds according to their effect on the per- formance of the engine. First, there are those grades whose drop (vertical height) is sufficiently small that a locomotive can operate without shutting off steam, or, in other words, will not reach a dangerous speed due to the down-grade while exerting a constant pull on the train. The effect of these grades on train operation is negligible, both as regards the effort required of the engine and the time required for the train to traverse a given distance, since the kinetic energy acquired in descending one grade is utilized in ascending the opposite grade. The second kind of minor grades consists of those whose drop is so great that steam must be shut off to prevent the train from acquiring a dangerous velocity, and hence the effort of the engine must be increased on the opposite up-grade. Therefore there is a loss of power, since the engine is working intermit- tently and not at its greatest efficiency. The third kind of minor grades consists of those which require the use of brakes in descending them. This causes an enormous loss of energy because part of the kinetic energy gained by the descent is absorbed by the brakes, and hence is not available on the next up-grade. Further, the engine is working at a dis- advantage, and there is considerable wear on equipment due to the action of the brakes. 54. Ruling grades are those which limit the weight of the train which can be hauled by one engine. Cars can not be picked up or dropped off along the line to make up a train in DISTANCE, CURVATURE, AND GRADES 19 accordance with the grades met with, and hence a train must run through from one terminal of a division to the other. There- fore, excluding pusher grades, and such short steep grades which can always be operated by momentum, the maximum grade is the ruling grade on that division. 55. A pusher grade is one which is so steep that one or more extra engines are required to haul the train which one engine can handle on the remainder of the division. It will nearly always happen that some grades on each division will be con- siderably greater than the majority of the grades, and unless these can be reduced to the general average at a reasonable cost or a pusher engine is used, they limit the weight of train over the division and become the ruling grades. A pusher grade adds enormously to the cost of operation since the auxiliary engines must be maintained; and, further, they pass twice over the line for each train handled, and hence do not operate at maximum efficiency. Obviously, therefore, a pusher grade must be considerably in excess of the ruling grade on the remainder of the division before its extra cost of operation would be justifiable. 56. Choice of grades. The total energy available to carry a moving train up a grade is the sum of the kinetic energy of the train due to its velocity (momentum) and that developed by the engine. The maximum amount of energy which the engine can develop is a fixed quantity, but the amount of kinetic energy which can be utilized depends on the initial speed of the train. A given engine can just pull a train of given weight up such a grade that the sum of the frictional resistances and the grade resistances (including curvature resistances) equals the total force exerted by the engine. Or, conversely, the weight of train which a given engine can pull up a given grade is such that the sum of frictional and grade resistances again equals the effort of the engine. The frictional resistances are practically constant at about 8 pounds per ton for freight trains, while the grade resistance is 20 pounds per ton for each per cent, of grade. The length of such a grade is limited only by the endurance of the engine. If the train approaches this grade at considerable speed, its kinetic energy may be utilized to overcome additional grade 20 TEXT BOOK ON RAILROAD SURVEYING resistance, hence the engine can pull the train up a much steeper grade to the point where the additional vertical rise has absorbed all the kinetic energy. If the end of the grade is reached first, the train will pass over easily, suffering only a reduction in speed; if the grade extends further than this point, the train will be stalled. Thus it is seen that for minor grades it is more the vertical height than the rate of slope which affects the operation. And also the ruling grade the one which limits the train weight may not be the one of steepest elope, provided that the heaviest trains can always approach it at sufficient speed to reach its top with the aid of momentum. Therefore if a grade line can be established with alternate de- scents and ascents of such small amount that the train can always make use of its momentum, the rate of slope is of small importance. But, if the slopes are so long that momentum can not be utilized all the way, or the grade must be entered at very low speed, the rate of slope limits the train weight. The first step in projecting a grade line on a profile is to determine approximately the ruling grade. The lowest value of the ruling grade is that of a uniform slope between terminals. The maximum value depends on the type and weight of locomo- tives, the weight of trains, etc., and can never be exactly de- termined, and therefore it is more usual to simply choose a maximum or ruling grade which will probably fit the territory through which the line runs. In assuming this maximum value, due consideration should be given to the direction of heaviest traffic, and easier grades secured if possible for trains in this direction. A grade line is then laid out on the profile, keeping below this maximum if possible in such a way that the fills will balance the cuts and the total amount of earthwork will be kept as low as possible in order to reduce the cost of construction. The laying out of this line will require several trials, and each trial should be carefully studied as to its effect on operation as well as on first cost. After such a line has been laid out it may be found that the assumed maximum grade has been exceeded at some point, and study must be given as to whether it is possible to reduce this grade or whether it is justifiable to perhaps increase it suf- ficiently to be operated as a pusher grade. Or it may be found that only one grade approaches the chosen maximum, and again DISTANCE, CURVATURE, AND GRADES 21 the line must be studied to determine if it is possible to reduce this grade to the general average. For example, the maximum grade on a division is found to be 1 per cent., while there are no others over 0.8 per cent., and possibly only two of these, the next lower being 0.5 per cent. The problem then is to decide the advisability of reducing the 1 per cent, grade to 0.8 per cent., and then possibly the three 0.8 per cent, grades to the average of 0.5 per cent. And this process of reducing the ruling grade could be continued until the additional cost of construction equals the sum which can profitably be spent for the purpose of reducing the ruling grade. 57. VERTICAL CURVES. The intersections of the several portions of the grade line must be rounded off to avoid undue stress in the drawbars. The curve used for this purpose is a parab- ola because of its convenient characteristics. Such curves are termed vertical curves to distinguish them from the curves in the alinement. The length of vertical curves depends on the total change in the grade which is the algebraic difference of the intersecting grades, Gi and G?., and on the rate of change of the vertical curve. Ascending grades are plus, and descending grades are minus. The rate of change per station, r, of the vertical curve as rec- ommended by the American Railway Engineering Association is 0.1 foot for summits and 0.05 foot for sags for first-class rail- ways; and double these amounts for second-class and electric railways. It is to be noted, however, that a large number of first-class roads use the second-class rating. Since the change at the ends of the curve is from tangent to chord instead of from chord to chord, the rate of change of the first and last stations on the curve is just one-half the rate used on the remainder of the curve. In solving a vertical curve problem, the first step is to find the length of the curve. This is done by dividing the algebraic difference of the grades by the assumed rate of change per station. This will usually give an odd length which is inconvenient to handle, and therefore the next higher even number of stations is chosen as the length of the curve. The second step is to divide the algebraic difference of the grades by the length of curve as found in step one. The quotient 22 TEXT BOOK ON RAILROAD SURVEYING will be the rate of change per station which is to be actually used. The third step is to find the station numbers and elevations of the T.C. and C.T. (beginning and end of curve). The fourth step is to find the elevations of the intermediate stations on the curve. This is done by starting at the T.C. and adding (or subtracting as the case may be) the changing grades successively until the C.T. is reached. The grade from the T.C. to the first station will be GI \ r, from the second to the third station it will be GI -f r, from the third to the fourth it will be GI r, and so on till the C. T. is reached. The elevation thus found for the C. T. should check with that determined from step three. Example. A +0.8 per cent, grade meets a 0.6 per cent, grade at station 30 + 00, whose elevation is 750.50. It is de- sired to connect these grades by a vertical curve whose rate of change per station is approximately 0.2 foot. GI G 2 (algebraically) Length of curve = 2 v * ' . . . f L,W.s4,-,7 r - - - CD Making L = 8, r = -^ = 0.175 o Taking everything into consideration it is better to make L = 800 feet (400 feet each side of apex), and to use a value of r = 0.175 than it is to make L = 700 ft. and use a value of r = 0.2. Sta. of T.C. = 30 - 4 = 26 + 00. Sta. of C.T. = 30 + 4 = 34 + 00. Elev. of T.C. = Elev. P.I. - \LGi = 750.50 - 3.20 = 747.30 Elev. of C.T. = Elev. P.I. - \LGi = 750.50 - 2.40 = 748.10 Elevation Station 26 = 747.30 = 747.30 " 27 = 747.30 + (0.8 - 0.0875) = 747.30 + 0.7125 = 748.01 DISTANCE, CURVATURE, AND GRADES 23 Elevation Station 28 = 748.01 + (0.7125 - 0.175) = 748.01 + 0.5375 = 748.55 " " 29 = 748.55 + (0.5375 - 0.175) = 748.55 -f 0.3625 = 748.91 " 30 = 748.91 -f (0.3625 - 0.175) = 748.91 +0.1875 = 749.10 " 31 = 749.10 + (0.1875 - 0.175) = 749.10 -h 0.0125 = 749.11 " " 32 = 749.11 + (0.0125 - 0.175) = 749.11 - 0.1625 = 748.95 " 33 = 748.95 + (-0.1625 - 0.175) = 748.95 - 0.3375 = 748.61 " " 34 = 748.61 + (-0.3375 - 0.175) = 748.61 - 0.5125 = 748.10 It should be noted that the highest point on the vertical curve comes at Sta. 31 instead of Sta. 30. The elevation of each station on the vertical curve is shown on the location profile, and is used in setting slope stakes. If the distance from the sub-grade to the top of rail (from 1.5 to 2.0ft.) is added to the above elevations the results are the eleva- tions of the final top of rail. These elevations are used after the grading has been completed for setting grade stakes for sur- facing the track. Note. For a further discussion on the Economics of Railway Loca- tion, the student is referred to Wellington's "Economic Theory of the Location of Railways," or Webb's Economics of Railroad Construc- tion." CHAPTER IV CURVES Part I. Circular Curves RAILWAY alinement consists of 1. Tangent, or straight track. 2. Curves, which unite the tangents. Curves are of two kinds; (1) arcs of circles, and (2) arcs of spirals. The circle is employed for the body of all curves since it is more easily located with the transit and tape than any other form of curve. Spirals are used for easements at the ends of the circular arcs. Circular curves are classified as Simple, Compound, and Reversed. Simple Curves 58. FUNCTIONS. A simple curve is an arc of a circle which unites two tangents differing in direction. The functions of a simple curve are shown in Fig. 2. 24 CURVES 25 Point of Intersection P.I. is the point where the two tangents intersect. Tangent to Curve T.C. is the end of the tangent and the beginning of the curve. Curve to Tangent C.T. is the end of the curve and the beginning of the tangent. Intersection Angle 7 is the deflection angle between the two tangents, and is equal to the angle at the center. The Radius of the curve is denoted by R. Tangent Distance T is the distance from the T.C. or the C.T. to the P.I. T = R tan \I. . ..... (2) Long Chord L.C. is the chord from the T.C. to the C.T. L.C. = 2fl sin i/ ....... (3) Middle Ordinate M is the ordinate to the curve from the middle of the Long Chord. M = R - R cos i/ = R vers JJ ... (4) External Distance E is the distance from the middle of the curve to the P.I. E = ~ jR Degree of Curve D is the angle at the center subtended by one 100-ft. chord, two 50-ft. chords, or four 25-ft. chords, de- pending upon the length of the radius. The intention is to make the difference between the length of the chord and the arc so small that it may be neglected. Length of Curve L is the distance along the curve between the T.C. and the C.T. L (in stations) = -yr ..... (6) Deflection Angle \D From geometry the angle between two chords, or a chord and a tangent, intersecting on the circum- 26 TEXT BOOK ON HAILROAD SURVEYING ference of a circle is measured by one-half the intercepted arc. For a chord 100 feet in length this angle is called the deflection angle, and therefore is JD. Subdeflection Angle d is a similar angle for a chord of less than 100 feet. If C is the length of the chord in feet, Q d (in degrees) =^3^5 (7) d (in minutes) = 0.3CD (8) Total Deflection Angle is the angle at the T.C. between the tangent and a chord to any point on the curve. Total deflection angles are taken to the nearest 0.5 minute. 59. The relation between R and D. If the degree of curve be defined as the angle at the center subtended by a 100-ft. arc, an exact relation between R and D may be found, as follows. The circumference of a complete circle in terms of the radius is O(\(\ 2 nR' } and in terms of the degree of curve is 100 -~- , hence 100 X 360 5729.58 R= 2*D =-D~ ^ From (9) it is evident that R varies inversely as D. Therefore knowing the radius of a 1 degree curve, the radius of any other degree of curve can be found by simple proportion. Since the chord is used instead of the arc, the value of R from FIG. 3. (9) can not be used, and a new value of R must be found which will agree with the chord-definition of the degree of curve. CURVES 27 In Fig. 3, R sin JD = |C Whence R = -^75 = iC cosec D .... (10) Dill 2 *-' When C is 100 ft., # = 50 cosec il> (11) The difference in the values of R as computed from (9) and (11) becomes about 0.5 ft. when D is 7 degrees. Since a difference greater than this is objectionable, curves above 7 degrees are staked out with 50-ft. chords. When C is 50 ft., R = 25 cosec JD (12) When D becomes 14 degrees, the difference in R as computed from (9) and (12) again become objectionable, and 25-ft. chords are used. When C is 25 ft., R = 12.5 cosec I D (13) When D becomes 28 degrees, 10-ft. chords should be used. Table 5 of radii was computed from (11), (12*), and (13). It is customary to consider the radius of a 1 degree curve as 5730 ft., and to assume that R varies inversely as D. Hence R = p- (approx.) (14) The difference between R as given in Table 5, and as computed from (14) is so small that it may be neglected. By substituting R in terms of D in (2) to (6) it is seen that the various functions of a curve for any given value of 7 are approximately inversely proportional to D. Therefore, if these functions are known for any degree of curve, they can be found for any other degree of curve by simple proportion. Table 6 gives the functions of a 1 degree curve for intersection angles from degree to 100 degrees. 60. FIELD WORK. In laying out a curve successive points are located by a meas- urement from the preceding point and by line. The line is 28 TEXT BOOK ON RAILROAD SURVEYING not the same for all points and is determined by the total de- flection angle calculated for each point. Thus in Fig. 4, point 1 is located by the chord T.C. 1 and the total deflection angle A; point 2 by the chord 1-2 and the total deflection angle B] and point 3 by the chord 2-3 and the angle C. FIG. 4. Stakes are usually placed 100 feet apart except on sharp curves where the interval is reduced to 50 or 25 feet. However, the stationing is continued unbroken around the curve. The T.C. and the C.T. will therefore usually fall at a plus station, hence there will usually be a subdeflection angle at each end of the curVe. The first step in the field work is to determine the station numbers of the T.C. and the C. T. and to compute the total de- flection angles for the entire curve. In Fig. 4, T.C. 1 and 4- C.T. are odd distances, and 1-2, 2-3, and 3-4 are full stations. Then A is the subdeflection angle (di) for the chord T.C.-l, and is computed by Eq. (8). A = di Then B = A + $D = di + \D And C = B + $D = d, + $D + |D E = C + *> = rfi + $D + $D + iD F = E + d 2 = di + \D -h JD + iD + & = \I The total deflection angles are thus computed by successive additions, and the entire series of computations is checked if the last value is \I. If stakes are placed 50 feet apart the in- crements are \D, and if 25 feet apart they are \ D, instead of JD as above. The form of notes is shown in Fig. 5. CURVES 29 On the ground the T.C. and the C.T. are located by measuring the tangent distance T from the P. 7. The curve can then be run in by means of angles and distances as explained above. Since in turning off an angle with a transit, an error as large as 0.5 minute may easily be made which amounts to 0.15 of a foot at 1000 feet from the transit the length of sight on a curve should never exceed 1000 feet. It has been found by experience that if the angle between the tape and the line of sight is more than about 30 degrees, the location of the point is inaccurate. Therefore the angle between the line of sight and the tangent at the transit station should never be more than 30 degrees, i.e., the product of the length of sight in stations and |D should be less than 30 degrees. It is good practice in any case to run a portion of the curve from the C.T., since the errors of surveying can be adjusted more satisfactorily on the curve than at the C. T. There should be no more error, however, in either line or distance than is permissible in good chaining under the particular conditions. If the entire curve can not be run in from the T.C. and the C.T., one or more intermediate set-ups on the curve will be necessary. There are, therefore, three possible positions of the transit in running in curves, viz.j the T.C., any intermediate point, and the C.T. The curve notes as computed above are used in all three cases, as follows: At the T.C. Orient the transit by a sight along the tangent with the plates set at zero. Turn off the total deflection angle of each station, successively, and chain the corresponding dis- tances between them. At any intermediate point. Orient the transit by a back- sight on the last transit station with the telescope inverted and the plates set at the total deflection angle of that station as recorded in the notes. To continue the curve, plunge the telescope * and set the plates at the total deflection angles of the succeeding stations and measure the corresponding distances between them. At the C.T. Orient the transit by a sight along the tangent with the telescope normal and the plates set at the total de- flection angle of the C.T., i.e., J7. To run in the curve, set the * The line of collimation must be in adjustment. TRANSIT NOTES FOR LINE L STATION . ALINE- MENT TOTAL DEFL. ANGLE CALC. BEAR. MAG. BEAR. REMARKS -{- 55.8 O C.T. 35-4l' -f NOO-39'W NOO-40V _*i 1=85-41' 62 34 -34 61 32 -34 10560 30 -34 D = 400' 59 28 -34 1 = 7122' 58 26-34 57 58 55 24-34 22-34 20-34 CO 8 -f 5 o g !o TH II 54 18-34 PU 53 ^ 16-34 52 JA bo 3 14-34 51 p 12-34 10550 ** 10-34 43 8 -34 48 6 -34 47 4-34 46 2-34 45 -34 H-71.6 O T.C. 0-00' N72-Ol'W N72-10'W 44 43 FIG. 5. J.E.Schmidt Inst. R.Carr-H.C. C.E.Agg R.C. G.W.Pix R.F. C. Wiles Ax June 12, 13 Clear-Warm C.T. )P.O.C. T.C. FIG. 5. 31 32 TEXT BOOK ON RAILROAD SURVEYING plates at the total deflection angle of each station as given in the notes, and measure the corresponding distances. 61. The FUNDAMENTAL PRINCIPLES are: When sighting at any station the plates must read the total de- flection angle of that station. When on tangent at any station the plates must read the total deflection angle of that station. If these principles are strictly observed a mistake in instrument work is impossible. Example. The P.I. of two tangents is at Sta. 10555 + 00.3 and / = 71 22'. It is desired to connect the tangents with a 4 curve. Tforal curve for / = 71 22' D 4114.9 (from Table 6) = 1028.7 ft. Sta. T.C. = Sta. P.I. - T = (10555 + 00.3) - (10 + 28.7) = 10544 + 71.6 / 71 22' 71.367 L = -~- = j = j = 17.842 Stations. Sta. C.T. = Sta. T.C. + L = (10544 + 71.6) + (17 + 84.2) = 10562 + 55.8 The distance from the T.C. to the first Sta. on the curve is 28.4 ft., hence di = 0.3 CD = 0.3 X 28.4 X 4 = 34' Deflection angle = JD = 2 00' The distance from the last Sta. to the C.T. is 55.8 ft., hence d 2 = 0.3 X 55.8 X 4 = 67' = 1 07' \ Total Deflection Angle of T.C. = 00' " " Sta. 10545 = 00' + 34' = 34' " "" 10546 = 34' + 2 00' = 2 34' " " " " " 10547 = 2 34' + 2 00' = 4 34' CURVES 33 Total Deflection Angle of Sta. 10562 = = 34 34' " " " " C.T. = 34 34' + 107' = 35 41' = i/ (check) These results are recorded in the form shown in Fig. 5. 62. Offsets from chords produced. In addition to his chain- ing duties the head chainman also sets the stakes on line as directed by the transitman. Obviously this work is much facilitated if the head chainman can quickly approximate the position of the stakes. This can be done by an offset from the line through the last two stations. For example, in Fig. 6, A and B are the last two stations set. Produce AB to E the distance C from B. F is located by the offset y from E and the distance C from B. y can be found by Eq. (16). If E is care- fully lined in and C and y measured with the tape, the point can be located with considerable accuracy; and, if a transit is not at hand, this method can be employed in locating an entire FIG. 6. curve. However, this method is most valuable as an aid to the head chainman in approximating the position of the stakes, and it should always be used. If E is only roughly lined in and y is paced, the points can be located within less than one foot, which is sufficient for the purpose. From Fig. 6, \y : C: : J C : R Whence \y = p (C and R expressed in feet) 34 TEXT BOOK ON RAILROAD SURVEYING Expressing R in terms of Z), we have * ~ 1)1460 If n represents C expressed in stations, then \y = 0.873 n*D (15) \y is the tangent offset to the point F. For C = 100 ft., n = 1, and \y = 0.873 D, and y = 1.746 D = if D (approx.) (16,) For C = 50 ft., n = \, and Jy = 0.22 D, and y = 0.44 D 63. Chord Offset Method. There are some cases in which the deflection angle method of running in curves can not be used to advantage. For instance, in heavily wooded country it may not be economical to locate every station by the deflection method on account of the large amount of clearing necessary for the lines of sight. Fig. 7 illustrates this condition. If the stakes B, C, E, and F were all set from A it would necessitate the clearing FIG. 7. of a line to each or practically the entire area between the curve and the chord AF. If, on the other hand, F were set from A by its total deflection angle and the chord AF, and B, C, and E located by offsets from this chord, the only clearing required would be along AF and the short offsets. The transit would then be moved to F and the process repeated to J, and so on. The chord AF is the L.C. of a curve whose value of / is the central angle of the curve ABCEF, and therefore can be de- termined from Eq. (3) or from Table 6. In chaining this chord, temporary stakes are set at the full station distances, although perpendicular offsets from these will not give the exact location CURVES 35 of the curve stations since the chord AF is shorter than the line ABCEF. This error may be neglected for the following reasons: (1) the error at any point in practice will rarely exceed one foot; (2) the perpendicu- larity of the offsets is estab- lished by eye, hence is not exact; and (3) the approxi- mate location of these stations is sufficient since this method would be used only on first location to determine the profile. The offsets are com- puted as follows: From geometry the pro- ducts of the segments of two intersecting chords are FlG 8 equal. Then from Fig. 8, or Si XS 2 S 1 X 2R-S, Since $ 3 is small compared with 2R it may be neglected in the denominator, and since in practice the difference between $ 3 and the offset will rarely -reach 0.05 ft., we may write Si XS 2 Si XS. 2 XD , A Offset = nn = ^^,r^ (Approx.) 2R 2 X 5730 Reducing Si and $2 to stations, we have Offset (in feet) = 0.873 SiS 2 D (Slide Rule) (17) 64. Example. The T.C. of a 4 curve is at Sta. 67 + 82.7. Sta. 72 is set by its total deflection angle (8 21') and the L.C. of a 4 curve for / = LD = 4.173 X 4 = 16 41.5', which from Table 6 = 415.8 ft. While chaining this chord, temporary stakes are set at 17.3, 117.3, 217.3, and 317.3 ft. from the T.C. From Eq. (17), The offset at Sta. 68 = 0.873 X 0.173 X 4 X 4 = 2.4 ft. " " " 69 = 0.873 X 1.173 X 3 X 4 = 12.3 ft. " " " 70 = 0.873 X 2.173 X 2 X 4 = 15.2 ft. " " " 71 = 0.873 X 3.173 X 1 X 4 = 11.1 ft. 36 TEXT BOOK ON RAILROAD SURVEYING The stations on the curve are located by measuring these offsets from the temporary stakes already set. Change of Location Problems After the line is located according to the paper projection and a profile made, it may be found that, owing to inaccuracies in the platted topography, a considerable cut or fill comes on a hillside and may be eliminated by shifting a portion of the alinement. Occasionally other considerations than earthwork make euch a shift desirable. 65. Problem 1. In Fig. 9, the line as originally located (shown dotted) involves considerable earthwork which could FIG. 9. be eliminated by shifting the curve the distance p at or near its middle point. The problem is to find the new degree of curve and the change in the position of the T.C. and the C. T. p is the difference in the external distances of the two curves, or f^new curve ~ -^old curve i P The new degree of curve may be computed from the new external by Eq. (5) or by means of Table 6, which is the quicker CURVES 37 and more common method. The estimated value of p will usually give an odd value of D which is undesirable; hence the value of D to be used is taken only to the nearest ten minutes. The distance that the T.C. and the C.T. are shifted is equal to the difference in T for the two curves. 66. Problem 2, Case 1. In Fig. 10, the line as originally located (shown dotted) in- volved considerable earthwork between A and B, which could FIG. 10. largely be eliminated by shifting the tangent the distance p parallel to its original position, which involves a change in the location of TWO curves. The problem is to find the change in the T.C.s and the C.T.s of the two curves, the degree of curve remaining unchanged. In triangle 1, the sides m and I can be readily computed since / and p are known, m = n = the required change in the T.C. and P.I. p will rarely exceed 100 feet, hence the new tangent is best located by offsets from the old tangent. The new C.T. is then located from the old C.T. by rectangular coordinates one of which is p and the other is the side I of triangle 1. The second curve is located similarly using triangle 2. 38 TEXT BOOK ON RAILROAD SURVEYING 67. Problem 2, Case 2. Owing to some special cause, for example a stream-crossing as shown in Fig. 11, it may be undesirable to move the C.T. for- ward as in Case 1. This condition can usually be met by placing FIG. 11. the new C.T. approximately on the same radial line as the old C.T. This involves a change in the degree of curve and in the position of the T.C. In triangle 1, 7 and p are known, and I and m are computed. 7 new curve = 1 old curve ' whence the new D is easily found from Table 6 and, as in Problem 1, is taken only to the nearest 10 minutes. (The approximation in the position of the C.T. is due to thus not using the exact value of D.) A new value of T is then found which agrees with the D that is used. Then ^ = -* old curve ~{~ Wl 2 new curve 68. Problem 2, Case 3. This case is similar to Case 2, except that the C.T. is on the fixed tangent instead of the one being changed, hence the solution is somewhat different. Solve triangle 2 for the side m, then -* new curve = Told curve W whence the new degree of curve is obtained from Table 6. CURVES 39 Solve triangle 3 for the side 1. Then the new T.C. can be located from the old T.C. by the co-ordinates I and p. As in the previous problem, D should be taken only to the nearest ten minutes. This necessitates a slight change in both the T.C. and the C.T., since T for this value of D is not the same as that used in the abpve computations. It must be remembered that the T.C. becomes the C.7 7 ., or vice versa, when the line is run in the opposite direction. 69. Problem 3. Sometimes curves are run in beginning at the T.C. without previously locating the P. I. and the C.T., and checking the direction of the forward tangent. On reaching the C.T. and projecting the forward tangent, it is found that the tangent does not pass through some controlling point. It is therefore necessary to change the direction of the tangent by shifting the C.T. This is most readily accomplished by measuring the necessary change of direction with the transit, and then changing the length of the curve to correspond. For example, in Fig. 12, the curve is run to B as a C.T. and the tangent is found to pass through x, whereas it is necessary to pass through A. The angle e is measured with the transit. The length of curve corresponding to e is BC, whence the C.T. is moved back to C. The forward tangent then takes the position CA', parallel to BA, and misses A by the distance m, which is the middle ordinate of the curve BE. 40 TEXT BOOK ON RAILROAD SURVEYING BE = 2BC = m = -g-X X D = 0.875-- m will rarely reach 5 ft., hence the line CA' meets the require- ments. Problems 1. If the P.I. is in a stream or is otherwise inaccessible, how can / and the positions of the T.C. and the C.T. be determined? 2. If the T.C. is inaccessible, how can the curve be run in and checked? 3. If the line of a curve passed through a building or similar obstruction, how can the curve be run in and checked? 4. Given I = 63 43'. Find T, L, R, and E for each of the following values of D. 30', 2 00', 5 00', and 7 30'. Solve by equation and check by Table 6. 5. Given the P.I. at Sta. 118 + 60.0, / = 57 48', and D = 5 00'. Find the station numbers of the T.C. and the C.T. 6. Write transit notes for the preceding problem. 7. Given, on original location, a 6 00' curve with / = 78 21' and the T.C. at Sta. 1041 + 72.6. Topography requires the middle point of the curve to be moved toward the center about 45 ft. What is the new degree of curve and the station number of its T.C.? 8. Given an established curve with 7 = 69 38', D = 3 00', and the T.C. at Sta. 982 + 41.1. It is desired to move the forward tangent outward a distance of 50 ft. Find the new station numbers of the T.C. and the C.T 7 ., and the coordinates from the old C.T. to the new C.T. Answer. T.C. = 982 + 94.4; C.T. = 1006 + 15.5; I = 18.6, and p = 50.0. 9. Data same as in Problem 8, except that the forward tangent is moved inward. Answer. T.C. = 981 + 87.8; C.T. = 1005 + 08.9; I = 18.6, and p = 50.0. 10. Given an established curve with / = 47 23', D = 2 30', and the T.C. at Sta. 1841 + 83.7. It is desired to move the CURVES 41 forward tangent outward a distance of 60 ft., keeping the C.T. on the same radial line approximately. Find the new degree of curve, the station number of the T.C., and the distance that the C.T. moves forward or backward due to using D to the near- est ten minutes only. Answer. D, computed = 2 38.7'; Z), used = 2 40'; T.C. ~ 1843 + 28.0; C.T. moves 7.6 ft. backward. 11. Data same as in Problem 10, except that the forward tangent is moved inward. Answer. D, computed = 2 22.2'; D, used = 2 20'; T.C. = 1840 + 30.3; C.T. moves 16.7 ft. forward. 12. Given an established curve with / = 50 35', D = 4 30', and the T.C. at Sta. 155 + 24.5. It is desired to move the initial tangent 35 ft. inward without changing the position of the C.T. (except the small amount due to using approx. value of D). Find the new degree of curve, the station numbers of the new T.C. and C.T., the coordinates of the new T.C. from the old T.C., and the small amount that the T.C. and the C.T. must shift due to the approx. value of D used. Answer. .D, computed = 4 52'; D, used = 4 50'; T.C., computed = 155 + 98.7; T.C., used = 155 + 94.9; C.T 7 ., com- puted = 166 + 38.1; C.T., used = 166 + 41.5; I = 70.3; p = 35.0; T.C. and C.T. shifted 3.8 ft. 13. Data same as in Problem 12 except that the tangent is moved outward. Answer. D, computed = 4 11'; D, used = 4 10'; T.C., computed = 154 + 50.3; T.C., used = 154 + 47.5; C.T., com- puted = 166 + 59.2; C.T., used = 166 + 61.5; I = 77.0; p = 35.0; T.C. and C.T. shifted 2.8 ft. 14. After running in a 3 00' curve to the right, using paper location notes, it is found that the forward tangent passes to the right of the governing point by 2 20' as measured at the C.T. (a) What distance must the C.T. be shifted? (b) How much does the new tangent miss the controlling point? Answer, (a) = 77.8ft.; (b) = 1.6ft. Compound Curves 70. A compound curve is a combination of two or more simple curves in the same direction with a common tangent at the 42 TEXT BOOK ON iAILROAD SURVEYING point of junction. A compound curve should never be used except under conditions where a simple curve will not meet the require- ments. In rough country it may happen that a large volume of earthwork can be avoided by making one part of a curve sharper than another, resulting in a compound curve. For example, in Fig. 13, the line is located as a compound curve ABC, requiring little earthwork and keeping on the bank of the stream. If the simple curve AB had been produced to F to end in a parallel tangent, it would have fallen in the river. Or if the simple curve CB had been prolonged to E, it would have pierced the cliff. The degree of curve and the point of compound curve, C.C., are chosen to fit the contours and other governing conditions, and the central angles of the branches are scaled from the map. This is one of many cases where a compound curve is ap- plicable. From the standpoint of operation a compound curve is better than two simple curves separated by a short tangent, hence it should be used in such cases. In flat country a compound curve is inexcusable on main line. If the degrees of curve of two ad- jacent branches of a compound curve differ by more than 2 00', an easement curve should be inserted between the branches, and provision for this should be made in locating the curve. (See Spirals, paragraph 84.) CURVES 43 The nomenclature and positions of the functions are shown in Fig. 14. 71. Problem 1. In new work / will be measured and /i, 7 2 , A, and A will be determined from the preliminary maps. It is desired to know the value of T\ and T 2 in order to set the T.C. and the C.T. and to run in the curve. In Fig. 14, BE is the common tangent at the C.C. BE = BC + CE BC = AB is the tangent distance for a A curve for an inter- section angle I\\ and CE = #G is the tangent distance for a A curve for an intersection angle 7 2 and are computed by Table 6. Then in the triangle BEF, one side and the adjacent angles are known from which the sides BF and EF are computed. T l = AB + BF, and T 2 = EF + # 72. Problem 2. Occasionally the positions of the T.C. and the C.T. are fixed within narrow limits, giving unequal tangent distances, which necessitates a compound curve. 44 TEXT BOOK ON EAILROAD SURVEYING The degree of curve of one branch must be chosen before the problem can be solved. The problem then is to determine the FIG. 15. degree of curve of the second branch and the central angles /i and 7 2 . In Fig. 15, /, Ti, Ti, and A are known. A H = HF = T = the tangent of A curve for the intersection angle /. In triangle 1, m = To - T l and hence the triangle can be solved for p and n. In triangle 2, I = T + n - T 2 Since p is known, the triangle can be solved for the angle at E, which is i/ 2 . /i = / - h In triangle 3, since I and 7 2 are known, the side Oi0 2 = Ri Rz can be solved for, whence Z> 2 is determined from Table 5. CURVES 45 73. Field Work. A point on one branch of a compound curve can not be located with the transit set on a point on another branch, since in order to run in a circular curve with the transit and tape, the transit and the points to be located must be on the same circumference. Therefore each branch of a compound curve is run in indepen- dently as a simple curve. Thus the notes for the first branch are computed and the curve run in to the C.C. The transit is then placed at the C.C. and oriented as at the C.T. of a simple curve except that the plates read zero instead of ^1 when sighting along the tangent* The notes for the second branch are then com- puted and the curve run in to the C. T. or to the next C.C. Change of Location Problems 74. Problem 1. This problem arises under conditions similar to those in Problem 2, Case 1, of simple curves, paragraph 66. Given, as shown in Fig. 16, a located compound curve ABC. It is desired to move the forward tangent laterally a distance p. FIG. 16. The simplest solution of this problem is to retain the same degrees of curve and to shift the C.C. from B to E. BE (in feet) = -^T 1Q L>i * To orient at the C.C., back-sight on the last transit station with the plates set to read the difference between the total deflection angle of the C.C.,% Ji, and the total deflection angle of the back-sight sta- tion; turn the plates to zero and the line of sight is on tangent. 46 TEXT BOOK ON RAILROAD SURVEYING Therefore the problem resolves itself into solving for the angle b. In triangle 1, since a is known and the hypotenuse is the difference in the two radii, the base m can be computed. Then n = m p. In triangle 2, the hypotenuse and the base are known, whence the angle c is computed. b = c a In triangle 3, e = 90 J6 a (prove) whence the coordinates of M, the new C.T., from the old C.T. can be determined. Three other cases of this problem arise, viz.: (1) the same as above, except that the tangent is moved outward; (2) the curve of longer radius may fall on the tangent to be moved inward; and (3) the curve of longer radius may fall on the tangent to be moved outward. The solution of each of these cases is the same as that given above except that the various points take different relative positions. If the same relative construction lines are drawn and the corresponding points are used, no confusion should arise. 75. Problem 2. This problem arises under conditions similar to those in Problem 2, Case 2, of simple curves, paragraph 67. \ FIG. 17. Given, as shown in Fig. 17, a located compound curve ABC. It is desired to move the forward tangent laterally a distance p, CURVES 47 and to keep the C.T. approximately on the same radial line, which involves a change in the degree of curve of the second branch and in the position of the C.C. In triangle 1, a is known and the hypotenuse is the difference in the two given radii. Therefore the base m and the altitude n can be computed. In triangle 2, the base n is now known and the altitude I can be determined since R\ y R^ p, and m are known. The angle at p = |6 (Why?) can therefore be determined. Then c = b a and the C.C. is shifted the distance BE (in feet) = -- 100. DI In triangle 3, the angle b and the altitude n being known, the hypotenuse r can be computed from which R x is determined, and the corresponding degree of curve D x is found from Table 5, and is taken to the nearest ten minutes. Then since E and b are already fixed, this will change the value of p slightly and will shift the C.T. a short distance along the line EG from F, its theoretical location. The coordinates of this new position are given by the equations x = h sin b, and y ' x tan J6, in which x is the movement along the tangent, y is the change in p, and h is the difference between the computed value of R x and the value actually used. Three other cases of this problem arise, depending on whether the longer or the shorter radius curve is on -the tangent to be moved, and on whether the tangent moves inward or outward. In all of these cases the solution is relatively the same. Problems 1. Given the P. I. at Sta. 837 + 00, 7 = 64 44', A = 29 00', 7 2 = 35 44', A = 4 00', and D 2 = 5 30'. Find the station numbers of the T.C., C.C., and C.T. Answer. T.C. = 828 + 73.4; C.C. = 835 + 98.4; C.T. = 842 + 73.4. 2. Given the P.I. at Sta. 1846 + 50.0, / = 57 18', Ti = 835.0, 48 TEXT BOOK ON RAILROAD SURVEYING T 2 = 687.0, and A = 3 00'. Find /i, 7 2 , A, and the station numbers of the T.C., C.C., and C.T. Answer, h = 16 17'; 7 2 = 41 01'; Z) 2 = 4 47.5'; T.C. = 1838 + 15.0; C.C. = 1843 + 57.8; C.T. = 1852 + 13.5. 3. A 2 30' curve compounds with a 4 00' curve at Sta. 8792 -f 27.6. The central angle of the 4 00' curve = 26 C 45'. It is desired to move the forward tangent inward 30 ft., but to re- tain the same degree of curve. Find the station number of the new C.C. and the coordinates of the new C.T. referred to the old C.T. Answer. C.C. = 8790 + 61.6; I = 54.5 ft.; p = 30.0 ft. 4. Data the same as in Problem 3, except that the tangent is to be moved outward. Answer. C.C. = 8794 + 22.0; I = 66.4 ft.; p = 30.0 ft. 5. A 5 00' curve compounds with a 3 00' curve at Sta. 147 + 63.3. The central angle of the 3 00' curve is 19 00'. The forward tangent is to be moved 25 ft. inward, but the same degree of curve is to be retained. Find the station number of the new C.C. and the coordinates of the new C.T. referred to the old C.T. Answer. C.C. = 149 + 04.0; I = 90.2 ft.; p = 25.0 ft. 6. Data same as in Problem 5, except that the tangent is to be moved outward. Answer. C.C. = 146 + 61.3; I = 63.3ft.; p = 25.0ft. 7. A 4 30' curve compounds with a 7 30' curve at Sta. 999 + 67.0. The central angle of the 7 30' curve is 39 24'. The forward tangent is to be moved inward 50 ft. and the C. T. is to be kept approximately on the same radial line. Find the new degree of curve for the second branch, the station number of the new C.C., and the coordinates of the new C.T. (actual) referred to the old C.T. Answer. D, computed = 6 33'; D, used = 6 30'; C.C. = 996 + 36.2; x = 5.4 ft. forward; p' = 47.2 ft. 8. Data same as in Problem 7, except that the tangent is to be moved outward. Answer. D, computed = 12 51.3'; D, used = 12 50'; C.C. = 1003 + 31.4; x = 0.3 ft. forward; P' = 50.0. 9. A 5 30' curve compounds with a 3 40' curve at Sta. 1888 + 36.2. The central angle of the 3 40' curve is 27 50'. The CURVES 49 forward tangent is to be moved inward 30 ft. and the C.T. is to be kept approximately on the same radial line. Find the new degree of curve for the second branch, the station number of the new C.C., and the coordinates of the new C.T. (actual) referred to the old C.T. Answer. D, computed = 2 49.0 r ; D, used = 2 50'; C.C. = 1890 + 84.4; x = 3 ft. backward; p' = 30.4 ft. 10. Data the same as in Problem 9, except that the tangent is to be moved outward. Answer. D, computed = 4 03.0'; D, used = 4 00'; C.C. = 1886 + 01.5; x = 11.7 ft. forward; p f = 34.3 ft. Reversed Curves 76. A reversed curve is a combination of two simple curves of opposite curvature with a common tangent at the point of junction. A reversed curve should never be used on main lines on account of the shock due to the sudden reversal of curvature and also due to the fact that it is impossible to superelevate properly the outer rail at or near the point of reversal. If conditions require two curves of contrary curvature close together, they should be separated by sufficient tangent to run out the superelevation of each, or it should be provided with easement curves in which case the " points of spiral" may be coincident. In yards and connections, reversed curves may be permissible, since the speed is low, and in some cases may be imperative. In nearly every case, however, they should be employed in con- junction with turnouts, therefore further discussion of reversed curves will be included with the problems of turnouts and con- nections in Chapter VI. Part 2. Spirals 77. A spiral is a curve of varying radius which is used at the ends of circular curves in order that the change from rectilinear to circular motion, or vice versa, may be gradual and without shock. It is also used between the branches of a compound curve for similar reasons. On tangents the track should be level transversely, but on curves the outer rail is elevated above the inner one to counter- 50 TEXT BOOK ON RAILROAD SURVEYING ict the effect of centrifugal force. If this were carried out lit- erally, there would be a vertical jog in the outer rail at the T.C. and the C.T., which is impossible from the standpoint of opera- tion. It is evident then that the superelevation must be at- tained gradually; the distance in which this is done is called the " run-off." It is the custom on unspiraled curves to have the run-off either wholly or in part on the tangent with the result that there is a disagreeable tipping of the train upon entering and leaving curves. When a spiral is used this objec- tion is overcome since the superelevation is attained in its length, and at every point is correct for the degree of curve at that point. Thus the spiral may be defined as a curve whose degree of curve increases uniformly from zero at Us beginning to the degree of the main curve at its end. Application to Simple Curves 78. The functions of the spiral are shown in Fig. 18, and are defined as follows: T.S. the point of change from tangent to spiral. S.C. the point of change from spiral to circular curve. FIG. 18. C.S. the point of change from circular curve to spiral. S.T. the point of change from spiral to tangent. a the angle between the tangent at the T.S. and a chord to any point on the spiral the spiral deflection angle. CURVES 51 A a for the S.C. the spiral total deflection angle. b the angle at any point on the spiral between a tangent at that point and a chord to the T.S. Bb at the S.C. d degree of curve of the spiral at any point. D degree of curve of the central circular curve. / the angle at any point on the spiral between a tangent at that point and a chord to any other point (b is a special case of /). 7 the total central angle of the spiraled curve. d the central angle of the spiral from the T.S. to any point. A the central angle of the spiral from the T.S. to the S.C. k the rate of change in the degree of curve of the spiral per station. I the length of the spiral in feet from the T.S. to any point. L the length of the spiral in feet from the T.S. to the S.C. s the length of the spiral in stations from the T.S. to any point = JQJJ. 8 the length of the spiral in stations from the T.S. to the r the radius of curvature of the spiral at any point (radius corresponding to d). R the radius of curvature of the central circular curve. x the abscissa of any point referred to the T.S. X the abscissa of the 'S.C. referred to the T.S. y the ordinate, or tangent offset, of any point. Y the ordinate, or tangent offset, of the S.C. T.C. the point of curve of central curve produced back to a tangent parallel to the tangent at the T.S. o the ordinate of the T.C. t the abscissa of the T.C. referred to the T.S. P.I. the point of intersection of the tangents of the spiraled curve. T s the tangent distance of the spiraled curve (T.S. to P./.). 52 TEXT BOOK ON RAILROAD SURVEYING T the tangent distance of an unspiraled curve of the same D and 7. E s the external distance of the spiraled curve. E the external distance of the unspiraled curve of the same D and /. 79. Formulas. From definition, kl d = ks = loo The derivations of the following formulas require the use of the calculus and are given in the Appendix. d (in degrees) = Jks 2 , A = |kS 2 ....... (19) a (in degrees) = J d = | ks 2 ...... ) a (in minutes) = lOks 2 , > (20) A (in degrees) = f A = J kS 2 ) b (in degrees) = f d = 2a, B (in degrees) = f A = 2 A ' y = 0.291 ks 3 - .00000158kV [ ( ^ Y = 0.291 kS 3 - .00000158 k 3 S 7 f x = 1 - .000762 k 2 s ) /x X = L - .000762 k 2 S 5 f o = 0.0727 kS 3 .......... . . (24) t = iL - .000127 k 2 S 5 ......... (25) T s = T + otan JI + t = (R +o) tanfl + t . . . (26) E S = E + ^~Tf = (R + o) exsec JI + o . . . . (27) COS 2-*- 80. Length of Spiral. Since the function of a spiral is to ease the entrance to a cir- cular curve, D will always be known. Then from Eq. 18 it is only necessary to choose a value of k or L, and the spiral is fixed. Obviously any value of k could be chosen and the spiral would fit, but such a spiral may be so short as to require an excessive CURVES 53 rate of superelevation, or, on the other hand, it may be needlessly long. It is therefore more logical to determine the length of spiral necessary to give the desired rate of superelevation and to make k to correspond. The length of the spiral is a direct function of the rate of superelevation. The total superelevation depends upon the speed and the degree of curve and is found from the formula, *e = 0.00069 DV 2 (28) in which e is the superelevation in inches and V the velocity in miles per hour, e should never exceed 8 inches on account of the effect on slowly moving trains, but the track should be super- elevated for the fastest tram up to this limit. Therefore the maximum train speed for any given curve is limited by this maximum superelevation. According to the American Railway Engineering Association, curves requiring less than 2 inches of superelevation are not generally spiraled, as the run-off may be placed on the tangent without objection. The maximum rate at which the superelevation may be at- tained without discomfort to passengers is about 1 1-6 inches per second. f Therefore the minimum length of spiral is, L = 6/7 ev where v is in feet per second. Reducing v to miles, per hour, L = 1.26 eV Substituting the value of e from Eq. 28, L = . 00087 DV 8 .... (29) Substituting the value of D from Eq. 18, and solving for k, we have 115000., , QrA k = V3 (Approx.) .... (30) The diagram, Fig. 19, is platted from values computed by the above formulas, and gives the length, superelevation, and k for * For derivation, see Appendix, paragraph 143. t American Railway Engineering Association, Bulletin 108, Feb., 1911. TEXT BOOK ON RAILROAD SURVEYING CURVES 55 curves from 30' to 10 00', and for speeds from 20 to 80 miles per hour. In this diagram, with any given value of D and F, the length of spiral, the corresponding value of ft, and the re- quired superelevation can be readily determined. The consensus of opinion of 31 railroads of the United States as reported to the American Railway Engineering Association is that the theoretical speed for which the outer rail is superelevated, or the curve spiraled, may be exceeded about 20 per cent, without discomfort to passengers. Further, the maximum speed is never accurately known. Therefore, it is not necessary to use the exact values of k and L as determined for the assumed speed. Obviously, even values of k are desirable, and since considerable variation is permissible, such values may be chosen, provided the change does not increase the speed more than 20 per cent. An examination of the diagram, Fig. 19, will show that values of A; of i, 1, 2, and 4 will give a satisfactory length of spiral for practically all cases met in practice. 81. Field Work. The first step in the field work is to determine the station numbers of the T.S., S.C., C.S., and S.T. The T.S. and the FIG. 20. S.T. are located on the ground by measuring the distance !Fs from the P.I. From Fig. 20, T s = T + o tan ^1 + t (31) 56 TEXT BOOK ON RAILROAD SURVEYING T is found from Table 6 (or Eq. 2); o is found from Table 1 (or Eq. 24); and t is found from Table 1 (or Eq. 25). Example. Given: P.I. = 21 + 21.1, D = 3 40', / = 51 20', and maximum speed about 45 miles per hour. From diagram, Fig. 19. Superelevation = 5 inches, k = 1, and L = 366.7 ft. Sta. P.I = 21 +21.1 From Table 6, T = 750.9 From Table 1, o = 3.58, o tan \l = 1.7 From Table 1, t = 183.3 T s = 9 35.9 Sta. T.S. = 11 +85.2 S = 3 66.7 Sta. S.C. = 15 + 51.9 Length of circular curve = - = r S = 10 33.3 Sta. C.S. = 25 + 85.2 S= 3. 66.7 Sta. S.T. = 29 + 51.9 Spirals are staked out either by deflection angles or by offsets. 82. Deflection Angles. The process of staking out spirals by deflection angles is the same as that for simple curves. Spirals rarely exceed 600 ft. in length, and the total deflection angle to the S.C. rarely exceeds 5 00 ', therefore the entire spiral can be run in from the T.S* Since a spiral is more difficult to aline than a simple curve, stakes should be placed closer together; and since the spiral becomes sharper as it increases in length, stakes should be placed more closely together at the end than at the beginning. * For running in spiral from an intermediate point on spiral, see paragraph 85. CURVES 57 RULE: Place stakes 50 ft. apart on all spirals up to the point where the degree of curve (d) becomes about 3; beyond this point place them 25 ft. apart. For ease in computation, stakes are placed the above distances apart beginning at the T.S.', hence in general all the stakes will fall at plusses. The spiral deflection angles are a (in minutes) = 10 ks 2 (20) Note that the spiral deflection angles vary with the square of the distance instead of the first power as in simple curves. In Fig. 21 point 1 is located by a 50-ft. chord from the T.S. and the angle a\\ point 2 by a 50-ft. chord from 1 and the angle T.S. s.c. FIG. 21. 02', and so on, noting the change in chord length when d becomes 3. From Eq. 20, and ai = 10' k (0.5 ) 2 a* = 10' k (1.0 ) 2 = 4 ai a 3 = 10' k (1.5 ) 2 = 9ai a 4 = 10'k(1.75) 2 = 49/4ai a 5 = 10' k (2.0 ) 2 = 16ai A = a 6 = 10' k (2.25) 2 = 81/4ai = 1/3A It is thus seen that the deflection angle for the first station only needs to be determined from Eq. 20. The other values are found by multiplying this value by the square of the ratio of the first distance to the other distances. TRANSIT NOTES FOR LINE L STATION ALINE- MENT TOTAL DEFL. ANGLE CALC. BEAR MAG. BEAR REMARKS 1027 Tan. S 64-14'W S 64-10'W 26+23.1 S.T. o-oo' +73.1 25+23.1 +73.1 k =1- in from S.T. 0-02.o 0-10 0-22.5 24+23.1 +73.1 CQ C3 0-40 1-02.5 + 23.1 C.S. 1-30 +23.1 c.s. 8-45' -Ho (check) 23 8-24 22 6-54 P.I. 1020+40 21 .0 5-24 1 = 26-30' 1020 co 3-54 D= 3-00' 19 2-24 T 8 = 600 '2 18 0-54 17+39.8 s.c. 0-00' 17+39.8 1-30 k - 1- + 89.8 1-02.5 L= 300.'0 16+39.8 7 0-40 A =4- 30' +89.8 15+39.8 2 "3. CO 0-22.5 0-10 A= 1-30' B = 3_00' +89.8 0-02.5 +39,8 T.S. 0-00' 1014 Tan. S 37- 44'\V S37-50'W FIG. 22. 58 28 Mike McCarthy -last. Heine Heinrichson-H.C. 01ey01sen-R.C. June 17, 1913 S.Garibaldi- R.F. Clear-Hot Set up at S.T.plates at 00'; B.S.along Tangent; Back in Spiral to C.S. Ic=I-2A=17-30' B.S. at T.S., Plates at 3-00'. Plunge and turn to 0-00 'for Tangent. FIG. 22. 59 60 TEXT BOOK ON RAILROAD SURVEYING Example. Assume that in Fig. 21, T.S. is at Sta. 711 + 44.0, k = 2, and S = 2.25. Then D = 4 30'. Deflection angle of T.S., Sta. 711 + 44.0 = 00' " " 1 " 711 +94.0 = 005' " " " 2 " 712+44.0 = 020' " " 3 " 712 + 94.0 = 45' " " 4 " 713 + 19.0 = 101' (1 01. 25 'exact) " " " 5 " 713 + 44.0 = 1 20' " "S.C.(A)" 713 + 69.0 = 141' (1 41. 25' exact) A - 1/3A, A = 5-04' (5 03.75' exact) B = 2/3A = 2A =3 22.5' To orient the transit at the S.C., back-sight on the T.S. with the plates set at 3 22.5' (angle B)', then turn the plates to zero and the telescope is on tangent and the circular curve is run in as from the T.C. of a simple curve, stakes being placed at the regular stations (in the above problem at 714 + 00, 714 + 50, etc.). After running in about half of the circular curve, move the transit to the S.T. and run in the second spiral with the same deflection angles. Then move to the C.S. and back-in the remainder of the circular curve, thus placing the adjustment of the errors of surveying at the center of the curve instead of at the end of the spiral as is usually but unwisely done. The form of notes is shown in Fig. 22. This method is particularly applicable for locating spirals after the construction is completed and the track is to be brought to exact line and surface. For setting stakes during construction the offset method is preferable. 83. Offsets. Evidently the entire spiral can be located by means of the coordinates x and y (tangent offsets), and this is a satisfactory method for short flat spirals where y is less than about 10 feet. When y becomes greater than this, the spiral can not be located with sufficient accuracy unless the offsets are turned off with an instrument. On location, since it is usually desirable to advance the line as rapidly as possible, the best method is to run-in the circular curve from the T.C. to the C.T. (offsetted curve, see Fig. 20), CURVES 61 and to insert the spiral later by offsets from both the tangent and the circular curve. Since the spiral departs from an osculating circle at any point at the same rate that it departs from the tangent at the T.S. (see Appendix, paragraph 142), it follows that the offsets from the circular curve to the spiral are the same as the offsets from the tangent at the T.S. for the same distances. Since the offset o and the spiral bisect each other (see Appendix, para- graph 141), it is evident that the maximum offset is Jo, and there- fore will be small, and also that it is necessary to compute offsets for half the spiral only. Half of the spiral is then located by offsets y from the tangent at the T.S., and the other half by the same offsets measured normal to the circular curve. From Eq. 22 it is seen that the offsets vary approximately as the cube of the distances, and for this method of location can be taken so with inappreciable error. Since the maximum ordinate is o, and o is determined to offset the T.C., it is seen that the offsets can be determined directly from o instead of from Eq. 22. Application to Compound Curves At the C.C. of a compound curve as at the T.C. of a simple curve there is a change in the rate of curvature and in the amount of superelevation, and if this is great enough to be ob- jectionable, a spiral should be inserted between the two branches. Evidently only that part of the spiral of curvature intermediate between the degrees of the two curves is required, and as at the T.C. of a simple curve the two curves must be offsetted at the C.C. 84. Problem 1. In Fig. 23, HBC is a A curve and EFJ is a D 2 curve having parallel tangents at C and E (the position of the C.C. if the curve were unspiraled). It is desired to con- nect the two curves by the spiral BF. Consider the spiral run backwards to its T.S. at A. Since the degree of curve of the spiral at B must be A and at F must be Z) 2 , then the spiral from B to F is that part of a regular spiral from where d = Di to where d = D 2 . The value of k depends upon the maximum speed permissible on the sharper curve and upon the difference in the two degrees of curve. Since the spiral departs from every osculating circle at the 62 TEXT BOOK ON RAILROAD SURVEYING same rate as from the tangent at the T.S., BC = CF, the spiral bisects CE, and CE is equal to o-for a spiral whose D = D 2 D\ for the chosen value of k. FIG. 23. To insert a spiral between the curves, find o for a Z> 2 A curve and make the offset from C to #. The length of the spiral in stations is D 2 - D! Locate B and F by measuring J$i from C and E. The curves may be staked out by continuing the first branch to C, offsetting to E, and running-in the second branch; and then inserting the spiral by offsets from the circular curves in ex- actly the same way as explained in paragraph 83. 85. Deflection angles may also be used; but, since the transit is at a point on the spiral, B or F, and not at the T.S., the reflec- tion angles to be used are values of / instead of a. From a tangent at B in Fig. 23, the deflection angle n to any point N' on the circular curve BCM is JA X BN'. Since the spiral departs from the osculating circle at the same rate as from the tangent at the T.S., the angle between the circular curve and a point N on the spiral is a = 1/6 k(BN)*. But BN and BN' are equal. Then f = n + a = (BN) + 1/6 k (BN) 2 . CURVES 63 If the transit were set at F, the deflection angle to N would be f = JD 2 (FN) - 1/6 k (FN) 2 86. Problem 2. Fig. 24 shows a compound curve to be spiraled at each end and between the two branches. The FIG. 24. tangent distances T s i and T S2 are required, and the solution is fully indicated in the figure. Application to Existing Curves To insert a spiral in an existing track, it is necessary to shift the ends of the curve inward to provide room for- the spiral. If the same degree of circular curve were retained, this would shift the entire curve inward. Since the amount of such shifting may be considerable (o sec. JJ at its center) the new alinement may not be on the old roadbed and considerable cost of earth- work would be entailed in making the change. To obviate this, the degree of curve may be changed in such a way that the new alinement will permit the insertion of the spiral and at the same time require little or no additional earthwork. 64 TEXT BOOK ON RAILROAD SURVEYING 87. SIMPLE CURVES. Existing simple curves may be spiraled in two ways. (1) By shifting the center of the curve outward * a small amount, and by sharpening the curve sufficiently to give the desired offset. This method is particularly applicable for curves whose lengths are less than about four times the length of the spiral to be used. (2) By sharpening the curve slightly at the ends to give the necessary offset, leaving the middle portion of the curve un- changed. This method is preferable for long curves. 88. First Method. In Fig. 25, ABC is the existing unspiraled curve with B as its middle point. It is desired to shift the track outward at B FIG. 25. the assumed distance p, and to sharpen the curve to give the necessary offset o. From the figure, p is the difference between the external dis- tances of the existing curve and the spiraled curve. Therefore from Eqs. 5 and 27, we have p = R exsec JI [(Ri -f o)exsec JI + o] = (R - Ri - o) exsec JI - o ... (32) *The shifting must be outward to make the change in alinement a minimum. CURVES 65 The minimum change in radius (R Ri) is that which will make p equal to zero. Then from Eq. 32, Ri = R ^T- . (33) vers il (This is the maximum value of Ri that will satisfy the con- ditions.) In any given problem first determine approximately (mentally) the maximum value of Ri from Eq. 33. This is used simply as a guide in estimating the value of RI to use. Choose a value of Ri less than this (preferably to agree with an even 10 minutes of degree of curve) and solve Eq. 32 for p. If this value agrees sufficiently close with the desired shifting of the track at B t the solution is complete. If not, re-estimate Ri (using the previous value as a guide), and repeat until a satisfactory value of p is obtained. The distance from the T.C. of the original curve to the T.S. is the difference between T s for the new curve and T for the old curve. The T.S. and the S.T. are then located from the T.C. and the C.T., respectively, of the old curve, and the new curve is staked out as in new work. Example. / = 40 00', D = 4 00', k = 1, and p = about 1.5 ft. From Eq. 33, the maximum value of Ri = 1330 ft. (approx.) corresponding to DI = 4 18' (approx.). Choose a trial value of Ri = 1322.2 ft. for A = 4 20'. Then from Eq. 32, p = 0.8 ft. Choosing Ri = 1312.1 ft. for A =4 22', p becomes 1.3 ft., showing that a change of 2' in D! increases p about 0.5 ft. Then to make p = 1.5 ft., DI would have to be about 4 23'. Since an odd value of DI is undesirable, and since it is not necessary to make p exactly 1.5 ft., it is sufficient to take DI = 4 22'. In the foregoing discussion it was assumed that the degee of the existing curve was known and that the T.C. and C.T. were monumented. It will generally happen, however, that the degree of curve is unknown, that the track is in poor alinement, and that the T.C. and C.T. are not monumented. In this case, run out the tangents to an intersection and measure the inter- section angle. Then measure the external distance of the *o should correspond with Ri which is unknown, therefore use 'a value of o corresponding to about 0.9 R. 66 TEXT BOOK ON RAILROAD SURVEYING existing curve and from it determine the degree of the curve that will connect the tangents and pass through the middle point of the existing curve. Use the value of R corresponding to this value of D in solving Eqs. 32 and 33. 89. Second Method. Case 1. In Fig. 26, ABC is the exist- ing unspiraled curve, whose degree of curve, D, is known, whose alinement is good, and whose T.C. and C.T. are monumented. At some point on this curve, such as B, it is desired to compound with an assumed curve of slightly shorter radius, Ri, which, when run to a tangent parallel to the initial tangent, will be a distance from it equal to o for a Ri curve, thus providing room for the spiral. In the figure, EFB is the Ri curve, and EG is o corresponding to T.S. A T.C. G FIG. 26. it. It is required to find the location of the T.S., of the S.C., and of the C.C., B. From the figure, o = EG = GL - EL = (R - R x ) vers Ii vers Ii = ^ _ R (34) The T.S. is located by measuring back from the old T.C. T.C. to T.S. = t - (R - Ri - o) tan Ii. . . (35) The S.C. is located by measuring the spiral length from the T.S. The C.C., J5, is located by measuring the distance FB from the S.C. FB (in feet) = --^ 10 - ( 36 ) CURVES 67 The location of the point B may be checked by measuring the distance AB along the old curve. AB (in feet) = -g- 100 The distance from the T.S. to the C.C. along the new aline- ment is shorter than along the old alinement which will require the rails to be cut. This shortening is equal to the difference in the Sta. numbers of the C.C. as computed along the respective alinements. The limits of RI are such as will make the point B come at the middle of the original curve, or will make B and F coincide. In general, R\ should be chosen between 0.8 R and 0.9 R. Example. D = 3 00', A (assumed) = 4 00', k = 1, and o = 4.65. From Eq. 34, h = 12 14.3'. From Eq. 35, the distance from the old T.C. to the T.S. is 156.5 ft. The spiral is 400 ft. long. From Eq. 36, the distance from the S.C. to the C.C. is 121.1 ft. 90. Second Method. Case 2. In Fig. 27, ABC is the existing unspiraled curve, whose degree of curve is unknown, whose alinement may be poor, and whose T.C. and C. T. are no*, monumented. First set-up in the center of the track at some _A_ G , O (P L --^F^fec^ -TN~~T V/LB FIG. 27. point C near the middle of the curve and by trial deflection angles find the degree of the curve, D, that will most nearly conform to the existing track. Then run-in this D curve to a point B which is about 500 ft. from the end of the curve. If this curve were continued, its T.C. would fall at / on a tangent parallel to the initial tangent QL and at a distance p from it. 68 TEXT BOOK ON RAILROAD SURVEYING At B run-out the tangent to the D curve to an intersection with the initial tangent at L and measure the intersection angle IB and the distance BL. BN is the tangent distance of the curve corresponding to IB- Then, p = NL sin IB The D curve is to be compounded with a chosen D\ curve at some point F to give the required offset o. EK = o p = (R Ri) vers IB, whence o p (37) The T.S. is located at Q by measurement from L. QL = QG + GL = t + JN - JK - NL cos IB The S.C. and the C.C. are located as in Case 1, and the spirals are staked out as in new work. J may fall outside of QL, in which case p and o are numerically added; again, p may be greater than o, and in this case Ri must be greater than R. 91. COMPOUND CURVES. An existing compound curve can be spiraled in the following manner. FIG. 28. In Fig. 28, ABC is the existing compound curve with its C.C. Sit B. To make room for the spiral it is necessary to compound CURVES 69 the sharper branch, D 2 , at some point C with a still sharper curve of chosen degree, D 3 , and to continue the first branch, DI, to the point E where its tangent is parallel to the tangent of the D 3 curve at G. EG is the offset o for a spiral whose D = D 3 - DI. The problem is to find BE = HI and CG = n 2 . For small central angles the ordinates between two circular curves are equal to the difference in their tangent offsets. There- fore EF can be considered as the difference in the tangent offsets of the DI and D 2 curves in the distance HI] similarly FG is the difference in the tangent offsets of the D 2 and D 3 curves for the distance n 2 . Then from Eq. 15, o = EF + FG = 0.873 (D 2 - DI) ni 2 + 0.873 (D 3 - D 2 ) n 2 2 . Since the central angle of the arc EC equals the sum of the cen- tral angles of the arcs BE and CG, D 2 (ni + n 2 ) = Dini + D 3 n 2 or (D 2 - DI) ni = (D 3 - D 2 ) n 2 . Solving these simultaneous equations for HI and n 2 we have n - 1 07 I (D 3 - P2) O /OON L07 \' (D 2 - Dl ) (Da-DJ- n, = 1 07 !~~~ (E>2 - Di)o~ /o qi \(D 3 -D 2 )(D 3 -D 1 ) C is located by measuring the distance n\ + n z along the D 2 curve from B. The C.S. is located by running the DI curve from B the distance HI t, and the S.C. is located by running the D 3 curve from C the distance HZ (Si t). The spiral is staked out as usual for compound curves. 70 TEXT BOOK ON RAILROAD SURVEYING 31^00 OrHCOiOl^OO prHCOiOt>-30 OC pCOt^OCOt>; pCOt^pCOO; O CO l> O W CO SOrH (NCO^COOOrH IO X CO 00 "tf rH C5 00 l> 00 Ci pCOt>pCO^ OCOt>pCOl> OCOCDO5(NCO lOCOrH O 00 CO' 1C* 00 rH O 00 CO IO CO rH O 00 CD Tj5 CO rH ^Oit^ Oi CO O 00 iO O CO CO O5 ^ rH I-H LQ C^ iO CO CO 1> pOrH C O5 CO 00 COO5CDiO^OCO 00 . O5 rH CO CO O5 CO CO rH O O OOO OOOOOO OOOrHrHrH rH W C^ CO CO ^ IO CD 00 O rH CO IO CD 00 O rH CO Tf CO 00 O5 rH CO ^ rH CO CO rH 00 00 O5 CO rH CO 00 t> rH O Th CO 00 O I> rHrHrH(N(M COCO'*-* pCOl> pCOt^OSC^CO OOOCO^iOCO lOr)HC OiCDOOOOO CO CD Oi Tt< rH rH r}H (M IO O OOrH C^j TjH I> O5 CO 00 COpCDlOiOCO 00 W l> ^ OOO O O O O rH rH C^ W CO T|< U3 CO l> Oi O C^ boooo boocobccoo boocoibcooo boocoibcooo t^co oocot^co'd oocot^coo oocoi>coo OOO OOOrHrHrH N CURVES 71 5iC^CNlCOTt< ^iCcOr^XOi oic^WCOr? TfiocOl>OOX OiC^-KNfNCO TMCCCCOl^OO Oi r-< CV t>xx cicid'-^'-^c oo T-H i I i < rH i ( FH i-H rH FH rH FH r-i (N (N oqqoi'-iqco TJJ rH(M(N(N(NCO COCO-^-tiCiC CO >COiHTH|>O> COOiCiCOcOO O co ccoo occoi>cod docoi>coQ oocot^cod QQeofceoo odcot^coo o ^(NOiC^ CO(Ni-iOOO OOOOi-nC^ CO-^iCO(N^ Ot^b-i>l>I> XXXO 5COCOOCOCO 3 TH CO iC COX Ji Oi Oi Oi Oi Oi bbbbb bbbbbb bbbbbb bbbbbb bbbbbb bbbbbb b HMCOTFiC O rH 722.50 m m > 722.00 13.0 13.5 14.0 14.5 4.1 4.6 5.1 5.6 6.1 6.6 Vs ^0.3 +9.0 +9.8 + 10.2 27.5 +7.3 +7.3 29.3 +7.8 728.88 +0.72 729.60 -10.54 25.0 +5.5 +6.0 25.7 +6.4 22.3 +4.2 +4.4 23.6 +4.4 20.3 +2.2 +2.7 20.6 + 3.4 19.1 + 1.4 17.3 + 1.0 + 1.2 719.06 +2.53 721.69 15.5 Grade -1.2 Point +65 -0.6 16.1 - 0.8 10.8 -3.2 -2.4 10.2 - 2.1 13.8 -4.8 -3.8 12.2 - 3.5 16.2 -6.0 -5.2 14.3 - 5.0 16.5 - 6.6 18.0 -6.8 -6.6 19.2 -8.5 -7.8 18.9 - 7.0 19.5 21.8 FIG. 34. 78 End Areas. Volumes. Mass Curve Volumes. Cut. Fill. Cut. Fill. 412.7 3653 1303 290.8 2350 221.0 948 1402 150.2 687 715 88.3 442 273 230 35.8 0.0 0.0 15.3 43 10 43 00 55.0 130 10 140 278 95.3 418 434 139.2 852 602 186.0 1454 230.8 772 2226 FIG. 34. 79 80 TEXT BOOK ON RAILROAD SURVEYING equal to the cut or fill at that point multiplied by the slope ratio and added to one-half the roadbed. This is necessarily a cut- and-try method, but proficiency is soon attained in approxi- mating the correct position of the slope stak'^, and it is seldom that more than three trials are required. The cut or fill is marked (a) Level Section ( t>) Three-Level Section (c) Five-Level Section (d) Irregulkr Section FIG. 35. FIG. 36. on the slope stake and is recorded in the notebook as the numera- tor of a fraction whose denominator is the distance out from the center. Three-level, jive-level, and irregular sections come under this heading (see Figs. 35b-c-d). 103. Sections at Grade Points. Fig. 36 shows the junction between a cut and a fill. Stakes, marked " grade," are driven at A, B, C, and D, which are grade points. A cross-section is taken at B and D, and the distances x and y are measured. EARTHWORK 81 Areas of Cross-Sections Before the volume of earthwork can be computed, the area of each cross-section must be found. This is done as follows: FIG. 37. The cuts at A, B, C, E, F, and G (i.e., the vertical distances from the ground surface to the level of the roadbed), and the distances of these several points from the center, C, have been recorded in the field notebook. To each section annex the fraction and write plus and minus signs on each side of each b numerator, thus: q_ +C 3 - + C 2 - b D 3 D 2 C + ^ RULE 1. To find the area (1) multiply each denominator by the algebraic sum of its adjacent numerators, using the sign facing the denominator; and (2) divide the algebraic sum of the resulting products by 2. or, IT b(C 3 + c 2 ) + D 3 C 2 + D 2 (d- C,) +-] 2 L D!(C O - C 2 ) + di(C - c 2 )+ d 2Cl J (Let the student prove this equation.) After the student becomes familiar with this formula, it will not be necessarjr for him to actually write down the plus and minus signs in the notes. 82 TEXT BOOK ON RAILROAD SURVEYING Eq. (40) can be used for either cut or fill, and for level, three- level, five-level, and irregular sections. 104. Side-hill Sections. When a section is partly in cut and partly in fill as shown in Fig. 38, the field work and the com- putation of the area are slightly modified. \\Slope Stake Slope Stake 3 FIG. 38. In this case, after the fill at the center has been determined, the grade point, A, is found by trial. Then all horizontal dis- tances are measured from A. The notes would be recorded as follows : _0_ x +Ci - + + - c 2 d 2 Di di < Cut - X Fill - > Notice that x and y are used as the denominators of the an- nexed fractions instead of b as in the former case. The areas of the cut and fill are computed separately, and are recorded separately in the notes (see Fig. 34). 105. Uncompleted Sections. It is sometimes necessary to find the area of an uncompleted section for example, for monthly estimates or of an irregular-shaped borrow pit. The field work for this does not require that elevations be carried EARTHWORK 83 from station to station, but at each set-up the plane of the horizon- tal line of sight is taken as the reference plane. For example, in Fig. 39, A B is the reference plane whose elevation is not FIG. 39. necessary for the determination of the area of the section. The notes should be written thus: + c 4 - +C 3 - + C 2 ^- +Ci - + C + - ci + -_c 2 d 2 - c 4 -c. d, Notice carefully the end fractions annexed in this case. The area is found by Rule 1. Volumes of Earthwork Earthwork is paid for by the cubic yard, and in order to compute the cost it is necessary to find the volume of the material handled. The total volume is the sum of the -volumes of the prismoids formed by adjacent cross-sections. The volume of the prismoid between two adjacent cross-sections may be found (1) by the end area method, and (2) by the prismoidal method. 106. End Area Method. In this method the prismoid is treated as a prism, whose cross-section is the mean of the two end areas of the prismoid. Z v (cu. yds.) = 27 (41) where v is the volume, Ai and Ai the end areas in sq. ft., and I is the distance between cross-sections in feet. 84 TEXT BOOK ON RAILROAD SURVEYING For I = 100 ft., v = 1.85 (Ai + A 2 ) (41a) This method is exact when the end areas are equal, but is only approximate when they are unequal. The result is usually too large and the maximum error occurs when one end area is zero, and is about 16 per cent. Owing to its simplicity, and to the fact that sections usually do not change rapidly, this method is almost universally used. Table 12 gives the cubic yards per 100 ft. for center depths from 1 to 60 ft. Table 13 gives the cubic yards per 100 ft. for end areas from 1 to 1000 sq. ft. 107. Prismoidal Method. When the changes from cut to fill are frequent and abrupt or the work is in rock, greater accuracy may be required; in this case the prismoidal method is used. From Simpson's rule, . Ai + 4A m + A 2 , Mean area ol a pnsmoiu = ^ , where A\ and A 2 are the end areas and A m the area of a section midway be- tween the ends. Then, I Z A I + 4 Am -h A 2 V = 27 X " -6~ (42) This is called the prismoidal formula. This form however is in- convenient to use since it requires the area of a mid-section. It is therefore more convenient to find the difference between the prismoidal and end area formulas and to apply this difference to the latter. Subtracting the prismoidal formula from the end area formula and reducing, Prismoidal Correction = ^j (Ai 2A m + A 2 ) Putting AI, A 2 , and A m in terms of their slope stake dimen- sions remembering that the dimensions of A m are the average of those of AI and A 2 and reducing, we have Pris. Cor. = (d - C 2 ) (W 2 - WO . . . (43) EARTHWORK 85 in which Ci and Cz are the center cuts or fills at AI and A 2 , and Wi and Wz are the corresponding distances between slope stakes. When 1= 100 ft., Pris. Cor. = 0.31 (Ci - C 2 ) (W 2 - Wi) (43a) This correction is added algebraically to Eq. (41). The result is exact for three-level sections and is sufficiently accurate for any other form of cross-section. 108. Earthwork on Curves. The volume on curves is gen- erated by a section moving perpendicular to the center line of the curve. The section considered is the mid-section of the solid. The distance traveled by this section is the distance traveled by its center of gravity, and it is only when the center of gravity lies in a vertical line passing through the center line of the curve that the volume as computed by the prismoidal method is exact. If the center of gravity is inside the center line, the results ob- tained by the prismoidal method are too large; if outside, they are too small. The difference between the true volume and the volume determined by the prismoidal method is called the curvature correction. If the volumes are computed by the end area method, it is inconsistent to apply the curvature correction ; but if the pris- moidal method is used, the curvature correction should be ap- plied, especially in rock work. The curvature correction is computed as follows: Fig. 40 represents the mid-section of the solid. Construct FIG. 40. BC with the same slope as CF. Then the center of gravity of the part shown shaded is in a vertical lins through C and there- 86 TEXT BOOK ON RAILROAD SURVEYING fore there is no correction to be applied to this portion. The center of gravity of the triangle ABC is at G, which is the dis- tance x from the center, C. The volume generated by the triangle ABC is v = ABC X (R H-x) X I in which / is the angle at the center expressed in radians. The volume computed by the prismoidal method is vi = ABC X R X I The curvature correction then is C. C. = v - vi = ABC X x X I 2 1 But x = g X -^ X (Di+di). Substituting this value of x and expressing / in degrees, we have C. C. = 0.006 ABCX (D x + di) X 1 (44) For 100 ft., 7 becomes D, the degree of the curve. Haul and Overhaul 109. Haul. Haul is the work done in transporting the ma- terial from the excavation to its place in the embankment. Its unit is the yard-station, or the work done in transporting one cubic yard 100 ft. The term haul is frequently used to indicate either the yardage or the distance, but this must not be confused with the true meaning. Contract prices are usually based on a maximum length of haul generally 500 ft. called the free-haul distance. If the material is transported a longer distance the extra work in- volved is termed overhaul and the contractor is allowed extra compensation at a certain price per yard-station. The con- tractor is therefore paid for all excavation and for overhaul, but he is not paid for making the fills nor for haul inside the free-haul distance. The amount of excavation is readily determined from the cross-section notes, hence this portion of the contractor's re- EARTHWORK 87 muneration presents no particular difficulties except when the excavation involves several different kinds of materials to be paid for at different prices. The amount of overhaul, however, depends on the plan of distributing the excavated material, and the problem is to determine an economical distribution. This problem is not of great difficulty, but unfortunately it has often been neglected, resulting in uneconomical work, misunder- standings with the contractor, and frequent law-suits. The limit of profitable haul is that distance at which the cost of overhaul equals the cost of excavating a yard of earth. For example, if the contract price is 24 cents a yard for excavation and If cents a yard-sta. for overhaul, and the free-haul distance 24 is 500 ft., the limit of profitable haul is p -f 5 = 21 Sta.* 110. Case 1. Fig. 41 represents a fill between two cuts. It is readily seen that there is more than enough earth in the cuts to make the fill, and the question is how much of the fill shall be made from each cut. The points B and D are the free-haul distance apart and are located on the profile so that the volume of cut between B and C equals the volume of fill between C and D. F and H are simi- larly located with respect to G. Since the cut to the right is the deeper, it is evident that the fill can be made with the minimum haul if the greater volume of earth is taken from the right-hand cut. The problem is to locate A, E, and /, which are the limits of the distribution from each of the two cuts. This can not be done by inspection nor * A yard of earth excavated, hauled 700 ft., and dumped into the fill costs the railroad company 24 cts. plus 3 cts. overhaul. If the yard excavated had been hauled only a few feet to one side of the cut and dumped, and another yard of earth had been taken from a borrow pit along side of the fill and placed hi the fill, the cost would have been 48 cts. It is evident then that it is much cheaper for the railroad to pay the additional amount for overhaul. In the above example, the saving was 21 cts. If the haul had been longer, the saving would have been less. But the railroad saves money up to the point where the cost of overhaul equals the cost of making a yard of fill from a borrow pit. Beyond this point there is an increasing loss, and when twice the limit of profitable haul is reached, this loss equals the previous saving. The limit of profitable haul is considered as that distance at which the profit is the largest. 88 TEXT BOOK ON RAILROAD SURVEYING can it be accomplished easily from the cross-section notes. It can be done most easily and accurately by constructing a dis- tribution diagram. The first step in making the distribution diagram is to construct the mass curve, which is done in the following manner : FIG. 41. Choose a horizontal base line near the bottom of the profile sheet, and project the grade points C and G onto it at C" and G f as shown in Fig. 41. Then beginning at C' plat to the chosen scale usually 1,000 cu. yds. to the inch a curve such that the ordinate to it at any point is equal to the total volume of earthwork between that point and C". From G' plat a similar curve to the same scale. The ordinates to these curves are computed at the regular stations from the cross-section notes. Allowance for shrinkage or growth must be made by increasing the fill volumes by the amount of shrinkage or decreasing them by the amount of growth. The second step is to determine the limits of free haul. This is done by locating a horizontal line equal in length to the free- haul distance in such a position that its ends are in the mass curve as at B'D' and F' H'. These points are projected up to locate B, D, F, and H. The ordinate to B'D' is the free-haul yardage past C, and the ordinate to F'H' is the free-haul yardage past G. The third step is to find the positions of A, E, and J so that the haul will be a minimum. For all practical purposes this occurs EARTHWORK 9 when A E = EJ. Therefore in the distribution diagram find by trial the positions of the two horizontal lines A'E' and E"J r of equal length, with E r and E" on the same vertical line. These points are projected onto the profile to locate A, E, and /. These points should be located in the field and distinctly marked for the guidance of the contractor. The ordinates between B'D' and A'E' and also between F'H' and E"J' give the total volumes on which there is an overhaul charge. The total volume moved is given by the ordinates from the base line to A'E' and E"J'. Since the mass curve is a curve between distance and total volumes, it follows that the area inside the curve represents haul in yard-stations. Consequently the overhaul is given by the shaded portion between the curves and the lines A'E' and E"J'. If the ordinate B 'b is bisected dividing the overhaul yardage into two equal parts and the horizontal line M N is drawn, the two points M and N will be directly under the centers of fnass of the cut and the fill. Therefore M N minus the free-haul distance is the overhaul distance corresponding to B '&; and the overhaul can be found by multiplying B'b by this distance. This method is frequently easier of application than that of determining the areas direct. The overhaul past G can be determined similarly. C' 1 3 FIG. 42. G' 111. Case 2. In Fig. 42, C and G are so far apart and the cuts are so long that it is evident that the limit of profitable haul will be reached. Construct the mass curve and locate the free-haul 90 TEXT BOOK ON RAILROAD SURVEYING limits as before. Lay of A'E' and K'J' each equal to the limit of profitable haul. The centers of mass, M, N, P, and Q, are found as in Case 1. The points A', E', K f , and /' projected onto the profile give the points A, #, K, and J \ which should be marked in the field. The fill between E and K must be borrowed and the yardage required is given by the difference of the ordinates to E r and K" . The amount of overhaul is determined as in the former case. The complement of this case is the determination of waste in a long cut. 112. Case 3. In case earth can not be moved past some point, the distribution diagram is constructed as before except that the horizontal distribution lines (A'E' and E"J' in Fig. 41) must start at the fixed point. The remainder of the work is done as before. This condition frequently occurs usually at stream crossings, CHAPTER VI TURNOUTS, CONNECTIONS, AND CROSSINGS 113. The subjects treated in this chapter are as follows. , / Switches 1. Turnouts < ^ ( Frogs 2. Connections Diverging / From Tangents Tracks \ From Curves ( From c,.,. Tangents Sidings { ^ 1 From Parallel Curves Tracks From Crossovers Tangents From Curves Intersecting (Tangents Tracks \ Curves 3. Crossings 114. Definitions. 1. Turnouts A switch is a device to deflect at will the wheels of a train from the track on which they are running. A switch stand is a device for operating a switch (see Fig. 45). A frog is a device at the intersection of two rails to permit the passage of the wheel flanges. A guard rail is a short section of rail placed opposite the frog to guide the wheels so that the flanges will pass through the frog properly. A turnout is a combination of a switch, a frog, and the necessary connecting. rails, etc., to permit rolling stock to pass from one track to another (see Fig. 45). A siding is a secondary track parallel to the main track which is used for the storage of rolling stock or for the passing of trains. 91 92 TEXT BOOK ON RAILROAD SURVEYING It is connected to the main track by a turnout at one or both ends. A crossover is a connection at an intermediate point between two parallel adjacent tracks (see Fig. 45). 115. Switches. Switches are of three types, viz., Split Switches, Tongue Switches, and Stub Switches. The Split Switch, Fig. 43a, is the form universally used on steam roads. It consists of two wedge-shaped sections of rails connected by tie rods and sliding on metal plates placed on the ties. The broad end or heel of each point rail acts as a pivot by being fastened to the lead rails by means of the usual splice bars. The point moves through a distance of about 5J inches, which is called the throw (t), and is controlled by a switch stand placed outside the track on a long tie (sometimes two) called a head- block. The distance between the gage lines at the heel is called the heel spread (h). The angle between the gage lines of the switch and the main rail is called the switch angle (s). The width of the point is usually \" (when new). From Fig. 43a it is seen that sin s = *~ ...... (45) where p is the width of the point, and I the length of the switch rail. Split switches vary in length from 10 to 33 feet, depending on the flatness of the turnout desired. The point rails are cut from rails of standard length so as to have no waste. For 30-ft. rails, 10, 12, 15, 18, 20, and 30-ft. switch points are used, and for 33-ft. rails, 11, 13, 16J, 20, 22, and 33-ft. The most common lengths are 15 and 16J ft. The Tongue Switch, Fig. 43b, consists of a steel wedge or tongue pivoted at one end and moving in a heavy cast-iron frame so arranged that pavement may be built around it. This is the common form on street railways, where it is usually used singly with a rigid fitting called a mate in the opposite rail. Steam roads use them only for turnouts in pavement, and then usually in pairs, where the action is the same as in a split switch. A Stub Switch, Fig. 43c, consists of a pair of ordinary rails fastened together with tie rods. The rails are spiked for part of their length, the remainder being thrown to match the stub ends of the lead rails. A stub switch differs from a split switch in that the toe is fixed and the heel moves. This form of switch may be TURNOUTS, CONNECTIONS, AND CROSSINGS 93 rmnnnnnn HjS.fi [fil J!L 15L__ J5L ALJ3LL _U3, - LI U U U LJ FIG. 43a. MATE SWITCH FIG. 43b. FIG. 43c. 94 TEXT BOOK ON RAILROAD SURVEYING employed to advantage on construction and mine tracks owing to its cheapness and ease of installation. A stub switch should never be used on a standard road. 116. Frogs. Frogs are of two types, viz., Rigid Frogs and Spring-rail Frogs. Frogs of either type are universally made with straight gage lines (except on street railways) . The point rails are finished with a blunt point usually \" wide. The dis- tance P between this actual and the theoretical point or intersection of gage lines equals the width multiplied by the frog number. Fig. 44& shows the common form of rigid frog and the names of the various parts. The essential feature of the rigid frog is that both flangeways are open and must be jumped by passing wheel treads. Theoretical Point - ^Actual Point Flange way Short Point Rail Wing RaiK FIG. 44a. Rigid Frog. Main Point Rail FIG. 44b. Spring-rail Frog. Fig. 44b shows the common form of a Spring-rail Frog. In this type one wing rail (sometimes both) is made movable. It is normally held against the point rails by springs. A wheel on the main track therefore has an open flangeway and a con- tinuous support for the tread, while a wheel on the turnout must spring the wing-rail open to allow the flange to pass, and at the same time the treads lack continuous support as in a rigid frog. Guard rails are absolutely essential to the safe operation of a spring- FIG. 45. Crossover. 96 TEXT BOOK ON RAILROAD SURVEYING rail frog. Spring-rail frogs are used on main track turnouts, and rigid frogs in yards. A special type of frog called the movable point frog is used in connection with crossings particularly with " slip switches." 117. Frog Angle, F. The frog angle is the angle between the gage lines at their point of intersection. Obviously the frog angle may have any value, but only values of about 2 to 8 are practical for turnouts. For turnouts a few standard values of the frog angle can be chosen which will meet all the practical requirements. These angles are chosen to agree with simple values of the Frog Number which is a more convenient method of designating frogs than by their angles. 118. Frog Number, N. The Frog Number is the ratio of the a .ial length to the spread, i.e., it is the distance measured along the bisector of the angle in which the gage lines diverge a unit distance. Reducing this ratio to a trigonometric function we have, N = 1 cot JF (46) Values of N from 7 to 15 are commonly used, but for special purposes values from 3 to 27 are occasionally used. 119. Location of Turnouts. The first step in staking out a turnout (whether on straight or curved track) is to locate the frog. The heul or toe of the frog should come, if possible, at a regular rail joint in order to avoid short pieces of rail in the track. This fixes the position of the P.F. (point of frog). The next step is to locate the P.S. (point of switch), which is done by measuring along the main track the distance L (the lead). The next step is to locate the outside rail of the turnout curve. This is done by offsets from the outside main rail at the middle and quarter points of the turnout curve. The inner turnout rail is set by gage from the outer. 120. Turnouts from Straight Track. Since the frogs and switches are straight, the alinement of a turnout is not a simple curve from a tangent, but consists of two short pieces of tangent making a fixed angle with the main track rails connected by a curve. In triangle 1, Fig. 46, the hypotenuse and angles are known. Solve for m and n. In triangle 2, q = g h n and the angles TURNOUTS, CONNECTIONS, AND CROSSINGS 97 are known. Solve for j and c. c is the chord of the outside lead rail and the central angle (Fs) is known, hence the radius of the outside rail is - ..... (47) The distance along the straight track from the P.P. to the P.S. is called the lead, L. From the figure L = l+j+m + P ..... (48) P.S. h) (Vi I '=N-x Width of frog point FIG. 46. To compute the offsets for the lead rail, consider its gage line produced back to a tangent parallel to the main rail. The distance between these tangents is e = h (R + ig) vers s The offset from the main rail to the quarter point of the lead rail is then, yi = e + (R + ig) vers [s + J(F - s)] (Slide Rule) For the middle point, y 2 = e + (R + ig) vers [s + J(F - s)] " " (49) For the three-quarters point, y 3 = e + (R + ig) vers [s + f (F - s)] " " '^ 121. Turnouts from Curved Track. It can be shown (the solution is too involved to be included here) that the lead of a turnout from a curved track is almost identical with the lead of a turnout from straight track with the same frog number. And further, that the degree of curve of the lead rails is equal to the degree of curve of the turnout from straight track increased 98 TEXT BOOK ON RAILROAD SURVEYING or diminished by the degree of the main curve depending on whether the turnout is on the inside or the outside of the main curve. Therefore a turnout on a curve is staked out with the same dimen- sions as one with the same frog number on straight track. 122. Double Turnouts. Occasionally two turnouts to op- posite sides of the main track are located at the same point. This involves the use of two regular frogs in the main rails, a third or crotch frog at the intersection of the lead rails, and a three-throw switch. Such a switch is structurally weak and should never be used in a main track. And further, the condi- tions under which such arrangements are absolutely necessary are extremely rare. It is better practice to use two separate turnouts, placing one switch ahead of the other and if necessary using a sharper turnout to save distance, than to use a three-throw switch. Double turnouts will therefore not be further considered. 123. Practical Leads. Since it is undesirable to use short pieces or odd lengths of rails, it is the universal practice to modify the theoretical leads as computed above (Eq. 48), so as to use convenient lengths of lead rails. The difference between the theoretical leads and the practical leads is never very great, and the turnout is located as above except for this modification in the lead. This has the effect of changing the turnout curve from a true circle, but the amount is inappreciable. There is also a difference in the length of the straight and the curved lead rails. The best practice is to correct this by making the toe length of the frog on the turnout side longer than on the main-track side. Every, railroad has its standard turnouts, all dimensions of which are worked out and tabulated. The turnouts are then installed by the trackmen, the only duty required of the en- gineer being to locate the P.P. and occasionally the P.S. The engineer therefore has little occasion to use the functions of the turnout proper. His problem is to connect the turnout to the required track. Table 2 gives the practical switch leads recommended by the American Railway Engineering Association; also the rectangular co-ordinates to the quarter and center points on the gage side of the curved rail, referred to the point of switch as origin. TURNOUTS, CONNECTIONS, AND CROSSINGS 99 TABLE 2 Practical Leads as Recommended by the American Railway Engineering Association 8^ 1 0.1 || Rectangular Co-ordinates to the 8uarter and Center Points on I? y p age Side of Curved Rail, Re- (52) y 3 = p " x 3 = xi + R sin F Table 3 gives the values of x and y for track centers from 12 to 16 ft. and for frog numbers from 7 to 15. (b) Sometimes the connecting curve is commenced at the heel of the frog or a fixed distance beyond it. In this case x\ TABLE 3 105 Co-ordinates for Locating Parallel Sidings with Tape fc 3 fc H % O II Xl X2 X3 yi y 2 || Variation 1 EH a 2 7 12.0 23.31 50.45 77.73 8.12 11.03 15 00' 8 10' &&>& 12.5 26.79 53.93 81.21 8.62 11.53 13.0 30.28 57.42 84.70 9.12 12.03 _ 2*Ji 13.5 33.76 60.90 88.18 9.62 12.53 O OO 14.0 37.24 64.38 91.66 10.12 13.03 v 14.5 40.73 67.87 95.15 10.62 13.53 iH X >> 15.0 44.21 71.35 98.63 11.12 14.03 O .^ 15.5 47.70 74.84 102.12 11.62 14.53 t, ao a 16.0 51.18 78 . 32 105 . 60 12.12 15.03 fe.So8 8 12.0 25.33 57.73 90.26 7.94 10.98 11 00' 7 09' 0> 35 12.5 29.32 61.72 94.25 8.44 11.48 13.0 33.31 65.71 98.24 8.94 11.98 13.5 37.29 69.69 102 . 22 9.44 12.48 9 14, .0 41.28 73.68 106.21 9.94 12.98 v X > 14.5 45.26 77.66 110.19 10.44 13.48 T ~! 15.0 49.25 81.65 114.18 10.94 13.98 -v ' . 15.5 53.24 85.64 118.17 11.44 14.48 f_ ao a 16.0 57.22 89.62 122.15 11.94 14.98 fe.So 9 12.0 29.59 64.87 100.26 8.07 11.02 9 00' 6 22' 35 12.5 34.07 69.35 104 . 74 8.57 11.52 sf d 9 13.0 38.55 73.83 109 . 22 9.07 12.02 _rt Ji J3 13.5 43.03 78.31 113.70 9.57 12.52 yoy 14.0 47.51 82.79 118.18 10.07 13.02 v 14.5 51.99 87.27 122 . 66 10.57 13.52 r-J X r* 15.0 56.47 91.75 127.14 11.07 14.02 O -^' . 15.5 60.95 96.23 131.62 11.57 14.52 fc 16.0 65.43 100.71 136.10 12.07 15.02 fc.S6rf 10 12.0 31.23 72.08 113.04 7.90 10.98 7 00' 5 44' &3*a 12.5 36.21 77.06 118.02 8.40 11.48 13.0 41.19 82.04 123 . 00 8.90 11.98 J*li 13.5 46.17 87.02 127.98 9.40 12.48 14.0 51.15 92.00 132.96 9.90 12.98 ^ l ~ l fl 14.5 15.0 56.13 61.11 96.98 101.96 137.94 142 . 92 10.40 10.90 13.48 13.98 Sail 15.5 66.09 106.94 147 . 90 11.40 14.48 o' _ 3 16.0 71.07 111.92 152 . 88 11.90 14.98 fc.So 12 12.0 39.38 86.96 134 . 63 8.03 11.00 5 00' 4 46' 3 *a 12.5 45.38 92.96 140 . 63 8.53 11.50 C3 5 J 13.0 51.38 98.96 146.63 9.03 12.00 c3 Q 3 13.5 57.37 104 . 95 152 . 62 9.53 12.50 " N 14.0 63.37 110.95 158 . 62 10.03 13.00 ^ r " 1 CH 14.5 69.37 116.95 164.62 10.53 13.50 *" J3 15.0 75.36 122 . 94 170.61 11.03 14.00 ft r? W 15.5 81.36 128 . 94 176.61 11.53 14.50 ^^ 16.0 87.35 134 . 93 182 . 60 12.03 15.00 Sail 15 12.0 48.38 111.08 173.85 7.87 10.97 3 00' 3 46' 3*a 12.5 55.97 118.67 181.44 8.37 11.47 jn'^J . M 13.0 63.57 126.27 189 . 04 8.87 11.97 CS Q C3 13.5 71.16 133.86 196.63 9.37 12.47 w^^S, 14.0 78.76 141.46 204 . 23 9.87 12.97 X fl 14.5 86.35 149 . 05 211.82 10.37 13.47 '"1 ?H 15.0 93.95 156.65 219.42 10.87 13.97 o agw 15.5 101.54 164.24 227.01 11.37 14.47 o ^3 16.0 109.14 171.84 234.61 11.87 14.97 fe.Su'S 106 TEXT BOOK ON RAILROAD SURVEYING and 2/1 are known and the radius of the connecting curve must be computed. The other dimensions are then computed as before. Although this method flattens the connecting curve, it increases the distance from the P.P. to the T.C. There is nothing gained by making the connecting curve flatter than that of the turnout itself, and also the greater length of tangent is advantageous. (c) Sometimes the connection is made in the form of a reversed curve with radii about the same as the radius of the turnout curve under the false idea that this saves distance between the P.P. and the T.C. The actual difference in z 3 for such a connec- tion and for one of the form first given for a No. 10 turnout is only about 1.1 ft., an inappreciable amount considering the rel- ative riding qualities of the two layouts for heavy road engines. 127. Case 2. Curved Tracks. A turnout and a connection to a parallel track on a curve may be staked out by the co-ordinates given in Table 3. Such a layout will have the following characteristics. 1. The degree of curve of the turnout will be increased or diminished by the degree of the main curve, depending on FIG. 53. whether the siding is inside or outside of the main track curve. 2. The degree of the connecting curve will be diminished or increased by the degree of the main curve (approx.), depending TURNOUTS, CONNECTIONS, AND CROSSINGS 107 on whether the siding is inside or outside of the main track curve. 3. The track between the frog point and the beginning of the connecting curve will become a curve of the same degree (ap- prox.) as the main curve. On curves of about 1 this method is both satisfactory and convenient, but on sharper curves, or in case it is desired to maintain a frog tangent, it will be necessary to compute the connection for the particular case. Two general cases arise: (a) Sidings on the outside of the main curve. In Fig. 53, OA = (R + ig) cos F AB = (R + ig) sin F AC = AB + P + K = OiH OOi = R + p - Ri OH = OA - (R x - ig) In triangle 2, (OOO 2 = (OiH) 2 + (OH) 2 (R + p - RO 2 = (AC) 2 + (OA + ig - RO 2 expanding and reducing, (AC) 2 R ' = 2(OA + te 1 R _ p) + (OA + *8 + R + P) (53) then AC sm a = QQJ. Since the central angle a and the radius R\ are now known, the length of the connecting curve can be computed. The curve may be staked out' either by deflection angles from either end (orient how?) or by offsets from the main track. The latter method is the more rapid and convenient. Since the connecting curve and the siding curve have a common tangent and since the angles are small, the offset between them at any point is equal to the difference of their tangent offsets. Then the offset from the center line of the main track to the center line of the connecting curve at any point is 7 p 5- n 2 (Di D s ), in which DI and Z) s are the degrees of o curve of the connecting track and the siding, respectively, and n is the distance in stations along the connecting curve from the point where it joins the siding. These offsets should be measured 108 TEXT BOOK ON RAILROAD SURVEYING radially to the main curve and the distances should be measured along the connecting curve. (6) Sidings on the inside of the main curve. FIG. 54a. In Fig. 54a, OA = (R - Jg) cos F AB = (R - Jg) sin F AC = AB - K - P = OH OOi = Ri + p - R OiH = Ri + ^g - OA In triangle 2, (OOi) 2 = (OH) 2 + (OiH) 2 (Ri + p - R) 2 = (AC) 2 + (Ri + Jg - OA) 2 expanding and reducing, then AC OOI If Eq. 54 gives a positive value of R L , the layout is as shown in Fig. 54a. If a negative value of #1 is obtained, the layout is as shown in Fig. 54b and the value of RI is numerically correct. If (OA %g) - R p, Eq. 54 gives RI = infinity, and the TURNOUTS, CONNECTIONS, AND CROSSINGS 109 frog tangent produced is tangent to the siding as shown in Fig. 54c. These connections can be staked out by offsets as explained under case (a). It is to be noted, however, that if the layout is FIG. 54b. FIG. 54c. as shown in Fig. 54b, the sum of the tangent offsets must be used. PARALLEL TRACKS. CROSSOVERS. Since a crossover is equivalent to an entry to a siding, and therefore both turnouts will be run over at practically the same speed, there is no valid reason for making one turnout flatter than the other, even when the crossover is between a main track FIG. 55. and a siding. A crossover should therefore always have frogs of the same number at both ends. 128. Case 1. Straight Tracks. (a) Straight track between frogs of the same number. 110 TEXT BOOK ON RAILROAD SURVEYING From Fig. 55, AC = P cot F then x = AC - 2gN - 2P . . . . (55) To locate the crossover one P.F. is set so that the heel or toe will be at a regular rail joint. The other P.F. is fixed by the distance x. These are the only stakes ordinarily required. Table 4 gives the distance x between the frog points for track centers from 12 to 16 ft. for turnouts from No. 7 to 15. TABLE 4 Distances along main track between frog points for crossovers between parallel tracks. Frogs of same number, straight track between frogs. Track Centers N F 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 7 8 10' 17.14 20.63 24.11 27.59 31.08 34.66 38.05 41.53 45.02 8 7 09' 19.66 23.64 27.61 31.62 35.60 39.59 43 . 57 47.56 51 . 55 9 6 22' 22.05 26.53 31.05 35.49 39.97 44.45 48.93 53.42 57.90 10 5 44' 24.52 29.50 34.48 39.46 44.45 49.43 54.41 59.39 64.37 12 4 46' 29.90 35.90 41.90 47.90 53.89 59.89 65.89 71.88 77.88 15 3 49' 37.38 44.88 52 . 37 59.86 67.36 74.85 82.35 89.84 97.34 (6) Simple curve between frogs of different number. Sometimes unequal frogs are used on crossovers, and in this FIG. 56. TURNOUTS, CONNECTIONS AND CROSSINGS 111 case the best form of layout to use is a connecting curve of about the same degree as the sharper turnout, tangent at the heel of the flatter frog. This will place a short piece of tangent between the curve and the heel of the sharper frog which is desirable. In Fig. 56, F, F i} p, g, and Ki are known. R is chosen to agree with the turnout curve of F. In triangle 1, the hypotenuse and angle FI are known. Solve for the sides d\ and m. In triangle 2, the altitude n = p m and the angle F are known. Solve for the sides d and j. Then K = j - gN - R tan \ (F - FI) The distance along the main tracks between frog points is x = (d + dO - (gN + P + gNi + Pi) . . (56) 129. Case 2. Curved Tracks. It can be shown that, if a crossover is located between curved tracks with the same frog numbers and a simple curve tangent to the heels of the frogs, the distance along the main tracks be- tween frog points is practically the same as for a straight cross- over between straight tracks; and also that the degree of curve of the connection is practically the same as that of the main tracks. Therefore, a crossover on a curve is located with the same dimensions as one on tangent. It may be desirable, however, to locate points on the connecting curve. Usually one point at the middle is sufficient, and it will be seen that this point is midway between the frog points and on the center line between tracks. Crossovers should be avoided on curves. INTERSECTING TRACKS. 130. Case 1. Both Tracks Straight. (a) Connection in form of simple curve. Given in Fig. 57, the intersection angle, /; the frog angles, F and F\\ and the radius of the connecting curve. Required: The distances, A and B, from the intersection of the center lines to points opposite the frog points. 112 TEXT BOOK ON RAILROAD SURVEYING In triangle 1, the angles are known and one side, n, is readily obtainable. Solve for r and m. FIG. 57. In triangle 2, one side, .; + r, and the angles are now known, whence k and I can be computed. Then A = k - gNi - Pi B=Z+m -gN-P (57) (b) Connection in form of reversed curve. Given in Fig. 58, the intersection angle, 7; the frog angles, F and Fi' t the radii of the reversed curve, R and Ri' } and the location of one frog point, which is the distance, T, from the P.7. at A. Required: The central angles, a and 6, of the reversed curve, the distance, T\ t from the P.I. to the unknown frog point, and the distance x between frog points measured along the straight track. In triangle 1 (CDB) CD and angle F are given. Solve for CB and DB, whence AB and OB can be found. In triangle 2 (ABE) AB and the angles are known. Solve for BE and AE. In triangle 3 (OEG) OE and the angles are known. Solve for OG and GE. In triangle 4 (LHQ) LH and the angles are known. Solve for HQ and LQ = JM. TURNOUTS, CONNECTIONS, AND CROSSINGS 113 In triangle 5 ( HJOi) HOi and the angle FI are known. Solve for OiJ and HJ. In triangle 6 (OiOS) OiO = R + Ri and OiS = OJ + JM + GO are known. Solve for angles c and d, whence angles a and 6 are readily determined. T! = AE + EG - OS - LM + gNi + PI . (58) x = T - AL cos I - gNi - Pi . . . . (59) If Ti instead of T is known the plan of solution is the same. If the tracks do not run to an intersection or if it is inadvisable to run out the tangents and determine 7, locate the point D, set-up and turn off 90 from the frog tangent, and measure DE. Then set-up at E and measure DEL. With these data the prob- lem can be solved in the same manner as before. 131. Case 2. One Straight and One Curved Track. From Fig. 59, In triangle 1, the hypotenuse and the angle F are known. Solve for the two sides. Whence, In triangle 2, the two sides can be obtained by arithmetic. Solve for the hypotenuse and the angle a. 114 TEXT BOOK ON RAILROAD SURVEYING FIG. 59. In triangle 3, the hypotenuse and the angle 7 are known. Solve for the two sides. In triangle 4, the altitude = Ri + \Q and the angle F\ are known. Solve for the hypotenuse and the base. In triangle 5, the hypotenuse and the angle FI are known. Solve for the sides I and m. Whence, In triangle 6, the altitude is found by arithmetic and the hypotenuse is known from triangle 2. Solve for the angle b. Then x = a - b + (90 - I) - (90 - F) and = a -b - I + F (curve) AB (in feet) (60) 100 D The P.P. is set at the distance AB - P from A. From triangles 3, 5, and 6, the distances AE, m, and OG are known. Then AC is found by arithmetic. TURNOUTS, CONNECTIONS, AND CROSSINGS 115 The P.F.i is set at a distance AC PI from A. The central angle of the connecting curve z = I -\- x F\ F. Obviously such a connection can be placed in any of the four quadrants, but the one here shown gives the best alinement and should be used when possible. In case a connection is to be placed in any other quadrant, the plan of solution is identical with that above, and if followed exactly no difficulty should arise. 132. Case 3. Both Tracks Curved. From Fig. 60, In triangle 1 (AOOi], the two sides, R and Ri, and the included angle / are known. Solve for the side 00 i and the angles a and 6. In triangle 2 (BEO), the hypotenuse and angle F are known. Solve for the sides BE and EO. Then in triangle 3 (OOaG), the two sides are found by arith- metic. Solve for the angle c and the side 00 2 . 116 TEXT BOOK ON RAILROAD SURVEYING In triangle 4 (CO\H) the hypotenuse and .the angle FI are known. Solve for the sides, CH and HOi. Then in triangle 5 (Oi0 2 /), the two sides are found by arith- metic. Solve for the angle d and the side OiOz. In triangle 6 (00i0 2 ), the three sides are now known. Solve for the angles e and /. Then x = b - f + c - (90 - F) = b-f + c + F-90 . . . . (61) and y = a - e + d + F! - 90 . . . (62) curve AB (in ft.) = ~ , and the'P.F. is set at a distance AB - P from A. curve AC (in ft.) = FT* an( ^ the P-f-i is se t at the dis- tance AC PI from A. Then the central angle of the connecting curve is z=I+x + y-F-Fi . . . (63) Obviously such a connection can be placed in any of the four quadrants, but the one here shown gives the best alinement and should be used when possible. In case a connection is placed in any other quadrant the plan of solution is identical with that above, and if followed exactly no difficulty should arise. 3. Crossings 133. A crossing is a device used at the intersection of two tracks. It consists of four frogs and the necessary connecting rails. Any one of the frogs is a crossing frog. The crossing angle is the angle between the center lines of the tracks at their point of intersection. Crossings are designated as single curve, double curve, or straight according as one, both, or neither of the tracks are curved. Crossings are usually made of rolled rails fitted together. When the crossing angle is greater than about 25, the various pieces are cut to fit against each other and are united by filling blocks and heavy straps well bolted. This is frequently termed TURNOUTS, CONNECTIONS, AND CROSSINGS 117 solid construction. For angles under about 25, regular frog point construction is used, and such crossings are termed frog crossings. Distinguish between frog crossing and crossing frog. On street railway work hard steel (manganese) centers for frogs have been used for many years, and since 1905 such con- struction has been growing in use on steam roads both for standard frogs and for crossings. (This type is usually designat- ed as " manganese frogs or crossings.") There have been numerous attempts to construct crossings with a revolving section or turntable at each frog so that the wheels will have no flange way to jump; but only one of these seems to give much promise of success. The end frogs of a frog crossing are similar to a standard rigid frog in that there is a single point on which the wheels run. The middle frogs, however, have two running points and are therefore frequently termed " double-pointed frogs." FIG. 61. Double Slip Switch. When " slip switches " are used, the crossing is made to a standard frog number, and if located at an interlocking plant the middle frogs are frequently made with movable points. That is, with movable points joined in pairs and moving together similar to a split switch in such a way that the wheels have a solid bearing and no flangeway to jump. A " slip switch " or " combination crossing " is a combination of a small angle crossing with a pair of connecting tracks placed entirely within the limits of the crossing. They are used in large yards and terminals (Fig. 61). 134. Crossing Data. Very few railroads construct their own crossings, but have them built by manufacturers who make a specialty of such work. The field engineer is rarely called on to compute the dimensions of a crossing, and to do so is a waste of time if the crossing is ordered from a manufacturer. 118 TEXT BOOK ON RAILROAD SURVEYING It is far more important that the makers have all the data, and the field engineer is frequently required to furnish the data. The information required is: 1. The crossing angle. 2. The gage of each track. 3. The curvature degree of curve, radii, or the equivalent. 4. The direction of curvature. 5. The length along each gage line from one gage line in- tersection (theoretical P.P.) to the nearest rail joint. 6. Length over all along each gage line. 7. The height, weight, and style of rail of which the crossing is to be made. 8. The height, weight, and style of rail in intersecting track if offset splices are to be furnished. 9. The spacing and size of holes for splice bars. 10. The type of crossing, etc., unless covered by general specifications. This information can best be given by means of a small sketch. Field dimensions should be taken to the nearest 1/8" (0.01 ft.). Occasionally the field engineer is called on to compute the di- mensions of a crossing. The values which are required are the frog angles, FI, F 2 , F 3 , F^ the length of sides along the gage lines, and the two diagonals. The computations should be made with sufficient accuracy to give results which are correct to the nearest 1/16", which is the working limit of the manufacturers. The solutions of the three kinds of crossings are outlined as follows : 135. Straight Crossings. FIG. 62. (a) Unequal Gages. In Fig. 62, g it g 2 , and I are given, and the sides and diagonals are required. TURNOUTS, CONNECTIONS, AND CROSSINGS 119 In triangle 1, the altitude gi and the angle 7 are known. Solve for si and 61. Similarly solve for $2 and 6 2 in triangle 2. In triangle 3, the altitude gi and the base s 2 + 61 are known. Solve for the hypotenuse, which is the long diagonal of the crossing. In triangle 4, the altitude 2 and the base si b 2 are known. Solve for the hypotenuse, which is the short diagonal of the crossing. (6) Equal Gages. In this case, triangles 1 and 2 are identical and all four sides are equal. The angle between the hypotenuse and the base of triangle 3 becomes J7, and of triangle 4 becomes 90 ^7 which simplifies the solution. 136. Single-curve Crossings. Given, in Fig. 63, 0i, 2 , and 7 to find the frog angles, length of sides, and both diagonals. FIG. 63 Project the radius on a perpendicular to the straight track. Then OB = R cos 7, whence OE and OC are known. In triangle FiCO, the base and hypotenuse are known. Solve for FiC and the angle FI. Similarly solve for F 2 and F Z C, F s and F S E, and F 4 and F*E from corresponding triangles. Then the straight sides F\F* and F z Fi can be found by subtraction. The curved sides FiFt and F 2 F 3 can be determined since their radii are known and their central angles can be found by subtraction. 120 TEXT BOOK ON RAILROAD SURVEYING The long diagonal is the hypotenuse of a triangle whose altitude is g\ and whose base is F$E FiC, and the short diagonal is the hypotenuse of a triangle whose altitude is g\ and whose base is F 2 C - F 4 E. 137. Double-curve Crossings. In Fig. 64, #1, R 2 , g l} g 2) and / are given, and it is required to find the frog angles, length of sides, and both diagonals. In triangle A0i0 2 , two sides and the included angle are known. Solve for the distance between centers, 0\0 2 . Then in the four triangles formed on Oi0 2 by the radii from Fi, F 2 , F S} and F 4 the three sides of each are known, and the o, FIG. 64. angles can be solved for. From the angles at 0\ and 2 determine the central angles of the arcs, FiF 2 , F 2 F 3 , F S F 4 , and F 4 F lt from which the lengths of the arcs can be computed since their radii are known. The long diagonal is found from the triangles or Fi0 2 Fz, and the short diagonal from the triangles or APPENDIX . THEORY OF SPIRAL 138. Derivation of o and A. From definition, d = ks] fortheS.C., D = kS . . . . (18) The radius of curvature, r = -r^ do Substituting for r its value, -j- , and then for d its value from Eq. 18, and solving, ad - kldl 5730 kl 2 ks 2 Integrating, d = in which o is in circular measure. Changing d to degrees, d = iks 2 ........ (19) For the S.C., A = JkS 2 139. Derivation of the Co-ordinates x and >. The differential equations for the co-ordinates of a spiral are: dx = v ds cos d dy = ds sin d Expanding the sin and cos into series, / fc 6 4 dx - ds 1 - " + T ~ /* <* 3 A 5 \ dy = ds (d - IT + IF - etc -| ? E. 12- ' ks 2 Substituting TTT-^ for d 114.6 J / k 2 s 4 dx = ds ^1 ; 121 122 TEXT BOOK ON RAILROAD SURVEYING . / ks 2 k 3 s 6 ^ dy = ds \U4f> ~ 6(114:6? ~ k 2 s 5 Integrating, x = s - T^-TJ-^ + etc. ks 3 k 3 s 7 "' 3(114.6) 42(114.6) 3 * in which x and y are in stations. Reducing to feet and dropping small terms, x = 100 s - 0.000762 k 2 s 5 . . . (23) y = 0.291 ks 3 - 0.00000158 k 3 s 7 . (22) 140. Derivation of a, A, b, and B. ks 3 . kV x 3X114.6 42X(114.6) 3 + etc ' tan a = = - V ^ 2o 5 + etc. f etc. 10 X (114.6) 2 ks 2 k 3 s 6 Substituting d for 3 X H4.6 (114.6) 3 X 105 ks 2 114.6 t I A 6 Is- tan-^- = + "gT" ^ 3 + etc. (tangent series). 1 24 Whence, tan a = tan -- o O z Reducing to degrees, tan a = tan -j d - 0.000000015 o 3 Investigating the second term, it is found that o must be about 25 before there is an error of one minute in a by considering it as ^ $; and since rarely reaches 25 and any error of direction is corrected at the S.C., it is sufficient to write a = -^- d = lOks 2 (in minutes) .... (20) A = -^-A = 10kS 2 (in minutes) <> J THEORY OF SPIRAL 123 and, b = y 3 = 2a, and B = J- A = 2A . . (21) 141. Derivation of O. From Fig. 65, o = Y - R vers A Substituting for Y its value, 0.291- kS 3 , for R its value, -j-g-, and expanding the vers series, o = 0.291 kS 3 - 0.00000158 k 3 S 7 - 5730 / k 2 S 4 k 4 S 8 \ 730 /_ kS \2 kS \2 X (114.6) 2 24(114.6) 4 / o = 0.291 kS 3 - 0.00000158 k 3 S 7 - 5730 / kS 3 k 3 S 7 \ \2(114.6) 2 24(114.6)V Reducing, o = 0.0727 kS 3 - 0.0000002 k 3 S 7 Evidently the second term is very small, hence, o = 0.0727 kS 3 (24) From figure, t = X - R sin A Substituting for X its value from Eq. 23, and for R its value, 5730 , ,. ., . ~r^-j and expanding the sin series, t = 100 s- 0.000762 k 2 S 5 - ^~ /A - ^ + etc. ) - 1008 - 0.000762 kS- - Reducing, t = 50 S - 0.000127 k 2 S 5 ..... (25) 124 TEXT BOOK ON RAILROAD SURVEYING Substituting J S in Eq. 23, the abscissa of the middle of the spiral is 50S - 0.000024 k 2 S 5 . The difference between this and Eq. 25 is only 0.0001 k 2 S 5 , showing that the spiral is practically bisected by the ordinate to the T.C. FIG. 65. Substituting J$ in Eq. 24, the ordinate for the middle of the spiral is 0.0364 kS 3 . Jo = 0.0363 kS 3 showing that o is bisected by the spiral. If the second term of Eq. 24 had been used the difference would have been still less. 142. Osculating Circle. The fact that o and the spiral mutually bisect shows that the deflection of the circle from the spiral between the S.C. and the T.C. is the same as the deflection of the spiral from the tangent in half its length. Since the S.C. can be at any point on a spiral of given k, and the circular curve has the same radius as the spiral at that point, it follows that in general the spiral departs from an osculating circle at the same rate as from the initial tangent. 143. Superelevation. From mechanics, F - Wv 2 2gR THEORY OF SPIRAL 125 where F is the centrifugal force, W is the weight, v is the velocity in feet per second, R is the radius of curvature, and g is the ac- celeration of gravity (equal to 16.08). In Fig. 66, AB represents the horizontal distance between rail heads, which for standard gage is about 4.9 ft.; CB, the FIG. 66. superelevation; EH, the weight W', and GE, the centrifugal f orce. Then from similar triangles, Substituting the above value of F in this equation, we have Wv 2 _ We 2gR ~ AB whence, e (in feet) = ABv 2 _ 4.9 v 2 2gR ~ 32.16 R 5730, Reducing v to miles per hour, substituting g- for R, and reducing, e (in inches) = 0.00069 D V 2 .... (28) TABLES 128 TABLE 5. RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 0' Infinite 1 0' 5729.65 2 0/ 2864.93 3 0' 1910.08 4 0' 1432.69 1 313:75. 3 5635.72 1 2841.26 1 i 1899.53 1 1426.74 2 171887. 2 5544.83 2 2817.97 2 1889.09 2 142(K85 3 114595. 3 5456.82 3 2795.06 3 1878.77 3 1415^01 4 85943.7 4 5371.56 4 2772.53 4 1868.56 4 1409.21 5 68754.9 5 5388.92 5 2750.35 5 1858.47 5 1403^46 6 57595.8 6 5208.79 6 2728 52 6 1848.48 6 1397176 7 49110.7 7 5131.05 7 2707.04 7 1838.59 7 1392.10 8 42971.8 8 5055.59 8 2685.89 8 1828.82 8 1386.49 9 38197.2 9 4982.33 9 2665.08 9 1819.14 9 1380 92 10 34377.5 10 4911.15 10 2644.58 10 1809.57 10 1375.40 11 31252.3 11 4841.98 11 2624.39 11 1800.10 11 1369.92 12 28G47.8 12 4774.74 12 2604.51 12 1790.73 12 1364.49 13 26444.2 13 4709.33 13 2584.93 13 1781.45 13 1359.10 14 24555.4 14 4645.69 14 2565.65 14 1772.27 14 1:353.75 15 22918.3 15 4583.75 15 2546.64 15 1763.18 15 1348.45 16 21485.9 16 4523.44 16 2527.92 16 1754.19 16 1343.15 17 20222.1 17 4464.70 17 2509.47 17 1745.26 17 1337.65 18 19098.6 18 4407.46 18 2491.29 18 1736.48 18 1332.77 19 18093.4 19 4351.67 19 2473.37 19 1727.75 19 1327 63 20 17188.8 20 4297.28 20 2455.70 20 1719.12 20 1322! 53 21 16370.2 21 4244.23 21 2438.29 21 1710.56 21 1317.46 2:2 15626.1 22 4192.47 22 2421.12 22 1702.10 22 1312.43 23 14946.7 23 4141.96 23 2404.19 23 1693.72 23 1307.45 24 14323.6 24 4092.66 24 2387.50 24 1685.42 24 1302 50 25 13751.0 25 4044.51 25 2371.04 25 1677.20 25 1297.58 26 13222.1 26 3997.49 26 2354.80 26 1669.06 26 1292 71 27 12732.4' 27 3951.54 27 2338.78 27 1661.00 27 1287.87 28 12277.7 28 3906.54 28 2322.98 28 1653.01 28 1283.07 29 11854.3 29 3862.74 29 2307.39 29 1645.11 29 1278 30 30 11459.2 30 3819.83 30 2292.01 30 1637.28 30 1273.57 31 11089.6 31 3777.85 31 2276.84 31 1629.52 31 1268.87 32 10743 32 3736.79 32 2261.86 32 1621.84 32 1264.21 33 10417.5 33 3696.61 33 2247.08 33 1614.22 33 1259^58 34 10111.1 34 3657.29 34 2232.49 34 1606.68 34 1254.98 35 9822.18 35 3618.80 35 2218.09 35 1599.21 35 1250.42 36 9549.311 36 3581.10 36 2203.87 36 1591.81 36 1245.89 37 9291.29 37 3544.19 37 2189.84 37 1584.48 37 1241.40 38 9046.75 38 3508.02 38 2175.98 38 1577.21 38 1236 94 39 8814.78 39 3472.59 39 2162.30 39 1570.01 39 1232.51 40 8594.42 40 3437.87 40 2148.79 40 1562.88 40 1228.11 41 8384.80 41 3403.83 41 2135.44 41 1555.81 41 1223 74 42 8185.16 42 3370.46 42 2122.26 42 1548.80 42 1219.40 43 7994.81 43 3337.74 43 2109.24 43 1541.86 43 1215 30 44 7813.11 44 3305.65 44 2096.39 44 1534.98 44 1210.82 45 7639.49 45 3274.17 45 2083.68 45 1528.16 45 1206.57 46 7473.42 46 3243.29 46 2071.13 46 1521.40 46 1202.36 47 7314.41 47 3212.98 47 2058.73 47 1514.70 47 1198.17 48 7162.03 48 31&3.23 48 2046.48 48 1508.06 48 1194.01 49 7015.8? 49 3154.03 49 2034.37 49 1501.48 49 1189 88 50 6875.55 50 3125.36 50 2022.41 50 1494.95 50 1185.78 61 6740.74 51 3097.20 51 2010.59 51 1488.48 51 1181.71 62 6611.12 52 3069.55 52 1998.90 52 1482.07 52 1177.66 53 6486.38 53 3042.39 53 1987.35 53 1475.71 53 1173.65 54 6366.26 54 3015.71 54 1975.93 54 1469.41 54 1169 66 55 6250.51 55 2989.48 55 1964.64 55 1463.16 55 1165.70 56 6138.90 56 2963.71 56 1953.48 56 1456.96 56 1161.76 57 6031.20 57 2938.39 57 1942.44 57 1450.81 57 1157.85 58 5927.22 58 2913.49 58 1931.53 58 1444.72 58 1153.97 59 5826.76 59 2889.01 59 1920.75 59 1438.68 59 1150 11 60 5729.65 60 2864.93 60 1910.08 60 1432.69 60 1146.28 TABLE 5. RADII. 129 1 Deg Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 60 1146.28 60 955.37 70 818. G4 80' 716.34 90 / G36.78 I 1142.47 1 952.72 1 816.70 1 714.85 1 635.61 2 1138.69 2 950.09 2 814.76 2 713.37 2 634.44 3 1134.94 3 947.48 3 812.83 3 711.90 8 633.27 4 1131.21 4 944 . 83 4 810.92 4 710.43 4 632.10 5 1127.50 5 942.29 5 809.01 5 708.96 5 630.94 6 1123.82 6 939.72 6 807.11 6 707.51 6 629.79 7 1120.16 7 937.16 7 805.22 7 706.05 7 628.64 8 1116.52 8 934.62 8 803.34 8 704.60 8 627.49 9 1112.91 9 932.09 9 801.47 9 703.16 9 626. 35 10 1109.33 10 929.57 10 799.61 10 701.73 10 625.21 11 1105.76 11 927.07 11 797.75 11 700.30 11 624.08 12 1102.22 12 924.58 12 795.91 12 G98.88 12 622.95 13 1098.70 13 922.10 13 794.07 13 697.46 13 621.82 14 1095.20 14 919.64 14 79^.24 14 696.05 14 620.70 15 1091.73 15 917.19 15 790.42 15 694.65 15 619.58 16 1088.28 16 914.75 16 788.61 16 693.24 16 618.47 17 1084.85 17 912.33 17 786.80 17 691.85 17 617.30 18 1081.44 18 909.92 18 785.01 18 690.46 18 616.25 19 107.05 19 907.52 19 783.22 19 689.08 19 615 15 20 1074.68 20 905.13 20 781.44 20 687.70 20 614.05 21 1071.34 21 902.76 21 779.67 21 686.33 21 612.96 22 1068.01 22 900 .40 22 777.91 22 684.96 22 611.87 23 1064.71 20 898.05 23 776.15 23 G83.60 23 610.78 24 1061.43 24 895.71 24 774.40 24 682.25 24- 609.70 25 1058.16 25 893.39 2:> 772.66 25 680.89 25 608.62 26 1054.92 26 891.08 26 770.93 26 679.55 26 607.55 27 1051.70 27 '888.78 27 769.21 27 678.21 27 606.48 28 1048.48 28 886.49 28 767.49 28 676.88 28 605.41 29 1045.31 29 884.21 29 765.78 29 675.54 29 604.35 30 1042.14 30 881.95 30 764.08 30 674.22 30 603.29 31 1039.00 31 879.69 31 762.39 31 672.90 31 602.23 32 1035.87 32 877.45 32 700.70 32 671.59 32 601.18 33 1032.76 33 875.22 33 759.02 33 670.28 33 600.13 34 1029.67 34 873.00 34 757.35 34 668.98 34 599.09 35 1026.60 85 870.80 35 755.69 35 607.68 35 598.04 36 1023.55 36 868.60 36 754.03 36 666.39 36 597.01 37 1020.51 37 866.41 37 752.38 37 665.10 37 595.97 38 1017.49 38 864.24 38 750.74 38 603.82 38 594.94 39 1014.50 39 862.08 39 749.10 39 662.54 39 593.91 40 1011.51 40 859.92 40 747.48 40 661.26 40 592.89 41 1008.55 41 857.78 41 745.86 41 659.99 41 591.87 42 1005.60 42 855.65 42 744.24 42 658.73 42 590.85 43 1002.67 43 853.53 43 742.63 43 657.47 43 589.84 44 999.76 44 851.42 44 741.03 44 656.22 44 588.83 45 996.87 45 849.32 45 739.44 45 654.97 45 587.83 46 993.99 46 847.23 46 737.80 46 653.72 1 46 586.82 47 991.13 47 845.15 47 736.28 47 652.48 47 585.83 48 988.28 48 843.08 48 734.70 48 651.25 48 584.83 49 985.45 49 841.02 49 733.14 49 650.02 49 583.84 50 982.64 50 838.97 50 731.53 50 648.79 50 582.85 51 979.84 51 836.93 51 730.03 51 647.57 51 581.86 52 977.06 52 834.90 52 728.48 52 646.35 52 580.88 53 974.29 53 832.89 53 726.94 53 645.14 53 579.90 54 971.54 54 830.88 54 725.41 54 643.94 54 578.92 55 968.81 55 828.88 55 723.88 55 642.73 55 577.95 56 966.09 56 826.89 56 722.36 56 641.53 56 576.98 57 963.39 57 824.91 57 720.85 57 640.34 57 576.02 58 960.70 58 822.93 58 719.34 58 639.15 58 575.06 59 958.03 59 820.97 59 717.84 59 637.96 59 574.10 60 955.37 60 819.02 60 716.34 60 636.78 60 573.14 130 TABLE 5. RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 10 0' 573.14 120' 477.68 14 0' 409.32 16 0' 358.17 18 0' 318.39 2 571.24 2 476.36 2 408.35 2 357.43 2 317.80 4 569.35 4 475.05 4 407.38 4 356.69 4 317.22 6 567.47 6 473.74 6 406.42 6 355.95 6 316.63 8 565.60 8 472.44 8 405.46 8 355.21 8 316.05 10 563.75 10 471.15 10 404.51 10 354.48 10 315.47 12 561.91 12 469.86 12 403.56 12 353.75 12 314.89 14 560.08 14 468.58 14 402.61 14 353.03 14 314.32 16 558.26 16 467.31 16 401.67 16 352.30 16 313.75 18 556.45 18 466.04 18 400.74 18 351.58 18 313.18 20 554.66 20 464.78 20 399.80 20 350.86 20 312.61 22 552.88 22 463.53 22 398.88 22 350.15 22 312.04 24 551.11 24 462.29 24 397.95 24 349.44 24 311.47 26 549.35 26 461.05 26 397.03 26 348.72 26 310.91 28 547.60 28 459.82 28 396.13 28 348.02 28 310.35 30 545.87 30 458.59 30 395.21 30 347.32 30 309.79 32 544.14 32 457.88 32 394.30 32 346.62 32 309.23 34 542.42 34 456.16 34 393.40 84 345.93 34 308.68 36 540.72 36 454.96 36 392.50 36 345.23 36 808.13 38 539.03 38 453.76 38 391.61 38 344.54 38 307.58 40 537.34 40 452.57 40 390.72 40 343.85 40 307.03 42 535 67 42 451.38 42 389.83 42 343.16 42 306.48 44 534.01 44 450.20 44 388.95 44 342.48 44 305.93 46 532.36 46 449.02 46 388.07 46 341.80 46 305.39 48 530.71 48 447.86 48 387.20 48 841.12 48 304.85 50 529.08 50 446.69 50 386.33 50 340.45 50 304.31 52 527.46 52 445.54 52 385.47 52 339.78 52 303.77 54 525.85 54 444.39 54 384.60 54 339.11 54 303.24 56 524.25 56 443.24 56 383.75 56 338.44 56 302.70 58 522.65 58 442.11 58 382.89 58 337.77 58 302.17 11 0' 521.07 18 0' 440.97 16 0' 382.04 170' 337.11 19 0' 301.64 2 519.50 2 439.85 2 381.19 2 336.45 2 301.12 4 517.93 4 438.73 4 380.35 4 335.80 4 800.59 6 516.38 6 437.61 6 379.51 6 335.14 6 800.07 8 514.84 8 436.50 8 378.68 8 334.49 8 299. E4 10 513.30 10 435.40 10 377.84 10 333.84 10 299.02 12 511.77 12 434.30 12 377.0:2 12 333.19 12 298.50 14 510.26 14 433.21 14 376.19 14 332.55 14 297.99 16 508.75 16 432.12 16 375.37 16 331.91 16 297.47 18 507.25 18 431.04 18 374.55 18 331. 27 18 296.96 20 505.76 20 429.96 20 373.74 20 330.63 20 296.45 22 504.28 22 428.98 22 372.93 22 330.00 22 295.94 24 502.80 24 427.82 24 372.12 24 329.37 24 295.43 26 501.34 26 426.76 26 371.32 26 328.74 26 294.92 28 499.88 28 425.71 28 370.52 28 328.11 28 294 42 30 498.43 30 424.66 30 369.72 30 327.48 30 293.91 32 496.99 32 423.61 32 368.93 32 326.86 32 293.41 34 495.56 34 422.57 34 368.14 34 326.24 34 292.91 36 494.14 36 421.54 36 367.35 3G 325.62 36 292.41 38 492.73 38 420.51 38 306.57 38 325.01 38 291.92 40 491.32 40 419.49 40 365.79 40 324.40 40 291.42 42 489.92 42 418.47 42 365.01 42 323.79 42 290.93 44 488.53 44 417.45 44 364.24 44 323.18 44 290.44 46 487.15 46 416.44 46 863.47 46 322.57 40 289.95 48 485.77 48 415.44 48 362.70 48 321.97 48 289.46 50 484.40 50 414.44 50 361.94 50 321.37 50 288.98 52 483.05 52 413.44 52 361.18 52 320.77 52 288.49 54 481.69 54 412.45 54 360.42 54 320.17 54 288.01 56 430.35 56 411.47 56 359.67 56 319.57 56 287.53 58 479.01 58 410.49 58 358.9-3 58 318.98 58 287.05 60 477.68 60 409.51 60 358.17 60 318.39 60 28(5.57 131 TABLE 6. FUNCTIONS OF A ONE-DEGREE CURVE. The Long Chords, Mid-Ordinates, Externals, and Tangent Distances of this table are for a curve of 5730 feet radius. To find the corresponding functions of any other curve divide the tabular values by the degree of curve. For metric curves having 20-metre chords, multiply tne degree by 5 and enter the table with the result as a value of Z>, the tabu- lar values being taken as metres instead of feet Thus for a 1 30' metric curve having I 45 the tangent dis- tance is T = - ' = 316.45 metres. Again, suppose J= 38 1 . o X 5 and the long chord = 373.1 m. known and D required. The tabular L. C. is 3731 m.; therefore D = 3731 - = 2 > 3.3.1 X 5 1 L. C. M. E. T. L. C. M. E. T. 0.00 0.000 0.000 0.00 100.00 0.218 0.218 50.00 2 3.33 0000 0.000 1.67 103.33 0.233 0.233 51.67 2 4 6.6? 001 0.001 3.33 106.66 0.248 0.248 53.33 4 6 10.00 0.002 0.002 5.00 110.00 0.264 0.264 55.00 6 8 13.33 0.004 0.004 6.67 113.33 0.280 0.280 56.67 8 10 16.67 0.006 0.006 8.33 116.66 0.297 0.297 58.33 10 12 20.00 0.009 0.009 10.00 120.00 0.314 0.314 60.00 12 14 23.33 0.012 0.012 11.67 123.33 0.332 0.332 61.67 14 16 26.67 0.015 0.015 13.33 126.66 0.350 0.350 63.33 16 18 30.00 0.019 0.019 15.00 130.00 0.368 0.368 65.00 18 20 33.33 0.024 0.024 16.67 133.33 0.388 0.388 66.67 20 22 36.67 0.029 0.029 18.33 136.66 0.407 0.407 68.33 22 24 40.00 0.035 0.035- 20.00 140.00 0.427 0.427 l ~0.00 24 26 43.33 0.041 0.041 21.67 143.33 0.448 0.448 "1.67 26 28 46.67 0.048 0.048 23.33 146.66 0.469 0.469 ^3.33 28 30 50.00 0.054 0.054 25.00 15000 0.491 0.491 "5.00 30 32 53 33 0.062 0.062 26.67 153.33 0.513 0.513 ^6.67 32 34 56 67 0.070 0.070 28.33 156.66 0.536 0.536 78.33 34 36 6000 0.079 0.079 30.00 160.00 0.559 0.559 80.00 36 38 63.33 0.088 0.088 31.67 163.33 0.582 0.582 81.67 38 40 66.67 0.097 0.097 33 33 16666 0.606 0.606 83.33 40 42 70.00 0.107 0.107 35.00 170.00 0.630 0.630 85.00 42 44 73.33 0.117 0.117 36.67 173.33 0.655 0.655 86.67 44 46 76.6? 0.128 0.128 38.33 176.66 0.681 0.681 88.33 46 48 80.00 0.140 0.140 40.00 180.00 0.706 0.706 90.00 48 50 83.33 0.151 0.151 41.67 183.33 0733 0.733 91.67 50 52 86 67 0.164 0.164 43.33 186.66 0.760 0.760 93.33 52 54 9000 176 O.J76 45.00 190.00 0.788 0.788 95.00 54 56 93 33 0.190 0.190 46.6? 193.33 0.815 0.815 96.67 56 58 96.67 0.204 0.204 4833 196.66 0.844 0.844 98.33 58 tJB 10000 0.218 0.218 5000 199.98 0.873 873 100.00 60 132 6. FUNCTIONS OF A ONE-DEGREE CURVE. 2 3 4 L. C. M. E. T. L. C. M. E. T. r "199.98 0.873 0.873 100.00 299.96 1.964 1.964 150.07 2 203.31 0.902 0.902 101.67 3U3.29 2.008 2.009 151.74 2 4 206.64 932 0.932 103.34 306.62 2.053 2.054 153.41 4 6 209.97 0.962 0.962 105.01 309.95 2.098 2.099 155.08 6 8 213.31 0.993 0.993 106.68 313.29 2.143 2.144 156.75 8 10 216.64 1.024 1.024 108.35 316.62 2.188 2.189 158.42 10 12 219.97 1.056 1.056 110.02 319.95 2.235 2.236 160.09 12 14 223.30 1.088 1.088 111.69 323.28 2.282 2.283 161.76 14 16 226.64 1.121 1.121 113.36 326.62 2.329 2.330 163.43 16 18 229.97 1.154 1.154 115.02 329.95 2.376 2.377 165.09 18 20 233.30 1.188 1.188 116.69 333.28 T.424 2.425 166.76 20 22 236.63 1.222 1.222 118.36 336.61 2.473 2.474 168.43 22 24 239.97 1.256 1.256 120.03 339.95 2.523 2.523 170.10 24 26 243.30 1.292 1.292 121.70 343.28 2.572 2.573 171.77 26 28 246.63 1.328 1.328 123.37 346.61 2.622 2.623 173 44 28 30 249.96 1.364 1.364 125.03 349.94 2.672 2.673 175.10 30 32 253.29 1.399 1.399 126.70 353.27 2.724 2.725 176.72 32 34 256.62 1.437 1.437 128.37 356.60 2.776 2.777 178.39 34 36 259.96 1.475 1.475 130.04 359.94 2.828 2.829 18006 36 38 263.29 1.513 1.513 131.71 363.27 2.880 2.881 181.73 38 40 266.62 1.552 1.552 133.38 366 60 2 933 2.934 183.40 40 42 269.96 1.592 1.592 135.05 369.94 2.987 2.988 185.07 42 44 273.29 1.632 1.632 136.72 373.27 3.042 3 043 186.74 44 46 276.62 1.672 1.672 138.38 376.60 3.096 3097 188.40 46 43 279.96 1.712 1.712 140.05 379.94 3.151 3.152 190.07 48 50 283.29 1.752 1.752 141.72 383.27 3.206 3.207 191.74 50 52 286.62 1.794 1.794 143.39 38660 3.263 3.264 193.41 52 54 289.96 1.836 1.836 145.06 389.94 3.320 3.321 195.08 54 56 293.29 1.878 1.878 146.73 393.27 3.377 3378 196.75 56 58 296.62 1.921 1.921 148.40 396.60 3.434 3.435 198.42 58 60 299.96 1.964 1.964 150.07 399.94 3.491 3.492 2uO 09 60 4 6 L. C. M. E. T. L. C. M. E. T. 399.94 3.491 3.492 200.09 499.88 5.454 5.459 250.17 2 403.27 3.550 3.551 201.70 503.21 5.527 5.o33 251.84 2 4 406.60 3.609 3.610 203.43 506.54 5.601 5.607 253.51 4 6 409.93 3.668 3.670 205.10 509.87 5.675 5.681 255.18 6 8 413.26 3.727 3.730 206.77 513.20 5.749 5.755 256.85 8 10 416.59 3.787 3.790 208.44 51653 5.823 5.829 258.52 10 12 419.92 3.848 3.851 210.11 519.86 5.899 5.905 260.20 12 14 423.26 3.910 3.913 211.77 523.19 5.975 5981 261.86 14 16 426.59 3.972 3.975 213.45 526.52 6.052 6.058 263.54 16 18 429.92 4.034 4.037 215.11 529.85 6.129 6.135 265.20 18 20 433.25 4.096 4.099 216.78 533.18 6.206 6.212 266.87 20 22 436.58 4.160 4.163 218.45 536.51 6.284 6.290 208 54 22 24 439.91 4.224 4.227 220.12 539. 81 6.362 6.369 270.21 ' 24 26 443.24 4.288 4.291 221.79 543.17 6.441 6.448 271.88 26 28 446.58 4.353 4.356 223.46 546.50 6.520 6.527 273.54 28 30 449.91 4.418 4.421 225.13 549.83 6.599 6.606 275.21 30 32 453.24 4.484 4.487 226.80 553 17 6.680 6.687 276.88 32 34 456.57 4.550 4.554 228.47 556.50 6.761 6.768 278.55 34 36 459.90 4.617 4.621 230.14 559.83 6.842 6.849 280.23 36 38 463.23 4.684 4.688 231.81 563.16 6.923 6.931 281.90 38 40 466.56 4.751 4.755 233.48 566.49 7.005 7.013 283.57 40 42 469.89 4.820 4.824 235.15 569.82 7.088 7.096 285.24 42 44 473.23 4.889 0.893 236.82 573.15 7 171 7.180 286.91 44 46 47656 4.958 4962 238.48 576.48 7.255 7.264 288.59 46 48 479.89 5.027 5.031 240.15 579.81 7.339 7.348 290.26 48 50 483.22 5.096 5100 241.82 583.14 7423 7.432 291.93 50 52 486.55 5.167 5.171 243.49 586.47 7.508 7.517 293.60 52 54 489.88 5.238 5.243 245.16 589.80 7.593 7.603 295.27 54 56 493.21 5.310 5.315 246.83 593.13 7.678 7 689 296 95 56 58 49654 5.382 5.387 248.50 596.46 7.764 7.775 298.62 58 60 499.88 5 454 5.459 250.17 599.80 7.850 7 861 300.30 60 6 FUNCTIONS OF A ONE-DEGREE CURVE. 133 i 6 7 L C. M. E T. L. C. M. E. T. 599.80 7.850 7.861 300.30 699.60 10.69 10.71 350.44 2 603.13 7.940 7.951 301.97 702.93 10.79 10.81 352.11 2 4 606 46 8.030 8.041 303.64 706.26 10.90 10.92 353.79 4 6 609.78 8.120 8.131 305. 31 709.58 11.00 11.02 355.46 6 8 613.11 8.210 8. 2:il 306.98 712.91 11.11 11.13 357.13 8 10 616 44 8.300 8.311 308.65 716.24 11.21 11.23 358.81 10 12 619.76 8.390 8.401 310.32 719.56 11.31 11.33 360.48 12 14 623.09 8.480 8,491 311.99 722.89 11.42 11.44 362.15 14 16 626.42 8.570 8.581 313.66 726.21 11.52 11.54 363 83 16 13 629.74 8.660 8.671 315.33 729.53 11.63 11.65 365.50 18 20 633.07 8.750 8.761 317.00 732.86 J1.73 11.75 367.17 20 '22 63(5.40 8.844 8.856 318.67 736.19 11.84 11.86 368.85 22 24 639.72 8.939 8.951 320.34 739.51 11.95 11.97 370.52 24 26 643.05 9.033 9.046 322.01 742.84 12.06 12.08 372.19 26 28 646.38 9.128 9.141 323.68 746.17 12.17 12.19 373.86 28 30 649.70 9.222 9.236 325.35 749.49 12.27 12.30 375.54 30 32 653.03 9.317 9.331 327.02 752.82 12.38 12.41 377.22 32 34 656.36 9.411 9.426 328.69 756.15 12.49 1252 378.89 34 36 659.69 9.506 9.521 330.37 759.47 12.60 12.63 380 57 36 38 663.02 9.600 9.616 332.04 762.80 12.71 12.74 382.24 38 40 666.34 9.695 9.712 333.71 766.13 12.82 12.85 383.92 40 42 669.67 9.794 9.812 335.38 769.45 12.93 12.96 385.60 42 44 673.00 9.894 9.912 337.05 772.78 13.04 13.08 387.27 44 46 676.32 9.993 10.01 338.73 776.11 13.15 13.19 388.95 46 48 679.65 10.09 10.11 340.40 779.43 13.26 13.31 390.62 48 50 682.98 10.19 10.21 342.07 782.76 13.37 13.42 392.30 50 52 686.30 10.29 10.31 343.74 786.09 13.48 13.53 393.98 52 54 689.63 10.39 10.41 345.41 789.41 13.59 13.65 395.65 54 56 692.96 10.49 10.51 347.08 792.74 13.70 13.76 397.33 56 58 696.28 10.59 10.61 348.76 796.07 13.81 13.88 399.01 58 60 699 60 10.69 10.71 850.44 799.40 13.96 13.99 400.70 60 8 9 L. C. M. E. T. L. C. M. E. T. / 799.40 13.96 13.99 400.70 899.10 17.66 17.71 450.95 2 802.72 14.07 14.10 402.37 902.42 17.79 17.84 452.63 2 4 806.04 14.19 14.22 404.05 905.74 17.92 17.98 454.31 4 6 809.37 14.31 14.34 405.72 909.07 18.06 18.11 455.98 6 8 812.69 14 43 14.46 407.39 912.39 18.19 18.25 457.66 8 10 816.01 14.55 14.58 409.06 915.71 18.32 18.38 459.34 10 12 819.34 14.66 14.70 410.74 919.04 18.46 18.52 461.02 12 14 822.66 14.78 14.82 412.41 922.36 18.59 18.65 462.70 14 16 825.98 14.90 14.94 414. OS 925.68 18.72 18.79 464.37 16 18 829.31 15.02 15.06 415.75 929.01 18.86 18.92 466.05 18 20 832.63 15.14 15.18 417.43 932.33 18.99 19.06 467.73 20 22 83595 15.26 15.30 419.10 935.65 19.12 19.19 469.41 22 24 839.28 15.38 15.43 400.77 938.98 19.26 19.33 471.08 24 26 842.60 15.51 15.55 422.45 942.30 19.40 19.47 472.76 26 28 845.92 15.63 15.68 424.12 945.62 19.54 19.61 474.43 28 30 849.25 15.75 15.80 425.79 948.95 19.68 19.75 476.10 30 32 852.57 15.88 15.93 427.47 952.27 19.82 19.89 477.78 32 34 855.89 16.00 16.05 429.15 955.59 19.96 20.03 479.46 34 36 859.22 16.12 16.18 430.82 958.92 20.10 20.17 481.14 36 38 862.54 16.25 16.30 432.50 962.24 20.24 20.31 482.83 38 40 865.86 16.38 16.43 434.18 965.56 20.38 20.45 484.51 40 42 869.19 16.50 16.55 435.86 968.89 20.52 20.59 486.19 42 1 44 872.51 16.63 16.68 437.54 972.21 20.66 20.74 487.87 44 46 875.83 16.76 16.81 439.21 975.53 20.80 20.88 489.56 46 1 48 879.16 16.89 16.94 440.89 97886 20.94 21.03 491.24 48 50 882.48 17.02 17.07 442.57 982.18 21.09 21.17 492.92 50 52 885.80 17.14 17.19 444.25 985.50 21.23 21.31 494.60 52 54 889.13 17.27 17.32 445.93 988.83 21.37 21.46 496.28 54 56 892.45 17.40 17.45 447.60 992.15 21.51 21.60 497.96 56 58 895.77 17.53 17.58 449.28 995.47 21.65 21.75 499.65 58 60 899.10 1766 17.71 450.95 998.80 21.80 21.89 501.32 60 134 6. FUNCTIONS OF A ONE-DEGREE CURVE. 10 11 L C. M. E. T. L. C. M. E. T. 998.8 21.80 21.89 501.32 1098.4 26.38 26.50 551.74 2 1002.1 21.94 22.03 503.00 1101.7 26.54 26.66 553.42 2 4 1005.4 22.09 22.18 504.68 1105.0 26.70 26.83 555.10 4 6 1008.8 22.24 22.33 506.36 1108.3 26.86 26.99 556.78 6 8 1012.1 22.39 22.48 508.04 1111.7 27.02 27.16 558.46 8 10 1015.4 22.54 22.63 509. ".2 1115.0 27.19 27.32 560.14 10 12 1018.7 22.68 22.78 511.40 1118.3 27.35 27.48 561.82 12 14 1022 22.83 22.93 513.08 1121.6 27.51 27.65 563.50 14 16 1025.4 22.98 23.08 514.76 1124.9 27.67 27.81 565.18 16 18 1028.7 23.13 23.23 516.44 1128.2 27.83 27.98 566.86 18 20 1032.0 23.28 23.38 518.12 1131.6 28.00 28.14 568.54 20 22 1035.3 23.43 23.53 519.80 1134.9 28.17 28.30 570.22 22 24 1038.6 23.58 23.68 521.48 1138.2 28.34 28.47 571.90 24 26 1042.0 23.73 23.84 523.16 1141.5 28.50 28.64 573.58 26 28 1045.3 23.88 23.99 524.85 1144.8 28.67 28.81 575.27 28 80 1048.6 24.04 24.14 526.53 1148.1 28.84 28.98 576.95 30 32 1051.9 24.19 24.30 528.21 1151.5 29.00 29.14 578.63 32 34 1055.2 24.34 24.45 529.89 1154.8 29.17 29.31 580.32 34 36 1058.6 24.49 24.60 531.57 1158.1 29.34 29.48 582.00 36 38 1061.9 24.64 24.76 533.25 1161.4 29.50 29.65 583.69 38 40 1065.2 24.80 24.91 534.93 1164.7 29.67 29.82 585.87 40 42 1068.5 24.95 25.06 536.61 1168.0 29.84 29.99 587.05 42 44 1071.8 25.11 25.22 538.29 1171.4 30.01 30.17 588.74 44 46 1075.2 25.27 25.38 539.97 1174.7 30.18 30.34 590.42 46 48 1078.5 25.43 25.54 541.65 1178.0 30.35 30.52 592.11 48 50 1081.8 25.59 25.70 543.33 1181.3 30.53 30.69 593.79 50 52 1085.1 25.74 25.86 545.01 11S4.6 30.70 30.86 595.47 52 54 1088.4 25.90 26.02 546. G9 1187.9 30.87 31.04 597.16 54 56 1091.8 26.06 26.18 548.37 1191.3 31.04 31.21 598.84 56 58 1095.1 26.22 26.34 550.06 1194.6 31.21 31.39 600.53 58 60 1098.4 26.38 26.50 551.74 1197.9 31.39 31.56 602.22 60 1*2 13 L. C. M. E. T. L. C. M. E. T. -0 1197.9 31.39 31.56 602.22 1297.3 36.83 37.07 652.87 2 1201.2 31.57 31.73 603.91 1300.6 37.02 37.26 654.56 2 4 1204.5 31.74 31.91 605.60 1303.9 37.21 37.46 656.25 4 6 1207.8 31.93 32.09 607.28 1307.2 37.40 37.65 657.93 6 8 1211.1 32.09 32.27 608.97 1310.5 37.59 37.85 659.62 8 10 1214.5 32.27 32.45 610.66 1313.8 37.79 38.04 661.31 10 12 1217.8 32.45 32.63 612.35 1317.2 37.98 38.23 663.00 12 14 1221.1 32.62 32.81 614.04 1320.5 38.17 38.43 664.69 14 16 1224.4 32.80 32.99 615.72 1323.8 38.36 38.62 666.37 16 18 1227.7 32.97 33.17 617.41 1327.1 38.55 38.82 668.06 18 20 1231.0 33.15 33.35 619.10 1330.4 38.75 39.01 669.75 20 22 1234.3 33.33 33.53 6v!0.79 1333.7 38.95 39.20 671.44 22 24 1237.7 33.51 33.72 622.48 1337.0 39.15 39.40 673.13 24 26 1241.0 33.69 33.90 624.16 1340.3 39.35 39.60 674.81 26 28 1244.3 33.87 34.09 6^5.85 1343.6 39.54 39.80 676.51 28 30 1247.6 34.06 34.27 627.55 1346.9 39.74 40.00 678.20 30 32 1250.9 34.24 34.45 629.24 1350.3 39.94 40.19 679.89 32 34 1254.2 34.42 34.64 630.93 1353.6 40.13 40.39 681.58 34 36 1257.5 34.60 34.82 632.61 1356.9 40.33 40.59 683.26 36 38 1260.8 34.78 35.01 634.30 1360.2 40.52 40.79 684.95 38 40 1264.2 3497 35.19 635.99 1363.5 40.71 40.99 686.64 40 42 1267.5 35.16 35.37 637.68 1366.8 40.91 41.19 68833 42 44 1270.8 35.34 35.56 639.37 1370.1 41.11 41.40 690.02 44 46 1274.1 35.53 35.75 641.05 1373.4 41.31 41.60 691.70 46 48 1277.4 35.71 35.94 642.74 1376.7 41.51 41.81 693.39 48 50 1280.7 35.90 36.13 644.43 1380.0 41.71 42.01 695.08 50 52 1284.0 36.09 36.31 646.12 1383.4 41.91 42.21 696.77 52 54 1287.4 36.27 36.50 647.81 1386.7 42.11 42.42 698.46 54 56 1290.7 36.46 36.69 649.49 1390.0 42.31 42.62 700.14 56 58 1294.0 36.64 36.88 651.18 1393.3 42.51 42.83 701.83 58 [60 1297.3 36.83 37.07 652.87 1396.6 42.71 43.03 703.53 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 135 14 15 L. C. M. E. T. L. C. M. E. T. 1396.6 42.71 43.03 ?03.53 1495.9 49.02 49.44 754.85 2 1399.9 42.92 43.23 705.23 1499.2 49.24 49.66 756.05 2 4 1403.2 43.12 43.44 706.92 1502.5 49.46 49.89 757.74 4 6 1406.5 43.33 43.65 708.62 1505.8 49.68 50.11 759.44 6 8 1409.8 43.53 43.86 710.31 1509.1 49.90 50.34 761.13 8 10 1413.1 48.74 44.07 712.01 1512.4 50.12 50.56 762.88 10 12 1416.5 43.94 44.28 713.71 1515.7 50.34 50.78 764.53 12 14 1419.8 44.15 44.49 715.40 1519.0 50.56 51.01 766.22 14 16 1423.1 44.35 44.70 717.10 1522.3 50.78 51.23 767.92 16 18 1426.4 44.56 44.91 718.79 1525.6 51.00 51.46 769.61 18 20 1429.7 44.77 45.12 720.49 1528.9 51.22 51.68 771.31 20 22 1433.0 44.98 45.33 722.20 1532.2 51.44 51.90 773.01 22 24 1436.3 45.19 45.54 723.89 1535.5 51.67 52.13 774.70 24 26 1439.6 45.40 45.76 725.59 1538.8 51.89 52.36 776.40 26 28 1442.9 45.61 45.97 727.28 1542.1 52.12 52.59 778.09 28 30 1446.2 45.82 46.18 728.97 1545.4 52.34 52.82 779.79 30 32 1449.6 46.03 46.40 730.66 1548.7 52.57 53.05 781.49 32 34 1452.9 46.24 46.61 732.35 1552.0 52.79 53.28 783.19 34 36 1456.2 46.45 46.82 734.05 1555.3 53.02 53.51 784.89 36 38 1459.5 46.66 47.04 735.74 1558.6 53.24 53.74 786.59 38 40 1462.8 46.87 47.25 737.43 1561.9 53.47 53.97 788.29 40 42 146G.1 47.08 47.46 739.12 1505.2 53.69 54.20 789.99 42 44 1469.4 47.30 47.68 740.81 1568.5 53.92 54.44 791.69 44 46 1472.7 47.51 47.90 742.51 1571.8 54.15 54.67 793.39 46 48 1476. 47.73 48.12 744.20 1575.1 54.38 54.91 795.09 48 50 1479.3 47.94 48.34 745.89 1578.4 54.61 55.14 796.79 50 52 1482.7 48.16 48.56 747.58 1581.7 54.84 55.37 798.49 52 54 1486.0 48.37 48.78 749.27 1585.0 55.07 55 61 800.19 54 56 1489.3 48.59 49.00 750.97 1588.3 55.30 55.84 801.89 56 58 1492.6 48.80 49.22 752.66 1591.6 55.53 56.08 803.59 58 60 1495.9 49.02 49.44 754.35 1594.9 55.76 56.31 805.29 60 16 17 L C. M. E. T. L. C. M. E. T. 1594.9 55.76 56.31 805.29 1693.9 62.94 63.64 856.35 2 1598.2 55.99 56.54 806.99 1697.2 63.18 63.89 858.05 2 4 1601.5 56.23 56.78 808.64 1700.5 63.43 64.15 859.76 4 6 1604.8 56.46 57.02 810.39 1703.8 63.68 64.40 861.46 6 8 1608.1 56.70 57.26 812.09 1707.1 63.93 64.66 863.16 8 10 1611.4 56.93 57.50 813.79 1710.4 64.18 64.91 864.87 10 12 1614.7 57.17 57.74 815.49 1713.7 64.42 65.16 866.57 12 14 1618.0 57.40 57.98 817.19 1716.9 64.67 65.42 868.27 14 16 1621.3 57.64 58.22 818.89 1720.2 64.92 65.67 869.98 16 18 1624.6 57.87 58.46 820.59 1723.5 65.17 65.93 871.68 18 20 1627.9 58.11 58.70 822.29 1726.8 65.42 66.18 873.38 20 22 1631.2 58.34 58.94 823.99 1730.1 65.67 66.43 875.09 22 24 1634.5 58.58 59.19 825.69 1733.4 65.93 66.69 876.79 24 26 1637.8 58.82 59.43 827.39 1736.7 66.18 66.95 878.49 26 28 1641.1 59.06 59.68 829.09 1740.0 66.44 67.21 880.20 28 30 1644.4 59.30 59.92 830.79 1743.3 66.69 67.47 881.90 30 32 1647.7 59.54 60.16 832.49 1746.6 66.94 67.72 883.61 32 84 1651.0 59.78 60.41 834.20 1749.9 67.20 67.98 885.32 34 36 1654.3 60.02 60.65 835.90 1753.2 67.45 68.24 887.02 36 38 1657.6 60.26 60.90 837.61 1756.5 67.71 68.50 888.73 38 40 1660.9 60.50 61.14 839.31 1759.8 67.96 68.76 890.44 40 42 1664.2 60.74 61.39 841.01 1763.1 68.21 69.03 892.15 42 44 1667.5 60.99 61.64 842.72 1766.3 68.47 69.29 893.86 44 46 1670.8 61.23 61.89 844.42 1769.6 68.73 69.56 895.56 46 48 1674.1 61.48 62.14 846.13 1772.9 68.99 69.82 897.27 48 50 1677.4 61.72 62.39 847.83 1776 2 69.25 70.09 898.98 50 52 1680.7 61.96 62.64 849.53 1779.5 69.50 70.36 900.69 52 54 1684.0 62.21 62.89 851.24 1782.8 69.76 70.62 902.40 54 56 1687.3 62.45 63.14 852.94 1786.1 70.02 70.89 904.10 56 58 1690.6 62.70 63 39 854.65 1789.4 70.28 71.17 905.81 58 60 1693.0 62.94 63.64 856.35 1792.7 70.54 71.42 907.52 60 136 6. FUNCTIONS OF A ONE-DEGREE CURVE. 18 19 / L. C. M. E. T. L. C. M. E. T. 1792.7 70.54 71.42 907.52 1891.5 78.58 79.65 958.86 2 1796.0 70.80 71.69 909.23 1894.8 78.86 79.94 960.57 2 4 1799.3 71.06 71.96 910.94 1898.1 79.13 80.22 962.30 4 6 1802.6 71.33 72.23 912.65 1901.3 79.41 80.51 964.00 6 8 1805.9 71.59 72.50 914.36 1904.6 79.68 80.79 965.72 8 10 1809.2 71.85 72.77 916.07 1907.9 79.96 81.08 967.43 10 12 1812.5 72.12 73.04 917.78 1911.2 80.24 81.37 969.15 12 14 1815.7 72.38 73.31 919.49 1914.5 80.51 81.65 970.86 14 16 1819.0 72.64 73.58 921.20 1917.8 80.79 81.94 972.58 16 18 1822.3 72 91 73.85 922.91 1921.0 81.07 82.22 974.29 18 20 1825.6 73.17 74.12 924.63 1924.3 81.35 82.51 976.01 20 22 1828.9 73.43 74.39 926.34 1927.6 81.63 82.80 977.72 22 24 1832 2 73.70 74.67 928.05 1930.9 81.91 83.09 979.44 24 26 1835.5 73.97 74.94 929.76 1934.2 82.20 8338 981.15 26 28 1838.8 74.24 75.22 931.47 1937.5 82.48 83.67 982.86 28 30 1842.1 74.51 75.49 933.18 1940.7 82.76 83.97 984.58 30 32 1845.4 74.77 75.77 934.89 1944.0 83.05 84.26 986.30 32 34 1848.7 75.04 76.04 936.60 1947.3 83.33 84.55 988.02 34 36 1852.0 75.31 76.32 938.32 1950.6 83.61 84.84 989.74 36 38 1855.3 75.58 76.59 940.03 1953.9 83.90 85.13 991.46 38 40 1858.6 75.85 76.87 941.74 1957.2 84.18 85.43 993.18 40 42 1861.9 76.12 77.14 943.45 19604 84.47 85.73 994.90 42 44 1865.1 76.39 77.42 945.16 1903.7 84.75 80.02 996.62 44 46 1868.4 76.67 77.70 946.88 1967.0 85.04 86 32 998.34 46 48 1871.7 76.94 77.98 948.59 1970.3 85.32 86.61 1000.0 48 50 1875.0 77.21 78.26 950.30 1973.6 85.61 86.91 1001.8 50 52 1878.3 77.49 78.53 952.01 1976.9 85.90 87.21 1003.5 52 54 1881.6 77.76 78.81 953.72 1980.1 86.19 87.50 1005.2 54 50 1884.9 78.03 79.09 955.44 1983.4 86.47 87.80 1006.9 56 58 1888.2 78.31 79.37 957.15 19S6.7 86.76 88.09 1008.6 58 60 1891.5 78.58 79. (',5 958 86 1990.0 87.05 88.39 1010.4 60 20 21 L. C. M. E. T. L, C. M. E. T. / 1990.0 87.05 88.39 1010.4 2088.5 95.95 97.58 1062.0 2 1993.3 87.34 88.69 1012.1 2091.8 96.26 97.90 1063.7 2 4 1996.6 87. G3 88.99 1013.8 2095.0 96.56 98.21 1065.4 4 6 1999.8 87.92 89.29 1015.5 2098.3 96.87 98.53 1067.2 6 8 003.1 88.21 89.59 1017.2 2101.6 97.17 98.84 1068.9 8 10 2006.4 88.50 89.89 1019.0 2104.9 97.48 99.16 1070.6 10 12 2009.7 88.79 90.19 1020.7 2108.1 97.79 99.48 1072.4 12 14 2013.0 89.08 90.49 1022 4 2111.4 98.09 99.79 1074.1 14 16 2016 3 89.37 90.79 1024.1 2114.7 98.40 100.1 1075.8 16 18 2019.5 89.66 91.09 1025.8 2118.0 98.70 100.4 1077.5 18 20 2022.8 89.96 91.40 1027.6 2121.2 99.00 J00.7 1079.3 20 22 2G26.1 90.25 91.71 1029.3 2124.5 99.30 J01.1 1081.0 22 24 2029-4 90.55 92.01 1031.0 2127.8 99.60 J01.4 1082.7 24 26 2032.7 90.85 92.32 1032.7 2131.0 99.90 101.7 1084.4 26 28 2036.0 91.15 92. G2 1034.4 2134.3 100.2 102.0 1086.2 28 30 2039.2 91.45 92.93 1036 1 2137.6 100.5 102.3 1087.9 30 32 2042.5 91.74 93.24 1037.9 2140.9 100.8 102.7 1089.6 32 34 2045.8 92.04 93.54 1039.6 2144.1 101.1 1030 1091.3 34 36 2049.1 92.34 93.85 1041.3 2147.4 101.4 103.3 1093.1 36 38 2052.4 92.64 94.15 1043.0 2150.7 101.7 103.6 1094.8 88 40 2055.7 92.94 94.46 1044.8 2154.0 102.1 104.0 1096.5 40 42 2058.9 93.24 94.78 1046.5 2157.2 102.4 104.3 1098.3 42 44 2062.2 93.54 95.09 1048.2 2160.5 102.7 104.6 1100.0 44 46 2065.5 93.84 95.40 1049.9 2163.8 103.0 104.9 1101.7 46 48 2068.8 94.14 95.71 1051.7 2167.1 103.3 105.3 1103.4 48 50 2072.1 94.44 96.03 1053.4 2170.3 103.6 105.6 1105.2 50 52 2075.4 94.74 96.34 1055.1 2173.6 103.9 105.9 1106.9 52 54 2078.6 95.04 96.65 1056.8 2176.9 104.2 106.3 1108.6 54 56 2081.9 95.34 96 96 1058.6 2180.1 104.5 106.6 1110.3 56 58 2085.2 95.64 97 27 10G0.3 2183.4 104.8 106.9 1112.1 58 60 2088.5 95.95 97.58 1062.0 2186.7 105.2 107.2 1118.8 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 137 22 23 L. C. M. E. T. L. C. M. E. T. 2186.7 103.2 107.2 1113.8 2284.8 115.0 117.4 1165.8 2 2190.0 105.6 107.6 1115.5 2288.1 115.3 117.7 1167.5 2 4 2193.2 105.9 107.9 1117.3 2291.3 115.-7 118.1 1169.2 4 6 2196.5 106.2 108.2 1119.0 2294.6 116.0 118.4 1171.0 6 8 2199.8 106.5 108.6 1120.7 2297.8 116.4 118.8 1172.7 8 10 2203.0 106.8 108.9 1122.4 2301.1 116.7 119.1 1174.4 10 12 2206.3 107.1 109.2 1124.2 2304.4 117.0 119.5 1176.2 12 14 2209.6 107.4 109.6 1125.9 2307.6 117.4 119.8 1177.9 14 16 2212.9 107.7 109.9 1127.6 2310.9 117.7 120.2 1179.7 16 18 2216.1 108.0 110.2 1129.4 2314.1 118.1 120.5 1181.4 18 20 2219.4 108.4 110.6 1131.1 2317.4 118.4 120.9 1183.1 20 22 2222.7 108.7 110.9 1132.8 2320.7 118.7 121.2 1184.9 22 24 2-225.9 109.0 111.2 1134.6 2323.9 119.1 121 6 1186.6 24 26 2229.2 109.4 111.6 1136.3 2327.2 119.4 121.9 1188.4 26 28 2232.5 109.7 111.9 1138.0 2330.4 119.8 122.3 1190.1 28 30 2235.7 110.0 112.3 1139.7 2333.7 120.1 122.6 1191.8 30 32 2239.0 110.4 112.6 1141.5 2337.0 120.4 123.0 1193.6 32 34 2242.3 110.7 112.9 1143.2 2340.2 120.8 123.3 1195.3 34 36 2245.6 111.0 113.3 1144.9 2343.5 121.1 123.7 1197.1 36 38 2248.8 111.4 113.6 1146.7 2346.7 121.5 124.1 1198.8 38 40 2252.1 111.7 113.9 1148.4 23500 121.8 124.4 1200.5 40 42 2255.4 112.0 114.3 1150.1 2353.3 122.1 124.8 1202.3 42 44 2258.6 112.3 114.6 1151.9 2356.5 122.5 125.1 1204.0 44 46 2261.9 112.7 115.0 1153.6 2359.8 122.8 125.5 1205.8 46 48 2265.2 113.0 115.3 1155.4 2303.0 123.2 125.8 1207.5 48 50 2268.4 113.3 115.7 1157.1 2306.3 123.5 126.2 1209.2 50 52 2271.7 113.7 116.0 1158.8 2369.6 123.8 126.6 1211.0 52 54 2275.0 114.0 116.3 1160.6 2372.8 124.2 126.9 1212.7 54 56 2278.3 114.3 116.7 1162.3 2376.1 124.5 127.3 1214.5 56 58 2281.5 114.7 117.0 1164.0 2379.3 124.9 127.6 1216.2 58 60 2284.8 115.0 117.4 1165.8 2382.6 125.2 128.0 1218.0 60 24 25 / L. 0. M. E. T. L C. M. E. T. 23S2.6 125.2 128.0 1218.0 2480.4 135.8 139.1 1270.3 2 2385.9 125.5 128.4 1219.7 2483.6 186.2 139.5 1272.0 2 4 2389.1 125.9 128.7 1221.4 2486.9 136.5 139.9 1273.8 4 6 2392.4 126.2 129.1 1223.2 2400.1 136.9 140.3 1275.5 6 8 2395.6 126.6 129.5 1224.9 2493.4 137.2 140.6 1277.3 8 10 2398.9 126.9 129.8 1226.7 2496.6 137.6 141.0 1279.0 10 12 2402.2 127.3 130.2 1228.4 2499.9 138.0 141.4 1280.8 12 14 2405.4 127.6 130.6 1230.2 2503.1 138.3 141.8 1282.5 14 Ifl 2408.7 128.0 130.9 1231.9 2506.4 138.7 142.2 1284.3 16 18 2411.9 128.3 131.3 1233/6 2509.6 139.0 142.5 1286.1 18 20 2415.2 128.7 131.7 1235.4 2512.9 139.4 142.9 1287.8 20 22 2418.5 129.0 132.0 1237.1 2516.1 139.8 143.3 1289.6 22 24 2421.7 129.4 132.4 123S.9 2519.4 140.1 143.7 1291.3 24 26 2425.0 129.7 132.8 1240.6 2522.6 140.5 144.1 1293.1 26 28 2428.2 130.1 133.1 1242.4 2525.9 140.8 144.5 1294.8 28 30 2431.5 130.4 133.5 1244.1 2529.1 141.2 144.9 1296.6 30 32 2434.8 130.8 133.9 1215.8 2532.4 141.6 145.3 1298.3 32 34 2433.0 131.1 134.2 1247.6 2535.6 142.0 145.6 1300.1 34 36 2441.3 131.5 134.8 1249. 3 2538.9 142.3 146.0 1301.8 36 38 2444.5 131.8 135.0 1251.1 2542.1 142.7 146.4 1303.6 38 40 2447.8 132.2 135.4 1252.8 2545.4 143.1 146 8 1305.3 40 42 2451.1 132.6 135.7 1254.6 2548. d 143.5 147.2 1307.1 42 44 2454.3 132.9 136.1 1256.3 2551.9 143.8 147.6 1308.8 44 46 2457.6 133.3 136.5 1258.1 2555.1 144.2 148.0 1310.6 46 48 2460. S 133.6 136.9 1259.8 2558.4 144.5 148.4 1312.4 48 50 2464.1 134.0 137.2 1261.5 2561.6 144.9 148.8 1314.1 50 52 2467.4 134.4 137.6 1263.3 2564.9 145.3 149.2 1315.9 52 54 2470.6 134.7 138.0 1265.0 2568.1 145.7 149.5 1317.6 54 56 2473.9 135.1 138.4 1266.8 2571.4 146.0 149.9 1319.4 56 58 2477.1 135.4 138.7 1263.5 2574.6 146.4 150.3 1321.1 58 60 2480.4 135.8 139.1 1270.3 2577.9 1468 150.7 1322.9 60 138 6. FUNCTIONS OF A ONE-DEGREE CURVE. 26 27* L. C. M. E. T. L. C. M. E. T. I 2577.9 146.8 150.7 1322.9 2675.3 158.3 162.8 1375.6 2 2581.1 147.1 151.1 1324.6 2678.5 158.6 163.2 1377.4 2 4 2584.4 147.5 151.5 1326.4 2681.8 159.0 163.7 1379.2 4 6 2587.6 147.9 151.9 1328.1 2685.0 159.4 164.1 1380.9 6 8 2590.9 148.3 152.3 1329.9 2688.2 159.8 164.5 1882.7 8 10 2594.1 148.7 152.7 1331.6 2691.5 160.2 164.9 1384.5 10 12 2597.4 149.1 153.1 1333.4 2694.7 160.6 165.3 1386.2 12 14 2600.6 149.4 153.5 1335.2 2698.0 161.0 165.7 1388.0 14 16 2603.9 149.8 153.9 1336.9 2701.2 161.4 166.1 1389.8 16 18 2607.1 1502 154.3 1338.7 2704.4 161.8 166.5 1391.5 18 20 2610.4 1506 154.7 1340.4 2707.7 162.2 167.0 1393.3 20 22 2613.6 151 155.1 1342.2 2710.9 162.6 167.4 1395.0 22 24 2616.9 151.4 155.5 1343.9 2714.1 163.0 167.8 1396.8 24 26 2620.1 151.7 155.9 1345.7 2717.4 163.4 168.2 1398.6 26 28 2623.4 152.1 156.3 1347.4 2720.6 168.8 168.6 1400.3 28 30 2626.6 152.5 156.7 1349.2 2723.8 164.2 169.1 1402.1 30 32 2629.8 152.9 157.1 1351.0 2727.1 164.6 169.5 1403.9 32 34 2633.1 153.3 157.5 1352.7 2730.3 165.0 169.9 1405.6 34 36 2636.3 153.7 157.9 1354 5 2733.6 165.4 170.3 1407.4 36 38 2639.6 154.0 158.3 1356.2 2736.8 165.8 170.8 1409.2 38 40 2642.8 154.4 158.7 1358.0 2740.0 166.2 171.2 1410.9 40 42 2646.1 154.8 159.1 1359.8 2743.3 166.6 171.6 1412.7 42 44 2649.3 155.2 159.5 1361.5 2746.5 167.0 172.0 1414.5 44 46 2652.6 155.6 160.0 1363.3 2749.7 167.4 172.5 1416.3 46 48 2655.8 156.0 160.4 1365.1 2753.0 167.8 172.9 1418.0 48 50 2659.1 156.3 160.8 1366.8 2756.2 168.2 173.3 1419.8 50 52 2662.3 156.7 161.2 1368.6 2759.5 168.6 173.7 1421.6 52 54 2665.6 157.1 161.6 13~0.4 2762.7 169.0 174.1 1423.3 54 56 2668.8 1575 162.0 13~2.1 2765.9 169.4 174.6 1425.1 56 58 2672.1 157.9 162.4 1373.9 2769.2 169.8 175.0 1426.9 58 60 2675.3 158.3 162.8 1375.6 2772.4 170.2 175.4 1428.6 60 28 29 1 / L. C. M. E. T. L. C. M. E. T. 2772.4 170.2 175.4 1128.6 2869.4 182.5 188.5 1481.9 2 2775.6 170.6 75.8 1430.4 2872.6 182.9 189.0 1483.7 2 4 2778.9 171.0 76.3 1432.2 2875.8 183 3 189.4 1485.4 4 6 2782.1 171.4 76.7 1434.0 2879.1 183.7 189.9 1487.2 6 8 2785.3 171.8 77.1 1435.7 2882.3 18^.2 190.3 1489.0 8 10 2788.6 172.2 77.6 1437.5 2885.5 184.6 190.8 1490.8 10 12 2791.8 172.6 78.0 1439.3 2888.7 185.0 191.2 1492.6 12 14 2795.0 173.0 78.4 1441.1 2892.0 185.4 191.7 1494.3 14 16 2798.3 173.4 78.9 1442.8 2895.2 185.8 192.1 1496.1 16 18 2801.5 173.8 79.3 1444.6 2898.4 186.3 192.5 1497.9 18 20 2804.7 74.3 179.7 1446.4 2901.6 186.7 193.0 1499.7 20 22 2808.0 74.7 180.2 1448.2 2904.8 187.1 193.5 1501.5 22 24 2811.2 75.1 180.6 1449.9 2908.1 187.5 193.9 1503.2 24 26 2814.4 75.5 181.0 1451.7 2911.3 188.0 194.4 1505.0 26 28 2817.7 75.9 181.5 1453.5 2914.5 188.4 194.8 1506.8 28 30 2820.9 76.3 181.9 1455.2 2917.7 188.8 195.3 1508-6 30 32 2824.1 176.7 182.3 1457.0 2921.0 189.2 195.7 1510.4 32 34 2827.4 177.1 182.8 1458.8 2924.2 189.7 196.2 1512.1 34 36 2830.6 177.5 183.2 1460.6 2927.4 190.1 196.7 1513.9 36 38 2833.8 177.9 J83.6 1462.3 2930.6 190.5 197.1 1515.7 38 40 2837.1 178.4 184.1 1464.1 2933.9 190.9 197.6 1517.5 40 42 2840 3 178.8 184.5 1465. 9 2937.1 191.4 198.0 1519.3 42 44 28435 179.2 185.0 H67.7 2940.3 191.9 198.5 1521.0 44 46 2846.8 179.6 185.4 1469.5 2943.5 192.4 198.9 1522.8 46 48 2850.0 1800 185.9 1471.2 2946.8 192.8 199.4 1524.6 48 50 2853.2 180.4 186.3 1473.0 2950.0 193.2 199.8 1526.4 50 52 2856.5 180.8 186.8 1474.8 2953.2 193.6 200.3 1528.2 52 54 2859.7 181.2 187.2 1476.6 2956.4 194.0 200.8 1530.0 54 56 2862.9 181.6 187.6 1478.3 2959.6 194.4 201.2 1531.7 56 58 2866.2 182.0 183.1 1480.1 2962.9 194.8 201.7 1533.5 58 60 2869.4 182 5 188.5 1481.9 2966.1 195.2 202.1 1535 3 60 1 6. FUNCTIONS OF A ONE-DEGREE CURVE. 139 30 31 L. C. M. E. T. L. C. M. E. T. 2966.1 195.2 202.1 1535.3 3062.6 208.4 216 3 1589.0 2 2969.3 195.6 202.6 1537.1 3065.8 208.8 216.8 1590.8 2 4 2972.5 196.1 203.1 1538.9 3069.0 209.3 217 2 1592.6 4 6 2975.7 196.5 203.5 1540.7 3072.2 209.7 217.7 1594.4 6 8 2979.0 197.0 204.0 1542.5 3075.4 210.2 218.2 1596.2 8 10 2982.2 197.4 204.5 1544 3 3078.6 210.6 218.7 1598.0 10 12 2985.4 197.8 204.9 1546.0 3081.8 211.1 219.2 1599.8 12 14 2988.6 198.2 205.4 1547.8 3085.0 211.5 219.6 1601.6 14 16 2991.8 198.6 2059 1549.6 3088.3 212.0 220.1 1603.4 16 18 29950 199.1 2063 1551.4 3091.5 212.4 220.6 1605.2 18 20 2998.3 199 5 206.8 1553.2 3094.7 212.9 221 1 1607.0 20 22 3001.5 199.9 207.3 1555.0 3097.9 213.3 221.6 1608.8 22 24 3004.7 200.4 207.7 1556.8 3101.1 213.8 222.1 1610.6 24 26 3007.9 200.8 208.2 1558.6 3104.3 214.2 222.6 1612.4 26 28 3011 1 201.3 208.7 1560.4 3107.5 214.7 223.0 1614.2 28 30 3014.3 201.7 209.1 1562.2 3110.7 215.1 223.5 1616.0 30 32 30176 2021 209.6 1564.0 3113.9 215.6 224.0 1617.8 32 34 3020.8 202 6 210 1 1565.7 3117.1 216.0 224.5 1619.6 34 36 3024.0 203.0 210.5 1567.5 3120.3 216.5 225.0 1621.4 36 38 3027.2 203.5 211.0 1569.3 3123.5 216.9 225.5 1623.2 33 40 3030.4 203.9 211 5 1571.1 3126.7 217.4 226.0 1625.0 40 42 3033.6 204.3 212 1572.9 3129.9 217.8 226.5 1626.8 42 44 3036.9 204.8 212.4 1574.7 3133.1 218.3 227.0 1628.6 44 46 3040.1 205.2 212.9 1576.5 3136.4 218.7 227.5 1630.5 46 48 3043.3 205.7 213.4 1578.3 3139.6 219.2 228.0 1632.3 48 50 3046.5 206.1 213.9 1580.1 3142.8 219.6 228.4 1634.1 50 52 3049.7 206.5 214.4 1581.9 3146.0 220.1 228.9 1635.9 52 54 3052.9 207 214.8 1583.7 3149.2 220.5 229.4 1637.7 54 56 3056.2 207.4 215.3 1585.5 3152.4 221.0 2.9.9 1639.5 56 58 3059.4 207.9 215.8 1587.2 3155.6 221.5 230.4 1641.3 58 60 3062 6 208.4 216 3 1589.0 3158.8 222.0 230.9 1643.1 60 32 33 L. C. M. E. T. L. C M. E. T. / 3158.8 222.0 230.9 1643.1 3254.0 236.0 246.1 1697 3 2 3162.0 222 5 231.4 1644 9 3258.1 236.4 246.6 1699.1 2 4 3165.2 222.9 231.9 1646.7 3261.3 236.9 247.1 17009 4 6 3168.4 223.4 232.4 1648.5 3264.5 237.4 247.7 1702.7 6 8 3171.6 223.8 232.9 1650.3 3267 7 237.9 248.2 1704.5 8 10 3174.8 224.3 233.4 1652.1 3270.8 238.4 248.7 1706.4 10 12 3178.0 224 8 233.9 1653.9 3274 238.9 249.2 1708 .-2 12 14 3181.2 225.2 234.4 1655 7 3277.2 2393 249.7 1710.0 14 16 3184.4 225.7 234.9 1657.5 3280.4 239.8 250.2 1711.8 16 18 3187.6 226.1 235.4 1659.3 3283.6 240.3 250.8 1713.6 18 20 3190.8 226.6 2359 1661.1 3286.8 240.8 251.3 1715.5 20 22 3194.0 227.1 236.4 1662.9 3290 241.2 251.8 1717.3 22 24 3197 2 227.5 236 9 1664.7 3293.2 241.7 252.3 1719.1 24 26 3200.4 228.0 237.4 1666 5 3296.4 242.2 252.9 1720.9 26 28 3203.6 228.4 2379 1668.3 3299.6 242.7 253.4 1722.7 28 30 3206.8 228 9 238.4 1670.1 3302.7 243 2 253.9 1724.6 30 32 3210.0 229.4 239.0 1671.9 3305 9 243.6 254.4 1726.4 32 34 3213.2 229 8 239.5 1673 7 33091 244.1 255.0 1728.2 34 36 3216 5 230.3 240.0 1675.5 3312 3 244.6 255.5 1730.0 36 88 8219.7 230.7 240.5 1677.4 3315 5 245.1 256.0 1731.8 38 40 32229 231 2 241.0 1679.2 3318.7 245.6 256.5 1733.6 40 42 3226 1 231.7 241.5 1681.0 3321.9 246.0 257.1 1735.5 42 44 3629.3 232.2 242.0 1682 8 3325.1 246.5 257.6 1737.3 44 46 3232.5 232.6 242.5 1684.6 3328.3 247.0 258.1 1739.1 46 48 3235.7 233.1 243.0 16864 3331.5 247 5 258.6 1740.9 48 50 3238.9 233.5 243.5 16882 3334.6 248.0 259 2 1742.7 50 52 3242.1 234.0 244.1 1690.0 3337.8 248.4 259.7 1744.6 52 54 3245 3 234.5 244.6 1691.8 3341.0 248.9 2602 1746.4 54 66 3248.5 235 245.1 1693.7 3344.8 249.4 260.8 1748.2 56 58 3251.7 235.5 245.6 1695 5 3347.4 249.9 261.3 1750,0 53 60 3254.9 236.0 246 1 1697.3 3350-6 250.4 261 8 1751,8 An 140 6. FUNCTIONS OF A ONE-DEGREE CURVE 34 85 L. C. M. E. T. L. 0. M. E. T. , 3350.6 250.4 261.8 1751.8 3446.1 265.2 278.1 1806.7 2 3353.8 250.8 262.3 1753.7 3449.3 265.7 278.6 1808.5 2 4 3357,0 251.2 262.9 1755.5 3452.5 266.2 279.2 1810.3 4 6 3360.1 251.7 263.4 1757.3 3455.6 266.7 279.7 1812.2 6 8 3363.3 252.2 2U4.0 1759.1 3458.8 267.2 280.3 1814.0 8 10 3366.5 252.7 264.5 1761.0 3462.0 267.7 280.8 1815.8 10 12 3369.7 253.2 265.0 1762.8 3465.2 268.2 281.4 1817.7 12 14 3372.9 253.7 265.6 1764.6 3468.3 268.7 281.9 1819.5 14 16 3376.1 254.2 266.1 1766.4 3471.5 269.2 282.5 1821.3 16 18 3379.2 254.7 266.7 1768.3 3474.7 269.7 283.0 1823.2 18 20' 3382.4 255.2 267.2 1770.1 34V7.9 270.2 283.6 1825.0 20 22 3385.6 255.7 267.7 1771.9 3481.0 270.7 284.2 1820.8 22 24 3388.8 256.2 268.3 1773.7 3484.2 271.2 2S4.7 1828.7 24 26 3392.0 256.7 268.8 1775.6 3487.4 271.7 285.3 1830.5 26 28 3395.2 257.2 269.3 1777.4 3490.6 272.2 285.9 1832.3 28 30 3398.3 257.7 269.9 1779.2 3493.7 272.7 286.4 1834.2 30 83 3401.5 258.2 270.4 1781.0 3496.9 273.2 287.0 1830.0 32 34 3404.7 258.7 271.0 1782.9 3500.1 273. Y 287.5 1837.8 34 36 3407.9 2592 271.5 1784.7 3503.3 274.2 288.1 1839.7 8tf 38 3411.1 259.7 272.0 1786.5 3506.5 274.7 288.7 1841.5 38 40 3414.3 260.2 272.6 1788.4 3509.6 275.2 289.2 1843.4 40 42 3417.4 260.7 273.1 1790.2 3512.8 275.7 289.8 1845.2 42 44 3420.6 261.2 273.7 1792.0 3516.0 276.2 290.4 1847.1 44 46 3423.8 261.7 274.2 1793.9 3519.2 -276.7 290.9 1848.9 46 48 3427.0 262.2 274.8 1795.7 3522.3 277.2 291.5 1850.7 48 50 3430.2 262.7 275.3 1797.5 3525.5 277.7 292.0 185;!. 6 50 52 3433.4 263.2 275.9 1799.3 3528.7 278.2 292.6 1854.4 52 54 3436.5 263.7 276.4 1801.2 3531.9 278.7 293.2 1856.3 54 56 3439.7 264.2 277.0 1803.0 3535.0 279.2 293.7 1858.1 56 58 3442.9 264.7 277.5 1804.8 3538.2 279.8 294.3 1859.9 58 00 3446 1 265.2 278.1 1806.7 3541.4 280.4 294.9 1801.8 60 36 37 L. C. M. E. T. L. C. M. E. T. 3541.4 280.4 294.9 1861.8 3636.3 296.1 312.3 1917.3 2 3544.6 280 9 295.4 1863.6 3639.5 296.6 312.8 1919.1 2 4 3547.7 281.4 296.0 1865.5 3642 6 297 1 313.4 1921.0 4 6 3550.9 281.9 296.6 1867.3 3645.8 297.7 314.0 1922.8 6 8 3554.0 282. 5 297.2 3869.2 3648.9 298.2 314.6 1924.7 8 10 3557.2 2830 297.7 1871.0 3652.1 298.7 315.2 1926.5 10 12 3560.4 283.5 298.3 1872.9 3655.2 299.3 315.8 1928.4 12 14 3563 5 284.0 298.9 1874.7 3658.4 299.8 316.4 1930 2 14 16 3566.7 284.6 299.5 1876.5 3661.6 300.3 317.0 1932.1 16 18 3569.9 285.1 300.0 1878.4 3064.7 300.9 317.5 1933.9 18 20 3573.0 285.6 300.6 1880.2 3G67.9 301.4 318.1 1935.8 20 22 3576.2 286.1 301.2 1882.1 3671.0 301.9 318.7 1937.6 22 24 3579.4 286.7 301.8 1883.9 3674.2 302.5 319.3 1939.5 24 26 3582.5 2S7.2 302.3 1885.8 3677.3 303.0 319.9 1941.3 26 28 3585.7 287.7 302.9 1887.6 3680.5 303.5 320.5 1943.2 28 30 3588.8 288.2 303.5 1889.5 3683.6 304.1 321.1 1945.0 30 32 3592.0 288.8 304.1 1891.3 3686.8 304.6 321.7 1946.9 32 34 3595.2 289.3 304.6 1893.2 3690.0 305.1 322.3 1948.8 34 36 3598.3 289 8 305.2 1895.0 3693.1 305.7 322.9 1950.6 36 38 3601.5 290.3 305.8 1896.9 3696.3 306.2 323.5 1952.5 33 40 3604.7 290.9 306.4 1898.7 3699.4 306.7 324.2 1954.4 40 42 3607.8 291.4 307.0 1900.6 3702.6 307.3 324.8 1956.2 42 44 3611.0 291.9 307.5 1902.4 3705.7 307.8 325.4 1958.1 44 46 3614.1 292.4 308.1 1904.3 3708.9 308.3 326.0 1960.0 46 48 3617.3 293.0 308.7 1906.1 3712.1 308.9 326.6 1961.8 48 50 3620.5 293.5 309.3 1908.0 3715.2 309.4 327 2 1963.V 50 52 3623 6 294.0 309.9 1909.8 3718.4 309.9 327.8 1965.5 52 54 3626.8 294.5 310.5 1911.7 37'21.5 310.5 328.4 1967.4 54 56 3630.0 295.1 311.1 1913.5 3724.7 311.0 329.0 1969.3 56 58 3633.1 295.6 311.7 1915.4 3727.8 311.6 329.6 1971.1 58 60 3636.3 296.1 312.3 1917.3 3731.0 312.2 330.2 1973.0 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 141 38 39 L. C. M. E. T. L. G. M E. T. 3731.0 312.2 330.2 19730 3825.5 328-7 348.7 2029.1 2 3734.1 312.7 330.8 1974.9 3828.6 329.2 349 3 2031.0 2 4 3737.3 313,3 331.4 1976-7 3831.8 329.8 349.9 20329 4 6 3740.4 313.8 332.0 1978-6 3834.9 330-3 350.8 2034.7 6 8 3743.6 314.4 332.6 1980.5 3838.0 330.9 351.2 2036.6 8 10 3746.7 314.9 333.2 1982.3 3841.2 331 5 351.8 2038.5 10 12 3749.9 315.5 333.8 1984.2 3844 3 332.0 352.4 2040 4 12 14 37530 3160 334-5 1986.1 3847.4 332.6 353.1 2042 3 14 16 37562 316.6 335.1 1987.9 3850.5 333.2 353.7 2044.1 16 18 3759.3 317.1 335 7 1989.8 3853 7 333 7 354.3 2046.0 18 20 3762 5 317 7 336 3 1991.7 3856.8 334.3 354.9 2047.9 20 22 3765.6 318 2 336.9 1993-6 3860 334.9 355.6 2049.8 22 24 3768 8 318.8 337.5 1995.4 3863.1 335.4 356.2 2051.7 24 26 3771.9 319.3 338.1 1997.3 3866.2 336-0 356.9 2053.5 26 28 3775.1 319.9 338.7 1999.2 3869.4 336.6 357 5 2055.4 28 30 3778.2 3204 339.4 2001.0 3872.5 337.1 358.1 2057-3 30 32 3781.4 321.0 340.0 2002.9 3875 6 337 7 358.8 2059.2 32 34 3784 5 321.5 340-6 2004.8 3878.8 338.3 359 4 2061.1 34 36 3787.7 322.1 341.2 2006.6 3881.9 338.8 360.1 2063.0 36 38 37308 322.6 341.8 2008.5 3885.0 339 4 360.7 2064.8 38 40 3794.0 3'.'3 2 342.4 2010 4 3888.2 340.0 361.3 2066.7 40 42 3797 1 323.7 343.1 2012.3 3891.3 340.5 362.0 2068 6 42 44 3800.3 3','4.3 343.7 2014.1 3894.4 341 1 362.6 20^0.5 44 46 3803.4 324.8 344.3 2016 3897.6 341.7 363.3 2072.4 46 48 3806.6 325.4 344.9 2017.9 3900 7 342 2 363.9 2074 2 48 50 3809.7 325 9 345.6 2019.7 3903.8 342.8 364.5 2076 1 50 52 3812.9. 326.5 346.2 2021.6 3907.0 343 4 365.2 2078.0 52 54 3816.0 327.0 346.8 2023.5 3910.1 343.9 365.8 2079 9 54 56 3819.2 3-27.6 347.4 2025 4 3913.2 344.5 366.5 2081.8 56 68 3822.3 323.1 3481 2027.2 3916.4 345 1 367.1 2083.7 58 60 3825.5 328.7 34S. 7 ' 2029.1 3919 5 345.6 3(57.7 2085-5 60 40 41 L. C. M. E. T. L. C. M. E. T. 3919.5 345.6 367.7 2085.5 4013.4 3G2.9 387.4 2142.3 2 3922.6 346.1 368.4 2087.4 4016.5 C63.4 388.1 2144.2 2 4 3925.8 346 7 369 2089.3 4019.6 364.0 388.8 2146.1 4 6 3928.9 347.2 369.7 2091.2 4022.7 36J.5 389.4 2148.0 6 8 3932 347 8 370.3 20^3.1 4025.9 365.1 390.1 2149.9 8 10 3935.1 348.4 371.0 2095 4029.0 365.6 390.7 2151.9 10 12 3938.3 348.9 371 6 2096 9 4032.1 366.2 391.4 2158.8 12 14 3941.4 349.5 372.3 2098.8 4035.2 366. 8 302.1 2155.7 14 16 3944.5 350.1 372.9 2i00.7 4038.3 307.4 392.7 2157.6 16 18 3947.7 350.7 373.6 21026 4041.4 368.0 393.4 2159.5 18 20 3950.8 351.3 374.3 2104.5 4044.6 3G8.6 394.1 2161.4 20 22 3953.9 351 8 374.9 2106.3 4047.7 369.2 394.7 2163.3 22 24 3957.1 352.4 375.6 2108.2 4050.8 369.8 395.4 2165.2 24 26 3960.2 353.0 376.2 2110.1 4053.9 370.4 396.1 2167.1 26 28 3963.3 353 6 376.9 21120 4057.0 371.0 396.8 2169.0 28 30 3966.4 354.2 377.5 2113 9 4060.1 371.6 397.5 21709 30 32 3969.6 354.7 378.2 2115.8 4063.3 372.2 398.1 2172.8 32 34 3972.7 355.3 378.8 21177 4066.4 372.8 309.8 2174.7 34 81 3975.8 355.9 379.5 2119 6 4069.5 373.4 399.5 2176.6 36 38 3979.0 356.5 380 1 2121.5 4072.6 374.0 400.2 2178.5 38 40 3982.1 357.1 380.8 2123.4 4075.7 374.6 400.9 2180.4 40 42 3985.2 357 6 381.4 2125 3 4078.8 375.2 401.5 2182.4 42 44 3988.4 358.2 382.1 2127.2 4082.0 375.8 402.2 2184.8 44 46 3991 5 358 8 882.8 2129.1 4085.1 376.4 402.9 2186.2 46 48 3994.6 359.4 383.4 2131.0 4088.2 377.0 403.6 2188.1 48 50 3997 7 860 384.1 2132.9 4091.3 377.6 404.3 2190.0 50 52 4000 9 360 5 384,8 2134.7 4094.4 378.2 404.9 2191.9 52 54 4004.0 361 1 385.4 2136.6 4097.5 378.8 405.6 2193.8 54 56 40071 361.7 386.1 2138.5 4100.7 379.4 406.3 2195.7 56 58 4010.3 362.3 386 8 2140 4 4103.8 380.0 407.0 2197.6 58 60 4013 4 3t52.9 387.4 2142.3 4106.9 380.6 407.7 2199.5 60 142 6. FUNCTIONS OF A ONE-DEGREE CURVE. 42 43 / L. C. M. E. T. L. C. M. E. T. / 4106.9 380.6 407.7 2199.5 4200.1 398.7 428.6 2257.1 2 4110.0 381.2 408.3 2201.4 4203.2 399.3 429.3 2259.0 2 4 4113.1 381.8 409.0 2203.3 4206.3 399.9 430.0 2261.0 4 6 4116.2 382.4 409.7 2205.3 4209.4 400.5 430.7 2262.9 6 8 4119.3 383.0 410.4 2207.2 4212.5 401.1 431.4 2264.8 8 10 4122.4 383.6 411.1 2209.1 4215.6 401.7 432.1 2266.7 10 12 4125.5 384.2 411.8 2211.0 4218.7 402.4 432.8 2268.7 12 14 4128.6 384.8 412.5 2212.9 4221.8 403.0 433.5 2270.6 14 16 4131.8 385 4 413.2 2214.9 4224.9 403.6 434.2 2272.5 16 18 4134.9 38b 413.9 2216.8 4228.0 404.2 434.9 2274.5 18 20 4138.0 386 3 414.6 2218.7 4231.1 404.8 435.6 2276.4 20 22 4141.1 387.2 415.3 2220.6 4234.2 405.4 430. 3 2278.3 22 24 4144.2 387.8 416.0 2222.5 4237.3 406.1 437.0 2280.2 24 26 4147 3 388.4 416.6 2224.4 4240.4 406.7 437.8 2282.2 26 28 4150.4 389.0 417.3 2226.4 4243.5 407.3 438.5 2284.1 28 30 4153.5 389.6 418.0 2228.3 4246.5 407.9 439.2 2286.0 30 32 4156.6 390.2 418.7 2230.2 4249.6 408.5 439.9 2288.0 32 34 4159.7 390.8 419.4 223?. 1 4252.7 409.1 440.6 2289.9 34 36 4162.8 391.4 420.1 2234.0 4255.8 409.8 441.4 2291.8 36 38 4165.9 392.0 420.8 2236.0 4258.9 410.4 442.1 2293.8 38 40 4169.0 392.6 421.5 2237.9 4262.0 411.0 442.8 2295.7 40 42 4172.1 393.2 422.2 2239.8 4265.1 411.6 443.5 2297.7 42 44 4175.2 393.8 422.9 2241.7 4268.2 412.2 444.2 2299.6 44 46 4178.4 394.4 423.6 2243.6 4271.3 412.8 445.0 2301.5 46 48 4181.5 395.0 424.3 2245.6 4274.4 413.5 445.7 2303.5 48 50 4184.6 395.6 425.0 2247.5 4277.5 414.1 446.4 2305.4 50 52 4187.7 396.2 425.7 2249.4 4280.6 414.7 447.1 2307.3 52 54 4190.8 396.8 426.4 2251.3 4283.7 415.3 447.8 2309.3 54 56 4193.9 397.4 427.1 2253.3 4286.8 415.9 448.6 2311.2 56 58 4197.0 398.0 427.8 2255.2 4289.9 416.5 449.3 2313.1 58 60 4200.1 398.7 428.6 2257.1 4293.0 417.2 450.0 2315.1 60 44 45 L. C. M. E. T. L. C. M. E. T. 4293.0 417.2 450.0 2315.1 4385.5 436.2 472.1 2373.4 2 4296 1 417.8 450.7 2317.0 4388.6 436.8 472.9 2375.4 2 4 4299.2 418.4 451.5 2319.0 4391.7 437.5 473.6 2377.3 4 6 4302.2 419.1 452.2 2320.9 4394.7 438.1 474.4 2379.3 6 8 4305.3 419.7 4529 2322.8 4397.8 438.8 475.1 2381.2 8 10 4308.4 420.3 453.7 2324.8 4400.9 439.4 475.9 2383.2 10 12 4311.5 421.0 454.4 2326.7 4404.0 440.0 4766 2385.2 12 14 4314.6 421.6 455.1 2328.7 4407.0 440.7 477.4 2387.1 14 16 4317.7 422.2 455.9 2330.6 4410.1 441.3 478.1 2389.1 16 18 4320.7 422.9 456.6 2332.6 4413.2 442.0 478.9 2391.0 18 20 4323.8 423 5 457.3 2334.5 4416.3 442.6 479.6 2393.0 20 22 4326.9 424.1 458.1 2336.4 4419.3 443.2 480.4 2394.9 22 21 4330.0 424 8 458.8 2338.4 4422.4 443.9 481.1 2396.9 24 26 4333.1 425.4 459.5 23403 4425.5 444.5 481.9 2398.8 26 28 4336.2 426.0 460.3 2342.3 4428.6 445.2 482.6 2400.8 28 30 4339.2 426.7 461 2344.2 4431.6 445.8 483.4 2402.8 30 32 4342.3 427.3 461.7 2346.1 443*. 7 446.4 484.2 2404.7 32 34 4345.4 427 9 462.5 2348.1 4437.8 447.1 484.9 2406.7 34 36 4348.5 428 6 463.2 2350.0 4440.9 447.7 485.7 2408.6 36 38 4351.6 429.2 463.9 2352.0 4444.0 448.3 486.5 2410.6 3u 40 4354.7 429.8 464.7 2353.9 4447.0 448.9 487.2 2412.6 40 42 4357.7 430.5 465.4 2355.9 4450.1 449.5 488.0 2414.5 42 44 4360.8 431.1 466.2 2357.8 4453.2 450.2 488.7 2416.5 44 46 4363.9 431.7 466.9 2359.8 4456.3 450.8 489.5 2418.5 46 48 4367.0 432.4 467.7 2361.7 4459.3 451.5 490. 3 2420.4 48 50 4370.1 433.0 4684 2363.7 4462.4 452.1 491.0 2422.4 50 52 4373.2 433.6 469.1 2365.6 4465.5 452.7 491.8 2424.4 52 54 43762 434.3 469.9 2367.6 4468.6 453.4 492.5 2426.3 54 56 43^9.3 434.9 470. 6 2369.5 4471.6 454.1 493.3 2428.3 56 5b 4382.4 435.6 471.4 23^1.5 4474.7 454.8 494.1 2430.2 58 60 4385.5 436.2 472.1 2373.4 4477.8 455.5 494.8 2432.2 60 6 FUNCTIONS OF A ONE-DEGREE CURVE. 143 46 47 L. C. M. E. T. L. C. M. E. T. / 4477.8 455.5 494.8 2432.2 4569.7 475.2 518.3 2491.5 "~0~ 2 4480.9 456.1 495.6 2434.2 4572.7 475.9 519.0 2493.4 2 4 4483.9 456.8 496.5 2436.1 4575.8 476.5 519.8 2495.4 4 6 4487.0 457.4 497.2 2438.1 4578.8 477.2 520.6 2497.4 6 8 4490.0 458.1 497.9 2440.1 4581.9 477.8 521.4 2499.4 8 10 4493.1 458.7 498.7 2442.1 4584.9 478.5 522.2 2501.4 10 12 4496.2 459.4 499.5 2444.0 4588.0 479.2 523.0 2508.4 12 14 4499.2 460.0 500.3 2446.0 4591.0 479.8 523.8 2505.4 14 16 4502.3 460.7 501.0 2448.0 45P4.1 480.5 524.6 2507.8 16 18 4505.4 461.3 501.8 2449.9 4597.1 481.1 525.4 2509.8 18 20 4508.4 462.0 502.6 2451.9 4600.2 481.7 526.2 2511.3 20 22 4511.5 462.7 503.4 2453.9 4603.2 482.3 527.0 2513.3 22 24 4514.6 463.3 504.1 2455.9 4606.3 483.0 527.8 2515.3 24 26 4517.6 464.0 504.9 2457.8 4609.3 483.7 528.6 2517.3 26 28 4520.7 464.6 505.7 2459.8 4612.4 484.8 529.4 2519.3 28 80 4523.7 465.3 506.5 2461.8 4615.4 485.0 530.2 2521.2 30 32 4526.3 466.0 507.3 2463. S 4618.5 485.7 531.0 2523.2 32 34 4529.9 466.6 508.0 2465.7 4621.5 486.3 531.8 2525.2 34 36 4532.9 467.3 508.8 2467.7 4624.6 487.0 532.6 2527.2 86 88 4536.0 467.9 509.6 2469.7 4627.6 487.7 533.4 2529.2 38 40 4539.1 468.6 510.4 2471.7 4630.7 488.4 534.2 2531.2 40 42 4542.1 469.3 511.1 2473.6 4633.7 489.1 535.0 2533.2 42 44 4545.2 469.9 511.9 2475.6 4G36.8 489.8 535.8 2535.2 44 46 4548.2 470.6 512.7 2477.6 4639.8 490.5 536.6 2537.2 46 48 4551.3 471.2 513.5 2479.6 4642.9 491.2 537.4 2539.2 48 50 4554.4 471.9 514.3 2481.6 4645.9 491.9 538.2 2541.2 50 52 4557.4 472.6 515.1 2483.5 4649.0 492.6 539.0 2543.1 52 54 4560.5 473.2 515.9 2485.5 4652.0 493.3 539.8 2545.1 54 56 4563.6' 473.9 516.7 2487.5 4655.1 494.0 540.6 2547.1 56 58 4566.6 474.5 517.5 2489.5 4658.1 494.7 541.4 2549.1 58 60 4569.7 475.2 518.3 2491.5 4661.2 495.4 542.3 2551.1 60 48 49 / L. C. M. E. T. L. C. M. E. T. 4661.2 495.4 542.3 2551.1 4752.3 515.9 567.0 2611.3 2 46C4.2 496.0 543.1 2553.1 4755.3 516.5 567.8 2613.3 2 4 4667.3 496.7 543.9 2555.1 4758.4 517.2 568.7 2615.3 4 6 4670.3 497.4 544.7 2557.1 4761.4 517.9 569.5 2617.3 6 8 4673.3 498.1 545.5 2559.1 4764.4 518.6 570.3 2619.3 8 10 4676.4 498.8 546.4 2561.1 4767.4 519.3 571.2 2621.4 10 12 4679.4 499.4 547.2 2563.1 4770.5 520.0 572.0 2623.4 12 14 4682.5 500.1 548.0 2565.1 4773.5 520.7 572.8 2625.4 14 16 4685.5 500.8 548.8 2567.1 4776.5 521.4 573.7 2627.4 16 18 4688.5 501.5 549.6 2569.1 4779.6 522.1 574.5 2629.4 18 20 4691.6 502.2 550.5 2571.1 4782.6 522.8 575.3 2631.4 20 22 4694.6 502.8 551.3 2573.1 4785.6 523.5 576.2 2633.5 22 24 4697.6 503.5 552.1 2575.1 4788.7 524.2 577.0 2635.5 24 26 4700.7 504.2 552.9 2577.1 4791.7 524.9 677.9 2637.5 26 28 4703.7 504.9 553.7 2579.1 4791.7 525.6 578.7 2639.5 28 30 4706.7 505.6 554.6 2581.1 4797.7 526.3 579.6 2641.5 30 32 4709.8 506.2 555.4 2583.1 4800.8 527.0 580.4 2643.5 32 34 4712.8 506.9 556.2 2585.1 4803.8 527.7 581.3 2645.6 34 36 4715.9 507.6 557.0 2587.2 4806.8 528.4 582.1 2647.6 36 38 4718.9 508.3 557.8 2589.2 4809.9 529.1 583.0 2649.6 38 40 4721.9 509.0 558.7 2591.2 4812.9 529.8 583.8 2651.6 40 42 47-25.0 509.6 559.5 2593.2 4815.9 530.5 584.7 2653.7 42 44 4728.0 510.3 560.3 2595.2 4819.0 531.2 585.5 2655.7 44 46 4731.0 511.0 561.2 2597.2 4822.0 531.9 586.4 2657.7 46 48 4734.1 511.7 562.0 2599.2 4825.0 532.6 587.2 2659.7 48 50 4737.1 512.4 562.8 2601.2 4828.0 533.3 588.1 2661.8 50 52 4740.2 513.1 563.7 2603.2 4881.1 534.0 588.9 2663.8 52 54 4743.2 513.8 564.5 2605.2 4834.1 534.7 589.8 2665.8 54 56 4746.2 514.5 565.3 2607.2 4837.1 535.4 590.6 2667.8 56 58 4749.3 515.2 566.2 2609.3 4840.2 536.1 591.5 2669.9 58 60 4752.3 515.9 567.0 2611.3 4843.2 586.8 592.4 2671.9 60 144 6. -FUNCTIONS OF A ONE-DEGREE CURVE. 60 61 L. C. M. E. T. L. C. M. E. T. / 4843.2 536.8 592.4 2671. 9 4933 6 558.2 618.5 2733.0 2 4846.2 537.5 593.2 2673 9 4936.6 558.9 619.3 2735.1 2 4 4849.2 538.2 594.1 2676.0 4939.6 559.7 620.2 2737.1 4 6 4852.2 538.9 594.9 2678.0 4942.6 560.4 621.1 2739.2 6 8 4855.2 539.6 595.8 2680.0 4945.6 561.1 622.0 2741.2 8 10 4858.3 540.3 596.7 2682.1 4948.6 561.8 622.9 2743.3 10 12 4861.3 541.0 597.5 2684. 1 41)51.6 562.5 623.7 2745.3 12 14 4864.3 541. V 598.4 2686.1 4954.6 563.3 624.6 2747.4 14 16 4867.3 542.4 599.3 2688.2 4957.6 584.0 625.5 2749.4 16 18 4870.3 543.1 600.1 2690.2 4960.6 564.7 626.4 2751.5 18 20 4873.3 543.9 601.0 2692.3 49G3.6 565.4 627.3 2753.5 20 22 4876.3 544.6 601.9 2694.3 4966.6 566.2 628.2 2755.6 22 24 4879.4 545.3 602.7 2696.3 4969.6 566.9 629.9 2757.7 24 26 4882.4 546.0 603.6 2698.4 4972.6 567.6 630.0 2759.7 26 28 4885.4 546.7 604.5 2700.4 4975.6 568.3 630.9 2761.8 28 30 4888.4 547.4 605.3 2702.4 4978.6 569.1 631.8 2763.8 30 32 4891.4 548.1 606.2 2704.5 4981.6 569.8 632.7 2765.9 32 34 4894.4 548.8 607.0 2706.5 4984.6 570.5 633.6 2767.9 34 36 4897.4 549.5 607.9 2708.6 4987.7 571.2 634.5 2770.0 36 38 4900.4 550.2 608.8 2710.6 4990.7 572.0 635.3 2772.0 38 40 4903.5 551.0 609.7 2712.6 4993.7 572.7 636.2 2774.1 40 42 4906.5 551.7 610.5 2714.7 4996.7 573.4 637.1 2776.2 42 44 4909.5 552.4 611.4 2716.7 4999.7 574.1 638.0 2778.2 44 46 4912.5 553.1 612.3 2718.8 5002.7 574.9 638.9 2780.3 46 48 4915.5 553.8 613.2 2720.8 5005.7 575.6 639.8 2782.3 48 50 4918.5 554.5 614.1 2722.8 5008.7 5T6.3 640.7 2784.4 50 52 4921.5 555.2 614.9 2724.9 5011.7 577.0 611.6 2786.4 52 54 4924.6 555.9 615.8 2726.9 5014.7 577.8 642.5 2788.5 54 56 4927.6 556.6 616.7 2729.0 5017.7 578.5 643.4 2790.6 56 58 4930.6 557.4 617.6 2731.0 5020.7 579.2 644.3 2792.6 58 60 4933.6 558.2 618.5 2733.0 5023.7 579.9 645.2 2794.7 60 52 53 L. C. M. E. T. L. C. M. E. T. 5023.7 579.9 645.2 2794.7 5113.5 602.0 672.7 2856.9 2 5026.7 580.6 6J6.1 2796.8 5116.5 602.8 673.7 2858.9 2 4 5029.7 581.3 647.0 2798.8 5119.4 603.5 674.6 2861.0 4 6 5032.7 582.1 647.9 2800.9 5122.4 604.3 675.5 2863.1 6 8 5035.7 582.8 648.9 2803.0 5125.4 605.0 676.4 2865.2 8 10 5038.7 583.5 649.8 2805.0 5128.4 605.8 677.4 2867.3 10 12 5041.7 584.3 650.7 2807.1 5131.3 606.5 678.3 2869.4 12 14 5044.7 585.0 651.6 2809.2 5134.3 607.3 6r9.2 2871.5 14 16 5047.7 585.7 652.5 2811.2 5137.3 608.0 680.2 2873.5 16 18 5050.7 586.5 653.4 2813.3 5140.3 608.8 681.1 2875.6 18 20 5053.6 587.2 654.3 2815.4 5143.2 609.5 682.0 2877.7 20 23 5056.6 587.9 655.2 2817.4 5146.2 610.3 683.0 2879.8 22 24 5059.6 588.7 656.2 2819.5 5149.2 611.0 683.9 2881.9 24 26 5062.6 589.4 657.1 2821.6 5152.1 611.8 684.9 2884.0 26 28 5065.6 590.1 658.0 2823.6 5155.1 612.5 685.8 2886.1 28 30 5068.6 590.9 658.9 2825.7 5158.1 613.3 686.7 2888.1 30 32 5071.6 591.6 659.8 2827.8 5161.1 614.0 687.7 2890.2 32 34 5074.6 592.3 660.7 2829.8 5164.0 614.8 688.6 2892.3 34 36 5077.6 593.1 661,6 2831.9 5167.0 615.5 689.6 2894.4 36 ' 38 5080.6 593.8 662.5 2834.0 5170.0 616.3 690.5 2896.5 38 40 5083.6 594.5 663.5 2836.1 5173.0 617.0 691.5 2898.6 40 42 5086.6 595.3 664.4 2838.2 5175.9 617.8 692.4 2900.7 42 44 5089.6 596.0 665.3 2840.2 5178.9 613.5 693.4 2902.8 44 46 5092.6 596.7 666.2 2842.3 5181.9 619.3 694.3 2904.9 46 48 5095.6 597.5 667.2 2844.4 5184.9 620.1 695.3 2907.0 48 50 5098.6 598.2 668.1 2846.5 5187.8 620.8 696.2 2909.1 50 52 5101.6 598.9 669.0 2848.5 5190.8 621.5 697.1 2911.2 52 54 5104.6 599.7 669.9 2850.6 5193.8 622.3 698.1 2913.3 54 56 5107.6 600.4 670.9 2852.7 5196.7 623.0 699.0 2915.4 56 58 5110.6 601.2 671.8 2854.8 5199.7 623.8 700.0 2917.5 58 60 5113.5 602.0 672.7 2856.9 5202.7 624.6 700.9 2919.5 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 145 64 65 L. C. M. E. T. L. C, M. E. T. 5202.7 624.6 700.9 2919.5 5291.7 647.4 729.9 2982.8 2 5205.7 625.4 701.9 2921.6 5294.6 648.1 730.9 2984.9 2 4 5208.6 626.1 702.8 2923.8 5297.6 648.9 731.9 2987.1 4 6 5211.6 626.9 703.8 2925.9 5300.5 649.6 732.9 2989.2 6 8 5214.6 627.6 704.8 2928.0 5303.5 650.4 733.8 2991.3 8 10 5217.5 628.4 705.7 2930.1 5306.4 651.2 734.8 2993.4 10 12 5220.5 629.2 706.7 2932.2 5309.4 652.0 735.8 2995.5 12 14 5223.5 629.9 707.7 2934.3 5312.3 652.7 736.8 2997.7 14 16 5226.4 630.7 708.6 2936.4 5315.3 653.5 737.8 2999.8 16 18 5229.4 631.4 709.6 2938.5 5318.2 654.3 738. 7 3001.9 18 20 5232.4 632.2 710.5 2940.6 5321.2 655.1 739.7 3004.0 20 22 5235.3 633.0 711.5 2942.7 5324.1 655.8 740.7 3006.2 22 24 5238.3 633.7 712.5 2944.8 5327.1 656.6 741.7 3008.3 24 26 5241.3 634.5 713.4 2946.9 5330.0 657.4 742.7 3010.4 26 28 5244.2 635.2 714.4 2949.0 5333.0 658.2 743.7 3012.5 28 30 5247.2 636.0 715.3 2951.1 5335.9 658.9 744.7 3014.7 30 32 5250.2 636.8 716.3 2953.2 5338.8 659.7 745.7 3016.8 32 34 5253.1 637.5 717.3 2955.3 5341.8 660.5 746.7 3018.9 34 36 5256.1 638.3 718.2 2957.5 5344.7 661.3 747.7 3021.1 36 38 5259.1 639.0 719.2 2959.6 5347.7 662.0 748.7 3023.2 38 40 5262.0 639.8 720.2 2961.7 5350.6 662.8 749.7 3025.3 40 42 5265.0 640.6 721.1 2963.8 5353.6 663 6 750.7 3027.5 42 44 5268.0 641.3 722.1 2965.9 5356.5 664.4 751.7 3029.6 44 46 5270.9 642.1 723.1 2968.0 5359.5 665.1 752.6 3031.7 46 48 5273.9 642.8 724.1 2970.1 5362.4 665.9 753.6 3033.8 48 50 5276.9 643.6 725.0 2972.2 5365.4 666,7 754.6 3036.0 50 52 5279.8 644.4 726.0 2974.4 5368.3 667.5 755.6 3038.1 52 54 5282.8 645.1 727.0 2976.5 5371.3 668.3 756.6 3040.2 54 56 5285.8 645.9 728.0 2978.6 5374.2 669.1 757.6 3042.4 56 58 5288.7 646.6 729.0 2980.7 5377.2 669.9 758.6 3044.5 58 60 5291.7 647.4 729 9 2932.8 5380.1 670.7 759.6 3046.6 60 66 57 L. C. M. E. T. L. C. M. E. T. / 5380.1 670.7 759.6 3016.6 5468.2 694.4 790.2 3111.1 2 5383.0 671.4 760.6 3048.8 5471.1 695.2 791.2 3113.3 2 4 5386.0 672.2 761.6 3050.9 5474.0 696.0 792.2 3115.4 4 6 5388.9 672.9 762.7 3053.1 5477.0 696.8 793.3 3117.6 6 8 5391.8 673.7 763.7 3055.2 5479.9 697.6 794.3 3119.7 8 10 5394.8 674.4 764.7 3057.4 5482.8 698.4 795.3 3121.9 10 12 5397.7 675.2 765.7 3059.5 5485.7 699.2 796.3 3124.1 12 14 5400.7 676.0 766.7 3061.6 5488.7 700.0 797.4 3126.2 14 16 5403.6 676.8 767.7 3063.8 5491.6 700.8 798.4 3128.4 16 18 5406.5 677.6 768.7 3065.9 5494.5 701.6 799.4 3130.6 18 20 5409.5 678.4 769.7 3068.1 5497.4 702.4 800.5 3132.7 20 22 5412.4 679.2 770.8 3070.2 5500.3 703.2 801.5 3134.9 22 24 5415.3 680.0 771.8 30^2.4 5503.3 704.0 802.6 3137.0 24 26 5418.3 680.8 772.8 3074.5 5506.2 704.8 803.6 3139.2 26 28 5421.2 681.6 773.8 3076.6 5509.1 705.6 804.7 3141.4 28 30 5424.1 682.4 774.8 3078.8 5512.0 706.4 805.7 3143.5 30 32 5427.1 683.2 775.8 3080.9 5515.0 707.2 806.8 3145.7 32 34 5430.0 684.0 776.8 3083.1 5517.9 708.0 807.8 3147.9 34 36 5433.0 684.8 777.8 3085.2 5520.8 708.8 808.8 3150.0 36 38 5435.9 685.6 778.9 3087.4 5523.7 709.6 809.9 3152.2 38 40 5438.8 686.4 779.9 3089.6 55-26.7 710.4 810.9 3154.4 40 42 5441.8 687.2 780.9 3091.7 5529.6 711.2 812.0 3156.6 42 44 5444.7 688.0 781.9 3093.9 5532.5 712.0 813.0 3158.7 44 46 5447.6 688.8 783.0 3096.0 5535.4 712.8 814.1 3160.9 46 48 5450.6 689.6 784.0 3098.2 5538.4 713.6 815.1 3163.1 48 50 5453.5 690.4 785.0 3100.3 5541.3 714.4 816.2 3165.3 50 52 5456.5 691.2 786.0 3102.5 5544.2 715.2 817.2 3167.4 52 54 5459.4 692.0 787.1 3104.6 5547.1 716.0 818.3 3169.6 54 56 5462.3 692.8 788.1 3106.8 5550.0 716.8 819.3 3171.8 56 58 5465.3 693.6 789.1 3108.9 5553.0 717.6 820.4 3174.0 58 60 5468.2 694.4 790.2 3111.1 5555.9 718.4 821.4 3176.1 60 146 6. FUNCTIONS OF A ONE-DEGREE CURVE. 68 69 L. C. M. E. T. L. C. M. E. T. 5555.9 718.4 821.4 3176.1 5643.1 742.8 853.5 3241.9 2 5558.8 719.2 822.5 3178.3 5646.0 743.6 854.6 3244.1 2 4 5561.7 720.0 823.5 3180.5 5648.9 744.4 855.7 3246.3 4 6 5564.6 720.8 824.6 3182.7 5651.8 745.3 856.8 3248.5 6 8 5567.5 721.6 825.7 3184.9 5654.7 746.1 857.9 3250.7 8 10 5570.4 7224 826.7 3187.1 5657.6 746.9 859.0 3252.9 10 12 5573.3 723.2 827.8 3189.2 5660.5 747.7 860.0 3255.1 12 14 5576.2 724.0 828.9 3191.4 5663.4 748.6 861.1 3257.3 14 16 5579.2 724.8 829.9 3193.6 5666.3 749.4 862.2 3259.5 16 18 5582.1 725.6 831.0 3195.8 5669.2 750.2 863.3 3261.7 18 20 5585.0 726.5 832.1 3198.0 5672.1 751.1 864.4 3263.9 20 22 5587.9 727.3 833.1 3200.2 5675.0 751.9 865.5 3266.1 22 24 5590.8 728.1 834.2 3202.4 5677.9 752.7 866.6 3268 3 24 26 5593.7 728.9 835.3 3204.5 5680.8 753.5 867.7 3270.5 26 28 5596.6 729.7 836.3 3206.7 5683.7 754.4 868.8 3272.7 28 30 5599.5 730 5 837.4 3208.9 5686.5 755.2 869.9 3274.9 30 32 5602.4 731.3 838.4 3211.1 5689.4 756.0 871.0 3277.1 32 34 5605.3 732.1 839.5 3213.3 5692.3 756.9 872.1 3279.4 34 36 5608.2 732.9 840.6 3215.5 5695.2 757.7 873.2 3281 6 36 38 5611.1 733.7 841.6 3217.7 5698.1 758.5 874.3 3283.8 38 40 5614.0 734.6 842 7 3219.9 5701.0 759.4 875.4 3286.0 40 42 5616.9 735.4 843.8 3222.1 5703.9 760.2 876.5 3288.2 42 44 5619.8 736.2 844.9 3224.3 5706.8 761.0 877.6 3290.5 44 46 5622.8 737.0 846.0 3226.5 57097 761.9 878.7 3292.7 46 48 5625.7 737.8 847.0 3228.7 5712 6 762.7 879.8 3294.9 48 50 5628.6 738.6 848.1 3230.9 5715.5 763.5 880.9 3297.1 50 52 5631.5 739.4 849.2 3233.1 5718.4 764.4 882.0 3299.3 52 54 5634.4 740.2 850.3 3235.3 5721.3 765.2 883.1 3301.5 54 56 5637.3 741.0 851.4 3237.5 5724.2 766.0 884.2 8303.8 56 58 5640.2 741.9 852.5 32-39.7 5727.1 766.8 885.3 3306.0 58 60 5643.1 742.8 853.5 3241.9 5730.0 767.7 886.4 3308.2 60 60 61 L. C. M, E. T. L. C. M. E. T. 5730.0 767.7 886.4 3303.2 5816.4 792.9 920.2 3375.2 2 5732.9 768.5 887.5 3310.4 5*19.3 793.7 921.4 3377.4 2 4 5735.8 769.4 888.7 3312.7 5822.1 794.6 922.5 3379.7 4 6 5738.6 770.2 889.8 3314.9 5825.0 795.4 923.6 3381.9 6 8 5741.5 771.1 890.9 3317.1 5827.9 796.3 924.8 3384,2 8 10 5744.4 771.9 892.0 3319.3 5830.7 797.1 9:25.9 3386. 4 10 13 5747.3 772.7 893.1 3321.6 5833.6 798.0 927.1 3388.7 12 14 5750.2 773.6 894.3 3323.8 5836.5 798.8 928.2 3390.9 14 16 5753.0 774.4 895.4 3326.0 5839.3 799.7 9'29.3 3393.2 16 18 5755.9 775.3 896.5 3328.3 5842.2 800.5 930.5 3395.4 18 20 5758.8 776.1 897.6 3330.5 5845.1 801.4 931.6 3397.7 20 22 5761.7 776.9 898.8 3332.7 5847.9 802.2 932.8 8399.9 22 24 5764.6 777.8 899.9 3334.9 5850.8 803.1 933.9 3402.2 24 26 5767.4 778.6 901.0 3337.2 5853.7 803.9 935.1 3404.4 26 28 5770.3 779.5 902.1 3339.4 5856.5 804.8 936.3 3406.7 28 30 5773.2 780.8 903.2 3341.6 5859.4 805.6 937.4 3408.9 30 32 5776.1 781.1 904.4 3343.9 5862.3 806.5 938.6 3411.2 32 34 5779.0 782.0 905.5 3346.1 5865.1 807.3 939.7 3413.5 34 36 5781.8 782.8 906.6 3348.3 5868.0 808.2 940.9 3415.7 36 38 5784.7 783.7 907.7 3350.6 5870.9 809.0 942.1 3418.0 38 40 5787.6 784.5 908.8 3352.8 5873.7 809.9 943.2 3420.3 40 42 5790.5 785.3 910.0 3355.0 5876.6 810.7 944.4 3422.5 42 44 5793.4 786.2 911.1 3357.3 5879.5 811.6 945.5 3424.8 44 46 5796.2 787.0 912.3 3359.5 5882.3 812.4 946.7 3427.1 46 48 5799.1 787.9 913.4 3361.8 5885.2 813.3 947.8 3429.3 48 50 5802.0 788.7 914.5 3364.0 5888.1 814.1 949.0 3431.6 50 52 5804.9 789.5 915.7 3366.2 5890.9 815.0 950.2 3433.9 52 54 5807.8 790.4 916.8 3368.5 5893.8 815.8 951.3 3436.1 54 56 5810.6 791.2 918.0 3370.7 5896.7 816.7 952.5 3438.4 56 58 5813.5 792.1 919.1 3373.0 5899.5 817.5 953.6 3440.7 58 L 60 5816.4 792.9 920.2 3375.2 5902.4 818.4 954.8 3442.9 60 , 6. FUNCTIONS OF A ONE-DEGREE CURVE. 147 62 63 / L. C. M. E. T. L. C. M. E. T. 5902.4 818.4 954.8 3442.9 5987.8 844.4 990.3 3511.3 2 5905.2 819.3 956.0 3445.2 5990.6 845.3 991.5 3513.6 2 4 5908.1 820.1 957.2 3447.5 5993.5 846.2 992.7 3515.9 4 6 5910.9 821.0 9583 3449.7 5996.3 847.1 993.9 3518.2 6 8 5913.8 821.8 959.5 3452.0 5999.1 847.9 995.1 3520.5 8 10 5916.6 822.7 960.7 3454.3 6002.0 848.8 996.3 3522.8 10 12 5919.5 823.6 961.9 3456.6 6004.8 849.7 997.5 3525.1 12 14 5922.3 824.4 963.0 3458.8 6007.7 850.6 998.7 3527.4 14 16 5925.2 825.3 964.2 3461.1 6010.5 851.4 999.9 3529.7 16 is 5928.0 826.1 965.4 3463.4 6013.3 852.3 1001.1 3532.0 18 20 5930.9 827.0 966.6 3465.7 6016.2 853.2 1002.3 3534.3 20 22 5933.7 827.9 967.8 3167.9 6019.0 854.1 1003.5 3536.6 22 24 5936.6 828.7 968.9 3470.2 6021.8 854.9 1004.7 3538.9 24 26 5939.4 829.6 970.1 3472.5 6024.7 855.8 1005.9 3541.2 26 28 5942.3 830.4 971.3 3474.7 6027.5 856.7 1007.1 3543.5 28 30 5945.1 831.3 972.5 3477.0 6030.3 857.6 1008.4 3545.8 30 82 5947.9 832.2 973.6 3479.3 6033.2 858.4 1009.6 3548.1 32 34 5950.8 833.0 974.8 3481.6 6036. 859.3 1010.8 3550.4 34 36 5953.6 833.9 976.0 3483.9 6038.9 860.2 1012.0 3552.7 36 38 5956.5 834.7 977.2 3486.2 6041.7 861.1 1013.2 3555.0 38 40 5959.3 835.6 978.4 3488.5 6044.5 861.9 1014.5 3557.3 40 42 5962.2 836.5 979.6 3490.7 6047.4 862.8 1015.7 3559.6 42 44 5965.0 837.4 980.8 3493.0 6050.2 863.7 1016.9 3562.0 44 46 5967.9 838.3 982.0 3495.3 6053.0 864.6 1018.1 3564.3 46 48 5970.7 839.1 983.2 3497.6 0055.9 865.4 1019.3 3566.6 48 50 5973.6 840.0 984.4 3499.9 60f>8.7 866.3 1020.6 3568.9 50 52 5976.4 840.9 985.5 3502.2 6061.6 867.2 1021.8 3571.2 52 54 5979.3 84f.7 986.7 3504.5 60(54.4 868.1 1023.0 3573.5 54 56 5982.1 842.6 9S7.9 3506.8 6067.2 868.9 1024.2 3575.8 56 58 5985.0 843.5 989.1 3509.0 6070.1 869.8 1025.4 3578.1 58 60 5987.8 844.4 990.3 3511.3 6072.9 870.7 1026.7 3580.4 60 64: 65 L. C. M. ^ E. T. L. C. M. E. T. 6072.9 870.7 1026.7 3580.4 6157.5 897.3 1064.0 3650.4 2 6075.7 871.5 1027.9 3582.8 6160.3 898.2 1065.2 3652.8 2 4 6078.5 872.4 1029.2 3585.1 6163.1 899.1 1066.5 3655.1 4 6 6081.4 873.3 1030.4 3587.4 6165.9 900.0 1067.7 3657.5 6 8 6084.2 874.2 1031.7 3589.7 6168.7 900.9 1069.0 3659.8 8 10 6087.0 875.1 1032.9 3592.1 6171.5 901.8 1070.2 3662.2 10 12 6089.8 875.9 1034.1 3594.4 6174.3 902.7 1071.5 3664.5 12 14 6092.6 876.8 1035.4 3596.7 6177.1 903.6 1072.7 3666.9 14 16 6095.5 877.7 1036.6 3599.1 6179.9 904.5 1074.0 3669.2 16 18 6098.3 878.6 1037.9 3601.4 6182.7 905.4 1075.2 3671.6 18 20 6101.1 879.5 1039.1 3603.7 6185.5 906.3 1076.6 3673.9 20 22 6103.9 880.3 1040.3 3606.0 6188.3 907.2 1077.8 3676.2 22 24 6106.7 881.2 1041.6 3608.4 6191.1 908.1 1079.1 3678.6 24 26 6109.6 882.1 1042.8 3610.7 6193.9 909.0 1080.4 3680.9 26 28 6112.4 883.0 1044.1 3613.0 6196.7 909.9 1081.7 3683.3 28 30 6115.2 883.9 1045.3 3615.3 6199.5 910.8 1083.0 3685.6 30 32 6118.0 884.7 1046.5 3617.7 6202.3 911.7 1084.2 3688.0 32 34 6120.8 8S5.6 1047.8 3620.0 6205.1 912.6 1085.5 3690.4 34 36 6123.7 886.5 1049.0 3622.3 6208.0 913.5 1086.8 3692.7 36 38 6126.5 887.4 1050.3 3624.7 6210.8 914.4 1088.1 3695.1 38 40 6129.3 888.3 1051.5 3627.0 6213.6 915.3 1089.4 3697.4 40 42 6132.1 889.2 1052.7 3629.4 6216.4 916.2 1090.6 3699.8 42 44 6134.9 890.1 1054.0 3631.7 6219.2 917.1 1091.9 3702.2 44 46 6137.8 891.0 1055.2 3634.0 6222.0 918.0 1093.2 3704.5 46 48 6140.6 891.9 1056.5 3636.4 6224.8 918.9 1094.5 3706.9 48 50 6143.4 892.8 1057.7 3638.7 6227.6 919.8 1095.8 3709-3 50 52 6146.2 893.7 1059.0 3641.1 6230.4 920.7 1097.0 3711.6 52 54 6149.0 894.6 1060.2 3643.4 6233.2 921.6 1098.3 3714.0 54 56 6151.9 895.5 1061.5 3645.7 6236.0 922.5 1099.6 3716.3 56 58 6154.7 896.4 1062.7 3648.1 6238.8 923.4 1100.9 3718.7 58 60 6157.5 897.3 1064.0 3650.4 6241.6 924.3 1102.2 3721.1 60 '148 6. -FUNCTIONS OF A ONE-DEGREE CURVE. ! , 66 67 L. C. M. E. T. L. C. M. E. T. 6241.6 924.3 1102.2 3721.1 6325.2 951.8 1141.5 3792.6 2 6244.4 925.2 1103.5 3723.4 6328.0 952.7 1142.8 3795.0 2 4 6247.2 926.1 1104.8 3725.8 6330.7 953.6 1144.1 3797.4 4 6 6250.0 927.0 1106.1 3728.2 6333.5 954.5 1145.4 3799.8 6 8 6252.7 927.9 1107.4 3730.6 6336.3 955.5 1146.7 3802.2 8 10 6255.5 928.8 1108.7 3732.9 6339.0 956.4 1148.1 3804.6 10 12 6258.3 929.8 1110.0 3735.3 6341.8 957.3 1149.4 3807.0 12 14 6261.1 930.7 1111.3 3737.7 6344.6 958.2 1150.7 3809.4 14 16 6263.9 931 6 1112.6 3740.1 6347.4 959.2 1152.0 3811.8 16 18 6266.7 932.5 1113.9 3742.4 6350.1 960.1 1153.3 3814.2 18 20 6269.5 933.4 1115.2 3744.8 6352.9 961.0 1154.7 3816.6 20 22 6272.3 934.3 1116.5 3747.2 6355.7 961.9 1156.0 3819.0 22 24 6275,0 935.3 1117.8 3749.6 6358.4 962.9 1157.4 3821.4 24 26 6277.8 936.2 1119.1 3751.9 6301.2 963.8 1158.7 3823.8 26 28 6280.6 937.1 1120.4 3754.3 6364.0 964.7 1160.1 3826.2 28 30 6283.4 938.0 1121.7 3756.7 6366.7 965.6 1161.4 3828.6 30 32 6286.2 938.9 1123.0 3759.1 6369.5 966.6 1162.8 3831.0 32 34 6289.0 939.8 1124.3 3761.5 0372.3 907.5 1164.1 3833.4 34 36 6291.8 940.8 1125.6 3763.9 6375.1 968.4 1165.5 3835.9 36 38 6294.5 941.7 1126.9 3766.3 6377.8 909.3 1166.8 3838.3 38 40 6297.3 942 6 1128.3 3768.7 6380.6 970.3 1168.2 3840.7 40 42 6300.1. 943.5 1129.6 3771.0 6383.4 971.2 1169.5 3843.1 42 44 6302.9 944.4 1130.9 3773.4 6386.1 972.1 1170.9 3845.5 44 46 6305.7 945.3 1132.2 3775.8 6388.9 973.0 1172.2 3847.9 46 1.48 6308.5 946.3 1133.5 3778.2 6391.7 974.0 1173.6 3850.4 48 50 6311.3 947.2 1134.9 3780.6 6394.4 974.9 1174.9 3852.8 50 52 6314.1 948.1 1136.2 3783.0 6397.2 975.8 1176.3 3855.2 52 54 6316.8 949.0 1137.5 3785.4 6400.0 976.8 1177.6 3857.6 54 56 6319.6 949.9 1138.8 3787.8 6402.8 977.7 1179.0 3860.0 56 58 6322.4 950.8 1140.1 3790.2 6405.5 978.6 1180.3 3862.5 58 60 6325.2 951.8 1141.5 3792.6 6408.3 979.6 1181.6 3864.9 60 68 69 L. C. M. E. T. L. C. M. * E. T. 6408.3 979.6 1181.6 3864.9 6491.1 1007.7 1222.9 3938.1 2 6411.1 980.5 1183.0 3807.3 6493.8 1008.7 1224.3 3940.6 2 4 6413.8 981.4 1184.4 3869.7 6496.6 1009.6 1225.7 3943.0 4 6 6416. 6 982.4 1185.7 3372.2 6499.3 1010.6 1227.1 3945.5 6 8 G419.3 983.3 1187.1 3874.6 6502.1 1011.5 1228.5 3947.9 8 10 61-22.1 984.2 1188.5 3877.0 6504.8 1012.5 1229.9 3950.4 10. 12 0421. 9 985.2 1189.8 3879.5 6507.5 1013.4 1231.3 3952.9 12 14 6427.6 986.1 1191.2 3881.9 6510.3 1014.4 1232.7 3955.3 14 10 6430.4 987.0 1192.6 3884.3 6513.0 1015.3 1234.1 3957.8 16 18 6433.1 988.0 1193.9 3886.8 6515.8 1016.3 1235.5 3960.2 18 20 6435.9 988.9 1195.3 3889.2 6518.5 1017.2 1236.9 3962.7 20 22 6438.7 980.8 1196.7 3891.6 6521.2 1018.2 1238.3 3965.2 22 24 6141.4 990.8 1198.0 3894.1 6524.0 1019.1 1239.7 3967.6 24 26 6444.2 991.7 1199.4 3896.5 0520.7 1020.1 1241.1 3970,1 26 28 6140.9 992.6 1200.8 3898.9 6529.5 1021.0 1242.5 3972.5 28 30 6149.7 99J. 6 1202.1 3901.4 0532.2 1022.0 1243.9 3975.0 30 32 6452.5 994.5 1203.5 3903.8 0534.9 10. '2. 9 1245.3 3977.5 32 34 6455.2 995.4 1204.9 3906.3 0537.7 1023.9 1246.7 3980.0 34 36 0458.0 996.4 1206.2 3908.7 0540.4 1024.8 1248.1 3982.4 36 38 6400.7 997.3 1207.6 3911.2 6543.2 1025.8 1249.5 3984.9 38 40 6463.5 908.2 1209.0 3913.6 0545.9 1026.7 1250.9 3987.4 40 42 0466.3 999.2 1210.3 3916.1 6548.6 1027.7 1252.3 3989.9 42 44 6469.0 1000.1 1211.7 3918.5 6551.4 1028.6 1253.7 3993.3 44 46 6471.8 1001.0 1213.1 3921.0 CS54.1 1029.6 1255.1 3994.8 46 48 6474.5 1002.0 1214.5 3923.4 6550.9 1030.5 1256.5 3997.3 48 50 6477.3 1002.9 1215.9 3925.9 6559.6 1031.5 1257.9 3999.8 50 52 6480.1 1003.8 1217.3 3928.3 6562.3 1032.4 1259.3 4002.2 52 54 6482.8 1004.8 1218.7 3930.8 6565.1 1033.4 1200.7 4004.7 54 56 6485.6 1005.7 1220.1 3933.2 6567.8 1034.3 1262.1 4007.2 56 58 6488.3 1006.7 1221.5 3935.7 6570.6 1035.3 1263.5 4009.7 58 60 6491.1 1007.7 1222.9 3938.1 6573.3 1036.3 1265.0 4012.1 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 149 70 71 L. C. M. E. T. L. C. M. E. T. t 6573.3 1036.3 1265.0 4012.1 6654.9 1065.1 1308.4 4087.1 2 6576.0 1037.3 1266.4 4014.6 6657.6 1066.1 1309.9 4089.7 2 4 6578.7 1038.2 1267.9 4017.1 6660.3 1067.0 1311.3 4092.2 4 6 6581.5 1039.2 1269.3 4019.6 6663.0 1068.0 1312.8 4094.7 6 8 6584.2 1040.1 1270.8 4022.1 6605.7 1068.9 1314.2 4097.2 8 10 6586.9 1041.1 1272.2 4024.6 6668.4 1069.9 1315.7 4099.8 10 12 6589.6 1042.1 1273.6 4027.1 6671.1 1070.9 1317.2 4102.3 12 14 6592.3 1043.0 1275.1 4029.6 6673.8 1071.9 1318.6 4104.8 14 16 6595.1 1044.0 1276.5 4032.1 6676.6 1072.9 1320.1 4107.3 16 18 6597.8 1044.9 1278.0 4034.6 6679.3 1073.8 1321.5 4109.8 18 20 6600.5 1045.9 1279.4 4037.1 6682.0 1074.8 1323.0 4112.4 20 22 6603.2 1046.9 1280.8 4039.6 6684.7 1075.8 1324.4 4114.9 22 24 6605.9 1047.8 1282.3 4042.1 6687.4 1076.8 1325.9 4117.4 24 26 6608.7 1048.8 1283.7 4044.6 6690.1 1077.7 1327.4 4119.9 26 28 6611.4 1049.7 1285.2 4047.1 6692.8 1078.7 1328.9 4122.4 28 30 6614.1 1050.7 1286.6 4049.6 6695.5 1079.7 1330.4 4125.0 30 32 6616.8 1051.7 1288.0 4052.1 6098.2 1080.7 1331.8 4127.5 32 34 6619.5 1052.6 1289.5 4054.6 6700.9 1081.6 1333.3 4130.1 34 36 6622.3 1053.6 1290.9 4057.1 6703.6 1082.6 1334.8 4132.6 36 38 6625.0 1054.5 1292.4 4059.6 6706.3 1083.6 1336.3 4135.1 38 40 6627.7 1055.5 1293.8 4062.1 6709.0 1084.5 1337.8 4137.7 40 42 6630.4 1056.5 1295.3 4004.6 6711.7 1085.5 1339.2 4140.2 42 44 6633.1 1057.4 1296.7 4067.1 6714.4 1086.5 1340.7 4142.7 44 46 6635.9 1058.4 1298.2 4069.6 6717.2 1087.5 1342.2 4145.3 46 48 6638.6 1059.3 1299.6 4072.1 6719.9 1088.4 1343.7 4147.8 48 50 6641.3 1060.* 1301.1 4074.6 6722.6 1089.4 1345.2 4150.4 50 52 6644. C 1061.3 1302.6 4077.1 6725.3 1090.4 1346.7 4152.9 52 54 6646.7 1062.2 1304.0 4079.6 6728.0 1091.3 1348.2 4155.4 54 56 6649.5 1063.2 1305.5 4082.1 6730.7 1092.3 1349.7 4158.0 56 58 6652.2 1064.1 1306.9 4084.6 6733.4 1093.3 1351.2 4160.5 58 60 6654.9 1065.1 1308.4 4087.1 6736.1 1094.3 1352.7 4163.1 60 1 1. 73 L. C. M. E. T. L. C. M, E. T. 6736.1 1094.3 1352.7 4163.1 6816.6 1123.9 1398.1 4240.0 2 6738.8 1095.2 1354.2 4165.6 6819.3 1124.8 1399.6 4242.6 2 4 6741.5 1096.2 1355.7 4168.2 6821.9 1125.8 1401.2 4245.1 4 6 6744.1 1097.2 1357.2 4170.7 6824.6 1126.8 1402.7 4247.7 6 8 6746.8 1098.2 1358.7 4173.3 6827.3 1127.8 1404.2 4250.3 8 10 6749.5 1099.2 1360.2 4175.8 6830.0 1128.8 1405.8 4252.9 10 12 6752.2 1100.1 1361.7 4178.4 6832.6 1129.8 1407.3 4255.5 12 14 6754.9 1101.1 1303.2 4181.0 6835.3 1130.8 1408.8 4258.1 14 16 6757.6 1102.1 1364.7 4183.5 6838.0 1131.8 1410.4 4260.7 16 18 6760.2 1103.1 1366.2 '4186.1 6840.7 1132.8 1411.9 4263.2 18 20 6762.9 1104.1 1367.7 4188.6 6843.3 1133.8 1413.5 4265.8 20 22 6765.6 1105.1 1369.2 4191.2 6846.0 1134.8 1415.1 4268.4 22 24 6768.3 1106.0 1370.7 4193.7 6848.7 1135.8 1416.6 4271.0 24 26 6771.0 1107.0 1372.2 4196.3 6851.3 1136.8 1418.2 4273.6 26 28 6773.7 1108.0 1373.7 4198.8 6854.0 1137.8 1419.7 4276.2 28 30 6776.3 1109.0 1375.2 4201.4 6856.7 1138.8 1421.3 4278.8 30 32 6779.0 1109.9 1376.7 4204.0 6859.4 1139.8 1422.9 4281.4 32 34 6781.7 1110.9 1378.2 4206.5 6862.0 1140.8 1424.4 4284.0 34 36 6784.4 1111.9 1379.7 4209.1 6804.7 1141.8 1426.0 4286.6 30 38 6787.1 1112.9 1381.2 4211.7 6867.4 1142.8 1427.5 4289.2 38 40 6789.8 1113.9 3382.8 4214.3 6870.1 1143.8 1429.1 4291.8 40 42 6792.4 1114.9 1384.3 4216.8 6872.7 1144.8 1430.7 4294.4 42 44 0795.1 1115.9 1385.8 4219.4 6875.4 1145.8 1432.2 4297.0 44 46 6797.8 1116.9 1387.4 4222.0 6878.1 1146.8 1433.8 4299.6 46 48 6800.5 1117.9 1388.9 4224.5 6880.8 1147.8 1435.3 4302.2 48 50 6803.2 1118.9 1390.4 4227.1 6883.4 1148 8 1436.9 4304.8 50 52 6805.9 1119.9 1392.0 4229.7 6886.1 1149.8 1438.5 4307.4 52 54 6808.5 1120.9 1393.5 4232.3 68S8.8 1150.8 1440.0 4310.0 54 56 6811.2 1121.9 1395.0 4234.8 6891.4 1151.8 1441.6 4312.6 56 58 6813.9 1122.9 1396.6 4237.4 6894.1 1152.8 1443.1 4315.2 58 80 6816.6 1123.9 1398.1 4240.0 6896.8 1153.8 1444.7 4317.8 60 150 6. FUNCTIONS OF A ONE-DEGREE CURVE. 74 75 L. C. M. E. T. L. C. M. E. T. 6896.8 1153.8 1444.7 4317.8 6976.4 1184.1 1492.5 4396.7 2 6899.4 1154.8 1446.2 4320.5 6979.0 1185.1 1494.1 4399.4 2 4 6902.1 1155.8 1447.8 4323.1 6981.7 1186.1 1495.7 4402.1 4 6 6904.8 1156.8 1449.4 4325.7 6984.3 1187.1 1497.3 4404.7 6 8 6907.4 1157.8 1451.0 4328.3 6986.9 1188.1 1499.0 4407.4 8 10 6910.1 1158.8 1452.6 4330.9 6989.6 1189.2 1500.6 4410.0 10 18 6912.7 1159.8 1454.1 4333.6 6992.2 1190.2 1502.2 4412.7 12 14 6915.4 1160.8 1455.7 4336.2 6994.9 1191.2 1503.8 4415.3 14 16 6918.0 1161.8 1457.3 4338.8 6997.5 1192.2 1505.4 4418.0 16 18 6920.7 1162.8 1458.9 4341.4 7000.1 1193.2 1507.0 4420.7 18 20 6923.3 1163.9 1460.5 4344.0 7002.8 1194.3 1508.7 4423.3 20 22 6926.0 1164.9 1462.0 4346.7 7005.4 1195.3 1510.3 4426.0 22 24 6928.6 1165.9 1463.6 4349.3 7008.0 1196.3 1512.0 4428.6 24 26 6931.3 1166.9 1465!2 4351.9 7010.7 1197.3 1513.6 4431.3 26 28 6933.9 1167.9 1466.8 4354.5 7013.3 1198.3 1515.3 4434.0 28 30 6936.6 1168.9 1468.4 4357.1 7015.9 1199.4 1516.9 4436.6 30 32 6939.2 1169.9 1469.9 4359.8 7018.6 1200.4 1518.5 4439.3 32 34 6941.9 1170.9 1471.5 4362.4 7021.2 1201.4 1520.2 4442.0 34 36 6944.6 1171.9 1473.1 4365.1 7023.9 1202.4 1521.8 4444.6 36 88 6947.2 1172.9 1474.7 4367.7 7026.5 1203.4 1523.5 4447.3 38 40 6949.9 1174.0 1476.4 4370.3 7029.1 1204.5 1525.1 4450.0 40 42 6952.5 1175.0 1478.0 4373.0 7031.8 1205.5 1526.7 4452.7 42 44 6955.2 1176.0 1479.6 4375.6 7034.4 1206.5 1528.4 4455.3 44 46 6957.8 1177.0 1481.2 4378.3 7037.0 1207.5 1530.0 4458.0 46 48 6960.5 1178.0 1482.8 4380.9 7039.7 1208.5 1531.7 4460.7 48 50 6963.1 1179.0 1484.4 4383.5 7042.3 1209.6 1533.3 4463.4 50 52 6965.8 1180.0 1486.0 4386.2 7045.0 1210.6 1534.9 4466.0 52 54 6968.4 1181.0 1487.7 4388.8 7047.6 1211.6 1536.6 4468.7 54 56 6971.1 1182.0 1489.3 4391.5 7050.2 1212.6 1538.2 4471.4 56 58 6973.7 1183.0 1490.9 4394.1 7052.9 1213.6 1539.9 4474.1 58 60 6976.4 1184.1 1492.5 4396.7 7055.5 1214.7 1541.5 4476.7 60 76 77 L. C. M. E. T. L. C. M E. T. 7055.5 1214.7 1541.5 4476,7 7134.0 1245.6 1591.7 4557.8 2 7058.1 1215.7 1543.2 4479.4 7136.6 1246.6 1593.4 4560.5 2 4 7060.7 1216.7 1544.9 4482.1 7139.2 1247.7 1595.1 4563.3 4 6 7063.3 1217.8 1546.5 4484.8 7141.8 1248.7 1596.8 4566.0 6 8 7066.0 1218.8 1548.2 4487.5 7144.4 1249.8 1598.5 4568.7 8 10 7068.6 1219.8 1549.9 4490.2 7147.0 1250.8 1600.2 4571.5 10 12 7071.2 1220.9 1551.5 4492.9 7149.6 1251.8 1601.9 4574.2 12 14 7073.8 1221.9 1553.2 4495.6 7152.2 1252.9 1603.6 4576.9 14 16 7076.4 1222.9 1554.9 4498.3 7154.8 1253.9 1605.3 4579.7 16 18 7079.0 1224.0-1556.5 4501.0 7157.4 1255 1607.0 4582.4 18 20 7081.7 1225.0 1558.2 4503.7 7160.0 1256.0 1608.7 4585.1 20 22 7084.3 1226.0 1559.9 4506.3 7162.6 1257.0 1610.4 4587.9 22 24 7086.9 1227.1 1561.5 4509.0 71C5.2 1258.1 1612.1 4590.6 24 26 7089.5 1228.1 1563.2 4511.7 7167.8 1259.1 1613.8 4593.3 26 28 7092.1 1229.1 1564.9 4514.4 7170.4 1260.2 1615.5 4596.0 28 80 7094.7 1230.2 1560.5 4517.1 7173.0 1261.2 1617.3 45988 30 32 7097.4 1231.2 1568.2 4519.8 7175.6 1262.2 1619.0 4601.5 32 31 7100.0 1232.2 1569.9 4522.5 7178.2 1263.3 1620.7 4604.3 34 36 7102.6 1233.3 1571.5 4525.3 7180.8 1264.3 1622.4 4607.0 36 38 7105.2 1234.8 1573.2 4528.0 7183.4 1265.4 1624.1 4609.8 38 40 7107.8 1235.3 1574.8 4530.7 7186.0 1266.4 1625.9 4612.5 40 42 7110.4 1236.4 1576.4 4533.4 7188.6 1267.4 1627.6 4615.3 4-> 44 7113.1 1237.4 1578.1 4536.1 7191.2 1268.5 1629.3 4618.0 44 46 7115.7 1238.4 1579.8 4538.8 7193.8 1269.5 1631.0 4620.8 40 48 7118.3 1239.5 1581.5 4541.5 7196.4 1270.6 1632.7 4623.5 48 50 7120.9 1240.5 1583.2 4544.2 7199.0 3271.6 16345 4626.3 50 52 7123.5 1241.5 1584.9 4547.0 7201.6 1272.7 1636.2 4629.0 52 54 7126.1 1242.6 1586.6 4549.7 7204.2 1273.7 1637.9 4631.8 54 56 7128.8 1243.6 1588.3 4552.4 7206.8 1274.8 1639.6 4634.5 56 58 7131.4 1244.6 1590.0 4555.1 7209.4 1275.8 1641.3 4637.3 58 60 7134.0 1245.6 1591.7 4557.8 7212.0 1276.9 1643.1 4640.0 60 / 6 FUNCTIONS OF A ONE-DEGREE CURVE. 151 78 79 L. C. M. E. T. L. C. M. E. T. 7212.0 1276.9 1643.1 4640.0 7289.5 130S.5 1696.0 4723.4 o 7214.6 1278.0 1644.8 4642.8 7292.1 1309.5 1697.7 4726.2 2 4 7217.2 1279.0 1646.6 4645.6 7294.6 1310.6 1699.5 4729.0 4 6 7219.7 1280.1 1648.3 4648.3 7297.2 1311.7 1701.3 4731.8 6 8 7222.3 1281.1 1650.1 4651.1 7299.7 1312.7 1703.1 4734.7 8 10 7224.9 1282.2 1651.8 4653.9 7302.3 1313.8 1704.9 4737.5 10 12 7227.5 1283.2 1653.6 4656.7 7304.9 1314.9 1706.6 4740.3 12 14 7230.1 1284.3 1655.3 4659.4 7307.4 1315.9 1708.4 4743.1 14 16 7232.7 1285.3 1657.1 4662.2 7310.0 1317.0 1710.2 4745.9 16 18 7235.2 1286.4 1G58.8 4665.0 7312.6 1318.1 1712.0 4748.7 18 20 7237.8 1287.4 1660.6 4667.7 7315.1 1319.1 1713.8 4751.5 20 22 7240.4 1288.5 1662.3 4670.5 7317.7 1320*2 1715.6 4754.3 22 24 7243.0 1289.5 1664.1 4673.3 7320.3 1321.3 1717.4 4757.1 24 26 7245.6 1290.6 1665.8 4676.0 7322.8 1322.3 1719.2 4760.0 26 28 7248.2 1291.6 1667.6 4678.8 7325.4 1323.4 1721.0 4762.8 28 30 7250.7 1292.7 1669. 3 4681.6 7327.9 1324.5 1722.8 4765.6 30 32 7253.3 1293.7 1671.1 4684.4 7330.5 1325.5 1724.6 4768.4 32 34 7255.9 1294.8 1672.8 4687.2 7333.1 13:26.6 1726.4 4771.2 34 36 7258.5 1295.8 1674.6 4689.9 7335.6 1327.7 1728.2 4774.1 36 38 7261.1 1296.9 1676.3 4692.7 7338.2 1328.7 1730.0 4776.9 38 40 7263.7 1297.9 1678.2 4695.5 7340.8 1329.8 1731.9 4779.7 "40 42 7266.2 1299.0 1679.9 4698 3 7343.3 1330.8 1733.7 4782.6 42 44 7268.8 1300.0. 1681.7 4701.1 7345.9 1331.9 1735.5 4785.4 44 46 7271.4 1301.1 1683.5 4703.9 7348.4 1333.0 1737.3 4788.2 46 48 7274.0 1302.1 1685.3 470''.. 7 7351.0 1334.1 1739.1 4791.0 48 50 7276.6 1303.2 1687.1 4709.5 7353.6 1335.2 1740.9 4793.9 50 52 7279.2 1304.2 1688.8 4712.2 7356.1 1336.2 1742.7 4796.7 52 54 7281.7 1305.3 1690.6 4715.0 7358.7 1337.3 1744.5 4799.5 54 56 7284.3 1306.3 1692.4 4717.8 7361.3 1338.4 1746.3 4802.4 56 58 7286.9 13C7.4 1694.2 4720.6 73o3.8 1339.5 1748.1 4805.2 58 60 7289.5 1308.5 1696.0 4723.4 73G6.4 1340.6 1750.0 4808.0 60 80 81 L. C. M. E. T. L. C. M. E. T. 7366.4 1340.6 1750 4808.0 7442.7 1372.8 1805.5 4893.9 2 7368 9 1341.7 1751.8 4810.9 7445.2 1373.9 1807.3 4896.8 2 4 7371.5 1342.7 1753.7 4813.7 7447.7 1375.0 1809.2 4899.7 4 6 7374.0 1343.8 1755.5 4816.6 7450.3 1376.1 1811.1 4902.6 6 8 7376.6 1344.9 1757.4 4819.4 7452.8 1377.1 1813.0 4905.4 8 10 7379.1 1346.0 1759.2 4822.3 7455.3 1378.2 1814.9 4908.3 10 12 7381.7 1347.0 1761.0 4825.1 7457.8 1379.3 1816.8 4911.2 12 14 7384.2 1348.1 1762.9 4828.0 7460.4 1380.4 1818.6 4914.1 14 16 7386.7 1349.2 1764.7 4830.8 7462.9 1381.4 1820.5 4917.0 16 18 7389.3 1350.3 1766.6 4833.7 7465.4 1382.5 1822.4 4919.9 18 20 7391.8 1351.3 1768.4 4836.5 7467.9 1383.6 1824.2 4922.8 20' 22 7394.4 1352.4 1770.2 4839.4 7470.4 1384.7 1826.1 4925.7 22 24 7396.9 1353.5 1772.1 4842.2 7473.0 1385.7 1828.0 4928.6 24 26 7399.5 1354.6 1773.9 4845.1 7475.5 1386.8 1829.9 4931.5 26 28 7402.0 1355.6 1775.8 4847.9 7478.0 1387.9 1831.8 4934.4 28 30 7404.5 1356.7 1777.6 4850.8 7480.5 1389.0 1833.7 4937.2 30 32 7407.1 1357.8 1779.4 4853.7 7483.1 1390.1 1835.6 4940.2 82 34 7409.6 1358.9 1781.3 4856.5 7485.6 1391.2 1837.5 4943.1 34 36 7412.2 1359.9 1783.1 4859.4 7488.1 1392.3 1839.4 4946.0 36 38 7414.7 1361.0 1785.0 4862.3 7490.6 1393.4 1841.3 4948.9 38 40 7417.3 1362.1 1786.8 4865.1 7493.2 1394.5 1843.2 4951.8 40 42 7419.8 1363.2 1788.6 4868.0 7495.7 1395.6 1845.1 4954.7 42 44 7422.3 1364.2 1790.5 4870.9 7498.2 1396.7 1847.0 4957.6 44 46 7424.9 1365.3 1792.4 4873.8 7500.7 1397.8 1848.9 4960.6 46 48 7427.4 1366.4 1794 3 4876.6 7503.3 1398.9 1850.8 4963.5 48 50 7430.0 1367.5 1796.2 4879.5 7505.8 1400.0 1852.7 4966.4 60 52 7432.5 1368.5 1798.0 4882.4 7508.3 1401.1 18546 4969.3 52 54 7435.1 1369.6 1799.9 4885 3 7510.8 1402.2 1856.5 4972.2 54 56 7437.6 1370 7 1801.8 4888.1 7513.3 1403.3 1858.4 4975.1 56 58 7440.1 1371 .8 1803.7 4891.0 7515.9 1404.4 1860.3 4978.0 58 60 7442.7 1372.8 1805.5 4893.9 7518.4 1405.5 18623 4981. C 60 \ 152 6. FUNCTIONS OF A ONE-DEGREE CURVE. 82 83 L. C. M. E. T. L. C. M. E. T. 7518.4 1405.5 1862.3 4981.0 7593.6 1438.5 1920.6 5069.4 2 7520.9 1406.6 164.2 4983.9 7596.1 1439.6 1922.6 5072.4 2 4 7523.4 1407.7 1866.1 4986.8 7598.6 1440.7 1924.6 5075.4 4 6 7525.9 1408.8 1868.1 4989.8 7601.1 1441.8 1926.5 5078.4 6 8 7528.4 1409.9 1870.0 4992.7 7603.6 1442.9 1928.5 5081.4 8 10 7530.9 1411.0 1871.9 4995.7 7606.0 1444.0 1930.5 5084.4 10 12 7533.4 1412.1 1873 9 4998.6 7608.5 1445.1 1932.4 5087.3 12 14 7535.9 1413.2 1875.8 5001.5 7611.0 1446.2 1934.4 5090.3 14 16 7538.5 1414.3 1877.7 5004.5 7613.5 1447.3 1936.4 5093.3 16 18 7541.0 1415.4 1879.7 5007.4 7616.0 1448.4 1938.4 5096.3 18 20 7543.5 1416.5 1881.6 5010.3 7618.5 1449.6 1940.4 5099.3 20 22 7546.0 1417.6 1883.5 5013.3 7621.0 1450.7 1942.4 5102.3 22 24 7548.5 1418.7 1885.5 5016.2 7623.5 1451.8 1944.4 5105.2 24 26 7551.0 1419.8 1887.4 5019.2 7626.0 1452.9 1946.4 5108.2 26 28 7553.5 1420.9 1889.3 5022.1 7628.5 1454.0 1948.4 5111.2 28 30 7556.0 1422.0 1891.3 5025.0 7630.9 1455.1 1950.4 5114.2 30 32 7558.5 1423.1 1893.2 5028.0 7633.4 1456.2 1952.4 5117.2 32 34 7561.0 1424.2 1895.1 5031.0 7635.9 1457.3 1954.4 5120.2 34 36 7563.5 1425.3 1897.1 5033.9 7638.4 1458.4 1956.4 5123.2 36 38 7566.0 1426.4 1899.0 5036.9 7640.9 1459.5 1958.4 5126.2 38 40* 7568.5 1427.5 1901.0 5039.8 7643.4 1460.7 1960.4 5129.2 40 42 .7571.0 1428.6 1902.9 5042.8 7645.9 1461.8 1962.4 5132.2 42 44 7573.5 1429.7 1904.9 5045.8 7648.4 1462.9 1964.4 5135.2 44 46 7576.1 1430.8 1906.9 5048.7 7650.9 1464.0 1966.4 5138.2 46 48 7578.6 1431.9 1908.8 5051.7 7653.4 1465.1 1968.4 5141.2 48 50 7581.1 1433.0 1910.8 5054.6 7655.8 1466.2 1970.4 5144.3 50 52 7583.6 1434.1 1912.8 5057.6 7658.3 1467.3 1972.4 5147.3 52 54 7586.1 1435.2 1914.7 5060.6 7660.8 1468.4 1974.4 5150.3 54 56 7588.6 1436.3 1916.7 5063.5 7663.3 1469.5 1876.4 5153.3 56 58 7591.1 1437.4 1918.7 5066.5 7665.8 1470.6" 1978.4 5156.3 58 60 7593.6 1438.5 1920.6 5069.4 7668.3 1471.8 1980.5 5159.3 60 84 85 t L. C. M. E. T. L. C. M. E. 1 7668.3 1471.8 1980.5 5159.3 7742.4 1505.4 2041.8 5250.6 2 7670.8 1472.9 1982.5 5162.3 7744.8 1506.5 2043.9 5253.6 2 4 7673.2 1474.0 1984.5 5165.3 7747.3 1507.6 2046.0 5256.7 4 6 7675.7 1475.1 1986.6 5168.4 7749.7 1508.8 2048.0 5259.8 6 8 7678.2 1476.2 1988.6 5171.4 7752.2 1509.9 2050.1 5262.9 8 10 7680.6 1477.4 1990.6 5174.4 7754.6 1511.0 2052.2 5266 10 12 7683.1 1478.5 1992.7 5177.5 7757.1 1512.2 2054.2 5269.0 12 14 7685.6 1479.6 1994.7 5180.5 7759.5 1513.3 2056.3 5272.1 14 16 7688.1 1480.7 1996.7 5183.5 7762.0 1514.4 2058.4 5275.2 16 18 7690.5 1481.8 1998.8 5186.6 7764.4 1515.6 2060.5 5278.3 18 "20 7693.0 1483.0 2000.8 5189.6 7766.9 1516.7 2062.6 5281.4 20 22 7695.5 1484.1 2002.8 5192.6 7769.3 1517.8 2064.7 5284.4 22 24 7697.9 1485.2 2004.9 5195.6 7771.8 1519.0 2066.8 5287.5 24 26 7700.4 1486.3 2006.9 5198.7 7774.2 1520.1 2068.9 5290.6 26 28 7702.9 1487.4 2008.9 5201.7 7776.7 1521.2 2071.0 5293.7 28 30 7705.3 1488.6 2011.0 5204.7 7779.1 1522.4 2073.1 5296.7 30 32 7707.8 1489.7 2013.0 5207.8 7781.5 1523.5 2075.2 5299.8 32 84 7710.3 1490.8 2015.0 5210.8 7784.0 1524.6 2077 3 5302.9 34 36 7712.8 1491.9 2017.0 5213.9 7786.4 1525.8 2079.4 5306.1 36 38 7715.2 1493.0 2019.1 5216.9 7788.9 1526.9 2081.5 5309.2 38 40 7717.7 1494.2 2021.2 5220.0 7791.3 1528.0 2083.7 5312.3 40 42 7720.2 1495.3 2023.2 5223.1 7793.8 1529.2 2085.8 5315.4 42 44 7722.6 1496.4 2025.3 5226.1 7796.2 1530.3 2087.9 5318.5 44 46 7725.1 1497.5 2027.4 5229.2 7798.7 1531.4 2090.0 5321.6 46 48 7727.6 1498.6 2029.4 5232.2 7801.1 1532.6 2092.1 5324.7 48 50 7730.0 1499.8 2031.5 5235.3 7803.6 1533.7 2094.2 5327.8 50 52 7732.5 1500.9 2033.6 5238.3 7806.0 1534.8 2096.3 5330 9 52 54 7735.0 1502.0 2035.6 5241 4 7808.5 1536.0 2098.4 5334.0 54 56 7737.5 1503.1 2037.7 5244.5 7810.9 1537.1 21.00.6 5337.1 56 58 7739 9 1504.2 2039.8 5247.5 7813 4 1538.2 2102.7 5340.2 58 60 7742.4 15054 2041.8 5250.6 7815.8 1539.3 2104.8 5343.3 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 153 86 87 L. C. M. E. ,T. L. C. M. E. T, / 7815.8 1539.3 2104.8 5343.3 7888.5 1573.6 2169.5 5437.5 b 2 7818.2 1540.4 2106.9 5346.4 7890.9 1574.8 2171.6 5440.7 2 4 7820.6 1541.6 2109.1 5349.5 7893.3 1575.9 2173.8 5443.9 4 6 7823.1 1542.7 2111.2 5352.7 7895.7 1577.1 2176.0 5447.1 6 8 7825.5 1543.9 2113.4 5355.8 7898.1 1578.2 2178.2 5450.3 8 10 7827.9 1545.0 2115.5 5358.9 7900.5 1579.4 2180.4 5453.4 10 12 7830.3 1546.1 2117.6 5362.0 7903.0 1580.5 2182.5 5456.6 12 14 7832.8 1547.3 2119.8 5365.2 7905.4 1581.7 2184.7 5159.8 14 16 7835.2 1548.4 2121.9 5368.3 7907.8 1582.9 2186.9 5463.0 16 18 7837.6 1549.6 2124.1 5371.4 7910.2 1584.0 2189.1 5466.2 18 20 7840.0 1550.7 2126.2 5374.6 7912.6 1585.1 2191.3 5469.4 20 22 7842.4 1551.8 2128.3 5377.7 7915.0 1586.3 2193.5 5472.5 22 24 7844.9 1553.0 2130.5 5 r 80.8 7917.4 1587.4 2195.7 5475.7 24 26 7847.3 1554.1 2132.6 5383.9 7919.8 1588.6 2197.9 5478.9 26 28 7849.7 1555.3 2134.8 5387.1 7922.2 1589.7 2200.1 5482.1 28 30 7852.1 1556.4 2136.9 5390.2 7924.6 1590.9 2202.3 5485.3 30 32 7854.6 1557.5 2139.0 5393.4 7927.1 1592.0 2204.5 5488.5 82 34 7857.0 1558.7 2141.2 5396.5 7929.5 1593.2 2206.8 5491.7 34 36 7859.4 1559.8 2143.3 5399.7 7931.9 1594.3 2209.0 5494.9 36 38 7861.8 1561.0 2145.5 5402.8 7934.3 1595.5 2211.2 5498.1 38 40 7864.3 1562.1 2147.7 5406.0 7936.7 1596.6 2213.4 5501.3 40 42 7866.7 1563.2.2149.8 5409.1 7939.1 1597.8 2215.6 5504.5 42 44 7869.1 1564.4 2152.0 5412.3 7941.5 1598.9 2217.8 5507.7 44 46 7871.5 1565.5 2154.2 5415.4 7913.9 1600.1 2220.0 5510.9 46 48 7874.0 1566.7 2156.4 5418.6 7946.3 1601.2 2222.3 5514.1 48 50 7876.4 1567.8 2158.6 5421.8 7948.7 1602.4 2224.5 5517.3 50 52 7878.8 1568.9 2160.7 5424.9 7951.2 1603.5 2226.7 5520.5 52 54 7881.2 1570.1 2162.9 5428.1 7953.6 1604.7 2228.9 5523.7 54 56 7883.6 1571.2 2165.1 5431.2 7956.0 1605.8 2231.1 5526.9 56 58 7886.1 1572.4 2167.3 5434.4 7958.4 1607.0 2233.3 5530.1 58 60 7888.5 1573.6 2169.5 5437.5 7960.8 1608.2 2235.6 5533.3 60 88 89 L. C. M. E. T. L. C. M. E. T. 7960.8 1608.2 2235.6 5533.3 8032.4 1643.0 2303.6 5630.8 2 7963.2 1609.4 2237.8 5536.6 8034.8 1644.1 2305.9 5631.1 2 4 7965.6 1610.5 2240.1 5539.8 8037.1 1645.3 2308.2 5637.4 4 6 7968.0 1611.7 2242.3 5543.1 8039.5 1646.5 2310.5 5640.7 6 8 7970.3 1612.8 2244.6 5546.3 8041.9 1647.7 2312.8 5644.0 8 10 7972.7 1614.0 2246.8 5549.5 8044.2 1648.9 2315.1 5647.3 10 1? 7975.1 1615.2 2249.1 .5552.8 8046.6 1650.0 2317.4 5650.6 12 14 7977.5 1616.3 2251.3 5556.0 8049.0 1651.2 2319.7 5653.9 14 16 7979.9 1617.5 2253.6 5559.2 8051.4 1652.4 2322.0 5657.1 16 Ib 7982.3 1618.6 2255.8 5562.5 8053.7 1653.6 2324.3 5660.4 18 20 7984.7 1619.8 2258.1 5565.7 8056.1 1654.8 2326.7 5663.7 20 22 7987.1 1621.0 2260.4 5568.9 8058.5 1655.9 2329.0 5667.0 22 24 7989.4 1622.1 2262.7 5572.2 8060.8 1657.1 2331.3 5670.3 24 26 7991.8 1623.3 2264.9 5575.4 8063.2 1658.3 2333.7 5673.6 26 28 7994.2 1624.4 2267.2 5578.6 8065.6 1659.5 2336.0 5676.9 28 30 7996.6 1625.6 2269.5 5581.9 8067.9 1660.7 2338.3 5680.2 30 32 7999.0 1626.8 2271.7 5585.1 8070.3 1661.8 2340.7 5683.5 32 34 8001.4 1627.9 2273.9 5588.4 8073.7 1663.0 2343.0 5686.8 34 36 8003.8 1629.1 2276.2 5591.7 8075.1 1664.2 2345.3 5690.2 36 38 8006.1 1630.2 2278.5 5594.9 8077.4 1665.4 2347.7 5693.5 38 40 8008.5 1631.4 2280.8 5598.2 8079.8 1666.6 2350.0 5696.8 40 42 8010.9 1632.6 2283.0 5601.4 8082.2 1667.7 2352.3 5700.1 42 44 8013.3 1633.7 2285.3 5604.7 8084.5 1668.8 2354.7 5703.4 44 46 8015.7 1634.9 2287.6 5608.0 8086.9 1670.0 2357.0 5706.8 46 48 8018.1 1636.0 2289.9 5611.2 8089.3 1671.2 2359.3 5710.1 48 50 8020.5 1637.2 2292.2 5614.5 8091.6 1672.4 2361.7 5713.4 50 52 8022.9 1638.4 2294.4 5617.8 8094.0 1673.5 2364.0 5716.7 52 54 8025.2 1639.5 2296.7 5621.0 8096.4 1674.7 2366.3 5720.0 54 56 8027.6 1640.7 2299.0 5624.3 8098.8 1675.9 2368.7 5723.4 56 58 8030.0 1641.8 2301.3 56-37.5 8101.1 1677.1 2371.0 5726.7 58 60 8032.4 1643.0 2303.6 5630.8 8103.5 1678.3 2373.4 5730.0 60 154 6. FUNCTIONS OF A ONE-DEGREE CURVE. 90 91 L. C. M. E. T. L. C. M. E. T. 8103.5 1678.3 2373.4 5730.0 8173.9 713.8 2445.1 5830.9 2 8105.8 1679.5 2375.8 5733.3 8176.2 715.0 2447.5 5834.3 2 4 8108.2 1680.6 2378.2 5736.7 8178.5 716.2 2450.0 5837.7 4 6 8110.5 1681.8 2380.5 5740.0 8180.9 717.4 2452.4 5841.1 6 8 8112.9 1683.0 2382.9 5743.4 8183.2 718.6 2454.8 5844.5 8 10 8115.2 1684.2 2385.3 5746.7 8185.5 719.7 2457.2 5847.9 10 12 81176 1685.4 2387.6 5750.0 8187.9 720.9 2459.7 5851.3 12 14 8119.9 1686.5 2390.0 5753.4 8190.2 1722.1 2462.1 5854.7 14 16 8122.3 1687.7 2392.4 5756.7 8192.5 1723.3 2464.5 5858.1 16 18 8124.6 1688.9 2394.7 5760.1 8194.8 1724.5 2467.0 5861.5 18 20 8127.0 1690.1 2397.1 5763.4 8197.2 1725.7 2469.4 5864.9 20 22 8129.3 1691.3 2399.5 5766.8 8199.5 1726.9 2471.9 5868.3 22 24 8131.7 1692.5 2401.9 5770.1 8201.8 1728.1 2474.3 5871.8 24 26 8134.0 1693.6 2404.3 5773.5 8204.2 1729.3 2476.7 5875.2 26 28 8136.4 1694.8 2406.6 5776.9 8206.5 1730.5 2479.2 5878.6 28 30 8138.7 1696.0 2409.0 5780.2 8208.8 1731.7 2481.6 5882.0 30 32 8141.1 1697.2 2411.4 5783.6 8211.1 1732.9 2484.1 5885.4 32 34 8143.4 1698.4 2413.8 5787.0 8213.5 1734.1 2486.5 5888.9 34 36 8145.8 1699.6 2416.2 5790.3 8215.8 1735.3 2489.0 5892.3 36 38 8148.1 1700.7 2418.6 5793.7 8218.1 1736.4 2491.5 5895.7 38 40 8150.4 1701.9 2421.0 5797.1 8220.4 1737.6 2493.9 5899.2 40 42 8152.8 1703.1 2423.4 5800.4 8222.8 1738.8 2496.4 5902.6 42 44 8155.1 1704.3 2425.8 5803.8 8225.1 1740.0 2498.9 5906.0 44 46 8157.5 1705.5 2428.2 5807.2 8227.4 1741.2 2501.3 5909.4 46 48 8159.8 1706.7 2430.6 5810.6 8229.7 1742.4 2503.8 5912.9 48 50 8162.2 1707.9 2433.0 5814.0 8232.0 1743.6 2506.3 5916.3 50 52 8164.5 1709.0 2435.4 5817.3 8234.3 1744.8 2508.7 5919.8 52 54 8166.8 1710.2 2437.9 58:20.7 8236.7 1746.0 2511.2 5923.2 54 56 81G9.2 1711.4 2440.3 5824.1 8239.0 1747.2 2513.7 5926.7 56 58 8171.5 1712.6 2442.7 5827.5 8241.3 1748.4 2516.2 5930.1 58 60 8173.9 1713.8 2445.1 5830.9 8243.6 1749.6 2518.7 5933.6 60 92 93 / L. C M. E. T. L. C. M. E. T. 8243.6 1749.6 2518.7 5933.6 8312.8 1785.7 2594.2 603S.2 2 8245.9 1750.8 2521.2 5937.0 8315.1 1786.9 2596.8 6041.7 2 4 8248.2 1752.0 2523.6 5940.5 8317.4 1188.2 2599.3 6045.2 4 6 8250.6 1753.2 2526.1 5944.0 8319.7 1789.4 2G01.9 6048.7 6 \ 8252.9 1754.4 2528.6 5947.4 8322.0 1790.6 2604.4 6052.2 8 10 8255.2 1755.6 2531.1 5950.9 8324.3 1791.8 2G07.0 6055.8 10 12 8257.5 1756.8 2533.6 5954.4 8326.6 1793.0 2609.6 6059.3 12 14 8259.8 1758.0 2536.1 5957.8 8328.8 1794.2 2612.1 6062.8 14 16 8262.2 1759.2 2538.6 5961.3 8331.1 1795.4 2614.7 6066.4 16 18 8264.5 1760.4 2541.1 5964.8 8333.3 1796.6 2617.3 6069.9 18 20 8266.8 1761.6 2543.6 5968.2 8335.6 1797.8 2619.8 6073.4 20 22 8269.1 17G2.8 2546.1 5971.7 8337.9 1799.1 2622.4 6077.0 22 24 8271.4 1764.0 2548.6 5975.2 8340.2 1800.3 2625.0 6080.5 24 26 8273.7 1765.2 2551.2 5978.7 8342.5 1801.5 2627.6 6084.1 26 28 8276.0 1766.4 2553.7 5982.2 8344.8 1802.7 2630.2 6087.6 28 30 8278.3 1767.6 2556.2 5985.6 8347.1 1803.9 2632.7 6091.2 30 32 8280.6 1768.8 2558.7 5989.1 8349.4 1805.1 2635.3 6094.7 32 34 8282.9 17^0.0 2561.2 5992.6 8351.7 1806.3 2637.9 6098.3 34 36 8285.2 1771.2 2563.8 5996.1 8354.0 1807.6 2640.5 6101.8 36 .38 8287.5 1772.5 2566.3 5999.6 8356.3 1808.8 2643.1 6105.4 38 40 8289.8 1773.7 2568.8 6003.1 8358.5 1810.0 2645.7 6109.0 40 42 8292.1 1774.9 2571.3 6006.6 8360.8 1811.2 2648.3 6112.5 42 44 8294.4 1776.1 2573.9 6010.1 8363.1 1812.4 2650.9 6116.1 44 46 8296.7 1777.3 2576.4 6013.6 8365.4 1813.6 2653.5 6119.7 46 48 8299.0 1778.5 2578.9 6017.1 8367.7 1814.9 2656.1 6123.2 48 50 8301.3 1779.7 2581.5 6020.6 8369.9 1816.1 2658.7 6126.8 50 52 8303.6 1780.9 2584.0 6024.1 8372.2 1817.3 2661.3 6130.4 52 54 8305.9 1782.1 2586.6 6027.6 8374.5 1818.5 2663.9 6133.9 54 56 8308.2 1783.3 2589.1 6031.1 8376.8 1819.7 2666.6 6137.5 56 58 8310.5 1784.5 2591.7 6034.6 8379.1 1820.9 2669.2 6141.1 58 GO 8312.8 1785.7 2594.2 6038.2 8381.3 1822.2 2671.8 6144.7 60 6 FUNCTIONS OF A ONE-DEGREE CURVE. 155 94 95 L. C. M. E. T. L. C. M. E. T. 8381.3 1822.2 2671.8 6144.7 8449.2 1858.9 2751.5 6253.2 2 8383.6 1823.4 2674.4 6148.3 8451.5 1860.1 2754.2 6256.9 2 4 8385.9 1824.6 2677.0 6151.9 8453.7 1861.3 2756.9 6260.5 4 6 8388.1 1825.8 2679.7 6155.4 8456.0 1862.6 2759.6 6264.2 6 8 8390.4 1827.0 2682.3 6159.0 8458.2 1863.8 2762.3 6267.8 8 10 8392.7 1828.3 2684.9 6162.6 8460.4 1865.0 27650 6271.5 10 12 8395.0 1829.5 2687.6 6166.2 8462.7 1866.3 2767.7 6275.2 12 14 8397.2 1830.7 2690.2 6169.8 8464.9 1867.5 2770.4 6278.8 14 16 8399.5 1831.3 2692.8 6173.4 8467.2 1868.7 2773.1 6282.5 16 18 8401.7 1833.1 2695.6 6177.0 8469.4 1869.9 2775.8 6286.2 18 20 8404. C 1834.4 2698.1 6180.6 8471.7 1871.2 2778.5 6289.8 20 22 8406.3 1835.6 2700.8 6184.2 8473.9 1872.4 2781.2 6293.5 22 24 8408.5 1836.8 2703.4 6187.8 8476.2 1873.6 2784.0 6297.2 24 26 8410.8 1838.0 2706.1 6191.5 8478.4 1874.9 2786.7 6300.9 26 28 8413.1 18J9.3 2708.7 6195.1 8480.7 1876.1 2789.4 6304.6 28 30 8415.3 1840.5 2711.4 6198.7 8482.9 1877.3 2792.1 6308 2 30 32 8417.6 1841.7 2714.0 6202.3 8485.1 1878.6 2794.9 6311.9 32 34 8419.9 1842.9 2716.7 6205.9 8487.4 1879.8 2797.6 6315.6 34 36 8422.1 1844.2 2719.3 6209.5 8489.6 1881.0 280U.3 6319.3 36 38 8424.4 1845.4 2722.0 6213.2 8491.9 1882.3 2803.1 6323.0 38 40 8426.6 1846.6 2724.7 6216.8 8494.1 1883.5 2805.8 6326.7 40 42 8428.9 1847.8 itf27.3 6220.4 8496.3 1884.8 2808.6 6330.4 42 44 8431.2 1849.1 2730.0 6224.1 8498.6 1886.0 2811.3 6334.1 44 46 8433.4 1850.3 2732.7 6227.7 8500.8 1887.2 2814.1 6337.8 46 48 8435.7 1851.5 2735.4 6231.3 8503.0 1888.5 2816.8 6341.5 48 50 8437.9 1852.7 2738.0 6235.0 8505.3 1889.7 2819.6 6345.2 50 52 8440.2 1854.0 2740.7 6238.6 8507.5 1890.9 2822.3 6349.0 52 54 8442.4 1855.2 2743.4 6242.3 8509.8 1892.2 2825.1 6352.7 54 56 8444.7 1856.4 2746.1 6245.9 8512.0 1893.4 2827.8 6356.4 56 58 8447.0 1857.6 2748.8 6249.6 8514.2 1894.6 2830.6 6360.1 58 60 8449.2 1858.9 2751.5 6253.2 8516.4 1895.9 2833.4 6363.8 60 96 97 / L. C. M. E. T. L. C. M. E. T. t 8516.4 1895. S 2833.4 6363.8 8583.0 1933.2 2917.5 6476.6 2 8518.7 1897.1 2836.1 6367.5 8585.2 1934.4 2900.3 6480.4 2 4 8520.9 1898.4 2838.9 6371.3 8587.5 1935.7 2923.2 6484.2 4 6 8523.1 1899.6 2841.7 6375.0 8589.7 1936.9 2926.0 6488.0 6 8 8525.4 1900.8 2844.5 6378.7 8591.9 1938.2 2928.9 6491.8 8 10 8527.6 1902.1 2847.2 6382.5 8594.1 1939.4 2931.7 6495.6 10 12 8529.8 1903.3 2850.0 6386.2 8596.3 1940.7 2934.6 6499.4 12 14 8532.0 19046 2852.8 6389.9 8598.5 1941.9 2937.5 6503 2 14 16 8534.3 1905.8 2855.6 6393.7 8600.7 1943.2 2940.3 6507.1 16 18 8536.5 1907.0 2858.4 6397.4 8602.9 1944.4 2943.2 6510.9 18 20 8538.7 1908.3 2861.2 6401.2 8605.1 1945.7 2946.1 6514.7 20 22 8540.9 1909.5 2864.0 6404.9 8607.3 1946.9 2948.9 6518.5 22 24 8543.2 1910.8 2866.7 6408.7 8609.5 1948.2 2951.8 6522.3 24 26 8545.4 1912.0 2869.5 6412.4 8611.7 1949.4 2954.7 6526.2 26 28 8547. e 1913.3 2872.3 6416.2 8613.9 1950.7 2957.6 6530.0 28 30 8549.8 1914.5 2875.1 6419.9 8616.1 1952:0 29G0.4 6533.8 30 32 8652.0 1915.7 2877.9 6423.7 8618.3 1953.2 2963.3 6537.7 32 34 8554.3 1917.0 2880.8 6427.5 8620.5 1954.5 2966.2 6541.5 ' 34 36 8556.5 1918.2 2883.6 6431.2 8622.7 1955.7 2969.1 6545.3 36 38 8558.7 1919.5 2886.4 6435.0 8624.9 1957.0 2972.0 6549.2 38 40 8560.9 1920.7 2889.2 6438.8 8627.1 1958.2 2974.9 6553.0 40 42 8563.1 1922.0 2892.0 6442 5 8629.3 1959 5 2977.8 6556 9 42 44 8565 3 1923.2 2894.8 6446.3 8631.5 1960.7 2980.7 6560.7 44 46 8567 6 1924 5 2897.7 6450.1 8633.7 1962.0 2983.6 6564.6 46 48 8569 8 1925.7 2900.5 6453.9 8635.8 1963.2 2986.5 6568 4 48 50 8572.0 1927.0 2903.3 6457.6 8638.0 1964.5 2989.4 6572.3 50 52 8574.2 1928.2 2906.1 6461.4 8040.2 1965.8 2992.3 6576.2 52 54 8576.4 1929 4 2W9.0 6465.2 8642.4 1967.0 2995.2 6580.0 54 56 8578.6 1930.7 2911.8 6-169.0 8644.6 1968.3 2998.1 6583.9 56 58 8580.8 1931.9 2914.7 6472.8 8646.8 1969.5 3001.1 6587.7 58 60 8583.0 1933.2 2917.5 6476.6 8649 1970.8 3004 6591.6 60 156 6. FUNCTIONS OF A. ONE-DEGREE CURVE. 98 99 L. C. M. B. T. L. C. M. E. T. 8649.0 1970.8 3004.0 6591.6 8714.3 2008.7 3092.9 6709.0 2 8651.2 1972.0 3006.9 6595.5 8716.4 2009.9 3095.9 6712.9 2 4 8653.3 1973.3 3009.8 6599.4 8718.6 2011.2 3098.9 6716.9 4 6 8655.5 1974.6 3012.8 6603.2 8720.7 2012.5 3101.9 6720.8 6 8 8657.7 1975.8 3015.7 6607.1 8722.9 2013.7 3104.9 6724.8 8 10 8659.9 1977.1 3018.6 6611.0 8725.1 2015.0 3107.9 6728.8 10 12 8662.1 1978.3 3021.6 6614.9 8727.2 2016.3 3111.0 6732.7 12 14 8664.3 1979.6 3024.5 6618.8 8729.4 2017.5 3114.0 6736.7 14 16 8666.4 1980.9 3027.5 6622.7 8731.5 2018.8 3117.0 6740.7 16 18 8668.6 1982.1 3030.4 6626.6 8733.7 2020.1 3120.0 6744.6 18 20 8670.8 1983.4 3033.3 6630.5 8735.9 2021.4 3123.1 6748.6 20 22 8673.0 1984.6 3036.3 6634.4 8738.0 2022.6 3126.1 6752.6 22 24 8675.2 1985.9 3039.3 6638.3 8710.2 2023.9 3129.1 6756.6 24 26 8677.8 1987.2 3042.2 6642.2 8742.3 2025.2 3132.2 6760.6 26 28 8679.5 1988.4 3045.2 6646.1 8744.5 2026.4 3135.2 G764.6 28 30 8681.7 1989.7 3048.1 6650.0 8746.6 2027.7 3138.3 6768.6 30 32 8683.9 1991.0 3051.1 6653.9 8748.8 2029.0 3141.3 6772.6 32 34 8686.0 1992.2 3054.1 6657.8 8750.9 2030.3 3144.4 6776.6 34 36 8688.2 1993.5 3057.0 6661.7 8753.1 2031.5 3147.4 6780.6 36 38 8690.4 1994.7 3060.0 6665.7 8755.3 2032.8 3150.5 6784.6 38 40 8692.6 1996.0 3063.0 6669.6 8757.4 2034.1 3153.5 6788.6 40 42 8694.7 1997.3 3066.0 6673.5 8759.5 2035.4 3156.6 6792.6 42 44 8696.9 1998.5 3068.9 6677.4 8761.7 2036.6 3159.7 6796.6 44 46 8699.1 1999.8 3071.9 6681.4 8763.8 2037.9 3162.7 6800.6 46 48 8701.2 2001.1 3074.9 6685.3 8766.0 2039.2 3165.8 6804.6 48 60 8703.4 2002.3 3077.9 6689.2 8768.1 2040.5 3168.9 6808.6 50 52 8705.6 2003.6 3080.9 6693.2 8770.3 2041.7 3172.0 6812.6 52 54 8707.8 2004.9 3083.9 6697.1 8772.4 2043.0 3175.1 6816.7 54 56 8709.9 2006.1 3086.9 6701.1 8774.6 2044.3 3178.1 6820.7 56 58 8712.1 2007.4 3089.9 6705.2 8776.7 2045.6 3181.2 6824.7 58 60 8714.3 2008.7 3092.9 6709.0 8778.9 2046.8 3184.3 6828.8 60 100 101 L C. M. E. T. L. C. M. E. T. / " 8778.9 2046.8 3184.3 6828.8 8842.8 2085.3 3278.3 6951 2 8781.0 2048.1 3187.4 6832.8 8844.9 2086.6 3281.5 6955.2 2 4 8783.1 2049.4 3190.5 6836.8 8847.0 2087.8 3284.7 6959.3 4 6 8785.3 2050.7 3193.6 6840.9 8849.2 2089.1 3287.9 6963.4 6 8 8787.4 2051.9 3196.7 6844.9 8851.3 2090.4 3291.1 6967.6 8 10 8789.6 2053.2 3199.8 6,849.0 8853.4 2091.7 3294.3 6971.7 10 12 8791.7 2054.5 3:202.9 6853.0 8855.5 2093.0 3297.5 6975.8 12 14 8793.9 2055.8 3:206.0 6857.1 8857.6 2094.3 3300.7 6980.0 14 16 8796.0 2057.1 3209.1 6861.1 8859.8 2095.6 3303.9 6984 1 16 18 8798.9 2058.3 3212.2 6865.2 8861.9 2096.9 3307.1 6988.2 18 20 8800.3 2059.6 3215.4 6869.2 8864.0 2098.2 3310.3 6992.4 20 22 8802.4 2060.9 3218.5 6873.3 8866.1 2099.4 3313.5 6996.6 22 24 8804.5 2062.2 3221.6 6877.4 8868.2 2100.7 3316.7 7000.7 24 26 8806.7 2063.5 3224.7 6881.4 8870.3 2102.0 3319.9 7004.9 26 28 8808.8 2064.7 3227.9 6885.5 8872.4 2103.3 3323.1 7009.0 28 30 8810.9 2066.0 3231 6889.6 8874.5 2104.6 3326.4 7013.2 30 32 8813.1 2067.3 3234.1 6893.7 8876.7 2105.9 3329.6 7017.3 32 34 8815.2 2068.6 3237 3 6897.8 8878.8 2107.2 3332.8 7021.5 34 36 8817.3 2069.9 32404 6901.8 8880.9 2108.5 3336.0 7025.7 36 38 8819.5 2071.1 3243.5 6905.9 8883.0 2109.8 3339.3 7029.9 38 40 8821.6 2072.4 3246.7 6910.0 8885.1 2111.1 3342.5 7034.0 40 42 ' 8823.7 2073.7 3249.8 6914.1 8887.2 2112.4 3345.8 7038.2 42 44 8825.8 2075.0 3253.0 6918.2 8889.3 2113.6 3349.0 7042.4 44 46 8828.0 2076.3 3256.2 6922.3 8891.4 2114.9 3352.3 7046.6 46 48 8830.1 2077.6 3259.3 6926.4 8893.5 2116.2 3355.5 7050.8 48 50 8832.2 2078.9 3262.5 6930.5 8895.6 2117.5 3358.8 7055.0 50 52 8834.3 2080.1 3265.7 6934.6 8897.7 2118.8 3362.0 7059.2 52 54 8836.4 2081.4 3268.8 6938.7 8899.8 2120.1 3365.5 70634 54 56 8838.6 2082.7 3272.0 6942.8 8901.9 2121.4 3368.7 7067.6 56 58 8340.7 2084.0 3275.2 6946.9 8904.0 2122.7 3372.0 7071.8 58 60 88428 2085.3 3278.3 6951.0 8906.1 2124.0 3375.1 7076.0 60 6. FUNCTIONS OF A ONE-DEGREE CURVE. 157 102 103 L. C. M. E. T. L. C. M. E. T. 8906.1 2124.0 3375.1 7076.0 8968.7 2163.0 3474.6 7203.6 2 8908.2 2125.3 3378.3 7080.2 8970.8 2164.3 3478.0 7207.9 2 4 8910.3 2126.6 3381.6 7084.4 8972.9 2165.6 3481.4 7212.2 4 6 8912.4 2127.9 3384.9 7088.6 8974.9 2166.9 3484.7 7216.5 6 8 8914.5 2129.2 3388.2 7092.8 8977.0 2168.2 3488.1 7220.8 8 10 8916.6 2130.5 3391.5 7097.1 8979.1 2169.5 3491.5 7225.1 10 12 8918.7 2131.8 3394.7 7101.3 8981.1 2170.8 3494.9 7229.5 12 14 8920.8 2133.1 3398.0 7105.5 8983.2 2172.1 3498.3 7233.8 14 16 8922.9 2134.4 3401.3 7109.7 8985.3 2173.4 3501.6 7238.1 16 18 8925.0 2135.7 3404.6 7114.0 8987.3 2174.7 3505.3 7242.4 18 20 8927.0 2137.0 3407.9 7118.2 8989.4 2176.1 3508.4 7246.8 20 22 8929.1 2138.3 3411.2 7122.4 8991.5 2177.4 3511.8 7251.1 30 24 8931.2 2139.6 3414.5 7126.7 8993.5 2178.7 3515.2 7255.4 24 26 8933.3 2140.9 3417.9 7130.9 6995.6 2180.0 3518.7 7259.8 26 28 8935.4 2142.2 3421.2 7135.2 8997.7 2181.3 3522.1 7264.1 28 30 8937.5 2143.5 3424.5 7139.4 8999.7 2182.6 3525.5 7268.5 30 32 8939.6 2144 8 3427.8 7143.7 9001.8 2183.9 3528.9 7272.8 32 34 8941.6 2146.1 3431.1 7148.0 9003.9 2185.2 3532.3 7277.2 34 36 8943.7 2147.4 3434.5 7152.2 9005.9 2186.5 3535.7 7281.5 36 38 8945.8 2148.7 3437.8 7156.5 9008.0 2187.8 3539.2 7285.9 38 40 8947.9 2150.0 3441.1 7160.7 9010.0 2189.1 3542.6 7290.3 40 42 8950.0 2151.3 3444.4 7165.0 9012.1 2190.5 3546.0 7294.6 42 44 8952.1 2152.6 3447.8 7169.3 9014.2 2191.8 3549.5 7299.0 44 46 8951.1 2153.9 3451.1 7173.6 9016.2 2193.1 3552.9 7303.4 46 48 8956.2 2155.2 3454.5 7177.9 9018.3 2194.4 3556.3 7307.7 48 50 8958.3 2156.5 3457.8 7182.1 9020.3 2195.7 3559.8 7312.1 50 52 8960.4 2157.8 3461.2 7186.4 9022.4 2197.0 3563.2 7316.5 52 54 8962.5 2159.1 3464.5 7190.7 9024.5 2198.3 3566.7 7320.9 54 56 8964.5 2160.4 3467.9 7195.0 9026.5 2199.6 3570.2 7325.3 56 58 8966.6 2161.7 3471.2 7199.3 9028.6 2200.9 3573.6 7329.7 58 60 8968.7 2163.0 3474.6 7203.6 9030.6 2202.3 3577.1 7334.1 60 104 105 L. C. M. E. T. L. C. M. E. T. i _ 9030.6 2202.3 3577.1 7334.1 9091.8 2241.8 3682.6 7467.5 2 9032.7 2203.6 3580.5 7338.5 9093.9 2243.1 3686.1 7472.0 2 4 9034.7 2204.9 3584.0 7342.9 9095.9 2244.4 3689.7 7476.5 4 6 9036.8 2206.2 3587.5 7347.3 9097.9 2245.8 3693.3 7481.0 6 8 9038.8 2207.5 3591.0 7351.7 9099.9 2247.1 3696.9 7485.5 8 10 9040.9 2208.8 3594.4 7356.1 9102.0 2248.4 3700.4 7490.0 10 12 9042.9 2210.2 8597.9 7360.5 9104.0 2249.7 3704.0 7494.5 12 14 9045.0 2211.5 3601.4 7364.9 9106.0 2251.1 3707.6 7499.1 14 16 9047.0 2212.8 3604.9 7369.4 9108.0 2252.4 3711.2 7503.6 16 18 9049.1 2214.1 3608.4 7373.8 9110.1 2253.7 3714.8 7508.1 18 20 9051.1 2215.4 3611.9 7378.2 9112.1 2255.0 3718.4 7512.6 20 22 9053.1 2216.7 3615.4 7382.0 9114.1 2256.4 3722.0 7517.2 22 24 9055.2 2218.0 3618.9 7387.1 9116.1 2257.7 3725.6 7521.7 24 26 9057.2 2219.4 3622.4 7391.5 9118.1 22.-9.0 3729.3 7526.3 26 28 9059.3 2220.7 3625.9 7396.0 9120.2 2260.3 3732.9 7530.8 28 30 9061.3 2222.0 3629.4 7400.4 9122.2 2261.7 3736.5 7535.3 30 32 9063.3 2223.3 3G33.0 7404.8 9124.2 2263.0 3740.1 7539 9 32 34 9065.4 2224.6 3636.5 7409.3 9126.2 2264.3 3743.7 7544.4 34 36 9067.4 2226 3640.0 7413 8 9128.2 2265.7 3747.4 7549.0 36 38 9069.5 2227.3 3643.5 7418.2 9130.2 2267.0 3751.0 7553.6 88 40 9071.5 2228.6 3647.1 7422.7 9132.3 2268.3 3754.6 7558.1 40 42 9073.5 2229.9 3650.6 7427.1 9134.3 2269.6 3758.3 7562.7 42 44 9075.6 2231.2 3654.1 7431.6 9136.3 2271.0 3761.9 7567.3 44 46 9077.6 2232.6 3657.7 7436.1 9138.3 2272.3 3765.6 7571.8 46 48 9079.8 2233.9 3661.2 74406 9140.3 2273.6 3769.2 7576.4 48 50 9081.7 2235.2 3664.8 7445.0 9142.3 2275.0 3772.9 7581.0 50 52 9083.7 2236.5 3668.3 7449.5 9144.3 2276.3 3776.5 7585.6 52 54 9085.7 2237.8 3671.9 7454.0 9146.3 2277.6 3780.2 7590.2 54 56 9087.8 2239.2 3675.4 7458 5 9148.3 2278.9 3783.9 7594.8 56 58 9089.8 2240.5 3679.0 7463.0 9150.4 2280.3 3787.5 7599.4 58 60 9091.8 2241.8 3682.6 74675 9152.4 2281.6 3791.2 7604.0 60 158 TABLE 7. LOGARITHMS OF NUMBERS. N0123456789 100 1 2 3 4 5 6 7 8 9 110 1 4 5 6 7 8 9 120 1 2 3 4 5 6 7 8 9 130 1 2 3 4 5 6 7 8 9 140 1 2 3 4 5 6 7 8 9 150 00000 00043 00087 00130 00173 00217 00260 00303 00346 00389 0432 0475 0518 0561 0604 0647 0689 0732 0775 0817 0860 0903 0945 0988 1030 1072 1115 1157 1199 1242 1284 1326 1368 1410 1452 1494 1536 1578 1620 1662 1703 1745 1787 1828 1870 1912 1953 1995 2036 2078 2119 2160 2202 2243 2284 2325 2366 2407 2449 2490 2531 2572 2612 2653 2694 2735 2776 2816 2857 2898 2938 2979 3019 3060 3100 3141 3181 3222 3262 3302 3342 3383 3423 3463 3503 3543 3583 3623 3663 3703 3743 3782 3822 3862 3902 3941 3981 4021 4060 4100 04139 04179 04218 04258 04297 04336 04376 0441^ 04454 04493 4532 4571 4610 4650 4689 4727 4766 4805 4844 4883 4922 4961 4999 5038 5077 5115 5154 5192 5231 5269 6308 5346 5385 5423 6461 5500 5538 5576 5614 5652 5690 5729 5767 5805 5843 5881 5918 5956 5994 6032 6070 6108 6145 6183 6221 6258 6296 6333 6371 6408 6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 6819 6856 6893 693Q 6967 7004 7041 7078 7115 7151 7188 7225 7262 7298 7335 7372 7408 7445 7482 7518 7555 7591 7628 7664 7700 7737 7773' 7809 7846 7882 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 8636 8672 8707 8743 8778 8814 8849 8884 8920 8955 8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 9342 9377 9412 9447 9482 9517 9552 9587 9621 9656 9691 9726 9760 9795 9830 9864 9899 9934 9968 10003 10037 10072 10106 10140 10175 10209 10243 10278 10312 0346 0380 0415 0449 0483 0517 0551 0585 0619 0653 0687 0721 0755 0789 0823 0857 0890 0924 0958 0992 1025 1059 1093 1126 1160 1193 1227 1261 1294 1327 1361 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 2057 2090 2123 2156 2189 2222 2254 2287 2320 2352 2385 2418 2450 2483 2516 2548 2581 2613 2646 2678 2710 2743 2775 2808 2840 2872 2905 2937 2969 3001 3033 3066 3098 3130 3162 3194 3226 3258 3290 3322 3354 3386 3418 3450 3481 3513 3545 3577 3609 3640 3672 3704 3735 3767 3799 3830 3862 3893 3925 3956 3988 4019 4051 4082 4114 4145 4176 4208 4239 4270 4301 4333 4364 4395 4426 4457 4489 4520 4551 4582 14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 4922 4953 4983 5014 5045 5076 5106 5137 5168 5198 5229 5259 5290 5320 5351 5381 5412 5442 5473 5503 5534 5564 5594 5625 5655 5685 5715 5746 5776 5806 5836 5806 5897 5927 5957 5987 6017 6047 6077 6107 6137 6167 6197 6227 6256 6286 6316 6346 6376 6406 6435 6465 6495 6524 6554 6584 6613 6643 6673 6702 6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 7319 7348 7377 7406 7435 7464 7493 7522 7561 7580 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 TABLE 7. LOGARITHMS OF NUMBERS. 159 N0123456789 150 1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 180 1 2 3 4 5 6 7 8 9 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 7898 7926 7955 7984 8013 8041 8070 8099 8127 8156 8184 8213 8241 8270 8298 8327 8355 8384 8412 8441 8469 8498 8526 8554 8583 8611 8639 8667 8696 8724 8752 8780 8808 8837 8865 8893 8921 8949 8977 9005 9033 9061 9089 9117 9145 9173 9201 9229 9257 9285 9312 9340 9368 9396 9424 9451 9479 9507 9535 9562 9590 9618 9645 9673 9700 9728 9756 9783 9811 9838 9866 9893 9921 9948 99762000320030200582008520112 2014020167201942022220249 0276 0303 0330 0358 0385 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656 0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 0952 0978 1005.1032 1059 1085 1112 1139 1165 1192 1219 1245 1272 1299 1325 1352 1378 1405 1431 1458 1484 1511 1537 1564 1590 1617 1643 1669 1696 1722 1748 1775 1801 1827 1854 1880 1906 1932 1958 1985 2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 2272 2298 2324 2350 2376 2401 2427 2453 2479 2505 2531 2557 2583 2608 2634 2660 2686 2712 2737 2763 2789 2814 2840 2866 2891 2917 2943 -2968 2994 3019 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 3300 3325 3350 3376 3401 3426 3452 3477 3502 3528 3553 3578 3603 3629 3654 3679 3704 3729 3754 3779 3805 3830 3855 3880 3905 3930 3955 3980 4005 4030 4055 4080 4105 4130 4155 4180 4204 4229 4254 4279 4304 4329 4353 4378 4403 4428 4452 4477 4502 4527 4551 4576 4601 4625 4650 4674 4699 4724 4748 4773 4797 4822 4846 4871 4895 4920 4944 4969 4993 5018 5042 5066 5091 5115 5139 5164 5188 5212 5237 5261 5285 5310 5334 6358 5382 5406 5431 5455 5479 6503 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 5768 6792 5816 6840 5864 5888 6912 5935 5959 5983 6007 6031 6055 6079 6102 6126 6150 6174 6198 6221 6245 6269 6293 6316 6340 6364 6387 6411 6435 6458 6482 6505 6529 6553 6576 6600 6623 6647 6670 6694 6717 6741 6764 6788 6811 6834 6858 6881 6905 6951 6975 6998 7021 7045 7068 7091 7114 7138 7161 7184 7207 7231 7254 7277 7300 7323 7346 7370 7393 7416 7439 7462 7485 7508 7531 7554 7577 7600 7623 7646 7669 7692 7715 7738 7761 7784 7807 7830 7852 27875 27898 27921 27944 27967 27989 28012 28035 28058 28081 8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 8330 8353 8375 8398 8421 8443 8466 8488 8511 8533 8556 8578 8601 8623 8646 8668 8691 8713 8735 8758 8780 8803 8825 8847 8870 8892 8914 8937 8959 8981 9003 9026 9048 9070 9092 9115 9137 9159 9181 9203 9226 9248 9270 9292 9314 9336 9358 9380 9403 9425 9447 9469 9491 9513 9535 9557 9579 9601 9623 9645 9667 9688 9710 9732 9754 9776 9798 9820 9842 9863 9885 9907 9929 9951 9973 999430016300383006030081 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 160 TABLE 7. LOGARITHMS OF NUMBERS. N0123456789 200 1 2 3 4 5 6 7 8 9 210 1 2 3 4 5 6 7 8 9 220 1 2 3 4 5 6 7 8 9 280 1 2 3 4 5 6 7 8 9 240 1 2 3 4 5 6 7 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 0320 0341 0363 0384 0406 0428 0449 0471 0492 0514 0535 0557 0578 0600 0621 0643 0664 0685 0707 0728 0750 0771 0792 0814 0835 0856 0878 0899 0920 0942 0963 0984 1006 1027 1048 1069 1091 1112 1133 1154 1175 1197 1218 1239 1260 1281 1302 1323 1345 1366 1387 1408 1429 1450 1471 1492 1513 1534 1555 1576 1597 1618 1639 1660 1681 1702 1723 1744 1765 1785 1806 1827 1848 1869 1890 1911 1931 1952 1973 1994 2015 2035 2056 2077 2098 2118 2139 2160 2181 2201 32222 32243 32263 32284 32305 32325 32346 32366 32387 32408 2428 2449 2469 2490 2510 2531 2552 2572 2593 2613 2634 2654 2675 2695 2715 2736 2756 2777 2797 2818 2838 2858 2879 2899 2919 2940 2960 2980 3001 3021 3041 3062 3082 3102 3122 3143 3163 3183 3203 3224 3244 3264 3284 3304 3325 3345 3365 3385 3405 3425 3445 3465 3486 3506 3526 3546 3566 3586 3606 3626 3646 3666 3686 3706 3726 3746 3766 3786 3806 3826 3846 3866 3885 3905 3925 3945 3965 3985 4005 4025 4044 4064 4084 4104 4124 4143 4163 4183 4203 4223 34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 4439 4459 4479 4498 4518 4537 4557 4577 4596 4616 4635 4655 4674 4694 4713 4733 4753 4772 4792 4811 4830 4850 4869 4889 4908 4928 4947 4967 4986 5005 5025 5044 5064 5083 5102 6122 5141 5160 5180 5199 5218 5238 5257 5276 5295 5315 5334 5353 5372 5392 5411 5430 5449 5468 5488 5507 5526 5545 5564 5583 5603 5622 5641 5660 5679 5698 5717 5736 5755 5774 5793 5813 5832 5851 5870 5889 5908 5927 5946 5965 5984 6003 6021 6040 6059 6078 6097 6116 6135 6154 36173 36192 36211 36229 36248 36267 36286 36305 36324 36342 6361 6380 6399 6418 6436 6455 6474 6493 6511 6530 6549 6568 6586 6605 6624 6642 6661 6680 6698 6717 6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 6922 6940 6959 6977 6996 7014 7033 7051 7070 7088 7107 7125 7144 7162 7181 7199 7218 7236 7254 7273 7291 7310 7328 7346 7365 7383 7401 7420 7438 7457 7475 7493 7511 7530 7548 7566 7585 7603 7621 7639 7658 7676 7694 7712 7731 7749 7767 7785 7803 7822 7840 7858 7876 7894 7912 7931 7949 7967 7985 8003 38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 8382 8399 8417 8435 8453 8471 8489 8507 8525 8543 8561 8578 8596 8614 8632 8650 8668 8686 8703 8721 8739 8757 8775 8792 8810 8828 8846 8863 8881 8899 8917 8934 8952 8970 8987 9005 9023 9041 9058 9076 9094 9111 9129 9146 9164 9182 9199 9217 9235 9252 9270 9287 9305 9322 9340 9358 9375 9393 9410 9428 9445 9463 9480 9498 9515 9533 9550 9568 9585 9602 9620 9637 9655 9672 9690 9707 9724 9742 9759 9777 250 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 TABLE 7. LOGARITHMS OF NUMBERS. 161 O12345678 250 1 2 3 4 5 6 7 8 9 260 1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 280 1 2 3 4 5 6 7 8 9 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 4014040157 0175 0192 0209 0226 0243 0261 0278 0295 0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 0483 0500 0518 0535 0552- 0569 0586 0603 0620 0637 0654 0671 0688 0705 0722 0739 0756 0773 0790 0807 0824 0841 0858 0875 0892 0909 0926 0943 0960 0976 0993 1010 1027 1044 1061 1078 1095 1111 1128 1145 1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 1996 2012 2029 2045 2062 2078 2095 2111 2127 2144 2160 2177 2193 2210 2226 2243 2259 2275 2292 2308 2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 2488 2504 2521 2537 2553 2570 2586 2602 2619 263 2651 2667 -2684 2700 2716 2732 2749 2765 2781 2797 2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 2975 2991 3008 3024 3040 3056 3072 3088 3104 3120 43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 3457 3473 3489 3505 3521 3537 3553 3569 3584 3600 3616 3632 3648 3664 3680 3696 3712 3727 3743 3759 3775 3791 3807 3823 3838 3854 3870 3886 3902 3917 3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 4560 4576 4592 4607 4623 4638 4654 4669 4685 4700 44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 5025 5040 5056 5071 5086 5102 5117 5133 5148 5163 5179 5194 5209 5225' 5240 5255 5271 5286 5301 5317 5332 5347 5362 5378 5393 5408 5423 5439 5454 5469 5484 5500 5515 5530 5545 5561 5576 5591 5606 5621 5637 5652 5667 5682 5697 5712 5728 5743 5758 5773 5788 5803 5818 5834 5849 5864 5879 5894 5909 5924 5939 5954 5969 5984 6000 6015 6030 6045 6060 6075 6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 6538 6553 6568 6583 6598 6613 6627 6642 6657 6672 6687 6702 6716 6731 6746 6761 6776 6790 6805 6820 6835 6850 6864 6879 6894 6909 6923 6938 6953 6967 6982 6997 7012 7026 7041 7056 7070 7085 7100 7114 7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 7422 7436 7451 7465 7480 7494 7509 7524 7538 7553 7567 7582 7596 7611 7625 7640 7654 7669 7683 7698 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 162 TABLE 7. LOGARITHMS OF NUMBERS. O12345678 300 1 2 3 4 5 6 7 8 9 310 1 2 3 4 5 6 7 8 9 320 1 2 3 4 5 6 7 9 330 1 2 3 4 5 G 7 8 9 340 1 2 3 4 5 6 7 8 9 350 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 7857 7871 7885 7900 7914 7929 7943 7958 7972 7986 8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 8144 8159 8173 8187 8202 8216 8230 8244 8259 8273 8287 8302 8316 8330- 8344 8359 8373 8387 8401 8416 8430 8444 8458 8473 8487 8501 8515 8530 8544 8558 8572 8586 8601 8615 8629 8643 8657 8671 8686 87QO 8714 8728 8742 8756 8770 8785 8799 8813 8827 8841 8855 8869 8883 8897 8911 8926 8940 8954 8968 8982 8996 9010 9024 9038 9052 9066 9080 9094 9108 9122 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 9415 9429 9443 9457 9471 9485 9499 9513 9527 9541 9554 9568 9582 9596 9610 9624 9638 9651 9665 9679 9693 9707 9721 9734 9748 9762 9776 9790 9803 9817 9831 9845 9859 9872 9886 9900 9914 9927 9941 9955 9969 9982 9996 50010 50024 50037 50051 50065 50079 50092 501065012050133 0147 0161 0174 0188 0202 0215 0229 0243 0256 0270 0284 0297 0311 0325 0338 0352 0365 0379 0393 0406 0420 0433 0447 0461 0474 0488 0501 50515 50529 50542 50556 50569 50583 50596 50610 50623 50637 0651 0664 0678 0691 0705 0718 0732 0745 0759 0772 0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 1055 1068 1081 1095 1108 1121 1135 1148 1162 1175 1188 1202 1215 1228 1242 1255 1268 1282 1295 1308 1322 1335 1348 1362 1375 1388 1402 1415 1428 1441 1455 1468 1481 1495 1508 1521 1534 1548 1561 1574 1587 1601 1614 1627 1640 1654 1667 1680 1693 1706 1720 1733 1746 1759 1772 1786 1799 1812 1825 1838 51851 51865 51878 51891 51904 51917 51930 51943 51957 51970 1983 1996 2009 2022 2035 2048 2061 2075 2088 2101 2114 2127 2140 2153 2166 2179 2192 2205 2218 2231 2244 2257 2270 2284 2297' 2310 2323 2336 2349 2362 2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 2504 2517 2530 2543 2556 2569 2582 2595 2608 2621 2634 2647 2660 2673 2686 2699 2711 2724 2737 2750 2763 2776 2789 2802 2815 2827 2840 2853 2866 2879 2892 2905 2917 2930 2943 2956 2969 2982 2994 3007 3020 3033 3046 3058 3071 3084 3097 3110 3122 3135 53148 53161 53173 53186 53199 53212 53224 53237 53250 53263 3275 3288 3301 3314 3326 3339 3352 3364 3377 3390 3403 3415 3428 3441 3453 3466 3479 3491 3504 3517 3529 3542 3555 3567 3580 3593 3606 3618 3631 3643 3656 3668 3681 3694 3706 3719 3732 3744 3757 3769 3782 3794 3807 3820 3832 3845 3857 3870 3882 3895 3908 3920 3933 3945 3958 3970 3983 3995 4008 4020 4033 4045 4058 4070 4083 4095 4108 4120 4133 4145 4158 4170 4183 4195 4208 4220 4233 4245 4258 4270 4283 4295 4307 4320 4832 4345 4357 4370 4382 4394 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 TABLE 7. LOGARITHMS OF NUMBERS. 163 N 0123456789 350 1 2 3 4 5 6 7 8 9 360 1 2 3 4 5 6 7 8 9 370 1 2 3 4 5 6 7 8 9 380 1 2 3 4 5 6 7 8 9 390 1 2 3 4 5 6 7 8 9 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 4531 4543 4555 4568 4580 4593 4605 4617 4630 4642 4654 4667 4679 4691 4704 4716 4728 4741 4753 4765 4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 4900 4913 4925 4937 4949 4962 4974 4986 4998 6011 5023 5035 5047 5060 5072 5084 5096 5108 5121 5133 5145 5157 5169 5182 5194 5206 5218 5230 5242 5255 5267 5279 5291 5303 5315 5328 6340 5352 6364 5376 5388 5400 5413 5425 5437 6449 5461 5473 5485 5497 5509 5522 5534 5546 5558 6570 5582 6594 5606 6618 55630 55642 55654 55666 55678 65691 55703 55715 55727 55739 5751 5763 5775 6787* 5799 5811 5823 5835 5847 5859 5871 5883 5895 5907 5919 5931 5943 5955 5967 5979 5991 6003 6015 6027 6038 6050 6062 6074 6086 6098 6110 6122 6134 6146 6158 6170 6182 6194 6205 6217 6229 6241 . 6253 6265 6277 6289 6301 6312 6324 6336 6348 6360 6372 6384 6396 6407 6419 6431 6443 6455 6467 6478 6490 6502 6514 6526 6538 6549 6561 6573 6585 6597 6608 6620 6632 6644 6656 6667 6679 6691 6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 56820 56832 56844 56855 56867 56879 56891 56902 56914 56926 6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 7054 7066 7078 7089 7101 7113 7124 7136 7148 7159 7171 7183 7194 7206 7217 7229 7241 7252 7264 7276 7287 7299 7310 7322 7334 7345 7357 7368 7380 7392 7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 7519 7530 7542 7553 7565 7576 7588 7600 7611 7623 7634 7646 7657 7669 7680 7692 7703 7715 7726 7738 7749 7761 7772 7784 7795 7807 7818 7830 7841 7852 7864 7875 7887 7898 7910 7921 7933 7944 7955 7967 57978 57990 58001 58013 58024 58035 58047 58058 58070 58081 8092 8104 8115 8127 8138 8149 8161 8172 8184 8195 8206 8218 8229 8240 8252 8263 8274 8286 8297 8309 8320 8331 8343 8354 8365 8377 8388 8399 8410 8422 8433 8444 8456 8467 8478 8490 8501 8512 8524 8535 8546 8557 8569 8580 8591 8602 8614 8625 8636 8647 8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 8771 8782 8794 8805 8816 8827 8838 8850 8861 8872 8883 8894 8906 8917 8928 8939 8950 8961 8973 8984 8995 9006 9017 9028 9040 9051 9062 9073 9084 9095 59106 59118 59129 59140 59161 59162 59173 59184 59195 59207 9218 9229 9240 9251 9262 9273 9284 9295 9306 9318 9329 9340 9351 9362 9373 9384 9395 9406 9417 9428 9439 9450 9461 9472 9483 9494 9506 9517 9528 9539 9550 9561 9572 9583 9594 9605 9616 9627 9638 9649 9660 9671 9682 9693 9704 9715 9726 9737 9748 9759 9770 9780 9791 9802 9813 9824 9835 9846 9857 9868 9879 9890 9901 9912 9923 9934 9945 9956 9966 9977 9988 9999 60010 60021 60032 60043 60054 60065 60076 60086 60097 60108 0119 0130 0141 0152 0163 0173 0184 0195 400 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 164 TABLE 7. LOGARITHMS OF NUMBERS. NO123456789 400 1 2 3 4 5 6 7 8 9 410 1 2 3 4 5 6 7 8 9 420 1 2 3 4 5 6 7 8 9 430 1 2 3 4 5 6 7 8 9 440 1 2 3 4 5 6 7 8 9 450 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 0314 0325 0336 0347 0358 0369 0379 0390 0401 0412 0423 0433 0444 0455 0466 0477 0487 0498 0509 0520 0531 0541 0552 0563 0574 0584 0595 0606 0617 0627 0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 0746 0756 0767 0778 0788 0799 0810 0821 0831 0842 0853 0863 0874 0885 0895 0906 0917 0927 0938 0949 0959 0970 0981 0991 1002 1013 1023 1034 1045 1055 1066 1077 1087 1098 1109 1119 1130 1140 1151 1162 1172 1183 1194 1204 1215 1225 1236 1247 1257 1268 61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 1490 1500 1511 1521 1532 1542 1553 1563 1574 1584 1595 1606 1616 1627 1637 1648 1658 1669 1679 1690 1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 1805 1815 1826 1836 1847 1857 1868 1878 1888 1899 1909 1920 1930 1941 1951 1962 1972 1982 1993 2003 2014 2024 2034 2045 2055 2066 2076 2086 2097 2107 2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 2221 2232 2242 2252 2263 2273 2284 2294 2304 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 2428 2439 2449 2459 2469 2480 2490 2500 2511 2521 2531 2542 2552 2562 2572 2583 2593 2603 2613 2624 2634 2644 2655 2665 2675 2685 2696 2706 2716 2726 2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 2839 2849 2859 2870 2880 2890 2900 2910 2921 2931 2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 3043 3053 3063 3073 3083 3094 3104 3114 3124 3134 3144 3155 3165 3175 3185 3195 3205 3215 3225 3236 3246 3256 3266 3276 3286 3296 3306 3317 3327 3337 63347 63357 63367 63377 63387 63397 63407 63417 63428 63438 3448 3458 3468 3478 3488 3498 3508 3518 3528 3538 3548 3558 3568 3579 3589 3599 3609 3619 3629 3639 3649 3659 3669 3679 3689 3699 3709 3719 3729 3739 3749 3759 3769 3779 3789 3799 3809 3819 3829 3839 3849 3859 3869 3879 3889 3899 3909 3919 3929 3939 3949 3959 3969 3979 3988 3998 4008 4018 4028 4038 4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 4246 4256 4266 4276 4286 4296 4306 4316 4326 4335 64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 4444 4454 4464 4473 4483 4493 4503 4513 4523 4532 4542 4552 4562 4572 4582 4591 4601 4611 4621 4631 4640 4650 4660 4fc70 4680 4689 4699 4709 4719 4729 4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 4836 4846 4856 4865 4875 4885 4895 4904 4914 4924 4933 4943 4953 4963 4972 4982 4992 5002 5011 5021 5031 5040 5050 5060 5070 5079 5089 5099 5108 5118 5128 6137 5147 5157 5167 5176 5186 5196 5205 5215 5225 6234 5244 5254 5263 5273 5283 5292 5302 5312 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 TABLE 7. -LOGARITHMS OF NUMBERS. 165 0123456789 450 1 2 3 4 5 6 7 8 9 460 1 2 3 4 5 6 7 % 9 3 4 5 6 7 8 9 500 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 5418 5427 5437 5447 5456 5466 5475 5485 5495 5504 5514 5523 5533 5543 5552 5562 5571 5581 5591 5600 5610 5619 5629 5639 5648 5658 5667 5677 5686 5696 5706 5715 5725 5734 5744 5753 5763 6772 5782 5792 5801 5811 5820 5830 5839 5849 5858 5868 6877 6887 5896 5906 5916 5925 5935 5944 5954 5963 5973 5982 5992 6001 6011 6020 6030 6039 6049 6058 6068 6077 6087 6096 6106 6115 6124 6134 6143 6153 6162 6172 6181 6191 6200 6210 6219 6229 6238 6247 6257 6266 66276 66285 66295 66304 66314 66323 66332 66342 66351 66361 6370 6380 6389 6398 T6408 6417 6427 6436 6445 6455 6464 6474 6483 6492 6502 6511 6521 6530 6539 6549 6558 6567 6577 6586 6596 6605 6614 6624 6633 6642 6652 6661 6671 6680 6689 6699 6708 6717 6727 6736 6745 6755 6764 6773 6783 6792 6801 6811 6820 6829 6839 6848 6857 6867 6876 6885 6894 6904 6913 6922 6932 6941 6950 6960 6969 6978 6987 6997 7006 7015 7025 7034 7043 7052 7062 7071 7080 7089 7099 7108 7117 7127 7136 7145 7154 7164 7173 7182 7191 7201 67210 67219 67228 67237 67247 67256 67265 67274 67284 67293 7302 7311 7321 7330 7339 7348 7357 7367 7376 7385 7394 7403 7413 7422 7431 7440 7449 7459 7468 7477 7486 7495 7504 7514 7523 7532 7541 7550 7560 7569 7578 7587 7596 7605 7614 7624 7633 7642 7651 7660 7669 7679 7688 7697 7706 7715 7724 7733 7742 7752 7761 7770 7779 7788 7797 7806 7815 7825 7834 7843 7852 7861 7870 7879 7888 7897 7906 7916 7925 7934 7943 7952 7961 7970 7979 7988 7997 8006 8015 8024 8034 8043 8052 8061 8070 8079 8088 8097 8106 8115 68124 68133 68142 68151 68160 68169 68178 68187 68196 68205 8215 8224 8233 8242 8251 8260 8269 8278 8287 8296 8305 8314 8323 8332 8341 8350 8359 8368 8377 8386 8395 8404 8413 8422 8431 8440 8449 8458 8467 8476 8485 8494 8502 8511 8520 8529 8538 8547 8556 8565 8574 8583 8592 8601 8610 8619 8628 8637 8646 8655 8664 8673 8681 8690 8699 8708 8717 8726 8735 8744 8753 8762 8771 8780 8789 8797 8806 8815 8824 8833 8842 8851 8860 8869 8878 8886 8895 8904 8913 8922 8931 8940 8949 8958 8966 8975 8984 8993 9002 9011 69020 69028 69037 69046 69055 69064 69073 69082 69090 69099 9108 9117 9126 9135 9144 9152 9161 9170 9179 9188 9197 9205 9214 9223 9232 9241 9249 9258 9267 9276 9285 9294 9302 9311 9320 9329 9338 9346 9355 9364 9373 9381 9390 9399 9408 9417 9425 9434 9443 $452 9461 9469 9478 9487 9496 9504 9513 9522 9531 9539 9548 9557 9566 9574 9583 9592 9601 9609 9618 9627 9636 9644 9653 9662 9671- 9679 9688 9697 9705 9714 9723 9732 9740 9749 9758 9767 9775 9784 9793 9801 9810 9819 9827 9836 9845 9854 9862 9871 9880 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 166 TABLE 7. LOGARITHMS OF NUMBERS. 0123456789 500 1 2 3 4 5 6 7 8 9 510 1 2 3 4 5 6 7 8 9 520 1 2 3 4 5 6 7 8 9 530 1 2 3 4 5 6 7 8 9 540 1 2 3 4 5 G 7 8 9 550 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 9984 9992 70001 70010 70018 70027 70036 70044 70053 70062 70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 0157 0165 0174 0183 0191 0200 0209 0217 0226 0234 0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 0329 0338 0346 0355 0364 0372 0381 0389 0398 0406 0415 0424 0432 0441 0449 0458 0467 0475 0484 0492 0501 0509 0518 0526 0535 0544 0552 0561 0569 0578 0586 0595 0603 0612 0621 0629 0638 0646 065 0663 0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 0842 0851 0859 0868 0876 0885 0893 0902 0910 0919 0927 0935 0944 0952 0961 0969 0978 0986 0995 1003 1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 1096 1105 1113 1122 1130 1139 1147 1155 1164 1172 1181 1189 1198 1206 1214 1223 1231 1240 1248 1257 1273 1282 1290 1299 1307 1315 1324 1332 1341 1349 1357 1366 1374 1383 1391 1399 1408 1416 1425 1433 1441 1450 1458 1466 1475 1483 1492 1500 1508 1517 1525 1533 1542 1550 1559 1567 1575 1584 1592 71600 71609 71617 71625 71634 71642 71650 71659 71667 71675 1684 1692 1700 1709 1717 1725 1734 1742 1750 1759 1767 1775 1784 1792 1800 1809 1817 1825 1834 1842 1850 1858 1867 1875 1883 1892 1900 1908 1917 1925 1933 1941 1950 1958 1966 1975 1983 1991 1999 2008 2016 2024 2032 2041 2049 2057 2066 2074 2082 2090 2099 2107 2115 2123 2132 2140 2148 2156 2165 2173 2181 2189 2198 2206 2214 2222 2230 2239 2247 2255 2263 2272 2280 2288 2296 2304 2313 2321 2329 2337 2346 2354 2362 2370 2378 2387 2395 2403 2411 2419 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 2509 2518 2526 2534 2542 2550 2558 2567 2575 2583 2591 2599 2607 2616 2624 2632 2640 2648 2656 2665 2673 2681 2689 2697 2705 2713 2722 2730 2738 2746 2754 2762 2770 2779 2787 2795 2803 2811 2819 2827 2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 2916 2925 2933 2941 2949 2957 2965 2973 2981 2989 2997 3006 3014 3022 3030 3038 3046 3054 3062 3070 3078 3086 3094 3102 3111 3119 3127 313^ 3143 3151 3159 3167 3175 3183 3191 3199 3207 3215 3223 3231 73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 3400 3408 3416 3424 3432 3440 3448 3456 3464 3472 3480 3488 3496 3504 3512 3520 3528 3536 3544 3552 3560 3568 3576 3584 3592 3600 3608 3616 3624 3632 3640 3648 3656 3664 3672 3679 3687 3695 3703 3711 3719 3727 3735 3743 3751 3759 3767 377 3783 3791 3799 3807 3815 3823 3830 3838 3846 3854 3862 3870 3878 3886 3894 3902 3910 3918 3926 3933 3941 3949 3957 3965 3973 3981 3989 3997 400 4013 4020 4028 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 TABLE 7. LOGARITHMS OF NUMBERS 167 NO1234:56789 550 1 2 3 4 5 6 7 8 9 560 1 2 3 4 5 6 7 8 9 570 1 2 3 4 5 6 7 8 9 580 1 2 3 4 5 6 7 8 9 590 1 2 3 4 5 6 7 8 9 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 4115 4123 4131 4139 4147 4155 4162 4170 4178 4186 4194 4202 4210 4218 4225 4233 4241 4249 4257 4265 4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 4507 4515 4523 4531 4539 4547 4554 4562 4570 4578 4586 4593 4601 4609 4617 4624 4632 4640 4648 4656 4663 4671 4679 4687 4695 4702 4710 4718 4726 4733 4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 74819 74827 74834 74842 74$50 74858 74865 74873 74881 74889 4896 4904 4912 4920 4927 4935 4943 4950 4958 4966 4974 4981 4989 4997 5005 5012 5020 5028 6035 5043 5051 5059 5066 5074 5082 5089 5097 5105 5113 5120 5128 5136 5143 5151 5159 5166 5174 5182 5189 5197 5205 5213 5220 5228 5236 5243 5251 5259 5266 5274 5282 5289 5297 5305 5312 5320 5328 5335 5343 5351 5358 5366 5374 5381 5389 5397 5404 5412 6420 5427 5435 5442 5450 5458 5465 5473 5481 5488 5496 5504 5511 5519 5526 5534 5542 5549 5557 556$ 5572 5580 75587 75595 75603 75610 75618 75626 75633 75641 75648 75656 5664 5671 5679 5686 5694 5702 5709 5717 5724 5732 5740 5747 5755 5762 5770 5778 5785 5793 5800 5808 5815 5823 5831 5838 5846 5853 5861 5868 5876 5884 5891 5899 5906 5914 5921 5929 5937 5944 5952 5959 5967 5974 5982 5989 5997 6005 6012 6020 6027 6035 6042 6050 6057 6065 6072 6080 6087 6095 6103 6110 6118 6125 6133 6140 6148 6155 6163 6170 6178 6185 6193 6200 6208 6215 6223 6230 6238 6245 6253 6260 6268 6275 6283 6290 6298 6305 6313 6320 6328 6335 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 6418 6425 6433 6440 6448 6455 6462 6470 6477 6485 6492 6500 6507 6515 6522 6530 6537 6545 6552 6559 6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 6716 6723 6730 6738 6745 6753 6760 6768 677$ 6782 6790 6797 6805 6812 6819 6827 6834 6842 6849 6856 6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 6938 6945 6953 6960 6967 6975 6982 6989 6997 7004 7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 7159 7166 7173 7181 7188 7195 7203 7210 7217 722$ 7232 7240 7247 7254 7262 7269 7276 7283 7291 7298 7305 7313 7320 7327 7335 7342 7349 7357 7364 7371 7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 7452 7459 7466 7474 7481 7488 7495 7503 7510 7517 7525 7532 7539 7546 7554 7561 7568 7576 7583 7590 7597 7605 7612 7619 7627 7634 7641 7648 7656 7663 7670 7677 7685 7692 7699 7706 7714 7721 7728 7735 7743 7750 7757 7764 7772 7779 7786 7793 7801 7808 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 168 TABLE 7. LOGARITHMS OP NUMBERS. O123456789 600 1 2 3 4 5 6 7 8 9 610 1 2 3 4 5 6 7 8 620 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 640 1 2 3 4 5 8 7 8 650 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 7887 7895 7902 7909 7916 7924 7931 7938 7945 7952 7960 7967 7974 7981 7988 7996 8003 8010 8017 802$ 8032 8039 8046 8053 8061- 8068 8075 8082 8089 8097 8104 8111 8118 8125 8132 8140 8147 8154 8161 8168 8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 8247 8254 8262 8269 8276 8283 8290 8297 830$ 8312 8319 8326 8333 8340 8347 8355 8362 8369 8376 8383 8390 8398 8405 8412 8419 8426 8433 8440 8447 845$ 8462 8469 8476 8483 8490 8497 8504 8512 8519 8526 78533 78540 78547 78554 78561 78569 78576 78583 78590 78597 8604 8611 8618 8625 8633 8640 8647 8654 8661 8668 8675 8682 8689 8606 8704 8711 8718 8725 8732 8739 8746 8753 8760 8767 8774 8781 8789 8796 8803 8810 8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 8888 8895 8902 8909 8916 8923 8930 8937 8944 8951 8958 8965 8972 8979 8986 8993 9000 9007 9014 9021 9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 9099 9106 9113 9120 9127 9134 9141 9148 9155 9162 9169 9176 9183 9190 9197 9204 9211 9218 9225 9232 79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 9309 9316 9323 9335 9337 934 i 9351 9358 9365 9372 9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 9449 9456 9463 9470 9477 9484 9491 9498 9505 9511 9518 9525 9532 9539 9546 9553 9500 9567 9574 9581 9588 9595 9602 9609 9616 9623 9630 9637 9644 9650 9657 9664 9671 9678 9685 9692 9699 9706 9713 9720 9727 9734 9741 9748 9754 9761 9768 9775 9782 9789 9796 9803 9810 9817 9824 9831 9837 9844 9851 9858 9865 9872 9879 9886 9893 9900 9906 9913 9920 9927 79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 80003 80010 80017 80024 80030 80037 80044 80051 80058 80065 0072 0079 0085 0092 0099 0106 0113 0120 0127 0134 0140 0147 0154 0161 0168 0175 0182 0188 0195 0202 0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 0277 0284 0291 0298 0305 0312 0318 0325 0332 0339 0346 0353 0359 0366 0373 0380 0387 0393 0400 0407 0414 0421 0428 0434 0441 0448 0455 0462 0468 0475 0482 0489 0496 0502 0509 0516 0523 0530 0536 0543 0550 0557 0564 0570 0577 0584 0591 0598 0604 0611 80618 80625 80632 8063P 80645 80652 80659 80665 80672 80679 0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 0821 0828 0835 0841 0848 085$ 0862 0868 0875 0882 0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 0956 0963 0969 0976 0983 0990 0996 1003 1010 1017 1023 1030 1037 1043 1050 1057 1064 1070 1077 1084 1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 1224 1231 1238 124$ 1251 1258 126$ 1271 1278 128$ 81291 81298 8130$ 81311 81318 8132$ 81331 81338 8134$ 81351 TABLE 7 LOGARITHMS OF NUMBERS. 169 0123456789 650 1 2 3 4 5 6 7 8 9 660 1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 690 1 2 3 4 5 6 7 8 9 700 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 1358 1365 1371 1378 1385 1391 1398 1405 1411 1418 1425 1431 1438 1445 1451 1458 1465 1471 1478 1485 1491 1498 1505 1511 1518 1525 1531 1538 1544 1551 1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 1624 1631 1637 1644 1651 1657 1664 1671 1677 1684 1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 2086 2092 2099 2105 2112 2119 2125 2132 2138 2145 2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 2282 2289 2295 2302 2308 2315 2321 2328 2334 2341 2347 2354 2360 2367 2373 2380 2387 2393 2400 2406 2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 2478 2484 2491 2497 2504 2510 2517 2523 2530 2536 2543 2549 2556 2562 2569 2575 2582 2588 2595 2601 82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 2737 2743 2750 2750 2763 2769 2776 2782 2789 2795 2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 2995 3001 3008 3014 3020 3027 3033 3040 3046 3052 3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 3123 3129 3136 3142 3149 3155 3161 3168 3174 3181 3187 3193 3200 3206 3213 3219 3225 3232 3238 3245 83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 3315 3321 3327 3334 3340 3347 3353 3359 3366 3372 3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 3442 3448 3455 3461 3467 3474 3480 3487 3493 3499 3506 3512 3518 3525 3531 3537 3544 3550 3556 3563 3569 3575 3582 3588 3S94 3601 3607 3613 3620 3626 3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 3696 3702 3708 3715 3721 3727 3734 3740 3746 3753 3759 3765 3771 3778 3784 3790 3797 3803 3809 3816 3822 3828 3835 3841 3847 3853 3860 3866 3872 3879 83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 3948 3954 3960 3967 3973 3979 3985 3992 3998 4004 4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 4198 4205 4211 4217 4223 4230 4236 4242 4248 4255 4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 4448 4454 4460 4466 4473 4479 448 4491 4497 4504 84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 170 TABLE 7. LOGARITHMS OF NUMBERS. N0123456789 700 1 2 3 4 5 6 7 8 9 710 1 2 3 4 5 6 7 8 9 720 1 2 3 4 5 6 7 8 9 730 1 2 3 4 5 6 7 8 9 740 1 2 3 4 5 6 7 8 9 750 84510 84516 84522 84528 84535 84541 84547 84553 84559 84560 4572 4578 4584 4590 4597 4603 4609 4615 4621 4628 4634 4640 4646 4652 4658 4665 4671 4677 4683 4689 4696 4702 4708 4714 4720 4726 4733 4739 4745 4751 4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 4819 4825 4831 4837 4844 4850 4856 4862 4868 4874 4880 4887 4893 4899 4905 4911 4917 4924 4930 4936 4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 5003 5009 5016 5022 5028 5034 5040 5046 5052 5058 5065 5071 5077 5083 5089 5095 5101 5107 5114 5120 85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 5187 5193 5199 5205 5211 5217 5224 5230 5236 5242 5248 5254 5260 5266 5272 5278 . 5285 5291 5297 5303 5309 5315 5321 5327 5333 5339 5345 5352 5358 5364 5370 5376 5382 5388 5394 5400 5406 5412 5418 5425 5431 5437 5443 5449 5455 5461 *467 5473 5479 5485 5491 5497 5503 5509 5516 5522 5528 5534 5540 5546 5552 5558 5564 5570 5576 55C2 5588 5594 5600 5606 5612 5618 5625 5631 5637 5643 5649 5655 5661 5667 5673 5679 6685 5691 5697 6703 5709 5715 5721 5727 85733 85739 85745 85751 85757 85763 85769 85775 85781 85788 5794 5800 5806 5812 5818 5824 5830 5836 5842 5848 5854 5860 5866 5872 5878 5884 5890 5896 5902 5908 5914 5920 5926 5932 5938 5944 5950 5956 5962 5968 5974 5980 5986 5992 5998 6004 6010 6016 6022 6028 6034 6040 6046 6052 6058 6064 6070 6076 6082 6088 6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 6153 6159 6165 6171 6177 6183 6189 6195 6201 6207 6213 6219 6225 6231 6237 6243 6249 6255 6261 6267 6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 6392 6398 6404 6410 6415 6421 6427 6433 6439 6445 6451 6457 6463 6469 6475 6481 6487 6493 6499 6504 6510 6516 6522 6528 6534 6540 6546 6552 6558 6564 6570 6576 6581 6587 6593 6599 6605 6611 6617 6623 6629 6635 6641 6646 6652 6658 6664 6670 6676 6682 6688 6694 6700 6705 6711 6717 6723 6729 6735 6741 6747 6753 6759 6764 6770 6776 6782 6788 6794 6800 6806 6812 6817 6823 6829 6835 6841 6847 6853 6859 6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 6982 6988 6994 6999 7005 7011 7017 7023 7029 7035 7040 7046 7052 7058 7064 7070 7075 7081 7087 7093 7099 7105 7111 7116 7122 7128 7134 7140 7146 7151 7157 7163 7169 7175 7181 7186 7192 7198 7204 7210 7216 7221 7227 7233 7239 7245 7251 7256 7262 7268 7274 7280 7286 7291 7297 7303 7309 7315 7320 7326 7332 7338 7344 7349 7355 7361 7367 7373 7379 7384 7390 7396 7402 7408 7413 7419 7425 7431 7437 7442 7448 7454 7460 7466 7471 7477 7483 7489 7493 7500 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 TABLE 7. LOGARITHMS OF NUMBERS. 171 N0123456789 750 1 2 3 4 5 6 7 8 9 760 1 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 780 1 2 3 4 5 6 7 8 9 790 1 2 3 4 5 6 7 8 9 800 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 7564 7570 7576 7581 7587 7593 7599 7604 7610 7616 7622 7628 7633 7639 7645 7651 7656 7662 7668 7674 7679 7685 7691 7697 7703 .7708 7714 7720 7726 7731 7737 7743 7749 7754 7760 7766 7772 7777 7783 7789 7795 7800 7806 7812 7818 7823 7829 7835 7841 7846 7852 7858 7864 7869 7875 7881 7887 7892 7898 7904 7910 7915 7921 7927 7933 7938 7944 7950 7955 7961 7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 8024 8030 8036 8041 8047. 8053 8058 8064 8070 8076 88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 8195 8201 8207 8213 8218 8224 8230 8235 8241 8247 8252 8258 8264 8270 8275 8281 8287 8292 8298 8304 8309 8315 8321 8326 8332 8338 8343 8349 8355 8360 8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 8423 8429 8434 8440 8446 8451 8457 8463 8468 8474 8480 8485 8491 8497 8502 8508 8513 8519 8525 8530 8536 8542 8547 8553 8559 8564 8570 8576 8581 8587 8593 8598 8604 8610 8615 8621 8627 8632 8638 8643 88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 8705 8711 8717 8722 8728 8734 8739 8745 8750 8756 8762 8767 8773 8779 8784 8790 8795 8801 8807 8812 8818 8824 8829 8835 8840 8846 8852 8857 8863 8868 8874 8880 8885 8891 8897 8902 8908 8913 8919 8925 8930 8936 8941 8947 8953 S958 8964 8969 8975 8981 8986 8992 8997 9003 9009 9014 9020 9025 9031 9037 9042 9048 9053 9059 9064 9070 9076 9081 9087 9092 9098 9104 9109 9115 9120 9126 9131 9137 9143 9148 9154 9159 9165 9170 9176 9182 9187 9193 9198 9204 89209 89215 89221 89226 89232 89237 89243 89248 89254 89260 9265 9271 9276 9282 9287 9293 9298 9304 9310 9315 9321 9326 9332 9337 9343 9348 9354 9360 9365 9371 9376 9382 9387 9393 9398 1)404 9409 9415 9421 9426 9432 9437 9443 9448 9454 9459 9465 9470 9476 9481 9487 9492 9498 9504 9509 9515 9520 9526 9531 9537 9542 9548 9553 9559 9564 9570 9575 9581 9586 9592 9597 9603 9509 9614 9620 9625 9631 9636 9642 9647 9853 9653 9664 9669 9675 9680 9686 9691 9697 9702 9708 9713 9719 9724 9730 9735 9741 9746 9752 9757 89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 9818 9323 9829 9834 9840 9845 9851 9856 9862 9867 9873 9878 9883 9889 9894 9900 9905 9911 9916 9922 9927 9933 9938 9944 9949 9955 9960 9966 9971 9977 9982 9988 9993 9998900049000990015900209002690031 90037 90042 90048 90053 0059 0064 0069 0073 0080 0086 0091 0097 0102 0108 0113 0119 0124 0129 013$ 0140 0146 0151 0157 0162 0168 0173 0179 0184 0189 0195 0200 0206 0211 0217 02'22 0227 0233 0238 0244 0249 0255 0260 0266 0271 0276 0282 0287 0293 0298 0304 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 172 TABLE 7. LOGARITHMS OF NUMBERS. N0123456789 800 1 4 5 6 7 8 9 810 1 2 3 4 5 6 7 8 9 820 1 2 3 4 5 6 7 8 9 830 1 2 3 4 5 6 7 8 9 840 1 2 3 4 5 6 7 850 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 0417 0423 0428 0434 0439 0445 0450 0455 0461 0466 0472 0477 0482 0488 0493 0499 0504 0509 0515 0520 0526 0531 0536 0542 0547 0553 0558 0563 0569 0574 0580 0585 0590 0596 0601 0607 0612 0617 0623 0628 0634 0639 0644 0650 0655 0660 0666 0671 0677 0682 0687 0693 0698 0703 0709 0714 0720 0725 0730 0736 0741 0747 0752 0757 0763 0768 0773 0779 0784 0789 0795 0800 0806 0811 0816 0822 0827 0832 0838 0843 90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 0902 0907 0913 0918 0924 0929 0934 0940 0945 0950 0956 0961 0966 0972 0977 0982 0988 0993 0998 1004 1009 1014 1020 1025 1030 1036 1041 1046 1052 1057 1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 1116 1121 1126 1132 1137 1142 1148 1153 1158 1164 1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 1275 1281 1286 1291 1297 1302 1307 1312 1318 1323 1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 1434 1440 1445 1450 1455 1461 1466 1471 1477 1482 1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 1540 1545 1551 1556 1561 1566 1572 1577 1582 1587 1593 1598 1603 1609 1614 1619 1624 1630 1635 1640 1645 1651 1656 1661 1666 1672 1677 1682 1687 1693 1698 1703 1709 1714 1719 1724 1730 1735 1740 1745 1751 1756 1761 1766 1772 1777 1782 1787 1793 1798 1803 1808 1814 1819 1824 1829 1834 1840 1845 1850 1855 1861 1866 1871 1876 1882 1887 1892 1897 1903 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 2012 2018 2023 2028 2033 2038 2044 2049 2054 2059 2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 2117 2122 2127 2132 2137 2143 2148 2153 2158 2163 2169 2174 2179 2184 2189 2195 2200 2205 2210 2215 2221 2226 2231 2236 2241 2247 2252 2257 2262 2267 2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 2324 2330 2335 2340 2345 2350 2355 2361 2366 2371 2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 2480 2485 2490 2495 2500 2505 2511 2516 2521 2526 2531 2536 2542 2547 2552 2557 2562 2567 2572 2578 2583 2588 2593 2598 2603 2609 2614 2619 2624 2629 2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 2737 2742 2747 2752 2758 2763 2768 2773 2778 2783 2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 2840 2845 2850 2855 2860 2865 2870 2875 2881 2886 2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 TABLE 7. LOGARITHMS OF NUMBERS. 173 NO123456789 850 1 2 3 4 5 6 7 8 9 860 1 2 3 4 5 6 7 8 9 870 1 2 3 4 5 6 7 8 9 880 1 2 3 4 5 6 7 8 9 890 1 2 3 4 5 6 7 8 9 900 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 2993 2998 3003 3008 3013 3018 3024 3029 3034 3039 3044 3049 3054 3059 3064 3069 3075 3080 3085 3090 3095 3100 3105 3110 3115 3120 3125 3131 3136 3141 3146 3151 3156 3161 3166 3171 3176 3181 3186 3192 3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 3247 3252 3258 3263 3268 3273 3278 3283 3288 3293 3298 3303 3308 3313 3318 3323 3328 3334 3339 3344 3349 3354 3359 3364 3369 3374 3379 3384 3389 3394 3399 3404 3409 3414 3420 3425 3430 3435 3440 3445 93450 93455 93460 93465 93470*93475 93480 93485 93490 93495 3500 3505 3510 3515 3520 3526 3531 3536 3541 3546 3551 3556 3561 3566 3571 3576 3581 3586 3591 3596 3601 3606 3611 3616 3621 3626 3631 3636 3641 3646 3651 3656 3661 3666 3671 3676 3682 3687 3692 3697 3702 3707 3712 3717 3722 3727 3732 3737 3742 3747 3752 3757 3762 3767 3772 3777 3782 3787 3792 3797 3802 3807 3812 3817 3822 3827 3832 3837 3842 3847 3852 3857 3862 3867 3872 3877 3882 3887 3892 3897 3902 3907 3912 3917 3922 3927 3932 3937 3942 3947 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 4052 4057 4062 4067 4072 4077 4082 4086 4091 4096 4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 4151 4156 4161 4166 4171 4176 4181 4186 4191 4196 4201 4206 4211 4216 4221 4226 4231 4236 4240 4245 4250 4255 4260 4265 4270 4275 4280 4285 4290 4295 4300 4305 4310 4315 4320 4325 4330 4335 4340 434 4349 4354 4359 4364 4369 4374 4379 4384 4389 4394 4399 4404 4409 4414 4419 4424 4429 4433 4438 4443 94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 4498 4503 4507 4512 4517 4522 4527 4532 4537 4542 4547 4552 4557 4562 4567 4571 4576 4581 4586 4591 4596 4601 4606 4611 4616 4621 4626 4630 4635 4640 4645 4650 4655 4660 4665 4670 4675 4680 4685 4689 4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 4743 4748 4753 4758 4763 4768 4773 4778 4783 4787 4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 4841 4846 4851 4856 4861 4866 4871 4876 4880 4885 4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 4988 4993 4998 5002 5007 5012 5017 5022 5027 5032 5036 5041 5046 5051 5056 5061 5066 5071 5075 5080 5085 5090 5095 5100 5105 5109 5114 5119 5124 5129 5134 5139 5143 5148 5153 5158 5163 5168 5173 5177 5182 5187 5192 5197 5202 5207 5211 5216 5221 5226 5231 5236 5240 5245 5250 5255 5260 5265 5270 5274 5279 5284 5289 5294 5299 5303 5308 5313 5318 5323 5328 5332 5337 5342 5347 5352 5357 5361 5366 5371 5376 5381 5386 5390 5395 5400 5405 5410 5415 5419 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 174 TABLE 7. LOGARITHMS OF NUMBERS. 0123456789 I 900 1 2 3 4 5 6 7 8 9 910 1 2 3 4 5 6 7 8 9 920 1 2 3 4 5 6 7 8 9 930 1 2 3 4 5 6 7 8 9 940 1 4 5 6 7 8 9 950 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 5472 5477 6482 5487 5492 5497 5501 5506 5511 5516 5521 5525 5530 5535 5540 5545 5550 5554 5559 5564 5569 5574 5578 5583 5588 ' 5593 5598 5602 5607 5612 5617 5622 5626 5631 6636 5641 5646 5650 5655 5660 5665 6670 5674 6679 6684 5689 5694 6698 5703 5708 5713 5718 5722 5727 5732 5737 5742 5746 5751 5756 5761 5766 5770 5775 5780 5785 5789 5794 5799 5804 5809 5813 5818 5823 5828 5832 5837 5842 5847 6852 5856 5861 6866 5871 5875 5880 5885 5890 5895 5899 95904 95909 95914 95918 95923 95928 95933 95938 95942 95947 5952 5957 5961 5966 5971 5976 5980 5985 5990 5995 5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 6047 6052 6057 6061 6066 6071 6076 6080 6085 6090 609^ 6099 6104 6109 6114 6118 6123 6128 6133 6137 6142 6147 6152 6156 6161 6166 6171 6175 6180 6185 6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 6237 6242 6246 6251 6256 6261 6265 6270 6275 6280 6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 6332 6336 6341 6346 6350 6355 6360 6365 6369 6374 96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 6426 6431 6435 6440 6445 6450 6454 6459 6464 6468 6473 6478 6483 6487 6492 6497 6501 6506 6511 6515 6520 6525 6530 6534 6539 6544 6548 6553 6558 6562 6567 6572 6577 6581 6586 6591 6595 6600 6605 6609 6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 6661 6666 6670 6675 6680 6685 6689 6694 6699 6703 6708 6713 6717 6722 6727 6731 6736 6741 6745 6750 6755 6759 6764 6769 6774 6778 6783 6788 6792 6797 6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 6988 6993 6997 7002 7007 7011 7016 7021 7025 7030 7035 7039 7044 7049 7053 7058 7063 7067 7072 7077 7081 7086 7090 7095 7100 7104 7109 7114 7118 7123 7128 7132 7137 7142 7146 7151 7155 7,160 7165 7169 7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 7220 7225 7230 7234 7239 7243 7248 7253 7257 7262 7267 7271 7276 7280 7285 7290 7294 7299 7304 7308 97313 97317 97322 97327 97331 97336 97340 97345 97350 97354 7359 7364 7368 7373 7377 7382 7387 7391 7396 7400 7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 7497 7502 7506 7511 7516 7520 7525 7529 7534 7539 7543 7548 7552 7557 7562 7566 7571 7575 7580 7585 7589 7594 7598 7603 7607 7612 7617 7621 7626 7630 7635 7640 7644 7649 7653 7658 7663 7667 7672 7676 7681 7685 -7690 7695 7699 7704 7708 7713 7717 7722 7727 7731 7736 7740 7745 7749 7754 7759 7763 7768 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 TABLE 7. LOGARITHMS OF NUMBERS. 175 0123456789 950 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 1 7818 7823 7827 7832 7836 7841 7845 7850 7855 7859 2 7864 7868 7873 7877 7882 7886 7891 7896 7900 7905 3 7909 7914 7918 7923 7928 7932 7937 7941 7946 7950 4 7955 7959 7964 7968 7973 7978 7982 7987 7991 7996 5 8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 6 8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 7 8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 8 8137 8141 8146 8150 8155 8159 8164 8168 8173 8177 9 8182 8186 8191 8195 8200. 8204 8209 8214 8218 8223 960 98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 1 8272 8277 8281 8286 8290 8295 8299 8304 8308 8313 2 8318 8322 8327 8331 8336 8340 8345 8349 8354 8358 3 8363 8367 8372 8376 8381 8385 8390 8394 8399 8403 4 8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 5 8453 8457 8462 8466 8471 8475 8480 8484 8489 8493 6 8498 8502 8507 8511 8516 8520 8525 8529 8534 8538 7 8543 8547 8552 8556 8561 8565 8570 8574 8579 8583 8 8588 8592 8597 8601 8605 8610 8614 8619 8623 8628 9 8632 8637 8641 8646 8650 8655 8659 8664 8668 8673 970 98677 98682 98686 98691 98695 98700 98704 98709 98713 98717 1 8722 8726 8731 8735 8740 8744 8749 8753 8758 8762 2 8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 3 8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 4 8856 8860 8865 8869 8874 8878 8883 8887 8892 8896 5 8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 6 8945 8949 8954 8958 8963 B967 8972 8976 8981 8985 7 8989 8994 8998 9003 9007 9012 9016 9021 9025 9029 8 9034 9038 9043 9047 9052 9056 9061 9065 9069 9074 9 9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 980 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 1 9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 2 9211 9216 9220 9224 9229 9233 9238 9242 9247 9251 3 9255 9260 9264 9269 9273 9277 9282 9286 9291 9295 4 9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 5 9344 9348 9352 9357 9361 9366 9370 9374 9379 9383 6 9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 7 9432 9436 9441 9445 9449 9454 9458 9463 9467 9471 8 9476 9480 9484 9489 9493 9498 9502 9506 9511 9516 9 9520 9524 9528 9533 9537 9542 9546 9550 9555 9559 990 99564 99568 99572 99577 99581 99585 99590 99594 99599 99603 1 9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 2 9651 9656 9660 9664 9669 9673 9677 9682 9686 9691 3 9695 9699 9704 9708 9712 9717 9721 9726 9730 9734 4 9739 9743 9747 9752 9756' 9760 9765 9769 9774 9778 5 9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 i 9826 9830 9835 9839 9843 9848 9852 9856 9861 9865 7 9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 8 9913 9917 9922 9926 9930 9935 9939 9944 9948 9952 9 9957 9961 9965 9970 9974 9978 9983 9987 9991 9996 1000 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 176 TABLE 8. LOGARITHMIC SINES AND COSINES. t 1 2 Sine Cosine Sine Cosine Sine Cosine 00 10.00000 8.24186 9.99993 8.54282 9 99974 60 1 6.46373 00000 24903 99993 54642 99973 59 2 76476 00000 25609 ' 99993 54999 99973 58 3 94085 00000 26304 99993 55354 99972 57 4 7.0G579 00000 26988 99992 55705 99972 56 5 16270 00000 27661 99992 56054 69971 55 6 24188 00000 28324 99992 56400 99971 54 7 30882 00000 28977 99992 56743 99970 51 8 36682 00000 29621 99992 57084 99970 52 9 41797 00000 30255 99991 57421 99969 51 10 7.46373 10.00000 8.30879 9.99991 8.57757 9.99969 50 11 50512 00000 31495 99991 58089 99968 49 12 54291 00000 32103 99990 58419 99968 48 13 57767 00000 32702 99990 58747 99967 47 14 60985 00000 33292 99990 59072 99967 46 15 63982 00000 33875 99990 59395 99967 45 16 66784 00000 34450 99989 59715 99960 44 17 69417 9.99999 35018 99989 60033 99966 43 18 71900 99999 35578 99989 60349 99965 42 19 74248 99999 36131 99989 60662 99964 41 20 7.76475 9.99999 8.36678 9.99988 8.60973 9.99964 40 21 78594 99999 37217 99988 61282 99963 39 22 80615 99999 37750 99988 61589 99963 38 23 82545 99999 38276 99987 61894 99962 37 24 84393 99999 38796 99987 62196 99962 36 25 86166 99999 39310 99987 62497 99961 35 26 87870 99999 39818 99986 62795 99961 34 27 89509 99999 40320 99986 63091 99960 33 28 91088 99999 40816 99986 63385 99960 32 29 92612 99998 41307 99985 63678 99959 31 30 7.94084 9.99998 8.41792 9.99985 8.63968 9.99959 30 31 95508 99998 42272 99982 64256 99958 29 32 96887 99998 42746 99984 64543 99958 28 33 98223 99998 43216 99984 64827 99957 27 34 99520 99998 43680 99984 65110 99956 26 35 8.00779 99998 44139 99983 65391 99956 25 36 02002 99998 44594 99983 65670 99955 24 37 03192 99997 45044 99983 65947 99955 23 38 04350 99997 45489 99982 66223 99954 22 39 05478 99997 45930 99982 66497 99954 21 40 8.06578 9.99997 8.46366 9.99982 8.66769 9.99953 20 41 07650 99997 46799 99981 67039 99952 19 42 08696 99997 47226 99981 67308 99952 18 43 09718 99997 47650 99981 67575 99951 17 44 10717 99996 48069 99980 67841 99951 16 45 11693 99996 48485 99980 68104 99950 15 46 12647 99996 48896 99979 68367 99949 14 47 13581 99996 49304 99979 68627 99949 13 48 14495 99996 49708 99979 68886 99948 12 49 15391 99996 50108 99978 69144 99948 11 50 8.16268 9.99995 8.50504 9.99978 8.69400 9.99947 10 51 17128 99995 50897 99977 69654 99946 9 52 17971 99995 51287 99977 69907 99946 8 53 18798 99995 51673 99977 70159 99945 7 54 19610 99995 52055 99976 70409 99944 6 55 20407 99994 52434 99976 70658 99944 5 56 21189 99994 52810 99975 70905 99943 4 57 21958 99994 53183 99975 71151 99942 3 58 22713 99994 53552 99974 71395 99942 2 59 23456 99994 53919 99974 71638 99941 1 60 24186 99993 54282 99974 71880 99940 / Cosine Sine Cosine Sine Cosine Sine / 89 88 87 TABLE 8. LOGARITHMIC SINES AND COSINES. 177 / 3 4 6 Sine Cosine Sine Cosine Sine Cosine 8.71880 9.99940 8.84358 9.99894 8.94030 9.99834 60 1 72120 99940 84539 99893 94174 99833 59 2 72359 99939 84718 99892 94317 99832 58 3 72597 99938 84897 99891 944C1 99831 57 4 72834 99938 85075 99891 94603 99830 66 5 73069 99937 85252 99890 94746 99829 55 6 73303 99936 85429 99889 94887 998*8 54 7 73535 99936 85605 99888 95029 99827 53 8 73767 99935 85780 99887 95170 99825 52 9 73997 99934 85955 99886 95310 99824 51 10 8.74226 9.99934 8.86128 9.99885 8.95450 9.99823 50 11 74454 99933 86301 99.884 95589 99822 49 12 74680 99932 86474 99*883 95728 99821 48 13 74906 99932 86645 99882 95867 99820 47 14 75130 99931 86816 99881 96005 99819 46 15 75853 99930 86987 99880 96143 99817 45 16 75575 99929 87156 99879 96280 99816 44 17 75795 99929 87325 99879 96417 99815 43 18 76015 99928 87494 99878 96553 99814 42 19 76234 99927 87661 99877 96689 99813 41 20 8.76451 9.99926 8.87829 9.99876 8.96825 9.99812 40 21 76667 99926 87995 99875 96960 99810 39 22 76883 99925 88161 99874 97095 99809 38 23 77097 99924 88326 99873 97229 99808 37 24 77310 99923 88490 99872 97363 99807 36 25 77522 99923 88654 99871 97496 99806 35 26 77733 99922 88817 99870 97629 99804 34 27 , 77943 99921 88980 99869 97762 99803 33 28 78152 99920 89142 99868 97894 99802 33 29 78360 99920 89304 99867 98026 99801 31 30 8.78568 9.99919 8.89464 9.99866 8.98157 9.99800 30 31 78774 99918 89625 99805 98288 99798 29 32 78979 99917 89784 99864 98419 99797 28 33 79183 99917 89943 99863 98549 99796 27 34 79386 99916 P0102 9986'* 98679 99795 26 35 79588 99915 90260 99861 98808 99793 25 36 79789 99914 90417 99860 98937 99792 24 37 79990 99913 90574 99859 99066 99791 23 38 80189 99913 90730 99858 99194 99790 22 39 80388 99912 90885 99857 99322 99788 21 40 8.80585 9.99911 8.91040 9.99856 8.99450 9.99787 20 41 8078-2 99910 91195 99855 99577 99786 19 42 80978 99909 91349 99854 99704 99785 18 43 81173 99909 91502 99853 99830 99783 17 44 81367 99908 91655 99852 99956 99782 16 45 81560 99907 91807 99851 9.00082 99781 15 46 81752 99906 91959 99850 00207 99780 14 47 81944 99905 92110 99848 00332 99778 13 48 82134 99904 92261 99847 00456 99777 12 49 82324 99904 92411 99846 00581 99776 11 50 8.82513 9.99903 8.92561 9.99845 9.00704 9.99775 10 51 82701 99902 92710 99844 00828 99773 9 52 82888 99901 92859 99843 00951 99772 8 53 83075 99900 93007 99842 01074 99771 7 54 83261 99899 93154 99841 01196 99769 6 55 83446 99898 93301 99840 01318 99768 5 56 83630 99898 93448 99839 01440 99767 4 57 83813 99897 93594 99838 01561 99765 3 58 83996 99896 93740 99837 01682 99764 2 59 84177 99895 93885 99836 01803 99763 1 60 84358 99894 94030 99834 01923 99761 / Cosine Sine Cosine Sine Cosine Sine / i 86 85 84 178 TABLE 8. LOGARITHMIC SINES AND COSINES. / 6 7 8 / Sine Cosine Sine Cosine Sine Cosine 9.01923 9.99761 9.08589 9.99675 9.14356 9.99575 60 1 02043 99760 08692 99674 14445 99574 59 2 02163 99759 08795 99672 14535 99572 58 3 02283 99757 08897' 99670 14624 99570 57 4 03402 99756 08999 99669 14714 99568 56 5 02520 99755 09101 99667 14803 99566 55 6 02639 99753 09202 99666 14891 99565 54 7 02757 99752 09304 99664 14980 99563 53 8 02874 99751 09405 99663 15069 99561 52 9 02992 99749 09506 99661 15157 99559 51 10 9.03109 9.99748 9.09606 9.99659 9.15245 9.99557 50 11 03226 99747 09707 99658 15333 99556 49 12 03342 99745 09807 99656 15421 99554 48 13 03458 99744 09907 99655 15508 99552 47 14 03574 99742 10006 99653 15596 99550 46 15 03690 99741 10106 99651 15683 99548 45 16 03805 99740 10205 99650 15770 99546 44 17 03920 99738 10304 99648 15857 99545 43 18 04034 99737 10402 99647 15944 99543 42 19 04149 99736 10501 99645 1C030 99541 41 20 9.04262 9.99734 9.10599 9.99643 9.16116 9.99539 40 21 04376 99733 10697 99642 16203 99537 39 22 04490 99731 10795 99640 16289 99535 38 23 04603 99730 10893 99638 16374 89533 37 24 04715 99728 10990 99637 16460 99532 86 25 04828 99727 11087 99635 16545 99530 35 26 04940 99726 11184 99633 16631 99528 34 27 05052 99724 11281 99632 16716 99526 33 28 05164 99723 11377 99630 16801 99524 32 29 05275 99721 11474 99629 16886 99522 31 30 9.05386 9.99720 9.11570 9.99627 9.16970 9.99520 30 31 05497 99718 11666 99625 17055 99518 29 32 05607 99717 11761 99624 17139 99517 28 33 05717 99716 11857 99622 17223 99515 27 34 05827 99714 11952 99620 J7807 99513 26 35 05937 99713 12047 99618 17391 99511 25 36 06046 99711 12142 99617 17474 99509 24 37 06155 99710 12236 99615 17558 99507 23 38 06264 99708 1233i 99613 17641 99505 22 39 06372 99707 12425 99612 17724 99503 21 40 9.06481 9.99705 9.12519 &. 99610 9.17807 9.99501 20 41 06589 99704 12612 99608 17890 99499 19 42 06696 99702 12706 99607 17973 99497 18 43 06804 99701 12799 99605 18055 99495 17 44 06911 99699 12892 99603 18137 99494 16 45 07018 99698 12985 99601 18220 99492 15 46 07124 99696 13078 99600 18302 99490 14 47 07231 99695 13171 99598 18383 99488 13 48 07337 99G93 13263 99596 18465 99486 19 49 07442 99692 13355 99595 18547 99484 11 50 9.07548 9.99690 9.13447 9.99593 9.18628 9.99482 30 51 07653 99689 13539 99591 18709 99480 9 52 07758 99687 13630 99589 18790 99478 8 53 07863 99686 13722 99588 18871 99476 7 54 07968 99684 13813 99586 18952 99474 6 55 08072 99683 18904 99584 19033 99472 5 56 08176 99681 13994 99582 19113 99470 4 57 08280 99680 14085 99581 19193 99468 3 58 08383 99678 14175 99579 19273 99466 2 59 08486 99677 14266 99577 19353 99464 1 60 08589 99675 14356 99575 19433 99462 I Cosine Sine Cosine Sine Cosine Sine I 88 82 81* TABLE 8. LOGARITHMIC SINES AND COSINES, 179 / 9 10 11 / Sine Cosine Sine Cosine Sine Cosine 9.19433 S. 99482 9.23967 9.99335 9.28060 9.99195 60 1 19513 99460 24039 99333 28125 99192 59 2 19592 99458 24110 99331 28190 99190 58 3 19672 99456 24181 99328 28254 99187 57 4 19751 99454 24253 99326 28319 99185 56 5 19830 9945* 24324 99324 28384 99182 55 6 19909 99450 24395 99322 28448 99180 54 7 19988 99448 24466 99319 28512 99177 53 8 20067 99446 24536 99317 28577 99175 52 9 20145 99444 24607 99315 28641 99172 61 10 9.20223 9.99442 9.24677 9.99313 9.28705 9.99170 50 11 20302 99440 24748 99310 2876 99167 49 12 20380 99438 24818 99308 28833 99165 48 13 20458 99436 24888 99306 28896 99162 47 14 20535 99434 24958 99304 28960 99160 46 15 20613 99432 25028 99301 29024 99157 45 16 20691 99429 25098 99299 29087 99155 44 17 207G8 99427 25168 99297 29150 99152 43 18 20845 99425 25237 99294 29214 99150 42 19 20922 99423 25307 99292 29277 99147 41 20 9.20999 9.99421 9.25376 9.99290 9.29340 9.99145 40 21 21076 99419 25445 99288 29403 99142 39 22 21153 99417 -25514 99285 29466 99140 38 23 21229 99415 25583 99283 29529 99137 37 24 21306 99413 25652 99281 29591 99135 36 25 21382 99411 25721 99278 29654 99132 35 2o 21458 99409 25790 99276 29716 99130 34 27 21534 99407 25858 99274 29779 99127 33 28 21610 99404 25927 99271 29841 99124 32 29 21685 99402 25995 99269 29903 99122 31 30 9.21761 9.99400 9.26063 9.99267 9.29966 9.99119 30 31 21836 99398 26131 99264 30028 99117 29 32 21912 99396 26199 99262 30090 99114 28 33 21987 99394 26267 99260 30151 99112 27 34 22062 99392 26335 99257 30213 99109 26 35 22137 99390 26403 99255 30275 99106 25 36 22211 99388 26470 99252 30336 99104 24 37 22286 99385 26538 99250 30398 99001 23 38 22361 99383 26605 99248 30459 99099 22 39 22435 99381 26672 99245 30521 99096 21 40 9.22509 9.99379 9.26739 9.99243 9.30582 9.99093 20 41 22583 99377 26806 99241 30643 99091 19 42 22657 99375 26873 99238 30704 99088 18 43 22731 99372 26940 99236 30765 99086 17 44 22805 99370 27007 99233 30826 99083 16 45 22878 99368 27073 99231 30887 99080 15 46 22952 99366 27140 99229 30947 99078 14 47 23025 99364 27206 99226 31008 99075 13 48 23098 99362 27273 99224 31068 99072 12 49 23171 99359 27339 99221 81129 99070 11 50 9.23244 9.99357 9.27405 9.99219 9.31189 9.99067 10 51 23317 99355 27471 99217 31250 99064 9 52 23390 99353 27537 99214 31310 99062 8 53 23462 99351 27602 99212 31370 99059 7 54 23535 99348 27668 99209 31430 99056 6 55 23607 99346 27734 99207 31490 99054 5 56 23679 99344 27799 99204 . 31549 99051 4 57 23752 9934? 27864 99202 31609 99048 3 58 23823 99340 27930 99200 31669 99046 2 59 23895 99337 27995 99197 31728 99043 1 60 23967 99335 28060 99195 31788 99040 t Cosine Sine Cosine Sine Cosine Sine / 80 79 78 180 TABLE 8. LOGARITHMIC SINES AND COSINES. I 2 13 14 Sine Cosine Sine Cosine Sine Cosine 9.31788 9.99040 9.35209 9.98872 9.38368 9.98690 60 1 31847 99038 35263 98869 38418 98687 59 2 31907 99035 35318 98867 38469 98684 58 3 31966 99032 35373 98864 38519 98681 57 4 32025 99030 35427 98861 38570 98678 56 5 32084 99027 35481 98858 38620 98675 55 6 32143 99024 35536 98855 38670 98671 54 7 32202 99022 35590 98852 38721 98668 53 8 32261 99019 35644 98849 38771 98665 52 9 32319 99016 35698 98846 38821 98662 51 10 9.32378 9.99013 9.35752 9.98843 9.38871 9.98659 50 11 32437 99011 35806 98840 38921 98656 49 12 32495 99008 35860 98837 38971 98652 48 13 32553 99005 35914 98834 39021 98649 47 14 32612 99002 35968 98831 39071 98646 46 15 32670 99000 36022 98828 39121 98643 45 16 32728 98997 36075 98825 39170 98640 44 17 32786 98994 ' 36129 98822 39220 98636 43 18 32844 98991 36182 98819 39270 98633 42 19 32902 98989 36236 98816 39319 98030 41 20 9.32960 9.98986 9.36289 9.98813 9.39369 9.98627 40 21 33018 98983 36342 98810 39418 98623 39 22 33075 98980 36395 98807 39467 98620 38 23 33133 98978 36449 98804 39517 98617 37 24 33190 98975 36502 98801 39566 98614 36 25 33248 98972 36555 98798 39615 98610 35 26 33305 98969 36608 98795 39664 98607 34 27 33362 98967 36660 98792 39713 98604 33 28 33420 98964 36713 98789 39762 98601 32 29 33477 98961 36766 98786 39811 98597 31 30 9.33534 9.98958 9.36819 9.98783 9.39860 9.98594 30 31 33591 98955 36871 98780 39909 98591 29 32 33647 98953 36924 98777 39958 98588 28 33 33704 98950 36976 98774 40006 98584 27 34 33761 98947 37028 98771 40055 98581 26 35 33818 98944 37081 98768 40103 98578 25 36 33874 98941 37133 98765 40152 98574 24 37 83931 98938 37185 98762 40200 98571 23 38 33987 98936 37237 98759 40249 98568 22 39 34043 98933 37289 98756 40297 98565 21 40 9.34100 9.98930 9.37341 9.98753 9.40346 9.98561 20 41 34156 98927 37393 98750 40394 98558 19 42 34212 98924 37445 98746 40442 98555 18 43 34268 98921 37497 98743 40490 98551 17 44 34324 98919 37549 98740 40538 98548 16 45 34380 98916 37600 98737 4058G 98545 15 46 34436 98913 37652 98734 40634 98541 14 47 34491 98910 37703 98731 40682 98538 13 48 34547 98907 37755 98728 40730 98535 12 49 34602 98904 37806 98725 40778 98531 11 50 9.34658 9.98901 9.37858 9.98722 9.40825 9.98528 10 51 34713 98898 37909 98719 40873 98525 9 52 34769 98896 37960 ' 98715 40921 98521 8 53 34824 98893 38011 98712 40968 98518 7 54 34879 98890 38062 98709 41016 98515 6 55 34934 98887 38113 98706 41063 98511 5 56 34989 98884 38164 98703 41111 98508 4 57 35044 98881 38215 98700 41158 98505 3 58 35099 98878 38266 98697 41205 98501 2 59 35154 98875 38317 98694 41252 98498 1 00 35209 98872 38368 98690 41300 98494 t Cosine Sine Cosine Sine Cosine Sine t 77 o 76 75 TABLE 8. LOGARITHMIC SINES AND COSINES. 181 t 15 16 17 / Sine Cosine Sine Cosine Sine Cosine 9.41300 9.98494 9.44034 9.98284 9.46594 9.98060 60 I 41347 98491 44078 98281 46635 98056 59 2 41394 98488 44122 98277 46676 98052 58 3 41441 98484 44166 98273 46717 98048 57 4 41488 98481 44210 98270 46758 98044 56 5 41535 98477 44253 98266 46800 98040 55 6 41582 98474 44297 98262 46841 98036 54 7 416-,>8 98471 44341 98259 46882 98032 53 8 41675 98467 44385 98255 46923 98029 52 9 41722 98464 44428 98251 46964 98025 51 10 9.41768 9.98460 9.44472 9.98248 9.47005 9.98021 50 11 41815 98457 44516 98244 47045 98017 49 12 41861 98453 44559 96240 47086 98013 48 18 41908 98450 44602 98237 47127 98009 47 14 41954 98447 44646 98233 47168 98005 46 15 42001 98443 44689 98229 47209 98001 45 16 42047 98440 44733 98226 47249 97997 44 17 42093 98436 44776 98222 47290 97993 43 18 42140 98433 44819 98218 47330 97989 42 19 42186 98429 44862 98215 47371 97986 41 20 9.42232 9.98426 9.44905 9.98211 9.47411 9.97982 40 21 42278 98422 44948 98207 47452 97978 39 22 42324 98419 44992 98204 47492 97974 38 23 42370 98415 45035 98200 47533 97970 37 24 42416 98412 45077 98196 47573 97966 36 25 42461 98409 45120 98192 47613 97962 35 26 42507 98405 45163 98189 41654 97958 34 27 42553 98402 45206 98185 47694 97954 33 28 42599 98398 45249 98181 47734 97950 32 29 42(344 98395 45292 98177 47774 97946 31 30 9.42690 9.98391 9.45334 9.98174 9.47814 9.97942 30 31 42735 98388 45377 98170 47854 97938 29 32 42781 98384 45419 98166 47894 97934 28 33 42826 98381 45462 98162 47934 97930 27 34 42872 98377 45504 98159 47974 97926 26 35 42917 98373 45547 98155 4h014 97922 25 36 42962 98370 45589 98151 48054 97918 24 37 4:3008 98366 45632 98147 48094 97914 23 38 43053 98363 45674 98144 48133 97910 22 39 43098 98359 45716 98140 48173 97906 21 40 9.43143 9.98356 9.45758 9.98136 9.48213 9.97902 20 41 43188 98352 45801 98132 48252 97898 19 42 43233 98349 45843 98129 48292 97894 18 43 43278 98345 4.1885 98125 48332 97890 17 44 43323 98342 45927 98121 48371 97886 16 45 43367 98338 45969 98117 48411 97882 15 46 43412 98334 46011 98113 48450 97878 14 47 43457 98331 46053 98110 48490 97874 18 48 43502 98327 46095 98106 48529 97870 12 49 43546 98324 46136 98102 48568 97866 |1 50 9.43591 9.98320 9.46178 9.98098 9.48607 9.97861 10 51 43635 98317 46220 98094 48647 97857 9 52 43680 98313 46-^62 98090 48686 97853 8 53 43724 98309 46303 9H087 48725 97849 7 54 43769 98306 46345 98083 48764 97845 6 55 43813 98302 46386 98079 48803 97841 5 56 43857 98299 46428 98075 48842 97837 4 57 43901 98295 46469 98071 48881 97833 3 58 43946 98291 46511 98067 4S920 97829 2 59 43990 98288 46552 98063 48959 97825 1 60 44034 98284 36594 98060 48998 97821 f Cosine Sine Cosine Sine Cosine Sine / 74 73 72 182 TABLE 8. LOGARITHMIC SINES AND COSINES. 18 19 20 Sine Cosine Sine Cosine Sine Cosine 9.48998 9.97821 9.51264 9.97567 9.53405 9 97299 60 1 1 49037 97817 51301 97563 53440 97294 59 ) 2 49076 97812 51338 97558 53475 97289 58 3 49115 97808 51374 97554 53509 97285 57 4 49153 97804 51411 97550 53544 97280 50 5 49192 97800 51447 97545 53578 97276 55 6 49231 97796 51484 97541 53613 97271 54 7 49269 97792 51520 97536 53647 97266 53 8 49308 97788 51557 97532 53682 97262 52 9 49347 97784 51593 97528 53716 97257 51 10 9.49385 9.97779 9.51629 9.97523 9.53751 9 97252 50 11 49424 97775 51666 97519 53785 97248 49 12 49462 97771 51702 97515 53819 97243 48 18 49500 97767 51738 97510 53854 97238 47 14 49539 97763 51774 97506 53888 97234 46 15 49577 97759 51811 97501 53922 97229 ; 45 16 49615 97754 51847 97497 53957 97224 44 17 49654 97750 51883 97492 53991 97220 43 18 49692 97746 51919 97488 54025 97215 42 19 49730 97742 51955 97484 54059 97210 41 20 9.49768 9.97738 9.51991 9.97479 9.54093 9 97206 40 21 49806 97734 52027 97475 54127 97201 39 22 49844 97729 52063 97470 54161 97196 38 23 49882 97725 52099 97466 54195 97192 37 24 49920 97721 52135 97461 54229 97187 36 25 49958 97717 52171 97457 54263 97182 35 26 49996 97713 52207 97453 54297 97178 34 27 50034 97708 52242 97448 54331 97173 33 28 50072 97704 52278 97444 54365 97168 32 29 50110 97700 52314 97439 54399 97163 31 30 9.50148 9.97696 9.52350 9.97435 9.54433 9 97159 30 31 50185 97691 52385 97430 54466 97154 29 32 50223 97687 52421 97426 54500 97149 28 33 50261 97683 52456 97421 54534 97145 27 34 50298 97679 52492 97417 54567 97140 26 35 50336 97674 52527 97412 54601 97135 25 36 50374 97670 52563 97408 54635 97130 24 37 50411 97666 52598 97403 54668 97126 23 38 50449 97662 52634 97399 54702 97121 22 39 50486 97657 52669 97394 54735 97116 21 40 9.50523 9.97653 9.52705 9.97390 9.54769 9 97111 20 41 50561 97649 52740 97385 54802 . 97107 19 42 50593 97645 52775 97381 54836 97102 18 43 50635 97640 52811 97376 54869 97097 17 44 50673 97636 52846 97372 54903 97092 16 45 50710 97632 52881 97367 54936 97087 15 46 50747 97628 52916 97363 54969 97083 14 47 50784 97623 52951 97358 55003 97078 13 48 50821 97619 52986 97353 55036 97073 12 49 50858 97615 53021 97349 55069 97068 11 50 9.50896 9.97610 9.53056 9.97344 9.55102 9 .97063 10 51 50933 97606 53092 97340 55136 97059 9 52 50970 97602 53126 97335 55169 97054 8 53 51007 97597 53161 97331 55202 97049 7 54 51043 97593 53196 97326 55235 97044 6 55 51080 97589 53231 97322 55068 97039 5 56 51117 97584 53266 97317 55301 97035 4 57 51154 97580 53301 97312 55334 97030 3 58 51191 97576 53336 97308 55367 97025 2 59 51227 97571 53370 97303 55400 97020 1 60 51264 97567 53405 9729!) 55433 97015 t Cosine Sine Cosine Sine Cosine Sine / 71 70 69 TABLE 8. LOGARITHMIC SINES AND COSINES. 183 / 21 22 23 o t Sine Cosine Sine Cosine Sine Cosine 9.55433 9.97015 9.57358 9.96717 9.59188 9.96403 60 1 55466 97010 57389 96711 59218 96397 59 2 55499 97005 57420 96706 59247 96392 58 3 5553vJ 97001 57451 96701 59277 963^7 57 4 55564 96996 57482 96696 59307 96381 56 5 55597 96991 57514 96691 59336 96376 55 6 55630 96986 57545 96686 59366 96370 54 7 55663 96981 57576 96681 59396 96365 53 8 55695 96976 57607 96676 59425 96360 52 9 55728 96971 57638 96670 59455 96354 51 10 9.55761 9.96966 9.57669 9.96665 9.59484 9.96349 50 11 55793 969(52 57700 96C60 59514 96343 49 12 55S26 96957 57731 96655 59543 96338 48 13 55858 96952 57762 96650 59573 96333 47 14 55891 96947 57793 96645 59602 96327 46 15 55923 96942 57824 96640 59632 96322 45 16 55956 96937 57855 96634 59661 96316 44 17 55988 96932 57885 96629 59690 96311 43 18 56021 96927 57916 96624 59720 96305 42 19 56053 90922 57947 96619 59749 96300 41 20 9.56085 9.96917 9.57978 9.96614 9.59778 9.96294 40 21 56118 96912 5800.8 96608 59808 96289 39 22 56150 96907 58039 96603 59837 96284 38 23 56182 96903 58070 96598 59866 96278 37 24 56215 96898 58101 96593 59895 96273 36 25 56-J47 96893 58131 96588 59924 96267 35 26 56279 96888 58162 96582 59954 96262 34 27 56311 96883 58192 96577 59983 96256 33 28 56343 96878 58223 96572 60012 96251 32 29 56375 96873 58253 96567 60041 96245 31 30 9.56408 9.96868 9.58284 9.96562 9.60070 9.96240 30 31 56440 96863 58314 96556 60099 96234 29 32 56472 96858 58345 96551 60128 96229 28 33 56504 96853 58375 96546 60157 962^3 27 34 56536 96848 58406 96541 60186 90218 26 35 56568 96843 58436 96535 60215 96212 25 36 56599 96838 58467 96530 60244 96207 24 37 56631 96833 58497 965^5 60273 96201 23 38 56663 96828 58527 96520 60302 96196 22 39 56695 96823 58557 96514 60331 96190 21 40 9.56727 9.96818 9.58588 9.96509 9.60359 9.96185 20 41 56759 90813 58618 96504 60^88 96179 19 42 56790 96808 58648 96498 60417 96174 18 43 668*2 96803 58678 96493 60446 96168 17 44 56854 96798 58709 96488 60474 96162 16 45 56886 96793 58739 96483 60503 i ! 6157 15 46 56917 96788 58769 96477 60532 96151 14 47 56949 96783 58799 96472 60561 96146 13 48 56980 96778 58829 96467 60589 96140 12 49 57012 90772 58859 96461 60618 96135 11 50 9.57044 9.96767 9.58889 9.96456 9.60646 9.96129 10 51 57075 96702 58919 96451 60675 96123 9 52 57107 96757 58949 96445 60704 96118 8 53 57138 96752 58979 96440 60732 96112 7 54 57169 90747 59009 96435 60761 96107 6 55 57201 96742 59039 96429 60789 96101 5 56 96737 59069 96424 60818 96095 4 57 57264 96732 59098 96419 60846 96090 8 58 57295 96727 59128 96413 60875 96084 2 59 57326 96722 59158 96408 60903 96079 1 60 57358 96717 59188 96403 60931 96073 t Cosine Sine Cosine Sine Cosine Sine i 68 67 66 184 TABLE 8. LOGARITHMIC SINES AND COSINES. 24 25 \ 6 Sine Cosine Sine Cosine Sine Cosine o 9.60931 9.96073 9.62595 9.95728 9.64184 9.95366 60 1 60960 96067 62622 95722 64210 95360 59 2 60988 96062 62649 95716 64236 95354 58 3 61016 96056 62676 95710 64262 95348 57 4 61045 96050 62703 95704 64288 95341 56 5 61073 96045 62730 95698 64313 95335 55 6 61101 96039 62757 95692 64339 95329 54 7 61129 96034 62784 95686 64365 95323 53 8 61158 96028 62811 95680 64391 95317 52 9 61186 96022 62838 95674 64417 95310 51 10 9.61214 9.96017 9.62865 9.95668 9.64442 9.95304 50 11 61242 96011 62892 95663 64468 95298 49 12 61270 96005 62918 95657 64494 95292 48 13 61298 96000 62945 95651 64519 95286 47 14 61326 95994 62972 95645 64545 95279 46 15 61354 95988 62999 95639 64571 95273 45 16 61382 95982 63026 95633 64596 95267 44 17 61411 95977 63052 95627 64622 95261 43 18 61438 95971 63079 95621 64647 95254 42 19 61466 95965 63106 95615 64673 95248 41 20 9.61494 9.95960 9.63133 9.95609 9.64698 9.95242 40 21 61522 95954 63159 95603 64724 95236 39 22 61550 95948 63186 95597 64749 95229 38 23 61578 95942 63213 95591 64775 95223 37 24 61606 95937 63239 95585 64800 95217 36 25 61634 95931 632G6 95579 64826 95211 35 26 61662 95925 63292 95573 64851 95204 34 27 61689 95920 63319 95567 64877 95198 33 28 61717 95914 63345 95561 64902 95192 32 29 61745 95908 63372 95555 64927 95185 31 30 9.61773 9.95902 9.63398 9.95549 9.64953 9.95179 30 31 61800 95897 63425 95543 64978 95173 29 32 618-28 95891 63451 95537 65003 95167 28 33 61856 95885 63478 95531 65029 95160 27 34 61883 95879 C3504 95525 65054 95154 26 35 61911 95873 63531 95519 65079 95148 25 86 61939 95868 63557 95513 65104 95141 24 37 61966 95862 63583 95507 65130 95135 23 38 61994 95856 63610 95500 65155 95129 22 39 62021 95850 63636 95494 65180 95122 21 40 9.62049 9.95844 9.63662 9.95488 9.65205 9.95116 20 41 62076 95839 63689 95482 65230 95110 19 42 62104 95833 63715 95476 65255 95103 18 43 62131 95827 63741 95470 65281 95097 17 44 62159 95821 63767 95464 65306 95090 16 45 62186 95815 63794 95458 65331 95084 15 46 62214 95810 63820 95452 65356 95078 14 47 62241 95804 63846 95446 65381 95071 13 48 62268 95798 63872 95440 65406 95065 12 49 62296 95792 63898 95431 65431 95059 11 50 9.62323 9.95786 9.63924 9.95427 9.65456 9.95052 10 51 62350 95780 63950 95421 65481 95046 9 52 62377 95775 63976 95415 65506 95039 8 53 62405 95769 64002 95409 65531 95033 7 54 62432 95763 64028 95403 65565 95027 6 55 62459 95757 64054 95397 65580 95020 5 56 62486 95751 64080 95391 65605 95014 4 57 62513 95745 64106 95384 65630 95007 3 58 62541 95739 64132 95378 65655 95001 2 59 62568 95733 64158 95372 65680 94995 1 60 62595 95728 64184 95366 65705 94988 / Cosine Sine Cosine Sine Cosine Sine / 66 64 63 TABLE 8. LOGARITHMIC SINES AND COSINES. 185 / 27 28 29 t Sine Cosine Sine Cosine Sine Cosine 9.65705 9.94988 9.67161 9.94593 9.68557 9.94182 60 1 65729 94982 67185 94587 68580 94175 59 2 65754 94975 67208 94580 68603 94168 58 3 65779 94969 67232 94573 68625 94161 57 4 65804 94962 67256 94567 68648 94154 56 5 65828 94956 07280 94560 68671 94147 55 6 65853 94949 67303 94553 68694 94140 54 7 65878 94943 67327 94546 68716 94133 53 8 65902 94936 67350 94540 68739 94126 52 9 65927 94930 67374 94533 68762 94119 51 10 9.65952 9.94923 9.67398 9.94526 9.68784 9.94112 50 11 65976 94917 67421 94519 68807 94105 49 12 66001 94911 67445 94513 68829 94098 48 18 66025 94904 67468 94506 68852 94090 47 14 66050 94898 67492 94499 68875 94083 46 15 66075 94891 67515 94492 68897 94076 45 16 66099 94885 67539 94485 68920 94069 44 17 66124 94878 67562 94479 68942 94062 43 18 66148 94871 67586 94472 68965 94055 42 19 66173 94805 67609 94465 68987 94048 41 20 9.66197 9.94858 9.67633 9.94458 9.69010 9.94041 40 21 66221 94852 67656 94451 69032 94034 39 22 66246 94845 67680 94445 69055 94027 38 23 66270 94839 67703 94438 69077 94020 37 24 66295 9J832 67726 94431 69100 94012 36 25 66319 94826 67750 94424 69122 94005 35 26 66343 94819 67773 91417 69144 93998 34 27 66368 94813 67796 94410 69167 93991 33 28 66392 94806 67820 94404 69189 S3984 82 29 66416 94799 67843 94397 69212 93977 31 30 9.66441 9.94793 9.67866 9.94390 9.69234 9.93970 30 31 66465 94786 67890 94883 69256 93963 29 32 66489 94780 67913 94376 69279 93955 28 33 66513 94773 67936 94369 69301 93948 27 34 66537 94767 67959 94362 69323 93941 26 * 35 66562 94760 67982 94355 69345 93934 25 36 665^6 94753 68006 94349 69368 93927 24 37 66610 94747 68029 94342 69390 93920 23 38 66634 94740 68052 94335 69412 93912 22 39 66G58 94734 68075 94328 69434 93905 21 40 9.66682 9.94727 9.68098 9.94321 9.69456 9.93898 20 41 66706 94720 68121 94314 69479 93891 19 42 66731 94714 68144 94307 69501 93884 18 43 66755 94707 68167 94300 69523 93876 17 44 66779 94700 68190 94293 69545 93869 16 45 66803 94694 68213 94286 69567 93862 15 46 66827 9468? 68237 94279 69589 93855 14 47 66851 94680 68260 94273 69611 93847 13 48 66875 94674 68283 94266 69638 93840 12 49 66899 94667 68305 94259 69655 93833 11 50 9.66922 9.94660 9.68328 9.94252 9.69677 9.93826 10 51 66946 94654 68351 94245 69699 93819 9 52 66970 94647 68374 94238 93811 8 53 66994 94640 68397 94231 69743 93804 7 54 67018 94634 68420 94224 69765 93797 6 55 67042 94627 68443 94217 69787 93789 5 66 67066 94620 68466 94210 69809 93782 4 57 67090 94614 68489 94203 69831 93775 3 58 67113 94607 68512 94196 69853 93768 2 59 67137 94GOO 68534 94189 69875 93760 1 60 67161 94593 68557 94182 69897 93753 / Cosine Sine Cosine Sine Cosine Sine / 62 61 o 60 186 TABLE 8. LOGARITHMIC SINES AND COSINES. $ 80 81 8 !2 Sine Cosine Sine Cosine Sine Cosine 9.69897 9.93753 9.71184 9.93307 9.72421 9.92842 60 1 69919 93746 71205 93299 72441 92834 59 2 69941 93738 71226 93291 72461 92826 58 3 69963 93731 71247 93284 72482 92818 57 4 69984 93724 71268 93276 72502 92810 50 5 70006 93717 71289 93269 72528 92803 55 6 70028 93709 71310 93261 72542 92795 54 7 70050 93702 71331 93253 7256-3 92787 53 8 70072 93595 71352 93246 72582 92779 52 9 70093 93687 71373 93238 72602 92771 51 10 9.70115 9.93680 9.71393 9.93230 9 7262 9.92703 50 11 70137 93673 71414 932,3 72048 92755 49 12 70159 93665 71435 93215 72063 92747 48 13 70180 93658 71456 93207 7208 J 92739 47 14 70202 93650 71477 93200 72703 92731 46 15 70224 93643 71498 93192 72723 92723 45 16 70245 93636 71519 93184 72743 92715 44 17 70267 93628 71539 93177 72763 92707 43 18 70288 936'>1 71560 93169 72783 92099 42 19 70310 93614 71581 93161 72803 92691 41 20 9.70332 9.93606 9.71602 9.93154 9.72828 9.92683 40 21 70353 93599 71622 93146 72843 92675 39 22 70375 93591 71643 93138 72863 92667 38 23 70396 93584 71664 93131 72883 92659 37 24 70418 93577 71685 93123 72902 92051 36 25 70439 93569 71705 93115 72922 92043 35 26 70461 93562 71726 93108 72942 92635 34 27 70482 93554 71747 93100 72962 92027 33 28 70504 93547 71767 93092 72982 92019 32 29 70525 93539 71788 93084 73002 92011 31 80 9.70547 9.93532 9.71809 9.93077 9.73022 9.92603 30 81 70508 93525 71829 93069 73041 92595 29 32 70590 93517 71850 93061 73061 92587 28 33 70611 93510 71870 93053 73081 92579 27 34 70633 93502 71891 93046 73101 92571 26 35 70654 93495 71911 93038 73121 92563 25 36 70675 93487 71932 93030 73140 92555 24 37 70697 93480 7195-J 93022 73160 92546 23 38 70718 93472 71973 93014 73180 92538 22 39 70739 93465 71994 93007 73200 92530 21 40 9.70761 9.93457 9.72014 9.92999 9.73219 9.92522 20 41 70782 93450 72034 92991 73239 92514 19 42 70803 93442 72055 92983 73259 92500 18 43 70824 934:35 72075 92976 73278 92498 17 44 70846 93427 72096 92968 73298 92490 16 45 70867 93420 72116 92960 73318 92482 15 46 70888 93412 72137 92952 73337 92473 14 47 70909 93405 72157 92944 73357 92465 13 48 70931 93397 72177 92936 73377 92457 12 49 70952 93390 72198 92929 73396 92449 11 50 9.70973 9.93382 9.72218 9.92921 9.73416 9.92441 10 51 70994 93375 72238 92913 73435 92433 9 62 71015 93367 72259 92905 73455 92425 8 53 71036 93360 72279 92897 73474 92416 7 54 71058 93352 72299 92889 73494 92408 6 55 71079 93344 72320 92881 73513 92400 5 56 71100 93337 72340 92874 73533 92392 4 57 71121 93329 72360 92866 73552 92384 3 58 71142 93322 72881 92858 73572 92376 2 59 71163 93314 72401 92850 73591 92367 1 60 71184 93307 72421 92842 73611 92359 / Cosine Sine Cosine Sine Cosine Sine / 69 58 67 TABLE 8. LOGARITHMIC SINES AND COSINES. 187 88 84 35 Sine Cosine Sine Cosine Sine Cosine 9.73611 9.92359 9.74756 9.91857 9.75859 9.91336 60 1 73630 92351 74775 91849 75877 91328 59 2 73650 92343 74794 91840 75895 91319 58 3 73669 92335 74812 91832 75913 91310 57 4 73689 92326 74&31 91823 75931 91301 56 5 73708 92318 74850 91815 75949 91292 55 6 73727 92310 74868 91806 75967 91288 54 7 73747 92302 74887 91798 75985 91274 53 8 73766 92293 74906 91789 76003 91266 52 9 73785 92285 74924 91781 76021 91257 51 10 9.73805 9.92277 9.74943 9.91772 9.76039 9.91248 50 11 73824 92269 74961 91763 . . 76057 91239 49 12 73843 922GO 74980 91755 76075 91230 48 13 73863 92252 74999 91746 76093 91221 47 14 73SS2 92244 75017 91738 76111 91212 46 15 73901 92235 75036 91729 76129 91203 45 16 73921 92227 75054 91720 76146 91194 44 17 73940 92219 75073 91712 76164 91185 43 18 73959 92211 75091 91703 76182 91176 42 19 73978 92202 75110 91695 76200 91167 41 20 9.73997 9.92194 9.75128 9. 91 f 86 9.76218 9.91158 40 21 74017 92186 75147 91677 76236 91149 39 22 74036 92177 75165 91669 76253 91141 38 28 74055 92169 75184 91660 76271 91132 37 24 74074 92161 75202 91651 76289 91123 36 25 74093 92152 75221 91643 76307 91114 35 26 74113 92144 75239 916^4 76324 91105 34 27 74132 92136 75258 91625 76342 91096 33 28 74151 92127 75276 91617 76360 91087 32 29 74170 92119 75294 91608 76378 91078 31 30 9.74189 9.92111 9.75313 9.91599 9.76395 9.91069 30 31 74208 92102 75331 91591 76413 91060 29 32 74227 92094 75350 91582 76431 91051 28 33 74246 92086 75368 91573 76448 91042 27 34, r ,4265 92077 75386 91565 76466 91033 26 35 74284 92069 75405 91556 76484 91023 25 36 74303 92060 75423 91547 76501 91014 24 sr 74322 92052 75441 91538 76519 91005 23 38 74341 92044 75459 91530 76537 90996 22 39 74360 92035 75478 91521 76554 90987 21 40 9.74379 9.92027 9.75496 9.91512 9.76572 9.90978 20 41 74398 92018 75514 91504 76590 90969 19 42 74417 92010 75533 91495 76607 90960 18 43 74436 92002 75551 91486 76625 90951 17 44 74455 91993 75569 91477 76642 90942 16 45 74474 91985 75587 91469 76660 90933 15 46 74493 91976 75605 91460 76677 90924 14 47 74512 91968 75624 91451 76695 90915 13 48 74531 91959 75642 91442 76712 90906 12 49 74549 91951 75660 91433 76730 90696 11 50 9.74568 9.91942 9.75678 9.91425 9.76747 9.90887 10 51 74587 91934 75696 91416 76765 90878 9 52 74606 91925 75714 91407 76782 90869 8 53 74625 91917 75733 91398 76800 90860 7 54 74644 91908 75751 91389 76817 90851 6 55 74662 91900 75769 91381 76835 90842 5 56 74681 91891 75787 91372 76852 90832 4 57 74700 91883 75805 91363 76870 90823 3 58 74719 91874 75823 91354 76887 90814 2 59 74737 91866 75841 91345 76904 90805 1 60 74756 91857 75859 91336 76922 90796 t Cosine Sine Cosine Sine Cosine Sine f 66 66 64 188 TABLE 8. LOGARITHMIC SINES AND COSINES. t 86 37 38 Sine Cosine Sine Cosine Sine Cosine 9.76922 9.90796 9.77946 9.90235 9.78934 9.89653 60 1 76939 90787 77963 90225 78950 89643 59 2 76957 90777 77980 90216 78967 89633 58 3 76974 90768 77997 90206 78983 89624 57 4 76991 90759 78013 90197 78999 89614 56 5 77009 90750 78030 90187 79015 89604 55 6 77026 90741 78047 90178 79031 89594 54 7 77043 90731 78063 90168 79047 89584 53 8 77061 90722 78080 90159 79063 89574 52 9 77078 90713 78097 90149 79079 89564 51 10 9.77095 9.90704 9.78113 9.90139 9.79095 9.89554 50 11 77112 90694 78130 90130 79111 89544 49 12 77130 90685 78147 90120 79128 89534 48 13 77147 90676 78163 90111 79144 89524 47 14 77164 90667 78180 90101 79160 89514 46 15 77181 90657 78197 90091 79176 89504 45 16 77199 90648 78213 90082 79192 89495 44 17 77216 90639 78230 90072 79208 89485 43 18 77233 90630 78246 90063 79224 89475 42 19 77250 90620 78263 90053 79240 89465 41 20 9.77268 9.90611 9.73280 9.90043 9.79256 9.89455 40 21 77285 90602 78296 90034 79272 89445 39 22 77302 90592 78313 , 90024 79288 89435 38 23 77319 90583 78329 90014 79304 89425 37 24 77336 90574 78346 90005 19319 89415 36 25 77353 90565 7836-2 89995 79335 89405 35 26 77370 90555 78379 89985 79351 89395 34 27 7738 r 90546 78395 89976 79367 89385 33 28 77405 90537 78412 89966 79383 89375 32 29 77422 905:27 78428 89956 Y9399 89364 31 30 9.77439 9.90518 9.78445 9.89947 9.79415 9.89354 30 31 77456 90509 78461 89937 79431 89344 29 38 77473 90499 78478 89927 79447 89334 28 33 77490 90490 78494 89918 79463 89824 27 34 77507 90480 78510 89908 79478 89314 26 35 77524 90471 785-27 89898 79494 89304 25 36 77541 90462 78543 89888 79510 89294 24 37 77558 90452 78560 89879 79526 89284 23 38 77575 90443 78576 89869 79542 89274 22 39 77592 90434 7859;! 89859 79558 89264 21 40 9.77609 9.90424 9.78609 9.89849 9.79573 9.89254 20 41 77626 90415 78625 89840 79589 89244 19 42 "7643 90405 7864-2 89830 79605 89233 18 48 "7060 90396 78658 89820 78621 89223 17 44 "7677 90386 78674 89810 79636 89213 16 45 "7694 90377 78691 89801 79652 89203 15 46 "7711 9036S 78707 89791 79668 89193 14 47 ""7728 90358 78723 89781 79684 89183 13 48 7744 90349 78739 89771 79699 89173 12 49 "7761 90339 78756 89761 79715 89162 11 50 9.77778 9.90330 9.78772 9.89752 9.79731 9.89152 10 51 77795 90320 78788 89742 79746 89142 9 52 77812 90311 78805 89732 79762 89132 8 53 77849 90301 78821 89722 79778 89122 7 54 77846 90292 78837 89712 79793 89112 6 55 77862 90282 78853 89702 79809 89101 5 f)6 77879 90273 78869 89693 79825 89091 4 57 77896 90263 78886 89683 79840 89081 3 58 77913 90254 78902 89673 79856 89071 2 59 77930 90244 78918 89663 79872 89060 1 GO 77946 90235 78934 89653 79887 89050 * Cosine Sine Cosine Sine Cosine Sine , 68 62 61 TABLE 8. LOGARITHMIC SINES AND COSINES. 189 / 39 40 41 / Sine Cosine Sine Cosine Sine Cosine 9.79887 9.89050 9.80807 9.88425 9.81694 9.87778 60 1 79903 89040 80822 88415 81709 87767 59 2 79918 89030 80837 88404 81723 87756 58 3 79934 89020 80852 88394 81738 87745 57 4 79950 89009 80867 88383 81752 87734 56 5 79965 88999 80882 88372 81767 87723 55 6 79981 88989 80897 88362 81781 87712 54 7 79996 88978 80912 88351 81796 87701 53 8 80012 88968 80927 88340 81810 87690 52 9 80027 88958 80942 88330 81825 87679 51 10 9.80043 9.88948 9.80957 9.88319 . 9.81839 9.87668 50 11 80058 88937 80972 88308 81854 87657 49 12 80074 88927 80987 88298 81868 87646 48 13 80089 88917 81002 88287 81882 87635 47 14 80105 88906 81017 88276 81897 87624 46 15 80120 88896 81032 88266 81911 87613 45 16 80136 88886 81047 88255 81926 87601 44 17 80151 88875 81061 88244 81940 87590 43 18 80166 88865 81076 88234 81955 87579 42 19 80182 88855 81091 88223 61969 87568 41 20 9.80197 9.88844 9.81106 9.88212 9.81983 9.87557 40 21 80213 88834 81121 88201 81998 87546 39 22 80228 88824 81136 88191 82012 87535 38 23 80244 88813 81151 88180 8-J026 87524 37 24 80259 8S803 81166 88169 82041 87513 36 & 80274 88793 81180 88158 82055 87501 35 26 80290 88782 81195 88148 82069 87490 34 27 80305 88772 81210 88137 82084 87479 33 28 80320 88761 81225 88*26 82098 87468 82 29 80336 88751 81240 88115 82112 87457 31 30 9.80351 9.88741 9.81254 9.88105 9.82126 9.87446 30 31 80366 88730 81269 88094 82141 87434 29 32 80382 88720 81284 88083 82155 87423 8 m 80397 88709 81299 88072 82169 87412 27 34 80412 88699 81314 88061 82184 87401 M 35 80428 88688 81328 88051 82198 87390 25 36 80443 88678 81343 88040 82212 87378 24 37 80458 88668 81358 88029 82226 87367 23 38 80473 88657 81372 88018 82240 87356 22 39 80489 88647 81387 88007 82255 87345 21 40 9.80504 9.88636 9.81402 9.87996 9.82269 9.87334 20 41 80519 88626 81417 87985 82283 87322 19 42 80534 88615 81431 87975 82297 87311 18 43 80550 88605 81446 87964 82311 87300 17 44 80565 88594 81461 87953 82326 87288 16 45 80580 88584 81475 87942 82340 87277 15 46 80595 8*573 81490 87931 82354 87266 14 47 80610 88563 81505 87920 82368 87255 13 48 80625 88552 81519 87909 82382 87243 12 49 80641 88542 81534 87898 82396 87232 11 50 9.80656 9.88531 9.81549 9.87887 9.82410 9.87221 10 51 80671 88521 81563 87877 82424 87209 9 52 80686 88510 81578 87866 82439 87198 8 53 80701 88499 81592 87855 824f3 87187 7 54 80716 88489 81607 87844 82467 87175 6 55 80731 88478 81622 87833 82481 87164 5 56 80746 88468 81636 87822 62495 87153 4 57 80762 88457 81651 87811 82509 87141 3 58 80777 88447 81665 87800 82523 87130 2 59 80792 88436 81680 87789 82537 87119 1 60 80807 88425 81694 87778 82551 87107 t Cosine Sine Cosine Sine Cosine Sine / 50 49 o 48 190 TABLE 8. LOGARITHMIC SINES AND COSINES. 42 48 44 Sine Cosine Sine Cosine Sine Cosine 9.82551 9.8710? 9.83378 9.86413 9.84177 9.85693 60 1 82565 87096 83392 86401 84190 85681 59 2 82579 87085 83405 86889 84203 85669 58 3 82593 87073 83419 86377 84216 85657 57 4 82607 87062 83432 86366 84229 85645 56 5 82621 87050 83446 86354 84242 85C32 55 6 82635 87039 83459 86342 84255 85620 54 7 82649 87028 83473 80330 84269 85608 53 8 82663 87016 83486 86318 84282 85596 52 9 82677 87005 83500 86306 84295 85583 51 10 9.82691 9.86993 9.83513 9.86295 9.84308 9.85571 50 11 82705 86982 83527 86283 84321 85559 49 12 82719 86970 83540 86271 84334 85547 48 13 82733 86959 83554 86259 84347 85534 47 14 82747 86947 83567 86247 84360 85522 46 15 sarei 86936 83581 86235 84373 85510 45 16 82775 86924 83594 86223 84385 85497 44 17 82788 86913 83608 86211 84398 85485 43 18 82802 86902 83621 86200 84411 85473 42 19 82816 86890 83634 86188 84424 85460 41 20 9.82830 9.86879 9.83648 9.86176 9.84437 9.85448 40 21 82844 86867 83661 86164 84450 85436 39 22 82858 86855 83674 86152 84463 85423 38 23 82872 86844 83688 86140 84476 85411 37 24 82885 86832 83701 86128 84489 85399 36 25 82899 86821 83715 86116 84502 85386 35 26 82913 86809 83728 86104 84515 85874 34 27 82927 86798 83741 86092 84528 85361 33 28 82941 86786 83755 86080 84540 85349 32 29 82955 86775 83768 86068 84553 85337 31 30 9.82968 9.86763 9.83781 9.86056 9.84566 9.85324 30 31 82982 86752 83795 86044 84579 85312 29 ' 32 82996 86740 83808 86032 84592 85299 28 33 83010 86728 83821 86020 84605 85287 27 34 83023 86717 83834 86008 84618 85274 26 a5 83037 86705 83848 85996 84630 85262 25 36 83051 86694 83861 85984 84643 85250 24 37 83065 86682 83874 85972 84656 85237 23 38 83078 86670 83887 85960 84669 85225 22 39 83092 86659 83901 85948 84682 85212 21 40 9.83106 9.86647 9.83914 9.85936 9.84694 9.85200 20 41 83120 86635 83927 85924 84707 85187 19 42 83133 86624 ' 83940 85912 84720 85175 . 18 43 83147 86612 83954 85900 84733 85162 17 44 83161 86600 83967 85888 84745 85150 16 45 83174 86589 83980 85876 84758 85137 15 46 88188 86577 83993 85864 84771 85125 14 47 83202 86565 84006 85851 84784 85112 13 48 83215 86554 84020 85839 84796 85100 12 49 832v!9 86542 84033 85827 84809 85087 11 50 9.83242 9.86530 9.84046 9.85815 9.84822 9.85074 10 51 83256 86518 84059 85803 84835 85062 9 52 a3270 86507 84072 85791 84847 85049 8 53 83283 86495 84085 85779 84860 85037 7 54 83297 86483 84098 85766 84873 85024 6 55 83310 86472 84112 85754 84885 85012 5 56 83324 86460 84125 85742 84898 84999 4 57 83338 86448 84138 85730 84911 84986 3 58 83351 86436 84151 85718 84923 84974 2 59 83365 86425 84164 85706 84936 84961 1 60 83378 86413 84177 85693 84949 84949 / Cosine Sine Cosine Sine Cosine Sine / 47 46* 46 TABLE 9. LOG. TANGENTS AND COTANGENTS. 191 1 2 / Tan Cotan Tan Cotan Tan Cotan o 00 00 8.24192 11.75808 8.54308 11.45692 60 1 6.46373 13.53627 24910 75090 54669 45331 59 2 76476 23524 25616 74884 55027 44973 58 3 94085 05915 26312 73688 55382 44618 57 4 7.06579 12.93421 26996 73004 55734 44266 56 5 16270 83730 27669 72331 56083 43917 55 6 24188 75812 28332 71668 56429 43571 54 7 80882 69118 28986 71014 56773 43227 53 8 36682 63318 29629 70371 57114 4tt86 62 9 41797 58203 30263 69737 57452 42548 51 10 7.46373 12.53627 8.30888 11.69112 8. 57788 11.42212 no 11 50512 49488 31505 68495 58121 41879 I 49 12 54291 45709 32112 67888 58451 41549 48 13 57767 42233 32711 67289 58779 41221 47 14 60986 39014 33302 66698 59105 40895 46 15 63982 36018 33886 66114 59428 40572 45 16 66785 33215 34461 65539 59749 40251 44 1? 69418 30582 35029 64971 60068 39932 43 18 71900 28100 35590 64410 60384 39616 42 19 74248 25752 36143 63857 60698 89302 41 20 7.76476 12.23524 8.36689 11.63311 8.610T9 11.38991 40 21 78595 21405 37229 62771 61319 38681 80 22 80615 19385 37762 62238 61626 38374 38 23 82546 17454 38289 61711 61931 38069 37 24 84394 15606 38809 61191 62234 37766 36 25 86167 13833 39323 60677 62535 37465 35 26 87871 12129 39832 60168 62834 87166 34 27 89510 10490 40334 59666 63131 36869 33 28 91089 08911 40830 59170 63426 36574 32 29 92613 07387 41321 58679 63718 36282 31 30 7.94086 12.05914 8.41807 11.58193 8.64009 11.35991 30 31 95510 04490 42287 57713 64298 35702 29 32 * 96889 03111 42762 57238 64585 35415 28 33 98225 01775 43232 56768 64870 35130 27 34 99522 00478 43696 56304 65154 34846 26 35 8.00781 11.99219 44156 55844 65435 34565 25 36 02004 97996 44611 55389 65715 34285 24 37 03194 96806 45061 54939 65993 34007 23 38 04353 95647 45507 54493 66269 33731 22 39 05481 94519 45948 54052 66543 33457 21 40 6.06581 11.93419 8.46385 11.53615 8.66816 11.38184 20 41 07653 92347 46817 53183 67087 82913 19 42 08700 91300 47245 52755 67356 32644 18 43 09722 90278 47669 52331 67624 32376 17 44 10720 89280 48089 51911 67890 32110 16 45 11696 88304 48505 51495 68154 31846 15 46 12651 87349 48917 51083 68417 31583 14 47 13585 86415 49325 50675 68678 31322 13 48 14500 85500 49729 50271 68938 31062 12 49 15395 84605 50130 49870 69196 30804 11 50 8.16273 11.83727 8.50527 11.49473 8.69453 11.30547 10 51 17133 82867 50920 49080 69708 20292 9 52 17976 82024 51310 48690 69962 30038 8 53 18804 81196 51696 48304 70214 29786 7 54 19616 80384 52079 47921 70465 29535 6 55 20413 79587 52459 47541 70714 29286 5 56 21195 78805 52835 47165 70962 29038 4 57 21964 78036 53208 46792 71208 28792 8 58 22720 77280 53578 46422 71453 2F547 2 59 23462 76538 53945 46055 71697 28303 1 60 24192 75808 54308 45692 71940 28060 / Cotan Tan Cotan Tan Cotan Tan / 89 88 87 192 TABLE 9. LOG. TANGENTS AND COTANGENTS 8 4 5 Tan Cotan Tan Cotan Tan Cotan 8.71940 11.28060 8.84464 11.15536 8.94195 11.05805 60 1 72181 27819 84646 15354 94340 05660 59 2 72420 27580 84826 15174 94485 05515 58 3 72659 27341 85006 14994 94630 05370 57 4 72896 27104 85185 14815 94773 05227 56 5 73132 26868 85363 14637 94917 05083 55 6 73366 26634 85540 14460 95060 04940 54 73600 26400 85717 14283 95202 04798 53 8 73832 2616S 85893 14107 95344 04656 52 9 74063 25937 86069 13931 95486 04514 51 10 8.74292 11.25708 8.86243 11.13757 8.95627 11.04373 50 11 74521 25479 86417 13583 95767 04233 49 12 74748 25252 86591 13409 95908 04092 48 18 74974 25028 86763 13237 96047 03953 47 14 75199 24801 86935 13065 96187 03813 46 15 75423 24577 87106 12894 96325 03675 45 16 75645 24355 87277 12723 96464 03536 44 17 75867 24133 87447 12553 96602 03398 43 18 76087 JJ3913 87616 12384 96739 03261 42 19 76306 23694 87785 12215 96877 02123 41 20 8.76525 11.23475 8.87953 11.12047 8.97013 11.02987 40 21 76742 23258 88120 11880 97150 02850 39 22 76958 23042 88287 11713 97285 02715 38 23 77173 22827 88453 11547 97421 02579 37 24 77387 22613 88618 11382 97556 02444 36 25 77600 22400 88783 11217 97691 02309 35 26 77811 22189 88948 11052 97825 02175 34 27 78022 21978 89111 10889 97959 02041 33 28 78232 21768 89274 10726 98092 01908 3* 29 78441 21559 89437 10563 98225 01775 31 30 8.78649 11.21351 8.89598 11.10402 8.98358 11.01642 30 31 78855 21145 89760 10240 98490 01510 29 32 79061 20939 89920 10080 98622 01378 28 83 79266 20734 90080 09920 98753 01247 27 34 79470 20530 90240 09760 98884 01116 26 35 79673 20327 90399 09601 99015 00985 25 36 79875 20125 90557 09443 99145 00855 24 37 80076 19924 90715 09285 99275 00725 23 38 80277 19723 90872 09128 99405 00595 22 39 80476 19524 91029 08971 99534 00466 21 40 8.80674 11.19326 8.91185 11.08815 8.99662 11.00338 20 41 80872 19128 91340 08660 99791 00209 19 42 81068 18932 91495 08505 99919 00081 18 43 81264 18736 91650 08350 9.00046 10.99954 17 44 81459 18541 91803 08197 00174 99826 16 45 81653 18347 91957 08043 00301 99699 15 46 81846 18154 92110 07890 00427 99573 14 47 82038 17962 92262 07738 00553 99447 13 48 82230 17770 92414 07586 00679 99821 12 49 82420 17580 92565 07435 00805 99195 11 50 8.82610 11.17390 8.92716 11.07284 9.00930 10.99070 10 51 82799 17201 92866 07134 01055 98945 9 52 82987 17013 93016 06984 01179 98821 8 53 83175 16825 93165 068&5 01303 98697 7 54 83361 16639 93313 06687 01427 98573 6 55 83547 16453 93462 06538 01550 98450 5 56 83732 16268 93609 06391 01673 98327 4 57 83916 16084 93756 06244 01796 98204 3 58 84100 15900 93903 06097 01918 98082 2 59 84282 15718 94049 05951 02040 97960 1 60 84464 15536 94195 05805 02162 97838 , Cotan Tan Cotan Tan Cotan Tan t 86' 85 84 TABLE 9. LOG. TANGENTS AND COTANGENTS. 193 / 6 7 8 Tan Cotan Tan Cotan Tan Cotan 9.02162 10.97838 9.08914 10.91086 9.14780 10.86220 60 1 02283 97717 09019 90981 14872 85128 59 2 02404 97596 09123 90877 14963 85037 58 3 02525 97475 09227 90773 15054 84946 57 4 02645 97355 09330 90670 15145 84855 56 5 02766 97234 09434 90566 15236 84764 55 6 02885 97115 09537 90463 15327 84673 54 7 C3005 96995 09640 90360 15417 84583 53 8 03124 96876 09742 90258 15508 84492 52 9 03242 96758 09845 90155 15598 84402 51 10 9.03361 10.96639 9.09947 10.90053 9.15688 10.84312 50 11 03479 96521 10049 89951 15777 84228 49 12 03597 96403 10150 89850 15867 84133 48 13 03714 96286 10252 89748 15956 84044 47 14 03832 96168 10353 89647 16046 83954 46 15 03948 96052 10454 89546 16135 83865 45 16 04065 95935 10555 89445 16224 83776 44 17 04181 95819 10656 89344 16312 83688 43 18 04297 95703 10756 89244 16401 83599 42 19 04413 95587 10856 89144 16489 83511 41 20 9.045C8 10.95472 9.10956 10.89044 9.16577 10.83423 40 21 04643 95357 11056 88944 16665 83335 39 22 04758 95242 11155 88845 16753 83247 38 23 04873 95127 11254 88746 16841 83159 87 24 04987 95018 11353 88647 16928 83072 86 25 05101 94899 11452 88548 17016 P2984 35 26 05214 94786 11551 88449 17103 82897 34 27 05378 94672 11649 88351 17190 82810 33 28 05441 94559 11747 882f.3 17277 82723 82 29 ^ 05553 94447 11845 88155 17363 82637 31 30 9.05666 10.94334 9.11943 10.88057 9.17450 10.82550 30 31 05778 94222 12040 87960 17536 82464 29 32 05890 94110 12138 87862 17622 82378 28 33 06002 93998 12235 87765 17708 82292 27 34 06113 93887 12332 87668 17794 82206 26 35 06224 93776 12428 87572 17880 82120 25 36 06335 93665 12525 87475 17965 82035 24 37 06445 93555 12621 87379 18051 81949 23 38 06556 93444 12717 87283 18136 81864 22 39 06666 93334 12813 87187 18221 81779 21 40 9.06775 10.93225 9.12909 10.87091 9.18306 10.81694 20 41 06885 93115 13004 86996 18391 81609 19 42 06994 93006 13099 86901 18475 81525 18 43 07103 92897 13194 86806 18560 81440 17 44 07211 92789 13289 86711 18644 81856 16 45 07820 92680 13384 86616 18728 81272 15 46 07428 92572 13478 86522 18812 81188 14 47 \536 92464 13573 86427 18896 81104 13 48 07643 92357 13667 86338 18979 81021 K 49 07751 92249 13761 86239 19063 80937 11 50 9.07858 10.92142 9.13854 10.86146 9.19146 10.80854 10 51 07964 92036 13948 86052 19229 80771 9 52 08071 91929 14041 85959 19312 80688 8 53 08177 91823 14134 85866 19395 80605 1 54 08288 91717 14227 85773 19478 80522 6 55 08389 91611 14320 85680 19561 80439 5 56 08495 91505 14412 85588 1964:3 80357 4 57 08600 91400 14504 85496 19725 80275 3 58 08705 91295 14597 85403 19807 80193 2 59 08810 91190 14688 85312 19889 80111 1 60 08914 91086 14780 85220 19971 800-^9 , Cotan Tan Cotan Tan Cotan Tan i 83 82 81 194 TABLE 9. LOG. TANGENTS AND COTANGENTS. t 9 10 11 / 1 Tan Cotan Tan Cotan Tan Cotan 9.19971 10.800-29 9.24632 10.75368 9.28865 10.71135 60 1 20053 79947 24706 75294 28933 71067 59 2 20134 79S66 24779 75221 29000 71000 58 3 20-216 79784 24853 75147 29067 70933 57 4 20297 79703 24926 75074 29134 70866 56 5 20378 79622 25000 75000 29201 70799 55 6 20459 79541 25073 74927 29268 70732 54 7 20540 79460 25146 74854 29335 70605 53 8 20621 79379 25219 74781 29402 70598 52 9 20701 79299 25292 74708 29468 70532 51 10 9.20782 10.79218 9.25365 10.74635 9.29535 10.70465 50 11 20862 79138 25437 74563 29601 70399 49 If 20942 79058 25510 74490 29668 70332 48 13 21022 78978 25582 74418 29734 70266 47 14 21102 78898 25655 74345 29800 70200 46 15 21182 78818 25727 74273 29866 70134 45 16 21261 78739 25799 74201 29932 70068 44 17 21341 78659 25871 74129 29998 70002 43 18 21420 78580 25943 74057 30064 69936 42 19 21499 78501 26015 73985 30130 69870 41 20 9.21578 10.78422 9.26086 10.73914 9.30195 10.69805 40 21 21657 78343 26158 73842 30261 69739 39 22 21736 78264 26229 73771 30326 69674 38 *3 21814 78186 26301 73699 30391 69609 37 24 21893 78107 26372 73628 30457 69543 36 25 21971 78029 26443 73557 30522 69478 35 26 22049 77951 26514 73486 30587 69413 34 27 22127 77873 26585 30652 69348 33 28 22205 77795 26655 30717 69283 32 29 22283 77717 26726 30782 69218 31 30 9.22361 10.77639 9.26797 10.73203 9.30846 10.69154 30 31 22438 7756C 26867 73133 30911 69089 29 33 22516 77484 26937 73063 30975 69025 28 33 225,98 77407 27008 72992 31040 68960 27 34 22670 77330 27078 72922 31104 68896 26 35 22747 77253 27148 72852 31168 68832 25 36 22824 77176 27218 72782 31233 68767 24 37 22901 77099 27288 72712 31297 68703 23 38 22977 77023 27357 72648 31361 68639 22 39 23054 76946 27427 72573 31425 68575 21 40 9.23130 10.76870 9.27496 10.72504 9.31489 10.68511 20 41 28206 76794 27566 72434 31552 68448 19 42 23283 76717 27635 72365 31616 68384 18 43 23359 76641 27704 72296 31679 68321 17 44 23435 76565 27773 72227 31743 68257 16 45 23510 76490 27842 72158 31806 68194 15 46 23586 76414 27911 72089 31870 68130 14 47 23661 76339 27980 72020 31933 68067 13 48 23787 76263 28049 71951 31996 68004 12 49 23812 76188 28117 71883 32059 67941 11 50 9.23887 10.76113 9.28186 10.71814 9.32122 10.67878 10 51 23962 76038 28254 71746 32185 67815 9 52 24037 75963 28323 71677 32248 67752 8 53 24112 75888 28391 71609 32311 67689 7 54 24186 75814 28459 71541 32373 67627 6 55 24261 75739 28527 71473 3243G 67564 5 56 24335 75665 28595 71405 32498 67502 4 57 24410 75590 28662 71338 32561 67439 3 58 24484 75516 28730 71270 32623 67377 2 59 24558 75442 28798 71202 32685 67315 1 60 24632 75368 28865 71135 32747 67253 / Cotan Tan Cotan Tan Cotan Tan 80* 79 78 TABLE 9. LOG. TANGENTS AND COTANGENTS. 195 12 18 14 / Tan Cotan Tan Co fan Tan Cotan o 9.32747 10.67253 9.36336 10.63664 9.39677 10.60323 60 1 32810 67190 36394 63606 39731 60269 59 2 32872 67128 36452 63548 39785 60?15 68 3 32933 67067 36509 63491 39838 60162 57 4 32995 67005 36566 63434 89892 60108 66 5 33057 66943 36624 68376 89945 60055 65 6 33119 66881 36681 63319 39999 60001 54 7 83180 66820 36738 63262 40052 59948 53 8 83->42 66758 36795 63205 40106 59894 62 9 83303 66697 36852 63148 40159 59841 61 10 9.33365 10.66635 9.36909 10.63091 9.40212 10.59788 50 n 33426 66574 36966 63034 40266 59734 49 12 33487 66513 37023 62977 40319 59681 48 13 33548 66452 37080 62920 40372 59628 47 14 33609 66391 37137 628C3 40425 59575 46 15 336^0 66330 37193 62807 40478 59522 45 16 33731 66269 37250 62750 40531 59469 44 17 33792 66208 37306 62694 40584 59416 43 18 33853 66147 37363 62637 40636 59364 42 19 83913 66087 37419 62581 40689 59311 41 20 9.33974 10.66026 9.37476 10.62524 9.40742 10.59258 40 21 34034 65966 37532 62468 40795 69205 39 22 34095 65905 37588 62412 40847 59153 38 23 84155 65845 37644 62356 40900 59100 37 24 34215 65785 37700 62300 40952 59048 36 25 34276 66724 377'56 62244 41005 58995 35 26 34336 65664 37812 62188 41057 58943 34 27 34396 65604 37868 62132 41109 58891 33 28 |I456 65544 87924 62076 41161 58839 3d 29 34516 65484 87980 62020 41214 58786 31 30 9.34576 10.65424 9.38035 10.61965 9.41266 10.58734 30 31 34635 65365 38091 61909 41318 58682 29 32 34695 65305 38147 61853 41370 58630 28 33 34755 65245 38202 61798 41422 58578 27 34 34814 65186 38257 61743 41474 58526 26 35 34874 65126 38313 61687 41526 68474 25 86 34933 65067 38368 61632 41578 58422 24 37 34992 65008 38423 61577 41629 58371 23 38 35051 64949 38479 61521 41681 58319 22 39 35111 64889 38534 61466 41733 58267 21 40 9.35170 10.64830 9.38589 10.61411 9.41784 10.58216 20 41 35229 64771 38644 61356 41836 58164 19 42 35288 64712 38699 61301 41887 58113 18 43 35347 64653 38754 61246 41939 58061 17 44 35405 64595 38808 61192 41990 58010 16 45 35464 64536 38863 61137 42041 57959 15 46 35523 64477 38918 61082 42093 57907 14 47 35581 64419 38972 61028 42144 57856 13 48 35640 64360 39027 60973 42195 57805 12 49 35698 64302 39082 60918 42246 57754 11 50 9.35757 10.64243 9.39136 10.60864 9.42297 10.57703 10 51 35815 64185 39190 60810 42348 57652 9 52 35873 64127 39245 60755 42399 57601 8 53 35931 64069 39299 60701 42450 57550 7 54 35989 64011 39353 60647 42501 57499 6 55 36047 63953 39407 60593 42552 57448 5 56 36105 63895 39461 60539 42603 57397 4 57 36163 63837 39515 60485 42653 57347 8 58 36221 63779 39569 60431 42704 57296 2 59 36279 63721 39623 60377 42755 57245 1 60 36336 63664 39677 60323 42805 57195 / Cotan Tan Cotan Tan Cotan Tan i 77 76 76* - 196 TABLE 9. LOG. TANGENTS AND COTANGENTS. 15 16 17 / Tan Cotan Tan Cotan Tan Cotan o 9.42805 10.57195 9.45750 10.54250 9.48534 10.51466 60 1 42856 57144 45797 54203 48579 51421 59 2 42906 57094 45845 54155 48624 51376 58 3 42957 57043 45892 54108 48669 51331 57 4 43007 56993 45940 54060 48714 51286 56 5 43057 56943 45987 54013 48759 51241 55 6 43108 56892 46035 53965 48804 51196 54 7 43158 56842 46082 53918 48849 51151 53 8 43208 56792 46130 53870 48894 51106 52 9 43258 56742 46177 53823 48939 51061 51 10 9.43308 10.56692 9.46224 10.53776 9.48984 10.51016 50 11 43358 56642 46271 53729 49029 50971 49 12 43408 56592 46319 53681 49073 50927 48 18 43458 56542 46366 53634 49118 50882 47 14 43508 56492 46413 53587 49163 50837 46 15 43558 56442 46460 53540 49207 50793 45 16 43607 56393 46507 53493 49252 50748 44 17 43657 56343 46554 53446 49296 50704 43 18 43707 56293 46601 53399 49341 50659 42 19 43756 56244 46648 53352 49385 50615 41 20 9.43806 10.56194 9.46694 10.53306 9.49430 10.50570 40 21 43855 56145 46741 53259 49474 50526 39 22 43905 56095 46788 53212 49519 50481 38 23 43954 56046 46835 53165 49563 50437 37 24 44004 55996 46881 53119 49607 50393 36 25 44053 55947 46928 53072 49652 50348 85 26 44102 55898 46975 53025 49696 50304 34 27 44151 55849 47021 52979 49740 50260 33 28 44201 55799 47068 52932 49784 50216 32 29 44250 55750 47114 52886 49828 50172 31 30 9.44299 10.55701 9.47160 10.52840 9.49872 10.50128 30 31 44348 55652 47207 52793 49916 50084 29 32 44397 55603 47253 52747 49960 50040 28 33 44446 55554 47299 52701 50004 49996 27 34 44495 55505 47346 52654 50048 49952 26 35 44544 55456 47392 52608 50092 49908 25 36 44592 55408 47438 52562 50136 49864 24 37 44641 55359 47484 52516 50180 49820 23 38 44690 55310 47530 52470 50223 49777 22 39 44738 55262 47576 52424 50267 49733 21 40 9.44787 10.55213 9.47622 10.52378 9.50311 10.49689 20 41 44836 55164 47668 52332 50355 49645 19 42 44884 55116 47714 52286 50398 49602 18 43 44933 55067 47760 52240 50442 49558 17 44 44981 55019 47806 52194 50485 49515 16 45 45029 54971 47852 52148 50529 49471 15 46 45078 54922 47897 52103 50572 49428 14 47 45126 54874 47943 52057 50616 49384 13 48 45174 54826 47989 52011 50659 49341 12 49 45222 54778 48035 51965 50703 49297 11 50 9.45271 10.54729 9.48080 10.51920 9.50746 10.49254 10 51 45319 54681 48126 51874 50789 49211 9 52 45367 54633 48171 51829 50833 49167 8 53 45415 54585 48217 51783 50876 49124 7 54 45463 54537 48262 51738 50919 49081 6 55 45511 54489 48307 51693 50962 49038 5 56 45559 54441 48353 51647 51005 48995 4 57 45606 54394 48398 51602 51048 48952 3 58 45654 54346 48443 51557 51092 48908 2 59 457C2 54298 48489 51511 51135 48865 1 60 45750 54250 48534 51466 51178 48822 / Cotan Tan Cotan Tan Cotan Tan / 74 73 72 TABLE 9. LOG. TANGENTS AND COTANGENTS. 197 18 19 20 / Tan Cotan Tan Cotan Tan Cotan 9.51178 10.48822 9.53697 10.46303 9.56107 10.43893 60 1 51221 48779 53738 46262 56146 43854 59 2 51264 48736 53779 46221 56185 43815 58 3 51306 48694 53820 46180 56224 43776 57 4 51349 48651 53861 46139 56264 43736 56 5 51392 48608 53902 46098 56303 43697 55 6 51435 48565 53943 46057 56342 43658 54 7 51478 48522 53984 46016 56381 43619 53 8 51520 48480 54025 45975 56420 48580 52 9 51563 48437 54065 45935 56459 43541 51 10 9.51606 10.48394 9.54106 10.45894 9.56498 10.43502 50 11 51648 48352 54147 45853 " 56537 43463 49 12 51691 48309 54187 45813 56576 43424 48 13 51734 48266 54228 45772 56815 43385 47 14 51776 48224 54269 45731 56654 445346 46 15 51819 48181 54309 45691 56693 43307 45 16 51861 48139 54350 45650 56738 43268 44 17 51903 48097 54390 45610 56771 43229 43 18 51946 48054 54431 45569 56810 43190 42 19 51988 48012 54471 45529 56849 43151 41 20 9.52031 10.47969 9.54512 10.45488 9.56887 10.43113 40 21 52073 47927 54552 45448 56926 43074 39 22 52115 47885 54593 45407 56965 43035 38 23 52157 47843 54633 45367 57004 42996 37 24 52200 47800 54673 45327 57042 42958 36 25 52242 47758 54714 45286 57081 42919 35 26 52284 47716 54754 45246 57120 42880 34 27 52326 47674 54794 45206 57158 42842 33 28 52368 47632 54835 45165 57197 42803 82 29 5241JJ 47590 54875 45125 57235 42765 31 30 9.52452 10.47548 9.54915 10.45085 9.57274 10.42726 30 31 52494 47506 54955 45045 57312 42688 29 32 52536 47464 54995 45005 57351 42649 88 33 52578 47422 55035 44965 57389 42611 27 34 52620 47380 55075 44925 57428 42572 26 35 52661 47339 55115 44885 57466 42534 25 36 52703 47297 55155 44845 57504 42496 24 37 52745 47255 55195 44805 57543 42457 23 38 52787 47213 55235 44765 57581 42419 22 39 52829 47171 55275 44725 57619 42381 21 40 9.52870 10.47130 9.55315 10.44685 9.57658 10.42342 20 41 52912 47088 55355 44645 57696 42304 19 42 52953 47047 55395 44605 57734 42266 18 43 52995 47005 55434 44566 57772 42228 17 44 53037 46963 55474 44526 57810 42190 16 45 53078 46922 55514 44486 57849 42151 15 46 53120 46880 55554 44446 57887 42113 14 47 53161 46839 55593 44407 57925 42075 13 48 53202 46798 55633 44367 57963 42037 12 49 53244 46756 55673 44327 56001 41999 11 50 9.53285 10.46715 9.55712 10.44288 9.58039 10.41961 10 51 53327 46673 55752 44248 58077 41923 9 52 53368 46632 55791 44209 58115 41885 8 53 53409 46591 55831 44169 58153 41847 7 54 53450 46550 55870 44130 58191 41809 6 55 53492 46508 55910 44090 58229 41771 5 56 53533 46467 55949 44051 58267 41733 4 57 53574 46426 55989 44011 58304 41696 3 58 53615 46385 56028 43972 58342 41658 2 59 53656 46344 56067 43933 58380 41620 1 60 53697 46303 56107 43893 58418 41582 > Cotan Tan Cotan Tan Cotan Tan / 71 70 69 198 TABLE 9. LOG. TANGENTS AND COTANGENTS, 1 21 22 23 / Tan Cotan Tan Cotan Tan Cotan 9.58418 10.41582 9.60641 10.39359 9.62785 10.37215 60 1 58455 41545 60677 39323 62820 37180 59 2 58493 41507 60714 39-386 62855 37145 58 3 58531 41469 60750 39250 62890 37110 57 4 58569 41431 60786 39214 62926 37074 56 5 58606 41394 60823 39177 62961 37039 55 6 58644 41356 60859 39141 62996 37004 54 7 58681 41319 60895 39105 63031 36.969 53 8 58719 41281 60931 39069 68066 36934 52 9 58757 41243 60967 39033 63101 36899 51 10 9.58794 10.41206 9.61004 10.38996 9.63135 10.36865 50 11 58832 41168 61040 38960 63170 36830 49 12 58869 41131 61076 38924 63205 36795 48 13 58907 41093 61112 38888 63240 36760 47 14 58944 41056 61148 38852 63275 36725 46 15 58981 41019 61184 38816 63310 36690 45 16 59019 40981 61220 38780 68345 36655 44 17 59056 40944 61256 38744 63379 36621 43 18 59094 40906 61292 38708 63414 36586 42 19 59131 40869 61328 38672 63449 36551 41 20 9.59168 10.40882 9.61304 10.38636 9.63484 10.36516 40 21 69205 40795 61400 3860C 63519 36481 39 22 59243 40757 61436 3856* 63553 36447 88 23 59280 40720 61472 38528 63588 36412 37 24 59317 40688 61508 38492 63623 36377 36 25 59854 40646 61544 38456 63657 36343 85 26 59391 40609 61579 38421 63692 36308 34 27 59429 40571 61615 38385 G3726 36274 33 28 59466 40534 61651 38349 63761 36239 32 29 59503 40497 61687 38313 63796 36204 31 80 9.59540 10.40460 9.61722 10.38278 9.63830 10.36170 30 31 59577 40423 61758 38242 63865 36135 29 32 59614 40386 61794 38206 63899 36101 28 33 59651 40349 61830 38170 63934 36066 27 34 69688 40312 61865 38135 63968 36032 26 35 59725 40275 61901 38099 64003 35997 25 36 69762 40238 61936 38064 64037 35963 24 87 59799 40201 61972 38028 64072 35928 23 38 59835 40165 62008 37992 64106 35894 22 39 59872 40128 62043 37957 64140 35860 21 40 9.59909 10.40091 9.62079 10.37921 9.64175 10.35825 20 41 59946 40054 62114 87886 64209 35791 19 42 59983 40017 62150 37850 64243 35757 18 43 60019 89981 62185 37815 64278 35722 17 44 60056 39944 62221 37779 64312 35688 16 45 60093 39907 62256 37744 64346 35654 15 46 60130 39870 62292 37708 64381 35619 14 47 60166 39834 62327 37673 64415 35585 13 48 60203 39797 62362 37638 64449 35551 12 49 60240 39760 62398 37602 64483 35517 11 50 9.60276 10.39724 9.62433 10.37567 9.64517 10.85483 10 51 60313 39687 62468 37532 64552 35448 9 52 60349 39651 62504 37496 64586 35414 8 53 60386 39614 62539 37461 64620 35380 IJr 54 60422 39578 62574 37426 64654 35346 6 55 60459 39541 62609 37391 64688 35312 5 56 60495 39505 62645 37855 64722 35278 4 57 60532 39468 62680 37320 64756 35244 8 58 605(58 39432 62715 37285 64790 35210 59 60605 89395 62750 37250 64824 35176 1 1 60 60641 39359 62785 87215 64858 85142 / Cotan Tan Cotan Tan Cotan Tan f 68 67 66 TA!BLE 9. LOG. TANGENTS AND COTANGENTS. 199 / 24 25 26 / Tan Cotan Tan Cotan Tan Cotan 9.64858 10.35142 9.66867 10.33133 9.68818 10.31182 60 1 64892 35108 66900 33100 68850 31150 59 2 64926 35074 66933 33067 68882 31118 58 3 64960 35040 66966 33034 68914 31086 57 4 64994 35006 66999 33001 68946 31054 56 5 65028 84972 67032 32968 68978 31022 55 6 65062 34938 67065 32935 69010 90990 54 7 65096 34904 67098 32902 69042 30958 53 8 65130 34870 67131 32869 69074 30926 52 9 65164 34836 67163 32837 69106 30894 51 10 9.65197 10.34803 9.67196 10.32804 9-. 69138 10.30862 50 11 65231 34769 67229 32771 69170 30830 i 49 12 65265 34735 67262 32738 69202 80798 48 13 65299 34701 67295 32705 69234 30766 47 14 65333 34667 67327 32673 69266 80734 46 15 65366 34634 67360 32640 69298 30702 45 16 65400 34600 67393 32607 69329 30671 44 17 65434 34566 67426 82574 69361 30689 43 18 65467 34533 67458 32542 69393 30607 42 19 65501 34499 67491 32509 69425 30575 41 20 9.65535 10.34465 9.67524 10.32476 9.69457 10.30543 40 21 65568 34432 67556 32444 69488 30512 39 22 65602 34398 67589 32411 69520 30480 38 23 65636 34364 67622 32378 69552 30448 37 24 656(59 34331 67654 32346 69584 30416 36 25 65703 34297 67687 32313 69615 30385 35 26 65736 34264 67719 32281 69647 30353 34 27 65770 34230 67752 32248 69679 30321 33 28 65803 ^ 34197 67785 32215 69710 30290 32 29 65837 34163 67817 32183 69742 30258 31 30 9.65870 10.34130 9.67850 10.32150 9.69774 10.30226 30 31 65904 34096 67882 32118 69805 30195 29 32 65937 34063 67915 32085 69837 30163 28 33 65971 34029 67947 32053 69868 30132 27 34 66004 33996 67980 32020 69900 30100 26 35 66038 33962 68012 31988 69932 80068 25 M 66071 83929 68044 31956 69963 30037 24 37 66104 33896 68077 31923 69995 30005 23 38 66138 33862 68109 31891 70026 29974 22 39 66171 33829 68142 31858 70058 29942 21 40 9.66204 10.38796 9.68174 10.31826 9.70089 10.29911 20 41 66238 33762 68206 31794 70121 29879 19 42 66271 33729 68239 31761 70152 29848 18 43 66304 33696 68271 31729 70184 29816 17 44 66337 33663 68303 31697 70215 29785 16 45 66371 33629 68336 31664 70247 29753 15 46 66404 33506 68368 31632 70278 29722 14 47 66437 33563 68400 31600 70309 29691 13 48 66470 33530 68432 31568 70341 29659 12 49 06503 33497 68465 31585 70372 29628 11 50 0.0(5537 10.33463 9.68497 10.31503 9.70404 10.29596 10 51 66570 33430 6a529 31471 70435 29565 9 52 66603 33397 68561 31439 70466 29534 8 f>3 66636 33364 68593 31407 70498 29502 7 54 66669 33331 68626 31374 70529 29471 6 55 66702 33298 68658 31342 70560 29440 5 56 6673") 33265 68690 31310 70592 29408 4 57 66768 33232 68722 31278 70623 29377 8 58 66801 33199 68754 31246 70654 29346 2 59 66834 33166 68786 31214 70685 29315 ! 1 fiO 66867 33133 68818 31182 70717 29283 i / Otan Tan Cotan Tan Cotan Tan / 65 64 63 200 TABLE 9. LOG. TANGENTS AND COTANGENTS. / 27 28 29 / Tan Cotan Tan Cotan Tan Cotan o 9.70717 10.29283 9.72567 10.27433 9.74375 10.25625 60 1 70748 29252 72598 27402 74405 25595 59 2 70779 29221 72628 27372 74435 25565 58 3 70810 29190 72659 27341 74465 25535 57 4 70841 29159 72689 27311 74494 25506 56 5 70873 29127 72720 27280 74524 25476 55 6 70904 29096 72750 27250 74554 25446 54 7 70935 29065 72780 27220 74583 25417 53 8 70966 29034 72811 27189 74613 25387 52 9 70997 29003 72841 27159 74643 25357 51 10 9.71028 10.28978 9.72872 10.27128 9.74673 10.25327 50 11 71059 28941 72902 27098 74702 25298 49 12 71090 28910 72932 27068 74732 25268 48 13 71121 28879 72963 27037 74762 25238 47 14 71153 28847 72993 27007 74791 25209 46 15 71184 28816 73023 26977 74821 25179 45 16 71215 28785 73054 26946 74851 25149 44 17 71246 28754 73084 26916 74880 25120 43 18 71277 28723 73114 26886 74910 25090 42 19 71308 28692 73144 26856 74939 25061 41 20 9.71339 10.28661 9.73175 10.26825 9.74969 10.25031 40 21 71370 28630 73205 26795 74998 25002 39 22 71401 28599 73235 26765 75028 24972 38 23 71431 28569 73265 26735 75058 24942 37 24 71462 28538 73295 26705 75087 24913 36 25 71493 28507 73326 26674 75117 24883 35 26 71524 28476 73356 26644 75146 24854 34 27 71555 28445 73386 26614 75176 24824 33 28 71586 28414 73416 26584 75205 24795 32 29 71617 28383 73446 26554 75235 24765 31 30 9.71648 10.28352 9.73476 10.26524 9.75264 10.24736 30 31 71679 28321 73507 26493 75294 24706 29 32 71709 28291 73537 26463 75323 24677 28 33 71740 28260 73567 26433 75353 24647 27 34 71771 28229 73597 26403 75382 24618 26 35 - 71802 28198 73627 26373 75411 24589 25 36 71833 28167 73657 26343 75441 24559 24 87 71863 28137 73687 26313 75470 24530 23 38 71894 28108 73717 26283 75500 24500 22 39 71925 28075 73747 26253 75529 24471 21 40 9.71955 10.28045 9.73777 10.26223 9.75558 10.24442 20 41 71986 28014 73807 26193 75588 24412 19 42 72017 27983 73837 26163 75617 24383 18 43 72048 27952 73867 26133 75647 24353 17 44 72078 27922 73897 26103 75676 24324 16 45 72109 27891 78927 26073 75705 24295 15 46 72140 27860 73957 26043 75735 24265 14 47 72170 27830 73987 26013 75764 24236 13 48 72201 27799 74017 25983 75793 24207 12 49 72231 27769 74047 25953 75822 24178 11 50 9.72262 10.27738 9.74077 10.25923 9.75852 10.24148 10 51 72293 27707 74107 25893 75881 24119 9 52 72323 27677 74137 25863 75910 24080 8 53 72354 27646 74166 25834 75939 24061 7 54 72384 27616 74196 25804 75969 24031 6 55 72415 27585 74226 25774 75998 24002 5 56 72445 27555 74256 25744 76027 23973 4 57 72476 27524 74286 25714 76056 23944 3 58 72506 27494 74316 25684 76086 28914 2 59 72537 27463 74345 25655 76115 23885 1 60 72567 27433 74375 25625 76144 23856 / Cotan Tan Cotan Tan Cotan Tan , 62 , 61' 60 TABLE 9. LOG. TANGENTS AND COTANGENTS. 201 f 30 31 32 t Tan Cotan Tan Cotan Tan Cotan 9.76144 10.23856 9.77877 10.22123 9.79579 10.20421 60 1 76173 23827 77906 22094 79607 20393 59 2 76202 23798 77935 22065 79635 20365 58 3 76231 23769 77963 22037 79663 20337 57 4 76261 23739 77992 22008 79691 20309 56 5 76290 23710 78020 21980 797J 9 20281 55 6 76319 23681 78049 21951 79747 20253 64 7 76348 23C52 78077 21923 79776 20224 53 8 76377 23623 78106 21894 79804 20196 52 9 76406 23594 78135 .21865 . 79832 20168 51 10 9.76435 10.23565 9.78163 10.21837 9.79860 10.20140 50 11 76464 23536 78192 21808 79888 20112 49 12 76493 23507 78220 21780 79916 20084 48 13 76522 23478 78249 21751 79944 20056 47 14 76551 23449 78277 21723 79972 20028 46 15 76580 23420 78306 21694 80000 20000 45 16 76609 23391 78334 21666 80028 19972 44 17 76639 23361 78363 21637 80056 19944 43 18 76668 23332 78391 21609 80084 19916 42 13 76697 23303 78419 21581 80112 19888 41 20 9.76725 10.23275 9.78448 10.21552 9.80140 10.19860 40 21 76754 23246 78476 21524 80168 19832 39 22 76783 23217 78505 21495 80195 19805 38 23 76812 23188 78533 21467 80223 19777 87 24 76841 2M59 78562 21438 80251 19749 86 25 76870 23180 78590 21410 80279 19721 35 26 76899 23101 78618 21382 80307 19693 34 27 76928 23072 78647 21353 80335 19665 33 28 76957 23043 78675 21325 80363 19637 82 29 76986 23014 78704 21296 80391 19609 31 30 9.77015 10.22985 9.78732 10.21268 9.80419 10.19581 30 31 77044 22956 78760 21240 80447 19553 29 32 77073 22927 78789 21211 80474 19526 28 33 77101 22899 78817 21183 80502 19498 27 31 77130 22870 78845 21155 80530 19470 26 35 77159 22841 78874 21126 80558 19442 25 36 77188 22812 78902 21098 80586 19414 24 37 77217 22783 78930 21070 80614 19386 23 38 77246 22754 78959 21041 80642 19358 22 39 77274 22726 78987 21013 80669 19331 21 40 9.77303 10.22697 9.79015 10.20985 9.80697 10.19303 20 41 77332 22668 79043 20957 80725 19275 19 42 77361 22639 79072 20928 80753 19247 18 43 77390 22610 79100 20900 80781 19219 17 44 77418 22582 79128 20872 80808 19192 16 45 77447 22553 79156 20844 80836 19164 15 46 77476 22524 79185 20815 80864 19136 14 47 77505 22495 79213 20787 80892 19108 13 48 77533 22467 79241 20759 80919 19081 12 49 77562 22438 79269 20731 80947 19053 11 50 9.77591 10.22409 9.79297 10.20703 9.80975 10.19025 10 51 77619 22381 79326 20674 81003 18997 9 52 77648 22352 79354 20646 81030 18970 8 53 77677 22323 79382 20618 81058 18942 7 54 77706 22294 79410 20590 81086 18914 6 55 77734 22266 79438 20562 81113 18887 5 50 77763 22237 79466 20534 81141 18859 4 57 77791 22209 79495 20505 81169 18831 3 58 77820 22180 79523 20477 81196 18804 2 59 77849 22151 79551 20449 81224 18776 1 60 77877 22123 79579 20421 81252 18748 / Cotan Tan Cotan Tan Cotan Tan / 59 68 67 202 TABLE 9. LOG. TANGENTS AND COTANGENTS. / 83 84 85 Tan Cotan Tan Cotan Tan Cotan o 9.81252 10.18748 9.82899 . 10.17101 9.84523 10.15477 60 1 81279 18721 82926 17074 81550 15450 59 2 81307 18693 82953 17047 84576 15424 58 3 81335 18665 82980 17020 84603 15397 57 4 81362 18638 83008 16992 84630 15870 56 5 81390 18610 83035 16965 84657 15343 55 6 81418 18582 83063 16938 84684 15316 54 7 81445 18555 83089 16911 84711 15289 53 8 81473 18527 83117 16883 84738 15262 52 9 81500 18500 83144 16856 84764 15236 51 10 9.81528 10.18472 9.83171 10.168-29 9.84791 10.15209 50 11 81556 18444 83198 16802 84818 15182 49 12 81583 18417 83-225 16775 84845 15155 48 18 81611 18389 83252 16748 84872 15128 47 14 81638 18362 83280 16720 84899 15101 46 15 81666 18334 83307 16693 84925 15075 45 16 81693 18307 83334 16666 84952 15048 44 17 81721 18279 83361 16639 84979 15021 43 18 81748 18252 83388 16612 85006 14994 43 19 81776 18224 83415 16585 85033 14967 41 20 9.81803 10.18197 9.83142 10.16558 9.85059 10.14941 40 21 81831 18169 83470 16530 85086 14914 39 22 81858 18142 83497 16503 85113 14887 38 23 81886 18114 83524 16476 85140 14860 37 24 81913 18087 83551 16449 85166 14834 36 25 81941 18059 83578 16423 85193 14807 35 26 81968 18032 83605 16395 85220 14780 34 27 81996 18004 83632 16368 85247 14753 33 28 82023 17'977 83659 16341 85273 14737 3.' 29 82051 17949 83686 16314 85300 14700 81 30 9.82078 10.17922 9.83713 10.16287 9.85327 10.14673 30 31 82106 17894 83740 16260 85354 14046 29 32 82133 1787 83768 16232 85380 14620 *8 83 82161 17839 83795 16205 85407 14593 34 82188 17812 83822 16178 85434 145G6 2(j 35 82215 17785 83849 16151 85460 14540 25 36 82243 17757 83876 16124 85487 14513 24 37 82270 17730 83903 16097 85514 14486 23 38 82298 17702 83930 16070 85540 14460 22 39 82325 17675 83957 16043 5567 14433 21 40 9.82352 10.17648 9.83984 10.16016 9.85594 10.14406 20 41 82380 17620 84011 15989 85620 14380 19 42 82407 17593 84038 15962 85647 14353 18 43 82435 17565 84065 15935 85674 14326 17 44 82462 17538 8409-2 15908 85700 14300 16 45 82489 17511 84119 15881 85727 14273 15 46 82517 17483 84146 15854 85754 14246 14 47 82544 17456 84173 15827 85780 14220 18 48 82571 17429 84200 15800 85807 14193 13 49 82599 17401 84227 15773 85834 141C6 11 50 9.82626 10.17374 9.84254 10.15746 9.85860 10.14140 10 51 82653 17347 84280 15720 85887 14113 9 52 82681 17319 84307 15693 85913 14087 8 53 82708 17292 84334 15666 85940 14060 7 54 82735 . 17265 84361 15639 85967 14033 6 55 82762 17238 84388 15612 85993 14007 5 56 82790 17210 84415 15585 86020 13980 4 57 82817 17183 84443 15558 86046 13954 3 58 83844 17156 84469 15531 86073 13927 2 59 82871 17129 84496 15504 86100 13900 1 60 82899 17101 84523 15477 86126 13874 / Cotan Tan Cotan Tan Cotan Tan / 56 65 64 TABLE 9. LOG. TANGENTS AND COTANGENTS. 203 / 80 87 88 / Tan Cotan Tan Cotan Tan Cotan 9.86126 10.13874 9.87711 10.12289 9.89281 10.10-719 60 1 86153 13847 87738 12262 89307 10693 59 2 86179 13821 87764 12236 89333 10667 58 3 86206 13794 87790 12210 89359 10641 57 4 86232 13768 87817 12183 89385 10615 56 5 86259 13741 87843 12157 89411 10589 55 6 86285 13715 87869 12131 89437 10563 54 7 86312 13688 87895 12105 89463 10537 53 8 8633S 13662 87922 12078 89489 10511 52 9 86365 13635 87948 12052 89515 10485 61 10 9.86392 10.13608 9.87974 10.12026 9.89541 10.10459 50 11 86418 13582 88000 12000 89567 10433 I 49 12 86445 13555 88027 11973 89593 10407 48 13 86471 13529 88053 11947 89619 10381 47 14 86498 13502 88079 11921 89645 10355 46 15 86524 13476 88105 11895 89671 10329 45 16 86551 13449 88J31 11869 89697 10303 44 17 86577 13423 88158 11842 89723 10277 43 18 86603 13397 88184 11816 89749 10251 42 19 86630 13370 88210 11790 89775 10225 41 20 9.86656 10.13344 9.88236 10.11764 9.89801 10.10199 40 21 . 86683 13317 88262 11738 89827 10173 39 22 86709 13291 88289 11711 89853 10147 38 23 86736 1334 88315 11685 89879 10121 37 24 86762 13238 88341 11659 89905 10095 36 25 86789 13211 88367 11633 89931 10069 35 26 86815 13185 88393 11607 89957 10043 34 27 86842 13158 88420 11580 89983 10017 33 28 86868 13132 88446 11554 90009 09991 32 29 86894 13106 88472 11528 90035 09965 31 30 9.86921 10.13079 9.88498 10.11502 9.90061 10.09939 30 31 86947 13053 88524 11476 90086 09914 29 32 86974 13026 88550 11450 90112 09888 28 33 87000 13000 88577 11423 90138 09862 27 34 87027 12973 88603 11397 90164 09836 26 35 87053 12947 88629 11371 90190 09810 25 36 87079 12921 88655 11345 90216 09784 24 37 87106 12894 88681 11319 90242 09758 23 38 87132 12868 88707 11293 90268 09732 22 39 87158 12842 88733 11267 90294 09706 21 40 9.87185 10.12815 9.88759 10.11241 9.90320 10.09680 20 41 87211 12789 88786 11214 90346 09654 19 42 87238 12762 88812 11188 90371 09629 18 43 87264 12736 88838 11162 90397 09603 17 44 87290 12710 88864 11136 90423 09577 16 45 87317 12683 88890 11110 90449 09551 15 46 87343 12657 88916 11084 90475 09525 14 47 87369 12631 88942 11058 90501 09499 13 48 87396 12604 88968 11032 90527 09473 12 49 87422 12578 88994 11006 90553 09447 11 50 9.87448 10.12552 9.89020 10.10980 9.90578 10.09422 10 51 87475 12525 89046 10954 90604 09396 9 52 87501 12499 89073 10927 90630 09370 8 53 87527 12473 89099 10901 90656 09344 7 54 87554 12446 89125 10875 90682 09818 6 55 87580 12420 89151 10849 90708 09292 5 56 87606 12394 89177 10823 90734 09266 4 57 87633 12367 89203 10797 90759 09241 3 58 87659 12341 89229 10771 90785 09215 2 59 87685 12315 89255 10745 90811 09189 1 60 87711 12289 89281 10719 90837 09163 / Cotan Tan Cotan Tan Cotan Tan , 63 62 61' 204 TABLE 9. LOG. TANGENTS AND COTANGENTS- 89 40 41 Tan Cotan Tan Cotan Tan Cotan o 9.90837 10.09163 9.92381 10.07619 9.93916 10.06084 60 1 90863 09137 92407 07593 93942 06058 59 2 90889 09111 92433 07567 93967 06033 58 3 90914 09086 92458 07542 93993 06007 57 4 90940 09060 92484 07516 94018 05982 56 5 90966 09034 92510 07490 94044 05956 55 6 90992 09008 92535 07465 94069 05931 54 7 91018 08982 92561 07439 94095 05905 53 8 91043 08957 92587 07413 94120 05880 52 9 91069 08931 92612 07388 94146 05854 51 10 9.91095 10.08905 9.92638 10.07362 9.94171 10.05829 50 11 91121 08879 92663 07337 94197 05803 49 12 91147 08853 92689 07311 94222 05778 48 13 91172 08828 92715 07285 94248 05752 47 14 91198 08802 92740 07260 94273 05727 46 15 9J224 08776 92766 07234 94299 05701 45 16 91250 08750 92792 07208 94324 05676 44 17 91276 08724 92817 07183 94350 05650 43 18 91301 08699 92843 07157 94375 05625 42 19 91327 08673 92868 07132 94401 05599 41 20 9.91353 10.08647 9.92894 10.07106 9.94426 10.05574 40 21 91379 08621 92920 07080 94452 05548 39 22 91404 08596 92945 07055 94477 05523 38 23 91430 08570 92971 07029 94503 05497 37 24 91456 08544 92996 07004 94528 05472 36 25 9148,! 08518 93022 06978 94554 05446 35 26 91507 08493 93048 06952 94579 05421 34 27 91533 08467 93073 06927 94604 05396 33 28 91559 08441 93099 06901 94G30 05370 32 29 91585 08415 93124 06876 94655 05345 3! 30 9.91610 10.08390 9.93150 10.06850 9.94681 10.05319 30 31 91636 08364 93175 06825 94706 05294 29 33 91662 08338 93201 06799 94732 05268 28 33 91683 08312 93227 06773 94757 05243 27 34 91713 08287 93252 06748 94783 05217 26 35 91739 08261 93278 06722 94808 05192 25 36 91765 08235 93303 06697 94834 05166 24 37 91791 08209 93329 06671 94859 05141 23 38 91816 08184 93354 06646 94884 05116 22 39 91842 08158 93380 06620 94910 05090 21 40 9.91868 10.08132 9.93406 10.06594 9.94935 10.05065 20 41 91893 08107 93431 06569 94961 05039 19 4-4 91919 08081 93457 06543 94986 05014 18 43 91945 08055 93482 06518 95012 04988 17 44 91971 08029 93508 06492 95037 04963 16 45 91996 08004 93533 06467 95062 04938 15 46 92022 07978 93559 06441 95088 04912 14 47 92048 07952 93584 06416 95113 04887 13 48 92073 07927 93610 06390 95139 04861 12 49 92099 07901 93636 06364 95164 04836 11 50 9.92125 10.07875 9.93661 10.06339 9.95190 10.04810 10 51 92150 07850 93687 06318 95215 04785 9 52 92176 07824 93712 06288 95240 04760 8 53 92202 07798 93738 06262 95266 04734 7 54 92227 07773 93763 06237 95291 04709 6 55 92253 07747 93789 06211 95317 04683 5 56 92279 07721 93814 06186 95342 04658 4 57 92304 07696 93840 06160 95368 04632 3 58 92330 07670 93865 06135 95393 04607 59 92356 07644 93891 06109 95418 04582 1 60 92381 07619 93916 06084 95444 04556 / Cotan Tan Cotan Tan Cotan Tan , 60* 49 48 TABLE 9. LOG. TANGENTS AND COTANGENTS. 205 42 43 440 Tan Cotan Tan Cotan Tan Cotan 9.95444 10.04556 9.96966 10.03034 9.98484 10.01516 60 1 95469 04531 96991 03009 98509 01491 59 2 95495 04505 97016 02984 98534 01466 58 3 95520 04480 97042 0-^958 98560 01440 57 4 95545 04455 97067 02933 98585 01415 56 5 95571 04429 97092 02908 98610 01390 55 6 95596 04404 97118 02882 98635 01365 54 95622 04378 97143 02857 98661 01339 53 8 95647 04353 97168 02832 98686 01314 52 9 95672 04328 97193 02807 98711 01289 51 10 9.95698 10.04302 9.97219 10.02781 9.98737 10.01263 50 11 95723 04277 97244 02756 98762 01238 49 12 95748 04252 97269 02731 98787 01213 48 13 95774 04226 97295 02705 98812 01188 47 14 95799 04201 97320 02680 98838 01162 46 15 95825 04175 97345 02655 98863 01137 45 16 95850 04150 97371 02629 98888 01112 44 1? 95875 04125 97396 02604 98913 01087 43 18 95901 04099 97421 02579 98939 01061 42 19 95926 04074 97447 02553 98964 01086 41 20 9.95952 10.04048 9.97472 10.02508 9.98989 10.01011 40 21 95977 04023 97497 02503 99015 00985 39 22 96002 03998 97523 02477 99040 00960 38 28 96028 03972 97548 02452 99065 00935 37 24 96053 03947 97573 02427 99090 00910 36 25 96078 03922 97598 0^402 99116 008S4 35 26 96104 03896 97624 02376 99141 00859 34 27 96129 03871 97649 02351 99166 00834 33 28 96155 03845 97674 02326 99191 00809 82 29 96180 03820 97700 02300 99217 00783 31 30 9.96205 10.03795 9.97725 10.02275 9.99242 10.00758 30 31 96231 03769 97750 02250 99267 00733 29 32 96256 03744 97776 02224 99293 00707 28 33 96281 03719 97801 02199 99318 00682 27 34 96307 03693 97826 02174 99343 00657 26 35 96332 03668 97851 02149 99368 00632 25 36 96357 03643 97877 02123 59394 00606 24 37 96383 03617 97902 02098 99419 00581 23 38 96408 0359-2 97927 02073 99444 00556 22 39 96433 03567 97953 0^047 99469 00531 21 40 9.96459 10.03541 9.97978 10.02022 9.99495 10.00505 20 41 96484 03516 9b003 01997 99520 00480 19 42 96510 03490 98029 01971 99545 00455 18 43 96535 03465 98054 C1946 99570 00430 17 44 96560 03440 98079 01921 99596 00404 16 45 965S6 03414 98104 01896 99621 00379 15 46 96611 03389 98130 01870 99646 00354 14 47 96636 03364 98155 01845 99672 00328 13 48 96662 03338 98180 01880 99697 00303 12 49 96687 03313 98206 01794 99722 00278 11 50 9.96712 10.03288 9.98231 10.01769 9.99747 10.00253 10 51 96738 03262 98256 01744 99773 00227 9 52 96763 03237 98281 01719 99798 00202 8 53 96788 03212 98307 01693 99823 00177 7 54 96814 03186 98332 01668 99848 00152 6 55 96839 03161 98357 01643 99874 00126 5 W 96864 03136 98383 01617 99899 00101 4 57 96890 03110 98408 01592 99924 00076 3 58 06915 03085 98433 01567 99949 00051 2 59 96940 03060 98458 01542 99975 00025 1 60 96966 03034 98484 01516 10.00000 00000 / Cotan Tan Cotan Tan Cotan Tan , 47 * 45<> I TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t o 2 3 4 .00000 .00029 .00058 .00087 .00116 One One One One One .00000 .00029 .00058 .00087 .00116 Infinite 3437-75 1718.87 1145.92 809.436 .01745 .01774 .01803 .01832 .01862 .99985 -99984 .99984 .99983 .99983 .01746 .01775 .01804 .01833 .01862 57.2900 56.3506 55-4415 54-5613 53-7086 60 59 58 57 56 5 6 8 g .00145 .00175 .00204 .00233 .00262 One One One One One .00145 .00175 .00204 .00233 .00262 687-549 572-957 491.106 429.718 381.971 .01891 .01920 .01949 .01978 .02007 .99982 .99982 .99981 .99980 .99980 .01891 .01920 .01949 .01978 .02007 52.8821 52-0807 51-3032 50.5485 49.8157 55 54 53 52 51 _ 10 11 12 13 14 .00291 .00320 .00349 .00378 .00407 One .99999 .99999 .99999 .99999 .00291 .00320 .00349 .00378 .00407 343-774 312-521 286.478 264.441 245.552 .02036 .02065 .02094 .02123 .02152 .99979 .99979 .99978 .99977 .99977 .02036 .02066 .02095 .02124 .02153 49*1039 48-4121 47.7395 47.0853 46.4489 5O 49 48 47 46 15 16 17 18 19 .00436 .00465 .00495 .00524 .00553 .99999 .99999 .99999 .99999 .99998 .00436 .00465 .00495 .00524 .00553 229-182 214.858 202-219 190-984 180-932 .02131 .02211 .02240 .02269 .02298 .99976 -99976 .99975 -99974 -99974 .02182 .02211 .02240 .02269 .02298 45-8294 45-2261 44-6386 44-0661 43-5081 45 44 43 42 41 20 21 22 23 24 .00582 .00611 .00640 .00669 .00698 .99998 .99998 .99998 .99998 .99998 .00582 .00611 .00640 .00669 .00698 171-885 163-700 156-259 149-465 143-237 .02327 .02356 .02385 .02414 .02443 .99973 .99972 .99972 .99971 .99970 .02328 .02357 .02386 .02415 .02444 42-9641 42-4335 41.9158 41.4106 40.9174 4O 39 38 37 36 25 26 27 28 29 .00727 .00756 $0785 .00814 .00844 .99997 .99997 .99997 .99997 .99996 .00727 .00756 .00785 .00815 .00844 137.507 132-219 127-321 122.774 118.540 .02472 .02501 .02530 .02560 .02589 .99969 .99969 .99968 .99967 .99966 .02473 .02502 .02531 .02560 .02589 40.4358 39-9655 39.5059 39.0568 38-6177 35 34 33 32 31 30 31 32 33 34 .00873 .00902 .00931 .00960 .00989 .99998 .99996 .99996 .99995 .99995 .00873 .00902 .00931 .00960 .00989 114-589 110.892 107.426 104.171 101.107 .02618 .02647 .02676 .02705 .02734 .99966 .99965 .99964 .99963 -99963 .02619 .02648 .02677 .02706 .02735 38-1885 37-7686 37.3579 36.9560 36.5627 30 29 28 27 26 35 36 87 38 39 .01018 .01047 .01076 .01105 .01134 .99995 .99995 .99994 .99994 .99994 .01018 .01047 .01076 .01105 .01135 98-2179 95.4895 92-9085 90.4633 88-1436 .02763 .02792 .02821 .02850 .02879 .99962 .99961 .99960 .99959 .99959 .02764 -02793 .02822 .02851 .02881 36.1776 35-8006 35.4313 35-C695 34-7151 25 24 23 22 -21 40 41 42 43 44 .01164 .01193 .01222- .01251 .01280 .99993 .99993 .99993 .99992 .99992 .01164 .01193 .01222 .01251 .01280 85-9398 83-8435 8L8470 79.9434 78.1263 02908 02938 02967 02996 .03025 -99958 -99957 -99956 .99955 .99954 02910 -02939 .02968 .02997 .03026 34-3678 34.0273 33-6935 33-3662 33-0452 20 19 18 17 .18 45 46 47 48 49 .01309 .01338 .01367 .01396 01425 .99991 .99991 .99991 .99990 .99990 01309 .01338 .01367 .01396 .01425 76-3900 74.7292 73-1390 71-6151 70-1533 03054 .03083 03112 -03141 .03170 .99953 .99952 .99952 .99951 .99950 .03055 -03084 .03114 -03143 .03172 32-7303 32-4213 32-1181 31-8205 31-5234 15 14 13 12 .11 50 51 52 53 54 .01454 01483 01513 .01542 .01571 .99989 .99989 .99989 .99988 .99988 .01455 .01484 .01513 .01542 .01571 68.7501 67.4019 66.1055 64-8580 63.6567 .03199 -03228 .03257 .03286 .03316 .99949 .99948 .99947 .99946 .99945 .03201 .03230 -03259 .03288 .03317 31.2416 30-9599 30-6833 30.4116 30-1446 10 9 8 7 6 55 56 57 58 59 .01600 01629 .01658 01687 .01716 .99987 .99987 .99986 .99986 .99985 .01600 .01629 .01658 .01687 .01716 62.4992 61.3829 60-3058 59.2659 58.2612 .03345 -03374 .03403 .03432 -03461 .99944 .99943 .99942 .99941 .99940 -03346 .03376 .03405 .03434 -03463 29.8823 29.6245 29.3711 29-1220 28.8771 5 4 3 2 1 60 .01745 .99985 .01746 57-2900 -03490 .99939 .03492 28-6363 - / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 1 19 2( )6 18 Inserted by permission from Webb's Railroad Construction. TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 2 3 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t 2 3 03490 03519 03548 03577 03606 99939 .99938 .99937 .99936 .99935 03492 03521 03550 03579 03609 28-6363 28-3994 28.1664 27-9372 27-7117 05234 05263 05292 05321 05350 99863 99861 99860 99858 99857 05241 05270 05299 05328 05357 19-0511 18-9755 18-8711 18-7678 18.6656 60 59 58 57 58 ~5~ 6 7 8 g 03635 03664 03693 03723 03752 .99934 .99933 .99932 .99931 .99930 03638 03667 03696 03725 03754 27-4899 27-2715 27-0566 26.8450 26-6367 05379 05408 05437 05466 05495 99855 99854 99852 99851 99849 .05387 .05416 .05445 .05474 .05503 18-5645 18-4645 18.3655 18-2677 18-1708 55 54 53 52 _51 10 11 12 13 14 03781 03810 03839 03868 03897 .99929 .99927 99926 99925 99924 03783 03812 03842 03871 03900 26.4316 26-2296 26-0307 25.8348 25-6418 05524 05553 05582 05611 05640 99847 99846 99844 99842 99841 .05533 .05562 .05591 .05620 .05649 18-0750 17-9802 17.8863 17-7934 17-7015 50 49 48 47 48 15~ 16 17 18 19 03926 03955 03984 04013 04042 99923 99922 99921 99919 99918 03929 03958 03987 04016 04046 25.4517 25-2644 25.0798 24-8978 24-7185 05669 05698 .05727 .05756 .05785 99839 99838 99836 99834 99ff33 .05678 .05708 .05737 .05766 .05795 17.6108 17.5205 17-4314 17-3432 17-2558 45 44 43 42 -M 20 21 22 23 24 04071 .04100 .04129 .04159 .04188 99917 99916 99915 99913 99912 04075 04104 04133 04162 04191 24-5418 24.3675 24.1957 24-0263 23-8593 .05814 .05844 .05873 .05902 .05931 99831 99829 99827 .99826 .99824 .05824 .05854 .05883 .05912 -05941 17-1693 17-0837 16-9990 16-9150 16-8319 40 39 38 37 _38 25~ 28 27 28 29 .04217 .04246 .04275 .04304 .04333 99911 99910 .99909 .99907 .99906 04220 04250 04279 04308 04337 23.6945 23-5321 23-3718 23-2137 23-0577 .05960 .05989 .06018 .06047 .06076 .99822 .99821 .99819 .99817 .99815 .05970 .05999 .06029 .06058 .06087 16.7498 16-6681 16-5874 16-5075 16-4283 35 34 33 32 31 30 31 32 33 34 .04362 .04391 .04420 .04449 .04478 .99905 .99904 .99902 .99901 .99900 04366 .04395 .04424 .04454 .04483 22-9038 22.7519 22-6020 22-4541 22-3081 .06105 .06134 .06163 .06192 .06221 .99813 .99812 .99810 .99808 .99806 .06116 .06145 -06175 -06204 .06233 16-3499 16-2722 16-1952 18.1190 16-0435 30 29 28 27 26 35 36 37 38 39 .04507 .04536 .04565 .04594 .04623 .99898 .99897 .99896 .99894 .99893 .04512 .04541 .04570 .04599 .04628 22-1640 22-0217 21-8813 21-7426 21-6056 .06250 .06279 .06308 .06337 .06366 .99804 .99803 .99801 .99799 .99797 -06262 .06291 .06321 .06350 .06379 15-9687 15-8945 15-8211 15-7483 15-6762 25 24 23 22 21 40 41 42 43 44 .04653 .04682 .04711 .04740 .04769 .99892 .99890 .99889 .99888 ,99886 .04658 .04687 .04716 .04745 04774 21-4704 21-3369 21-2049 21-0747 20-9460 .06395 .06424 .06453 .06482 .06511 .99795 .99793 .99792 .99790 .99788 .06408 .06437 .06467 .06496 .06525 15-6048 15-5340 15-4638 15-3943 15-3254 20 19 18 17 18 45 46 47 48 49 .04798 04827 .04856 .04885 .04914 .99885 .99883 .99882 99881 .99879 .04803 .04833 04862 04891 .04920 20-8188 20-6932 20-5691 20-4465 20-3253 .06540 .06569 .06598 .06627 06656 .99786 .99784 .99782 .99780 .99778 .06554 .06584 .06613 .06642 .06671 15-2571 15-1893 15-1222 15-0557 14-9898 Ib 14 13 12 11 50 51 52 53 54 .04943 .04972 .05001 .05030 .05059 .99878 .99876 .99875 .99873 .99872 .04949 .04978 .05007 .05037 .05066 20-2056 20-0872 19-9702 19-8546 19-7403 .06685 -06714 -06743 .06773 .06802 .99776 .99774 .99772 .99770 .99768 .06700 .06730 .06759 .06788 .06817 14-9244 14-8596 14-7954 14.7317 14-6685 1O 9 8 7 8 55 56 67 68 59 .05088 .05117 .05146 .05175 .05205 .99870 .99869 .99867 .99866 .99864 .05095 .05124 .05153 .05182 .05212 19-6273 19-5156 19-4051 19-2959 19-1879 .06831 .06860 .06889 .06918 .06947 .99766 .99764 .99762 .99760 .99758 .06847 .06876 .06905 .06934 .06963_ 14.6059 14-5438 14-4823 14-4212 14-3607 5 3 2 60 .05234 .99863 .05241 19-0811 .06976 .99756 .06993 14-3007 / Cos. Sin. Cot. an. Cos. Sin. Cot. Tan. * >7 207 86 TABLE 10. NATURAL SINES, COSINES, TANGENTS. AND COTANGENT& 4 5 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. ' 2 3 4 .06976 .07005 .07034 .07063 .07092 .99756 .99754 .99752 .99750 .99748 .06993 .07022 .07051 .07080 .07110 14.3007 14.2411 14-1821 14.1235 14.0655 .08716 .08745 .08774 .08803 .08831 99619 .99617 .99614 -99612 -99609 08749 08778 08807 .08837 .08866 11.4301 11.3919 11.3540 11.3163 11.2789 60 59 58 57 56 5 6 7 8 9 .07121 .07150 .07179 .07208 .07237 .99746 .99744 .99742 .99740 .99738 .07139 .07168 .07197 .07227 .07256 14-0079 13.9507 13.8940 13-8378 13.7821 .08860 .08889 .08918 .08947 .08976 -99607 -99604 .99602 .99599 -99596 .08895 -08925 -08954 .08983 .09013 11.2417 11.2048 11.1681 11.1316 11.0954 55 54 53 52 51 10 11 12 13 ,14 .07266 .07295 .07324 .07353 .07382 .99736 .99734 .99731 .99729 .99727 .07285 .07314 .07344 .07373 .07402 13.7267 13-6719 13.6174 13.5634 13.5098 .09005 .09034 .09063 .09092 .09121 .99594 .99591 .99588 -99586 .99583 -09042 .09071 -09101 .09130 .09159 11.0594 11.0237 10-9882 10-9529 10.9178 50 49 48 47 4B 15 16 17 18 19 .07411 .07440 .07469 .07498 .07527 .99725 .99723 .99721 .99719 .99716 .07431 .07461 .07490 .07519 .07548 13-4566 13.4039 13.3515 13-2996 13-2480 -09150 .09179 .09208 .09237 .09266 .99580 .99578 .9C575 .9&S72 .99570 .09189 .09218 .09247 .09277 .09306 10-8829 10-8483 10-8139 10-7797 10-7457 45 44 43 42 41 20 21 22 23 24 .07556 .07585 .07614 .07643 .07672 .99714 .99712 .99710 .99708 .99705 .07578 .07607 .07636 .07665 .07695 13.1969 13-1461 13-0958 13.0458 12-9962 .09295 .09324 .09353 .09382 .09411 .99567 .99564 .99562 .99559 .99556 .09335 .09365 .09394 -09423 -09453 10-7119 10.6783 10.6450 10-6118 10-5789 40 39 38 37 36 25 26 27 28 29 07701 .07730 .07759 .07788 .07817 .99703 .99701 .99699 .99696 .99694 .07724 .07753 .07782 .07812 .07841 12-9469 12-8981 12.8496 12.8014 12.7536 .09440 -09469 -09498 .09527 .09556 .99553 .99551 .99548 .99545 .99542 .09482 .09511 .09541 .09570 .09600 10-5462 10-5136 10.4813 10-4491 10-4172 35 34 33 32 31 30 31 32 33 34 .07846 .07875 .07904 .07933 .07962 .99692 .99689 .99687 .99685 .99683 .07870 .07899 .07929 .07958 .07987 12.7062 12.6591 12-6124 12.5660 12.5199 09585 .09614 .09642 .09671 .09700 .99540 -99537 -99534 -99531 -99528 -09629 .09658 .09688 .09717 .09746 10-3854 10-3538 10-3224 10.2913 10.2602 30 29 28 27 26 35 36 87 38 89 .07991 .08020 .08049 .08078 .08107 .99680 .99678 .99676 .99673 .99671 .08017 .08046 .08075 .08104 .08134 12.4742 12.4288 12-3838 12.3390 12-2946 .09729 -09758 .09787 .09816 .09845 -99526 .99523 .99520 .99517 .99514 -09776 .09805 09834 .09864 .09893 10.2294 10-1988 10.1683 i0.1381 10.1080 25 24 23 22 ,.21 40 41 42 43 44 .08136 .08165 .08194 .08223 .08252 .99668 .99666 .99664 .99661 .99659 08163 .08192 .08221 .08251 .08280 12-2505 12.2067 12.1632 12.1201 12.0772 .09874 .09903 .09932 09961 .09990 99511 99508 .99506 .99503 .99500 .09923 .09952 .09981 .10011 .10040 10-0780 10-0483 10-0187 9-98931 9.96007 20 19 18 17 16 45 46 47 48 49 08281 .08310 .08339 .08368 .08397 .99657 .99654 .99652 .99649 .99647 .08309 .08339 .08368 .08397 .08427 12.0346 11.9923 11.9504 11-9087 11-8673 -10019 .10048 -10077 .10106 .10135 .99497 .99494 .99491 .99488 .99485 .10069 .10099 .10128 .10158 -10187 9-93101 9-90211 9-87338 9 84482 9.81641 15 14 13 12 _!! 50 51 52 53 54 .08426 .08455 .08484 .08513 .08542 .99644 .99642 .99639 .99637 .99635 .08456 .08485 .08514 .08544 .08573 11.8262 11.7853 11.7448 11.7045 11.6645 -10164 -10192 -10221 .10250 .10279 -99482 -99479 .99476 .99473 .99470 .10216 -10246 -10275 .10305 .10334 9-78817 9-76009 9-73217 9.70441 9-67680 10 9 8 7 6 55 56 57 58 59 .08571 .08600 .08629 .08658 .08687 .99632 .99630 .99627 .99625 .99622 .08602 .08632 .08661 .08690 .08720 11.6248 11.5853 11.5461 11.5072 11.4685 .10308 .10337 .10366 .10395 .10424 .99467 -99464 .99461 -99458 -99455 .10363 .10393 .10422 10452 .10481 9.64935 9-62205 9.59490 9-56791 9-54106 5 4 3 2 60 .08716 .99619 .08749 11-4301 .10453 99452 .10510 9.51436 O 1 / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 86 208 84 TABLE 10 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 6 7 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t o 1 2 3 10453 10482 10511 10540 10569 .99452 .99449 .99446 .99443 .99440 10510 10540 10569 10599 10628 9.51436 9.48781 9.46141 9.43515 9.40904 12187 12216 12245 12274 12302 99255 99251 99248 99244 99240 12278 12308 .12338 .12367 .12397 8.14435 8.12481 8.10536 8.08600 8.06674 60 59 58 57 56 5 6 7 8 9 10597 10626 10655 .10684 .10713 .99437 .99434 .99431 .99428 .99424 10657 .10687 .10716 .10746 .10775 9.38307 9.35724 9.33155 9.30599 9.28058 .12331 .12360 12389 .12418 .12447 99237 .99233 .99230 .99226 99222 12426 12456 .12485 .12515 .12544 8-04756 8.02848 8.00948 7.99058 7-97176 55 54 53 52 51 10 11 12 13 14 .10742 .10771 .10800 .10829 .10858 .99421 .99418 .99415 .99412 .99409 .10805 .10834 .10863 .10893 .10922 9-25530 9.23016 9-20516 9.18028 9-15554 .12476 .12504 .12533 .12562 .12591 .99219 99215 .99211 .99208 .99204 .12574 .12603 .12633 .12662 .12692 7.95302 7-93438 7-91582 7-89734 7-87895 50 49 48 47 46 15 16 17 18 19 .10887 .10916 .10945 .10973 .11002 .99406 .99402 .99399 .99396 .99393 .10952 .10981 .11011 .11040 .11070 9.13093 9-10646 9.08211 9.05789 9-03379 .12620 .12649 .12678 .12706 .12735 .99200 .99197 .99193 .99189 .99186 .12722 .12751 .12781 .12810 .12840 7-86064 7-84242 7-82428 7-80622 7.78825 45 44 43 42 -il 2O 21 22 23 24 .11031 .11060 .11089 -11118 .11147 .99390 .99386 .99383 .99380 .99377 .11099 .11128 .11158 .11187 .11217 9.00983 8-98598 8-96227 8.93867 8.91520 .12764 .12793 .12822 .12851 .12880 .99182 .99178 99175 .99171 .99167 .12869 .12899 .12929 .12958 .12988 7-77035 7-75254 7-73480 7.71715 7.69957 40 39 38 37 36 25 26 7 28 29 .11176 11205 11234 .11263 .11291 .99374 .99370 .99367 .99364 .99360 -11246 11276 .11305 .11335 .11364 8-89185 8-86862 8.84551 8-82252 8-79964 .12908 .12937 .12966 .12995 .13024 99163 -99160 99156 .99152 .99148 .13017 .13047 .13076 .13106 .13136 7.68208 7.66466 7.64732 7.63005 7.61287 35 34 33 32 31 30 31 32 33 34 .11320 11349 .11378 11407 .11436 .99357 .99354 .99351 -99347 .99344 .11394 .11423 .11452 .11482 .11511 8-77689 8-75425 8-73172 8-70931 8.68701 .13053 .13081 .13110 .13139 .13168 .99144 .99141 .99137 .99133 .99129 .13165 .13195 .13224 .13254 .13284 7-59575 7-57872 7-56176 7-54487 7-52806 30 29 28 27 26 35 36 37 38 39 .11465 .11494 .11523 .11552 11580 .99341 .99337 .99334 .99331 .99327 .11541 .11570 .11600 .11629 .11659 8.66482 8-64275 8.- 62078 8.59893 8.57718 13197 .13226 .13254 .13283 .13312 .99125 .99122 .99118 .99114 99110 .13313 .13343 .13372 .13402 .13432 7.51132 7-49465 7-47806 7-46154 7-44509 25 24 23 22 21 40 41 42 43 44 .11609 11638 11667 .11698 .11725 .99324 .99320 .99317 .99314 .99310 .11688 .11718 .11747 .11777 .11806 8-55555 8-53402 8.51259 8-49128 8-47007 13341 .13370 .13399 .13427 .13456 .99106 .90102 .99098 .99094 .99091 .13461 .13491 .13521 .13550 .13580 7-42871 7-41240 7-39616 7-37999 7-36389 20 19 18 17 18 45 46 47 48 49 .11754 .11783 .11812 11840 .11869 .99307 .99303 .99300 99297 .99293 .11836 .11865 .11895 .11924 11954 8-44896 8-42795 8-40705 8-38625 8-36555 .13485 13514 .13543 .13572 .13600 .99087 .99083 .99079 .99075 .99071 .13609 .13639 .13669 .13698 .13728 7-34786 7-33190 7-31600 7-30018 7-28442 15 14 13 12 11 50 51 52 53 54 .11898 .11927 .11956 .11985 .12014 .99290 .99286 .99283 99279 .99276 .11983 =12013 .12042 12072 .12101 8.34496 8.32446 8-30406 8-28376 8.26355 .13629 .13658 13687 .13716 .13744 .99067 .99063 .99059 .99055 .99051 .13758 13787 .13817 .13846 .13876 7-26873 7-25310 7-23754 7-22204 7-20661 10 9 8 7 6 55 56 57 58 59 12043 12071 .12100 .12129 .12158 .99272 .99269 .99265 99262 .99258 .12131 .12160 .12190 12219 12249 8.24345 8-22344 8.20352 8.18370 8.16398 .13773 .13802 .13831 .13860 .13889 .99047 .99043 .99039 99035 .99031 .13906 .13935 .13965 .13995 .14024 7-19125 7-17594 7-16071 7-14553 7-13042 5 3 2 60 .12187 .99255 .12278 8.14435 .13917 .99027 .14054 7-11537 1 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. / S3? 209 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 8 9 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t 60 59 58 57 56 55 54 53 52 51 2 3 4 5 6 7 8 9 10 11 12 13 LL 15 16 17 18 !2_ 20 21 22 23 24 25 26 27 28 29 .13917 13946 13975 14004 .14033 99027 .99023 .99019 .99015 .99011 .14054 .14084 .14113 .14143 .14173 7.11537 7-10038 7-08546 7-07059 7-05579 .15643 .15672 .15701 .15730 .15758 .98769 .98764 .98760 .98755 .98751 .15838 .15868 .15898 .15928 .15958 6.31375 6.30189 6-29007 6.27829 6.26655 .14061 .14090 .14119 .14148 .14177 .99006 .99002 .98998 .98994 .98990 .14202 .14232 .14262 .14291 .14321 7-04105 7-02637 7.01174 6-99718 6-98268 -15787 15816 .15845 15873 15902 .98746 98741 .98737 98732 .98728 .15988 .16017 .16047 .16077 .16107 6-25488 6.24321 6-23160 6-22003 6-20851 .14205 .14234 .14263 .14292 .14320 .98986 .98982 .98978 .98973 .98969 .14351 .14381 .14410 . 14440 .14470 6-96823 6-95385 6-93952 6-92525 6-91104 -15931 .15959 .15988 .16017 -16046 98723 .98718 .98714 .98709 .98704 .16137 .16167 .16196 .16226 .16256 6.19703 6-18559 6.17419 6.16283 6-15151 5O 49 48 47 _46 45 44 43 42 -il 40 39 38 37 36 35 34 33 32 31 .14349 .14378 .14407 .14436 . 14464 .98965 .98961 .98957 .98953 .98948 .14499 .14529 .14559 .14588 .14618 6.89688 6-88278 6-86874 6-85475 6-84082 .16074 .16103 .16132 .16160 .16189 .98700 98695 -98690 -98686 .98681 .16286 -16316 .16346 -16376 -16405 6.14023 6.12899 6.11779 6.10664 6-09552 .14493 14522 .14551 .14580 .14608 .98944 .98940 .98936 .98931 .98927 .14648 .14678 .14707 .14737 .14767 6.82694 6-81312 6-79936 6-78564 6-77199 .16218 .16246 .16275 .16304 .16333 .98676 .98671 .98667 .98662 .98657 .16435 .16465 .16495 .16525 .16555 6.08444 6.07340 6-06240 6.05143 6-04051 .14637 .14666 .14695 .14723 .14752 .98923 .98919 .98914 .98910 .98906 .14796 .14826 .14856 .14886 .14915 6.75838 6-74483 6-73133 6-71789 6-70450 .16361 .16390 .16419 .16447 .16476 .98652 .98648 -98643 .98638 -98633 .16585 .16615 .16645 .16674 .16704 6.02962 6.01878 6-00797 5.99720 5.98646 30 31 32 33 34_ 35 36 37 38 39_ 40 41 42 43 44 .14781 .14810 .14838 .14867 .14896 .98902 .98897 .98893 .98889 .98884 .14945 .14975 .15005 .15034 .15064 6.69116 6-67787 6-66463 6-65144 6.63831 .16505 .16533 .16562 .16591 .16620 -98629 .98624 .98619 .98614 .98609 .16734 -16764 .16794 -16824 .16854 5-97576 5-96510 5.95448 5.94390 5-93335 30 29 28 27 26 25 24 23 22 21 .14925 .14954 .14982 .15011 .15040 .98880 .98876 .98871 .98867 .98863 .15094 15124 .15153 .15183 .15213 6-62523 6.61219 6-59921 6-58627 6.57339 .16648 .16677 .16706 -16734 -16763 .98604 .98600 .98595 .98590 .98585 .16884 .16914 .16944 .16974 .17004 5.92283 5.91236 5.90191 5.89151 5.88114 .15069 15097 .15126 .15155 .15184 .98858 .98854 .98849 .98845 .98841 15243 .15272 .15302 .15332 .15362 6.56055 6.54777 6.53503 6.52234 6.50970 .16792 .16820 .16849 .16878 -16906 .98580 .98575 .98570 .98565 -98561 .17033 .17063 .17093 -17123 .17153 5.87080 5-86051 5-85024 5-84001 5.82982 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 3 2 -0 45 46 47 48 49 .15212 .15241 .15270 .15299 15327 .98836 .98832 .98827 .98823 .98818 .15391 .15421 .15451 .15481 .15511 6.49710 6.48456 6-47206 6.45961 6.44720 .16935 -16964 16992 17021 .17050 .98556 -98551 -98548 -98541 .98536 .17183 .17213 .17243 .17273 .17303 5.81966 5.80953 5.79944 5-78938 5-77936 50 51 52 53 54 55 56 57 58 59 15356 .15385 .15414 .15442 .15471 98814 .98809 .98905 98800 98796 15540 .15570 .15600 .15630 .15660 6-43484 6.42253 6.41026 6.39804 6-38587 17078 .17107 .17136 .17164 17193 .98531 -98526 -98521 .98516 .98511 .17333 .17363 .17393 .17423 .17453 5-76937 5.75941 5.74949 5-73960 5.72974 15500 -15529 15557 15586 .15615 98791 .98787 .98782 .98778 .98773 15689 .15719 .15749 .15779 .15809 6-37374 6-36165 6-34961 6.33761 6.32566 .17222 17250 17279 17308 -17336 -98506 .98501 .98496 .98491 .98486 -17483 17513 -17543 .17573 .17603 5-71992 5-71013 5-70037 5-69064 5-68094 60. 15643 -98769 .15838 6.31375 .17365 .98481 .17633 5-67128 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 81 210 80 -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 10 11 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 2 3 ~5~ 8 7 8 JL_ 10 11 12 13 14 15 16 17 18 12- 20 21 22 23 24^ 25 26 27 28 29_ 30 31 32 33 3JL 35 36 37 38 82- 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 .17365 17393 -17422 17451 .17479 .98481 .98476 .98471 .98466 .98461 17633 -17663 17693 .17723 -17753 5-67128 5-66165 5-65205 5-64248 5-63295 -19081 -19109 -19138 19167 19195 .'98163 -98157 -98152 -98146 -98140 -19438 -19468 .19498 19529 .19559 5.14455 5-13658 5-12862 5-12069 5-11279 60 59 58 57 56 17508 17537 .17565 .17594 -17623 .98455 .98450 .98445 .98440 .98435 .17783 .17813 .17843 .17873 17903 5-62344 5-61397 5 . 60452 5-59511 5.58573 19224 19.252 -19281 -19309 -19338 98135 -98i29 -98124 -98118 98112 19589 19619 19649 .19680 .19710 5-10490 5.09704 5.08921 5-08139 5-07360 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 .17651 17680 .17708 .17737 17766 .98430 .98425 98420 .98414 .98409 17933 .17963 .17993 18023 .18053 5-57638 5.56706 5-65777 5.54851 5-53927 .19366 -19395 .19423 19452 -19481 -98UD7 -98101 -98096 -98090 -98084 .19740 19770 .19801 -19831 .19861 5-06584 5-05809 5-05037 5-04267 5-03495 .17794 .17823 .17852 .17880 .17909 .98404 .98399 .98394 .98389 .98383 18083 .18113 . 18143 18173 .18203 5-53007 5-52090 5-51176 5-50264 5-49356 .19509 19538 -19566 -19595 -19623 98079 .98073 .98067 .98061 .98056 19891 19921 19852 .19982 -20012 5.02734 5-01971 5-01210 5-00451 4-99695 17937 .17966 .17995 .18023 .18052 .98378 .98373 .98368 .98362 .98357 .18233 .18263 .18293 .18323 .18353 18384 .18414 .18444 .18474 .18504 5-48451 5-47548 5-46648 5-45751 5-44857 .19652 -19680 19709 19737 -19766 .98050 .98044 .98039 .98033 .98027 . 20042 -20073 -20103 -20133 -20164 4-98940 4-98188 4-97438 4.96690 4-95945 .18081 .18109 -18138 .18166 .18195 .98352 .98347 .98341 .98336 .98331 5-43966 5-43077 5-42192 5-41309 5-40429 .19794 -19823 -19851 .19880 -19908 .98021 .98016 .98010 98004 -97998 20194 .20224 20254 20285 -20315 4-95201 4.94460 4-93721 4-92984 4-92249 18224 .18252 .18281 .18309 .18338 .98325 .98320 .98315 .98310 .98304 18534 .18564 .18594 18624 .18654 5-39552 5-38677 5.37805 5-36936 5-36070 -19937 -19965 -19994 -20022 -20051 .97992 -97987 -97981 .97975 -97969 - 20345 -20376 -20406 -20436 20466 4-91516 4-90785 4-90056 4-89330 4-88605 3O 29 28 27 26 25 24 23 22 21 .18367 .18395 .18424 18452 .18481 .98299 .98294 98288 .982b3 .98277 .18634 18714 .18745 .18775 .18805 5-35206 5.34345 5-33487 5-32631 5 31778 .20079 -20108 -20136 -20165 20193 -97963 -97958 .97952 .97946 .97940 .20497 .20527 20557 -20588 -20618 4-87882 4-87162 4-86444 4-85727 4 85013 18509 18538 .18567 .18595 .18624 .98272 .98267 .98261 98256 98250 .18835 .18865 .18895 .18925 .18955 5-30928 5-30080 5-29235 5-28393 5.27553 -20222 20250 -20279 -20307 -20336 97934 .97928 -97922 .97916 -97910 -20648 -20679 -20709 -20739 -20770 4-84300 4-83590 4-82882 4-82175 4-81471 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 3 2 ~~0 .18652 18681 18710 18738 .18767 .98245 98240 98234 98229 .98223 .18986 -19016 .19046 .19076 .19106 5-26715 5-25880 5.25C48 5-24218 5.23391 -20364 20393 -20421 20450 20478 -97905 .97899 .97893 -97887 -97881 .20800 -20830 -20861 -20891 -20921 4-80769 4-80068 4-79370 4-78673 4-77978 -18795 .18824 .18852 .18881 18910 .98218 98212 .98207 98201 .98196 .19136 -19166 19197 .19227 .19257 5-22566 5.21744 5-20925 5.20107 5-19293 -20507 -20535 -20563 -20592 -20620 -97875 -97869 -97863 -97857 .97851 20952 -20982 -21013 -21043 -21073 4.77286 4-76595 4-75906 4-75219 4-74534 55 56 57 58 59 60_ .18938 .18967 18995 .19024 .19052 .98190 .98185 .98179 .98174 98168 .19287 19317 .19347 19378 .19408 5-18480 5.17671 5-16863 5-16058 5-15256 -20649 -206.77 -20706 -20734 20763 -97845 -97839 .97833 -97827 .97821 -21104 -21134 -21164 -21195 .21225 4.73851 4-73170 4-72490 4-71813 4-71137 -19081 .98163 19438 5-14455 -20791 -97815 -21256 4-70463 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 79 211 78 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 12 13 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 .20791 .20820 20848 .20877 . 20905 .97815 .97809 97803 .97797 97791 -21256 -21286 -21316 .21347 21377 4.70463 4-69791 4.69121 4.68452 4-67786 -22495 -22523 -22552 -22580 -22608 97437 -97430 -97424 -97417 -97411 -23087 -23117 -23148 -23179 23209 4-33148 4-32573 4-32001 4-31430 4-30860 60 59 58 57 5B 5 6 7 8 9 20933 .20962 . 2099-0 .21019 .21047 .97784 97778 .97772 .97766 .97760 21408 21438 21469 -21499 .21529 4-67121 4.66458 4-65797 4-65138 4-64480 -22637 -22665 -22693 -22722 .22750 -97404 .97398 97391 97384 -97378 -23240 -23271 -23301 23332 23363 4-30291 4-29724 4-29159 4-28595 4-28032 55 54 53 52 51 10 11 12 13 14 .21076 .21104 -2U32 .21161 .21189 .97754 97748 .97742 97735 .97729 -21560 21590 -21621 21651 -21682 4-63825 4-63171 4-62518 4-61868 4-61219 .22778 .22807 -22835 -22863 -22892 97371 97365 .97358 .97351 -97345 .23393 -23424 -23455 -23485 23516 4-27471 4-26911 4-26352 4-25795 4-25239 50 49 48 47 4B 15 16 17 18 19 .21218 2124S -21275 21303 -21331 .97723 97717 .97711 97705 97698 .21712 .21743 .21773 -21804 -21834 4-60572 4-59927 4-59283 4-58641 4-58001 22920 -22948 -22977 -23005 .23033 .97338 .97331 97325 .97318 -97311 -23547 -23578 .23608 -23639 -23670 4-24685 4.24132 4-23580 4.23030 4-22481 45 44 43 42 41 20 21 22 23 24 .21360 .21388 .21417 . 21445 21474 .97692 .97686 .97680 -97673 97667 .21864 -21895 -21925 -21956 21986 4-57363 4-56726 4.56091 4.55458 4-54826 -23062 -23090 -23118 -23146 .23175 97304 -97298 -97291 -97284 .97278 .23700 -23731 -23762 -23793 -23823 4-21933 4-21387 4-20842 4.20298 4-19756 40 39 38 37 36 25 26 27 28 29 21502 21530 21559 21587 .21616 .97661 .97655 97648 97642 -97636 .22017 22047 22078 22108 .22139 4.54196 4.53568 4.52941 4-52316 4-51693 .23203 .23231 -23260 23288 23316 97271 97264 -97257 .97251 97244 .23854 -23885 -23916 .23946 -23977 4-19215 4.18675 4.18137 4.17600 4-17064 35 34 33 32 31 30 31 32 33 34 21644 21672 21701 .21729 .21758 97630 97623 97617 .97611 .97604 .22169 -22200 -22231 22261 22292 4-51071 4-50451 4.49832 4-49215 4-48600 23345 -23373 23401 23429 23458 97237 97230 .97223 .97217 97210 . 24008 24039 .24069 .24100 .24131 4.16530 4-15997 4-15465 4-14934 4-14405 30 29 28 27 26 35 36 37 38 39 21786 .21814 21843 21871 21899 .97598 .97592 .97585 .97579 -97573 .22322 22353 -22383 22414 . 22444 4-47986 4.47374 4-46764 4-46155 4-45548 23486 23514 23542 23571 23599 97203 .97196 .97189 97182 -97176 24162 24193 24223 24254 .24285 4-13877 4-13350 4-12825 4-12301 4-11778 25 24 23 22 21 40 41 42 48 44 .21928 .21956 21985 22013 . 22041 97566 .97560 .97553 .97547 97541 .22475 22505 22536 22567 22597 4-44942 4-44338 4-43735 4-43134 4-42534 23627 .23656 23684 23712 .23740 -97169 97162 -97155 -97148 -97141 .24316 24347 24377 . 24408 . 24439 4-11256 4-10736 4-10216 4-09699 4-09182 20 19 18 17 16 45 46 47 48 49 22070 22098 22126 22155 .22183 97534 .97528 .97521 .97515 .97508 .22628 22658 22689 .22719 -22750 4.41936 4-41340 4.40745 4-40152 4-39560 .23769 23797 23825 23853 .23882 -97134 -97127 -97120 .97113 -97106 . 24470 .24501 .24532 24562 24593 4.08666 4-08152 4-07639 4-07127 4-06616 15 14 13 12 11 50 51 52 53 54 22212 22240 22268 .22297 22325 .97502 .97496 .97489 97483 97476 22781 .22811 .22842 22872 22903 4-38969 4-38381 4.37793 4-37207 4-36623 .23910 23938 23966 .23995 .24023 .97100 .97093 -97086 .97079 -97072 .24624 .24655 -24686 .24717 .24747 4-06107 4-05599 4-05092 4-04586 4-04081 10 9 8 7 6 55 56 57 58 59 .22353 .22382 22410 .22438 .22467 97470 -97463 97457 97450 -97444 .22934 .22964 22995 .23026 23056 4-36040 4-35459 4-34879 4-34300 4-33723 24051 24079 24108 24136 .24164 -97065 -97058 -97051 -97044 -97037 .24778 -24809 -24840 24871 - 24902 4-03578 4.03076 4.02574 4-02074 4-01576 5 4 3 2 1 60 22495 97437 .23087 4-33148 .24192 -97030 -24933 4-01078 ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. / 77 212 76 -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 14 15 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / o 1 2 3 4 .24192 .24220 . 24249 -24277 .24305 97030 97023 97015 -97008 .97001 24933 24964 .24995 25026 25056 4-01078 4-00582 4-00086 3-99592 3-99099 -25882 -25910 -25938 .25966 -25994 96593 .96585 -96578 -96570 -96562 -26795 -26826 -26857 -26888 -26920 3-73205 3-72771 3-72338 3-71907 3-71476 60 59 58 57 56 5 6 7 8 9 .24333 .24362 .24390 .24418 . 24446 -96994 .96987 -96980 -96973 .96966 25087 .25118 .25149 .25180 .25211 3-98607 3-98117 3.97627 3-97139 3-96651 -26022 .26050 -26079 -26107 -26135 -96555 -96547 96540 96532 -96524 -26951 -26982 -27013 -27044 -27076 3-71046 3-70616 3-70188 3-69761 3-69335 55 54 53 52 51 10 11 12 13 14 . 24474 .24503 .24531 .24559 .24587 .96959 .96952 .96945 -96937 -96930 .25242 25273 .25304 25335 .25366 3-96165 3-95680 3-95196 3-94713 3-94232 -26163 -26191 -26219 -26247 -26275 -965,17 96509 .96502 .96494 .96486 -27107 -27138 -27169 -27201 27232 3-68909 3-68485 3-68061 3-67638 3-67217 50 49 48 47 46 15 16 17 18 19 24615 24644 .24672 .24700 .24728 .96923 -96916 -96909 -96902 .96894 25397 .25428 25459 25490 .25521 3-93751 3-93271 3-92793 3-92316 3-91839 -26303 -26331 .26359 -26387 -26415 -96479 -96471 -96463 -96456 -96448 -27263 -27294 -27326 -27357 .27388 3-66796 3-66376 3-65957 3-65538 3-65121 45 44 43 42 -ii 20 21 22 23 24 .24756 .24784 24813 . 24841 .24869 96887 96880 -96873 .96866 -96858 25552 25583 25614 25645 25676 3-91384 3.90890 3-90417 3-89945 3-89474 -26443 -26471 -26500 -26528 -26556 -96440 -96433 .96425 -96417 .96410 .27419 27451 .27482 -27513 -27545 3-64705 3-64289 3-63874 3-63461 3 63048 4O 39 38 37 36 25 26 27 28 29 24897 24925 24954 24982 25010 -96851 -96844 -96837 -96829 -96822 25707 25738 25769 25800 .25831 3-89004 3-88536 3-88068 3-87601 3-87136 -26584 -26612 -26640 -26668 -266P8 -96402 -96394 -96386 -96379 -96371 -27576 -27607 -27638 -27670 -27701 3-62636 3-62224 3-61814 3-61405 3.60996 35 34 33 32 31 30 31 32 33 34 25038 25066 25094 25122 25151 -96815 -96807 -96800 -96793 -96786 25862 25893 25924 25955 25983 3-86671 3-86208 3-85745 3-85284 3-84824 .26724 26752 26780 26808 26836 -96363 96355 -96347 .96340 -96332 -27732 .27764 -27795 .27826 -27858 3-60588 3-60181 3-59775 3-59370 3-58966 30 29 28 27 28 35 36 37 38 39 25179 25207 25235 25263 25291 96778 96771 -96764 .96756 96749 26017 26048 26079 .26110 -26141 3-84364 3-83908 3-83449 3-82992 3-82537 26864 26892 26920 26948 26976 -96324 -96316 96308 -96301 -96293 -27889 27921 -27952 .27983 -28015 3-58562 3-58160 3-57758 3-57357 3-56957 25 24 23 22 21 40 41 42 43 44 25320 25348 25376 25404 25432 96742 96734 96727 96719 96712 26172 26203 26235 26268 28297 3-82083 3-81630 3-81177 3-80726 3-80278 27004 27032 27060 27088 -27116 -96285 96277 96269 96261 96253 -28046 28077 -28109 -28140 -28172 3-56557 3-56159 3-55761 3-55364 3-54968 20 19 18 17 18 45 46 47 48 49 25460 25488 25516 25545 25573 96705 .96697 96690 96682 96675 26328 .26359 .26390 .26421 .26452 3-79827 3.79378 3-78931 3-78485 3 - 78040 -27144 .27172 -27200 -27228 -27256 .96246 96238 96230 -96222 -96214 -28203 28234 -28266 -28297 -28329 3-54573 3-54179 3-53785 3-53393 3-53001 15 14 13 12 11 50 51 52 53 54 25601 25829 25657 25685 25713 96667 96660 96653 96645 96638 .26483 26515 26546 26577 26608 3-77595 3.77152 3.76709 3-76268 3-75828 -27284 -27312 -27340 27368 -27396 -96206 -96198 -96190 .96182 96174 -28360 .28391 .28423 .28454 -28486 3-52609 3-52219 3-518?9 3-51441 3-51053 10 9 8 7 6 55 56 57 58 59 25741 25769 25798 25826 25854 96630 .96623 .96615 96608 96600 .26639 .26670 26701 26733 26764 3.75388 3.74950 3.74512 3.74075 3-73640 . 27424 27452 27480 27508 27536 96166 96158 96150 96142 96134 -28517 .28549 .28580 28612 .28643 3-50666 3-50279 3-49894 3-49509 3-49125 5 4 3 2 1 GO 25882 96593 .26795 3-73205 27564 96126 -28675 3-48741 ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 75 213 74 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 16 17 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. ' o 1 2 3 27564 .27592 .27620 .27648 .27676 .96126 .96118 .96110 .96102 .96094 .28675 .28706 .28738 .28769 .28800 3-48741 3-48359 3-47977 3-47596 3-47216 .29237 -29265 .29293 .29321 .29348 .95630 -95622 -95613 -95605 .95596 30573 -30605 30637 .30669 .30700 3-27085 3-26745 3-26406 3-26067 3-25729 60 59 58 57 56 5 6 7 8 9 .27704 .27731 .27759 .27787 .27815 .96086 .96078 .96070 .96062 .96054 .28832 .28864 .28895 .28927 .28958 3-46837 3-46458 3-46080 3-45703 3.45327 29376 .29404 .29432 .29460 .29487 .95588 -95579 -95571 -95562 -95554 30732 .30764 -30796 .30828 .30860 3-25392 3-25055 3-24719 3-24383 3 24049 55 54 53 52 51 10 11 12 13 14 .27843 .27871 .27899 .27927 .27955 .96046 96037 .96029 .96021 -96013 .28990 .29021 .29053 .29084 .29116 3-44951 3-44576 3-44202 3-43829 3-43456 .29515 .29543 .29571 .29599 29626 .95545 95536 -95528 -95519 -95511 .30891 .30923 .30955 .30987 -31019 3-23714 3-23381 3-23048 3-22715 3-22384 5O 49 48 47 46 15 16 17 18 19 .27983 .28011 .28039 .28067 -28095 .96005 .95997 .95989 .95981 .95972 29147 29179 29210 .29242 .29274 3-43084 3-42713 3.42343 3-41973 3-41604 .29654 .29682 .29710 .29737 .29765 95502 95493 95485 -95476 -95467 .31051 .31083 -31115 -31147 .31178 3-22053 3 21722 3-21392 3-21063 3-20734 45 44 43 42 41 20 21 22 23 24 .28123 .28150 -28178 .28206 .28234 .95964 .95956 .95948 .95940 .95931 .29305 .29337 .29368 .29400 .29432 3-41236 3.40869 3-40502 3-40136 3-39771 .29793 .29821 .29849 .29876 .29904 -95459 -95450 -95441 95433 95424 .31210 -31242 -31274 -31306 .31338 3-20406 3-20079 3-19752 3-19426 3-19100 40 39 38 37 36 25 26 27 28 29 .28262 -28290 .28318 .28346 .28374 .95923 .95915 .95907 .95898 .95890 .29463 29495 .29526 .29558 .29590 3.39406 3-39042 3-38679 3-38317 3-37955 .29932 .29960 .29987 .30015 .30043 .95415 .95407 -95398 -95389 -95380 .31370 -31402 .31434 .31466 .31498 3-18775 3-18451 3-18127 3-17804 3-1748] 35 34 33 32 31 30 31 32 33 34 .28402 .28429 .28457 .28485 .28513 .95882 .95874 .95865 .95857 .95849 .29621 .29653 .29685 .29716 .29748 3.37594 3-37234 3-36875 3-36516 3-36158 .30071 -30098 -30126 -30154 .30182 .95372 -95363 95354 .95345 .95337 .31530 .31562 -31594 .31626 -31658 3-17159 3-16838 3-16517 3-16197 3-15877 30 29 28 27 26 35 36 37 38 39 .28541 28569 -28597 .28625 .28652 .95841 95832 .95824 .95816 .95807 .29780 .29811 .29843 .29875 .29906 3-35800 3-35443 3-35087 3-34732 3-34377 .30209 -30237 -30265 .30292 .30320 .95328 .95319 .95310 .95301 -95293 -31690 -31722 31754 31786 31818 3-15558 3-15240 3-14922 3-14605 3-14288 25 24 23 22 21 40 41 42 43 44 .28680 28708 .28736 .28764 .28792 .95799 .95791 .95782 95774 .95766 .29938 .29970 .30001 .30033 .30065 3-34023 3-33670 3-33317 3-32965 3-32614 .30348 30376 30403 -30431 -30459 .95284 95275 95266 95257 .95248 31850 .31882 .31914 .31946 .31978 3-13972 3-13656 3-13341 3-13027 3-12713 2O 19 18 17 18 45 46 47 48 49 .28820 28847 28875 28903 28931 95757 95749 .95740 95732 .95724 .30097 .30128 30160 .30192 30224 3-32264 3-31914 3-31565 3-31216 3-30868 30486 30514 -30542 -30570 .30597 95240 95231 95222 -95213 95204 .32010 .32042 -32074 32106 -32139 3-12400 3-12087 3-11775 3-11464 3-11153 15 14 13 12 11 50 51 52 53 54 28959 28987 29015 29042 29070 .95715 .95707 95698 .95690 .95681 30255 30287 30319 30351 30382 3-30521 3-30174 3-29829 3-29483 3-29139 30625 .30653 .30680 .30708 -30736 95195 -95186 .95177 .95168 95159 -32171 .32203 32235 .32267 32299 3-10842 3-10532 3-10223 3-09914 3-09606 10 9 8 7 6 55 56 57 58 59 29098 29126 29154 29182 29209 .95673 95664 .95656 95647 .95639 30414 30446 30478 30509 30541 3-28795 3-28452 3-28109 3-27767 3-27426 .30763 30791 30819 30846 30874 .95150 95142 .95133 95124 95115 32331 32363 32396 32428 32460 3-09298 3-08991 3-08685 3-08379 3-08073 5 3 2 1 0_ 29237 -95630 30573 3-27085 30902 .95106 32492 3-07768 O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. * 73 214 73 -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 18 19 ' Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / o 1 2 3 4 .30902 .30929 .30957 .30985 .31012 .95106 .95097 .95088 .95079 .95070 32492 .32524 32556 .32588 .32621 3-07768 3-07464 3-07160 3-06857 3-06554 -32557 32584 -32612 32639 -32667 94552 94542 94533 94523 94514 34433 -34465 -34498 -34530 .34563 2-90421 2-90147 2-89873 2-89600 2-89327 60 59 58 57 56 5 6 7 8 9 .31040 .31068 .31095 .31123 .31151 .95061 .95052 .95043 .95033 .95024 .32653 .32685 .32717 32749 .32782 3-06252 3-05950 3.05649 3-05349 3-05049 -32694 -32722 32749 32777 32804 -94504 94495 -94485 94476 94466 .34596 .34628 .34661 -34693 34726 2-89055 2-88783 2-88511 2-88240 2-87970 55 54 53 52 51 10 11 12 13 14 .31178 .31206 .31233 .31261 .31289 .95015 .95006 .94997 .94988 .94979 .32814 .32846 .32878 .32911 .32943 3 - 04749 3-04450 3-04152 3-03854 3-03556 -32832 .32859 -32887 32914 .32942 -94457 94447 94438 94428 .94418 34758 -34791 34824 34856 .34889 2-87700 2.87430 2.87161 2-86892 2.86624 5O 49 48 47 46 15 16 17 18 19 31316 .31344 31372 .31399 31427_ .94970 .94961 .94952 .94943 .94933 32975 .33007 .33040 .33072 .33104 3-03260 3-02963 3-02667 3-02372 3-02077 32969 .32997 -33024 ,33051 .33079, .94409 94399 94390 94380 .94370 .34922 .34954 34987 .35020 .35052 2.86356 2-86089 2-85822 2-85555 2-85289 45 44 43 42 41 30 21 22 23 24 .31454 .31482 .31510 .31537 .31565 .94924 .94915 .94906 .94897 .94888 .33136 .33169 .33201 .33233 .33266 3-01783 3-01489 3-01196 3-00903 3-00611 33106 33134 -33161 .33189 .33216 -94361 .94351 .94342 .94332 .94322 .35085 .35118 .35150 .35183 .35216 2-85023 2-84758 2-84494 2-84229 2-83965 4O 39 38 37 36 25 26 27 28 29 31593 .31620 31648 31675 .31703 .94878 .94869 94860 .94851 .94842 .33298 .33330 .33363 .33395 .33427 3-00319 3-00028 2-99738 2-99447 2-99158 33244 -33271 -33298 -33326 33353 .94313 .94303 .94293 .94284 .94274 .35248 .35281 .35314 .35346 .35379 2-83702 2-83439 2-83176 2-82914 2-82653 35 34 33 32 31 30 31 32 33 34 31730 31758 31786 31813 .31841 .94832 .94823 .94814 .94805 .94795 33460 .33492 33524 33557 .33589 2-98868 2-98580 2-98292 2-98004 2-97717 33381 33408 33436 33463 33490 -94264 -94254 .94245 .94235 94225 -35412 35445 -35477 -35510 35543 2-82391 2-82130 2-81870 2-81610 2-81350 30 29 28 27 36 35 36 37 38 39 .31868 .31896 .31923 31951 .31979 .94786 .94777 .94768 .94758 .94749 33621 33654 33686 -33718 33751 2-97430 2-97144 2-96858 2-96573 2-9fi288 -33518 33545 33573 33600 .33627 -94215 -94206 -94196 -94186 .94176 -35576 -35608 -35641 -35674 -35707 2-81091 2-80833 2-80574 2-80316 2-80059 25 24 23 22 -21 4O 41 42 43 44 .32006 .32034 .32061 .32089 .32116 .94740 94730 94721 94712 .94702 33783 33818 .33848 33881 .33913 2-96004 2-95721 2-95437 2-95155 2-94872 -33655 33682 33710 33737 33764 .94167 -94157 -94147 -94137 -94127 -35740 -35772 35805 35838 .35871 2-79802 2-79545 2-79289 2-79033 2-78778 20 19 18 17 16 45 46 47 48 49 .32144 .32171 .32199 .32227 32254 .94693 .94684 94674 94665 .94656 33945 33978 34010 . 34043 . 34075 2-94591 2-94309 2-94028 2-93748 2-93468 -33792 33819 33846 33874 33901 .94118 .94108 -94098 -94088 -94078 .35904 .35937 .35969 .36002 .36035 2-78523 2-78269 2-78014 2-77761 2-77507 15 14 13 12 11 50 51 52 53 54 32282 .32309 .32337 .32364 .32392 .94646 .94637 94627 94618 .94609 34108 34140 34173 34205 -34238 2-93189 2.92910 2.92632 2.92354 2-92076 -33929 33956 33983 -34011 -34038 -94068 -94058 94049 94039 94029 -36068 .36101 -36134 36167 -36199 2-77254 2-77002 2-76750 2-76498 2-76247 10 9 8 7 6 55 56 57 58 59 32419 32447 32474 32502 .32529 .94599 .94590 .94580 .94571 .94561 34270 34303 -34335 34368 . 34400 2.91799 2-91523 2-91246 2-90971 2-90696 -34065 34093 -34120 34147 34175 -94019 -94009 .93999 .93989 93979 -36232 36265 .36298 .36331 36364 2-75996 2-75746 2-75496 2.75246 2-74997 5 4 3 2 80 .32557 .94552 .34433 2-90421 .34202 -93969 .36397 2 - 74748 O Cos. Sin. Cot. Tan. COP. Sin. Cot. Tan. 71 215 70 TABLE 10 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 30 31 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t 60 59 58 57 56 55 54 53 52 51 o 1 2 3 4 -34202 .34229 .34257 .34284 34311 93969 93959 93949 93939 .93929 36397 36430 36463 36496 .36529 2-74748 2.74499 2-74251 2-74004 2-73756 35837 35864 .35891 .35918 .35945 -93358 -93348 -93337 .93327 .93316 -38386 -38420 -38453 -38487 38520 2-60509 2-60283 2-60057 2-59831 2-59606 5 6 7 8 9 10 12 13 14 15 16 17 18 19_ 30 21 22 23 24 .34339 .34366 .34393 .34421 .34448 93919 .93909 .93899 .93889 .93879 36562 36595 .36628 36661 .36694 2-73509 2-73263 2-73017 2-72771 2-72526 35973 -36000 -36027 -36054 -36081 93306 .93295 .93285 93274 -93264 -38553 -385S7 -38620 -38654 38687 2-59381 2-59156 2-58932 2.58708 2-58484 .34475 .34503 .34530 .34557 -34584 .93869 .93859 .93849 .93839 .93829 36727 .36760 .36793 .36826 .36859 2-72281 2-72036 2-71792 2-71548 2.71305 36108 36135 .36162 36190 36217 -93253 93243 .93232 -93222 -93211 38721 38754 -38787 -38821 .38854 2-58261 2-58038 2-57815 2.57593 2-57371 50 49 48 47 46 .34612 34639 -34666 .34694 .34721 .93819 .93809 93799 .93789 .93779 .36892 .36925 .36958 36991 .37024 2-71062 2-70819 2-70577 2-70335 2-70094 .36244 36271 -36298 .36325 .36352 .93201 .93190 93180 .93169 -93159 -38888 -38921 -38955 38988 .39022 2-57150 2.56928 2-56707 2-56487 2-56266 45 44 43 42 41 34748 .34775 .34803 .34830 .34857 .93769 .93759 .93748 .93738 .93728 -37057 37090 37123 .37157 .37190 2-69853 2-69612 2-69371 2-69131 2-68892 .36379 -36406 -36434 .36461 -36488 -93148 -93137 -93127 -93116 -93106 39055 -39089 -39122 -39156 -39190 2-56046 2-55827 2-55608 2-55389 2-55170 40 39 38 37 36 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 34884 34912 .34939 .34966 .34993 .93718 .93708 .93698 .93688 .93677 .37223 .37256 .37289 .37322 .37355 2-68653 2-68414 2-68175 2-67937 2-67700 -36515 -36542 .36569 .36596 .36623 .93095 93084 .93074 .93063 .93052 -39223 39257 .39290 -39324 -39357 2-54952 2.54734 2-54516 2.54299 2-54082 35 34 33 32 31 30 29 28 27 26 .35021 .35048 .35075 .35102 -35130 .93667 .93657 93647 93637 .93626 .37383 .37422 -37455 37488 -37521 2.67462 2.67225 2.66989 2.66752 2-66516 .36650 .36677 -36704 -36731 -36758 .93042 .93031 93020 93010 -92999 39391 -39425 39458 -39492 -39526 2-53865 2-53648 2-53432 2-53217 2-53001 35157 35184 35211 35239 .35266 .93616 93606 93596 93585 .93575 37554 37588 37621 37654 .37687 2-66281 2-66046 2.65811 2.65576 2-65342 -36785 -36812 -36839 -36867 -36894 .92988 -92978 -92967 -92956 -92945 -39559 -39593 -39626 -39660 -39694 2-52786 2-52571 2-52357 2.52142 2-51929 25 24 23 22 21 40 41 42 43 44 35293 .35320 35347 35375 35402 .93565 .93555 .93544 .93534 .93524 .37720 37754 -37787 -37820 -37853 2-65109 2-64875 2.64642 2-64410 2-64177 -36921 -36948 -36975 -37002 -37029 -92935 -92924 -92913 92902 .92892 -39727 -39761 39795 39829 .39862 2.51715 2.51502 2-51289 2.51076 2-50864 30 19 18 17 16 15 14 13 12 11 1O 9 8 7 6 5 4 3 2 1 - 45 46 47 48 49 35429 35456 .35484 35511 35538 93514 .93503 93493 93483 .93472 .37887 .37920 .37953 37986 -38020 2.63945 2-63714 2.63483 2-63252 2-63021 -37056 -37083 .37110 .37137 .37164 -92881 -928VO -92859 -92849 -92838 -39896 39930 -39963 39997 .40031 2-50652 2 - 50440 2-50229 2-50018 2-49807 50 51 52 53 54 55 56 57 58 59 60. 35565 35592 35619 35647 35674 93462 93452 93441 -93431 .93420 .38053 38086 .38120 .38153 38186 2.62791 2-62561 2-62332 2-62103 2-61874 .37191 .37218 .37245 .37272 -37299 92827 .92816 .92805 .92794 .92784 -40065 .40098 -40132 -40166 - 40200 2-49597 2-49386 2-49177 2-48967 2-48758 35701 35728 35755 35782 35810 93410 .93400 93389 .93379 .93368 38220 38253 38286 38320 -38353 2-61646 2.61418 2-61190 2-60963 2-60736 -37326 -37353 -37380 37407 37434 .92773 -92762 -92751 -92740 92729 -40234 40267 40301 40335 40369 2-48549 2-48340 2-48132 2-47924 2-47716 35837 93358 -38386 2-60509 37461 92718 40403 2-47509 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 216 68 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS, 22 23 1 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. , 2 8 4 .37461 .37488 37515 .37542 .37569 .92718 .92707 92697 .92686 .92675 .40403 .40436 .40470 .40504 .40538 2-47509 2.47302 2.47095 2-46888 2-46682 .39073 .39100 -39127 -39153 .39180 92050 .92039 .92028 -92016 -92005 42447 .42482 .42516 .42551 .42585 2-35585 2.35395 2-35205 2-35015 2-34825 60 59 58 57 5ft 5 6 7 8 9 37595 .37622 37649 .37676 .37703 .92664 .92653 .92642 .92631 92620 .40572 .40606 .40640 .40674 .40707 2-46476 2-46270 2-46065 2.45860 2.45655 39207 -39234 .39260 39287 .39314 91994 .91982 .91971 .91959 91948 .42619 .42654 42688 -42722 42757 2-34636 2-34447 2-34258 2-34069 2-33881 55 54 53 52 51 10 11 12 13 14 37730 .37757 37784 .37811 .37838 .92609 .92598 .92587 .92576 .92565 .40741 .40775 .40809 .40843 .40877 2-45451 2-45246 2-45043 2.44839 2.44636 39341 .39367 39394 -39421 .39448 .91936 -91925 .91914 .91902 .91891 -42791 42826 .42860 42894 .42929 2-33693 2-33505 2-33317 2-33130 2-32943 50 49 48 47 46 15 16 17 18 19 .37865 .37892 .37919 .37946 .37973 .92554 .92543 .92532 .92521 .92510 .40911 .40945 .40979 .41013 .41047 2-44433 2-44230 2-44027 2-43825 2-43623 .39474 .39501 .39528 .39555 .39581 .91879 .91868 .91856 .91845 .91833 42963 .42998 .43032 .43067 .43101 2-32756 2-32570 2-32383 2-32197 2.32012 45 44 43 42 41 20 21 22 23 24 .37999 .38026 .38053 .38080 .38107 .92499 .92488 .92477 .92466 .92455 .41081 .41115 .41149 .41183 .41217 2-43422 2-43220 2-43019 2-42819 2-42618 39608 39635 39661 .39688 .39715 -91822 .91810 .91799 .91787 .91775 .43136 .43170 .43205 -43239 -43274 2-31826 2.31641 2-31456 2-31271 2-31086 40 39 38 37 36 25 26 27 28 29 .38134 .38161 .38188 -38215 .38241 .92444 .92432 .92421 .92410 .92399 .41251 .41285 .41319 .41353 .41387 2-42418 2-42218 2-42019 2.41819 2-41620 -39741 .39768 .39795 .39822 .39848 .91764 .91752 .91741 .91729 .91718 -43308 .43343 43378 -43412 -43447 2-30902 2-30718 2-30534 2-30351 2-30167 35 34 33 32 31 30 31 32 33 34 .38268 .38295 .38322 .33349 .38376 .92388 .92377 92366 .92355 .92343 .41421 .41455 .41490 .41524 .41558 2 41421 2-41223 2-41025 2-40827 2-40629 39875 -39902 .39928 -39955 .39982 -91706 .91694 .91683 .91671 .91660 -43481 .43516 .43550 -43585 .43620 2.29984 2-29801 2.29619 2-29437 2-29254 30 29 28 27 26 35 36 37 38 39 -38403 .38430 .38456 .38483 .38510 .92332 .92321 .92310 .92299 .92287 -41592 .41626 .41660 .41694 .41728 2-40432 2-40235 2-40038 2-39841 2-39645 .40008 40035 40062 40088 .40115 .91648 .91636 .91625 .91613 .91601 .43654 .43689 .43724 .43758 .43793 2-29073 2-28891 2-28710 2-28528 2.28348 25 24 23 22 21 40 41 42 43 44 .38537 .38564 .38591 38617 .38644 .92276 .92265 .92254 .92243 .92231 .41763 .41797 .41831 .41865 .41899 2-39449 2.39253 2-39058 2-38863 2.38668 .40141 40168 40195 -40221 .40248 .91590 .91578 .91566 .91555 .91543 .43828 .43862 43897 -43932 .43966 2-28167 2-27987 2.27806 2-27626 2-27447 2O 19 18 17 16 45 46 47 48 49 -38671 .38698 .38725 .38752 -38778 .92220 .92209 .92198 .92186 .92175 .41933 .41968 .42002 .42036 .42070 2.38473 2-38279 2-38084 2-37891 2-37697 -40275 40301 -40328 .40355 .40381 .91531 .91519 .91508 .91496 .91484 -44001 .44036 -44071 .44105 .44140 2-27267 2-27088 2-26909 2-26730 2-26552 15 14 13 12 11 5O 51 52 53 54 .38805 .38832 .38859 .38886 .38912 .92164 .92152 .92141 .92130 .92119 .42105 .42139 .42173 .42207 .42242 2-37504 2-37311 2-37118 2-36925 2.36733 .40408 40434 .40461 .40488 40514 .91472 .91461 .91449 .91437 .91425 -44175 .44210 .44244 .44279 .44314 2-26374 2-26196 2-26018 2-25840 2-25663 10 9 8 7 6 55 56 57 58 59 38939 38966 38993 .39020 .39046 .92107 .92096 92085 .92073 .92062 42276 .42310 .42345 .42379 .42413 2-36541 2.36349 2.36158 2-35967 2-35776 .40541 40567 .40594 .40621 .40347 .91414 .91402 .91390 .91378 .91366 .44349 .44384 .44418 .44453 .44488 2-25486 2-25309 2-25132 2-24956 2-24780 5 "4 3 2 60 .39073 .92050 42447 2-35585 .40674 .91355 .44523 2-24604 O 1 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 67 217 66 -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 34 25 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. f 1 2 3 4 .40674 .40700 .40727 .40753 .40780 .91355 .91343 .91331 .91319 .91307 44523 44558 44593 44627 .44662 2.24604 2-24428 2.24252 2.24077 2.23902 -42262 .42288 .42315 .42341 -42367 90631 -90618 90606 90594 .90582 .46631 46666 46702 46737 46772 2-14451 2-14288 2.14125 2-13963 2-13801 60 59 58 57 56 5 6 7 8 9 .40806 .40833 .40860 -40886 .40913 .91295 91283 .91272 .91260 .91248 44697 44732 44767 .44802 44837 2-23727 2.23553 2.23378 2-23204 2-23030 .42394 42420 .42446 42473 -42499 -90569 90557 .90545 -90532 .90520 .46808 -46843 .46879 46914 .46950 2-13639 2.13477 2-13316 2.13154 2-12993 55 54 53 52 51 10 11 12 13 14 .40939 .40966 40992 .41019 .41045 .91236 .91224 .91212 .91200 .91188 44872 .44907 .44942 .44977 .45012 2-22857 2.22683 2-22510 2.22337 2.22164 42525 .42552 .42578 .42604 .42631 .90507 .90495 -90483 90470 .90458 .46985 .47021 .47056 .47092 -47128 2-12832 2.12671 2-12511 2-12350 2-12190 50 49 48 47 46 15 16 17 18 19 41072 .41098 .41125 .41151 .41178 .91176 .91164 .91152 .91140 .91128 .45047 .45082 .45117 45152 .45187 2.21992 2.21819 2.21647 2.21475 2-21304 -42657 .42683 -42709 -42736 -42762 . 90446 90433 90421 .90408 -90396 .47163 .47199 .47234 -47270 .47305 2-12030 2-11871 2-11711 2-11552 2-11392 45 44 43 42 41 20 21 22 23 24 41204 41231 .41257 .41284 41310 -91116 91104 .91092 .91080 .91068 .45222 45257 .45292 45327 .45362 2-21132 2.20961 2-20790 2-20619 2-20449 42788 -42815 .42841 -42867 .42894 .90383 90371 .90358 .90346 .90334 -47341 -47377 .47412 .47448 .47483 2-11233 2-11075 2-10916 2-10758 2-10600 40 39 38 37 36 25 26 27 28 29 .41337 41363 41390 41416 41443 91056 91044 91032 91020 .91008 45397 -45432 .45467 45502 .45538 2-20278 2.20108 2-19938 2-19769 2-19599 -42920 -42946 42972 -42999 .43025 -90321 .90309 .90296 -90284 .90271 -47519 -47555 .47590 .47626 -47662 2-10442 2-10284 2-10126 2-09969 2-09811 35 34 33 32 31 30 31 32 33 34 41469 .41496 .41522 .41549 .41575 .90996 .90984 .90972 .90960 .90948 .45573 45608 .45643 -45678 .45713 2.19430 2.19261 2-19092 2-18923 2-18755 .43051 .43077 43104 .43130 -43156 -90259 .90246 .90233 .90221 90208 .47698 .^7733 .47769 .47805 .47840 2-09654 2-09498 2-09341 2-09184 2-09028 30 29 28 27 26 35 36 37 38 39 41602 41628 .41655 41681 41707 90936 90924 -90911 90899 -90887 45748 45784 45819 45854 .45889 2-18587 2-18419 2-18251 2-18084 2-17916 43182 .43209 43235 43261 -43287 -90196 -90183 -90171 90158 .90146 .47876 .47912 .47948 .47984 .48019 2-08872 2-08716 2-08560 2-08405 2-08250 25 24 23 22 -21 40 41 42 43 44 41734 41760 41787 41813 41840 90875 .90863 -90851 -90839 -90826 45924 .45960 45995 46030 .46065 2-17749 2-17582 2-17416 2-17249 2.17083 -43313 .43340 43366 43392 .43418 .90133 .90120 .90108 90095 -90082 .48055 -48091 .48127 .48163 .48198 2-08094 2-07939 2-07785 2.07630 2-07476 20 19 18 17 16 45 46 47 48 49 41866 .41892 .41919 41945 .41972 90814 90802 .90790 90778 .90766 .46101 46136 .46171 46206 46242 2.16917 2-16751 2-16585 2-16420 2-16255 43445 43471 43497 43523 .43549 90070 90057 .90045 -90032 -90019 .48234 .48270 -48306 -48342 .48378 2-07321 2-07167 2-07014 2-06860 2-06706 15 14 13 12 11 50 51 52 53 54 41998 .42024 42051 .42077 .42104 90753 -90741 .90729 .90717 90704 .46277 46312 46348 46383 46418 2-16090 2-15925 2-15760 2-15596 2-15432 43575 43602 43628 43654 .43680 .90007 89994 89981 -89968 .89956 .48414 .48450 .48486 .48521 .48557 2-06553 2.06400 2-06247 2-06094 2-05942 10 9 8 7 6 55 56 57 58 59 42130 42156 .42183 .42209 .42235 90692 90680 90668 .90655 .90643 .46454 .46489 46525 46560 .46595 2-15268 2- 15104* 2-14940 2-14777 2-14614 43706 43733 -43759 43785 .43811 .89943 .89930 -89918 .89905 .89892 .48593 .48629 .48665 48701 48737 2-05790 2.05637 2-05485 2-05333 2-05182 5 4 3 2 1 60 .42262 .90631 -46631 2-14451 43837 .89879 .48773 2-05030 _o / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. / 65 218 64 -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 26 27 * Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t 2 3 4 43837 .43863 .43889 .43916 .43942 .89879 89867 .89854 .89841 .89828 .48773 .48809 .48845 .48881 .48917 2.05030 2.04879 2.04728 2-04577 2-04426 .45399 45425 .45451 .45477 .45503 -89101 -89087 -89074 -89061 .89048 -50953 -50989 -51026 -51063 -51099 1-96261 1-98120 1-95979 1.95838 1-95698 60 59 58 57 56 5 6 7 8 9 .43968 .43994 .44020 . 44046 .44072 .89816 .89803 .89790 .89777 .89764 .48953 .48989 .49026 .49062 .49098 2-04276 2.04125 2.03975 2.03825 2-03675 .45529 .45554 .45580 .45606 .45632 -89035 -89021 .89008 .88995 .88981 .51136 .51173 .51209 .51246 51283 1-95557 1-95417 1-95277 1-95137 1-94997 55 54 53 52 51 10 11 12 13 14 .44098 .44124 .44151 .44177 .44203 .89752 .89739 .89726 .89713 .89700 .49134 .49170 .49206 .49242 .49278 2.03526 2.03376 2-03227 2-03078 2-02929 .45658 -45684 -45710 -45736 .45762 88968 -88955 -88942 .88928 -88915 -51319 -51356 .51393 -51430 -51467 1-94858 1-94718 1-94579 1-94440 1-94301 5O 49 48 47 46 15 16 17 18 19 44229 44255 44281 .44307 .44333 .89687 .89674 .89662 .89649 .89636 .49315 .49351 .49387 49423 .49459 2.02780 2.02631 2-02483 2.02335 2-02187 .45787 .45813 .45839 .45865 .45891 .88902 .88888 .88875 -88862 .88848 -51503 -51540 -51577 -51614 .51651 1.94162 1.94023 1.93885 1.93746 1.93608 45 44 43 42 41 20 21 22 23 24 .44359 .44385 .44411 .44437 .44464 .89623 .89610 .89597 .89584 .89571 49495 .49532 .49568 .49604 .49640 2-02039 2-01891 2.01743 2-01596 2-01449 -45917 .45942 .45968 45994 .46020 -88835 88822 .88808 88795 .88782 51688 51724 51761 .51798 .51835 1.93470 1-93332 1.93195 1.93057 1-9?920 40 39 38 37 36 25 2fi 27 28 29 44490 44516 .44542 44568 .44594 .89558 .89545 .89532 .89519 .89506 49677 .49713 .49749 .49786 -49822 2.01302 2-01155 2.01008 2.00862 2-00715 .46046 .46072 -46097 -46123 -46149 -88768 -88755 .88741 .88728 .88715 .51872 -51909 -51946 -51983 -52020 1.92782 1.92645 1.92508 1.92371 1.92235 35 34 33 32 31 30 81 82 33 34 .44620 .44646 .44672 .44698 44724 89493 .89480 .89467 .89454 .89441 .49858 .49894 .49931 .49967 . 50004 2.00569 2-00423 2-00277 2-00131 1.99986 -46175 -46201 -46226 .46252 .46278 .88701 .88688 .88674 .88661 .88647 -52057 -52094 -52131 .52168 .52205 1.92098 1-91962 1.91826 1.91690 1.91554 30 29 28 27 26 35 36 37 38 39 44750 44776 44802 44828 44854 .89428 .89415 .89402 .89389 .89376 . 50040 .50076 50113 .50149 .50185 1.99841 1.99695 1.99550 1.99406 1.99261 .46304 -46330 -46355 -46381 .46407 .88634 88620 .88607 .88593 .88580 .52242 .52279 .52316 .52353 .52390 1.91418 1-91282 1.91147 1.91012 1-90876 25 24 23 22 -21 !i 42 43 44 44880 44906 44932 44958 44984 .89363 .89350 .89337 .89324 .89311 .50222 .50258 .50295 .50331 .50368 1-99116 1.98972 1.98828 1.98684 1.98540 -46433 46458 .46484 .46510 .46536 88566 88553 88539 .88526 .88512 .52427 .52464 -52501 -52538 -52575 1-90741 1-90607 1-90472 1-90637 1-90203 30 19 18 17 16 45 46 47 48 49 45010 45036 45062 45088 45114 .89298 .89285 .89272 .89259 .89245 . 50404 . 50441 50477 .50514 .50550 1-98396 1.98253 1.98110 1-97966 1-97823 .46561 .46587 46613 46639 .46664 -88499 .88485 .88472 .88458 .88445 -52613 52650 52687 52724 .52761 1-90069 1-89935 1-89801 1-89667 1.89533 15 14 13 12 11 50 51 52 53 54 45140 45166 45192 45218 45243 .89232 .89219 .89206 .89193 .89180 .50587 .50623 50660 .50696 .50733 1.97681 1.97538 1.97395 1.97253 1.97111 -46690 46716 46742 46767 46793 -88431 .88417 .88404 .88390 -88377 52798 52836 52873 52910 52947 1-89400 1-99266 1-89133 1.89000 1.88867 10 9 8 7 6 55 56 57 58 59 45269 45295 45321 45347 45373 89167 .89153 .89140 89127 .89114 50769 .50806 .50843 .50879 50916 1-96969 1-96827 1.96685 1.96544 1-96402 46819 46844 46870 46896 46921 88363 88349 88336 88322 88308 52985 53022 53059 53096 53134 1-88734 1-88602 1-88469 1-88337 1-88205 5 4 3 2 1 60 45399 .89101 .50953 1.96261 46947 88295 53171 1-88073 _2 "7~ Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ft * 21 9 G J -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 28 29 i 9 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 3 .46947 .46973 .46999 .47024 .47050 .88295 .88281 .88267 .88254 .88240 53171 .53208 53246 .53283 .53320 1-88073 1-87941 1-87809 1-87677 1.87546 48481 .48506 .48532 .48557 .48583 87462 .87448 87434 87420 .87406 .55431 55469 -55507 -55545 .55583 i 80405 1-80281 1.80158 1-80034 1-79911 60 59 58 57 56 5 6 7 8 9 .47076 .47101 47127 .47153 .47178 .88226 88213 .88199 .88185 .88172 53358 .53395 .53432 53470 .53507 1-87415 1-87283 1-87152 1-87021 1.86891 -48608 .48634 -48659 .48684 -48710 -87391 -87377 -87363 -87349 .87335 .55621 55659 55697 55736 .55774 1 79788 1 79665 1.79542 1-79419 1-79296 55 54 53 52 51 10 11 12 13 14 .47204 .47229 .47255 .47281 .47306 .88158 .88144 .88130 .88117 .88103 .53545 .53582 .53620 .53657 .53694 1.86760 1.86630 1.86499 1.86369 1.86239 .48735 48761 .48786 -48811 48837 .87321 .87306 .87292 .87278 .87264 55812 55850 .55888 .55926 55964 1-79174 1.79053 1-78929 1-78807 1.78685 50 49 48 47 46 15 16 17 18 J9 .47332 47358 .47383 .47409 .47434 .88089 .88075 .88062 .88048 .88034 53732 .53769 .53807 .53844 .53882 1-86109 1.85979 1.85850 1-85720 1-85591 .48862 .48888 .48913 .48938 -48964 .87250 .87235 .87221 .87207 .87193 .56003 -56041 -56079 .56117 -56156 1-78563 1 - 78441 1-78319 1-78198 1.78077 45 44 43 42 -41 20 21 22 23 24 .47460 .47486 .47511 .47537 .47562 .88020 .88006 .87993 .87979 .87965 .53920 .53957 .53995 .54032 .54070 1-85462 1-85333 1-85204 1-85075 1-84946 .48989 .49014 49040 49065 .49090 -87178 .87164 .87150 .87136 .87121 .56194 -56232 -56270 .56309 .56347 1-77955 1-77834 1-77713 1-77592 1.77471 40 39 38 37 36 25 26 27 28 29 .47588 47614 47639 .47665 .47690 .87951 87937 87923 .87909 .87896 .54107 .54145 54183 . 54220 .54258 1.84818 1-84689 1-84561 1-84433 1-84305 -49116 .49141 .49166 -49192 .49217 -87107 87093 -87079 -87064 .87050 -56385 -56424 -56462 -56501 .56539 1-77351 1-77230 1.77110 1-76990 1.76869 35 34 33 32 31 30 31 32 33 34 .47716 .47741 .47767 .47793 .47818 .87882 .87868 .87854 .87840 .87826 54296 .54333 .54371 . 54409 . 54446 1-84177 1 . 84049 1.83922 1-83794 1.83667 .49242 -49268 .49293 .49318 .49344 .87036 .87021 .87007 .86993 .86978 .56577 -56616 .56654 .56693 -56731 1 - 76749 1-76629 1-76510 1-76390 1-76271 30 29 28 27 26 35 36 37 38 39 .47844 .47869 .47895 .47920 .47946 .87812 .87798 .87784 .87770 .87756 . 54484 54522 .54560 .54597 .54635 1.83540 1-83413 1-83286 1-83159 1.83033 .49369 .49394 .49419 .49445 .49470 -86964 -86949 -86935 -86921 -86906 -56769 -56808 -56846 -56885 .56923 1-76151 1-76032 1-75913 1-75794 1.75675 25 24 23 22 -21 40 41 42 43 44 .47971 .47997 .48022 .48048 .48073 87743 87729 .87715 .87701 .87687 . 54673 .54711 .54748 ..54786 .54824 1-82906 1-82780 1.82654 1.82528 1-82402 -49495 -49521 .49546 -49571 .49596 .86892 -86878 .86863 -86849 .86834 56962 .57000 -57039 -57078 -57116 1.75556 1-75437 1-75319 1-75200 1-75082 20 19 18 17 16 45 46 47 48 49 .48099 .48124 .48150 .48175 .48201 .87673 .87659 .87645 .87631 .87617 54862 .54900 .54938 .54975 .55013 1-82276 1-82150 1-82025 1-81899 1.81774 -49622 -49647 -49672 -49697 .49723 -86820 -86805 -86791 -86777 -86762 -57155 -57193 -57232 -57271 .57309 1-74964 1-74846 1-74728 1-74610 1 . 74492 15 14 13 12 11 50 51 52 53 54 .48226 .48252 48277 .48303 .48328 87603 87589 .87575 87561 87546 55051 .55089 .55127 .55165 .55203 1-81649 1-81524 1-81399 1-81274 1.81150 -49748 -49773 -49798 .49824 .49849 .86748 .86733 .86719 .86704 .86690 -57348 -57386 -57425 .57464 .57503 1-74375 1-74257 1-74140 1 - 74022 1-73905 10 9 8 7 6 55 56 57 58 59 .48354 .48379 .48405 .48430 .48456 87532 87518 .87504 .87490 .87476 .55241 .55279 .55317 .55355 .55393 1-81025 1-80901 1-80777 1-80653 i-80529 -49874 .49899 .49924 .49950 .49975 .86675 -86661 -86646 -86632 .86617 -57541 -57580 -57619 -57657 .57696 1-73788 1-73671 1-73555 1-73438 1-73321 5 4 3 2 1 60 . 48481 87462 55431 1-80405 .50000 -86603 .57735 1-73205 O / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 61 220 60 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS 30 31 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. f 1 2 3 4 . 50000 .50025 .50050 .50076 .50101 86603 86588 86573 .86559 .86544 .57735 .57774 .57813 57851 57890 1.73205 1.73089 1.72973 1.72857 1-72741 51504 51529 .51554 .51579 .51604 .85717 .85702 .85687 .85672 .85657 -60086 -60126 .60165 .60205 60245 1.66428 1-66318 1.66209 1-66099 1.65990 60 59 58 57 56 5 6 7 8 9 .50126 .50151 .50176 .50201 .50227 .86530 .86515 .86501 .86486 .86471 .57929 .57968 58007 .58046 .58085 1-72625 1-72509 1-72393 1-72278 1-72163 .51628 51653 -51678 .51703 .51728 85642 85627 .85612 85597 85582 .60284 -60324 .60364 .60403 . 60443 1.65881 1-65772 1-65663 1-65554 1.65445 55 54 53 52 51 10 11 12 13 .50252 .50277 -50302 .50327 .50352 86457 86442 86427 .86413 86398 58124 .58162 .58201 .58240 -58279 1 . 72047 1-71932 1-71817 1-71702 1-71588 51753 51778 51803 51828 .51852 85567 85551 .85536 .85521 -85506 -60483 .60522 60562 60602 .60642 1-65337 1.65228 1-65120 1-65011 1 64903 5O 49 48 47 48 15 16 17 18 19 50377 . 50403 .50428 .50453 .50478 86384 86369 86354 86340 -86325 .58318 .58357 .58396 .58435 58474 1.71473 1.71358 1.71244 1.71129 1-71015 .51877 .51902 .51927 .51952 -51977 .85491 .85476 .85461 .85446 .85431 .60681 .60721 .60761 .60801 .60841 1.64795 1-64687 1-64579 1-64471 1.64363 45 44 43 42 41 20 21 22 23 24 50503 .50528 .50553 .50578 .50603 86310 86295 .86281 86266 86251 .58513 -58552 -58591 -58631 .58670 1-70901 1-70787 1.70673 1-70560 1 - 70446 52002 -52026 -52051 -52076 52101 85416 .85401 85385 .85370 -85355 .60881 .60921 .60960 .61000 61040 1-64256 1.64148 1-64041 1.63934 1.63826 40 39 38 37 36 25 23 27 28 39 50628 50654 .50679 .50704 -50729 .86237 .86222 .86207 .86392 .86178 58709 .58748 .58787 .58826 .58865 1-70332 1.70219 1.70106 1-69992 1-69879 52126 .52151 .52175 .52200 .52225 -85340 -85325 -85310 85294 .85279 -61080 -61120 .61160 -61200 -61240 1.63719 1-63612 1.63505 1-63398 1.63292 35 34 33 32 31 30 31 32 33 34_ .50754 .50779 .50804 50829 50854 .86163 86148 .86133 .86119 .86104 .58905 .58944 .58983 .59022 .59061 1-69766 1.69653 1-69541 1.69428 1.69316 52250 .52275 -52299 .52324 -52349 .85264 85249 .85234 .85218 .85203 .61280 61320 -61360 -61400 .61440 1-63185 1.63079 1.62972 1-62866 1.62760 30 29 28 27 26 35 36 37 38 39 .50879 50904 -50929 50954 50979 86089 .86074 86059 86045 86030 .59101 .59140 .59179 59218 .59258 1-69203 1-69091 1-68979 1.68866 1-68754 52374 52399 52423 52448 .52473 .85188 .85173 85157 85142 -85127 .61480 61520 61561 .61601 61641 1.62654 1-62548 1.62442 1.62336 1-62230 25 24 23 22 21 40 41 42 43 44 51004 .51029 .51054 51079 51104 86015 .86000 85985 85970 .85956 .59297 .59336 .59376 .59415 .59454 1.68643 1-68531 1.68419 1.68308 1.68196 52498 52522 52547 52572 -52597 .85112 .85096 85081 .85066 85051 .61681 .61721 .61761 .61801 61842 1.62125 ]. 62019 1.61914 1.61808 1.61703 20 19 18 17 16 45 46 47 48 49 .51129 51154 .51179 51204 51229 .85941 .85926 85911 .85896 .85881 .59494 .59533 .59573 59612 .59651 1-68085 1.67974 1-67863 1-67752 1-67641 52621 52646 52671 .52696 52720 85035 85020 .85005 .84989 .84974 .61882 61922 -61962 62003 -62043 1.61598 1.61493 1.61388 1.61283 1.61179 15 14 13 12 11 50 51 52 53 54 .51254 51279 51304 51329 51354 .85866 .85851 <85836 85821 8o806 59691 59730 59770 .59809 59849 1-67530 1.67419 1.67309 1-67198 1.67088 52745 52770 52794 52319 .52844 84959 - 84943 .84928 .84913 .84897 62083 62124 .62164 .62204 62245 1-61074 1.60970 1 60865 1-60761 1.60657 1O 9 8 7 6 55 56 57 58 59 51379 51404 .51429 51454 51479 85792 .85777 85762 85747 .85732 59888 .59928 59967 60007 . 60046 1.66978 1.66867 1.66757 1.66647 1.66538 -52869 -52893 -52918 52943 .52967 .84882 .84866 .84851 .84836 84820 .62285 62325 .62366 .62406 .62446 1.60553 1-60449 1-60345 1 60241 1-60137 5 4 3 2 1 ?0 51504 -85717 60086 1-66428 52992 .84805 62487 1 - 60033 O Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 59 221 58 TABLE 10. NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS 33 33 t Sin. Cos Tan. Cot Sin. Cos. Tan. Cot. < 2 3 .52992 .53017 .53041 .53066 .53091 84805 .84789 .84774 .84759 . 84743 62487 .62527 .62568 .62608 62649 1-60033 1-59930 1.59826 1.59723 1-59620 -54464 54488 54513 54537 .54561 .83867 83851 .83835 83819 83804 64941 64982 65024 65065 65106 1.53986 1.53888 1.53791 1-53693 1-53595 60 59 58 57 56 5 6 7 8 9 .53115 .53140 .53164 .53189 .53214 84728 .84712 84697 .84681 .84666 .62689 .62730 .62770 62811 .62852 1-59517 1.59414 1.59311 1-59208 1.59105 . 54585 -&4610 .54635 54659 .54683 83788 83772 83756 83740 83724 -65148 65189 65231 65272 65314 1.53497 1 53400 1-53302 1-53205 1.53107 55 54 53 52 51. 10 11 12 13 !!_ .53238 53263 53288 53312 .53337 .84650 .84635 .84619 .84604 .84588 62892 .62933 .62973 .63014 .63055 1.59002 1-58900 1.58797 1.58695 1.58593 54708 54732 54756 54781 54805 .83708 83692 83676 .83660 .83645 65355 65397 .65438 65480 -65521 1.53010 1.52913 1.52816 1.52719 1-52622 50 49 48 47 46 15 16 17 18 19 .53361 .53386 .53411 .53435 .53460 .84573 .84557 .84542 .84526 .84511 .63095 .63136 .63177 .63217 .63258 1.58490 1.58388 1-58286 1.58184 1.58083 -54829 .54854 54878 .54902 .54927 .83629 .83613 .83597 .83581 .83565 -65563 -65604 -65646 65688 .65729 1-52525 1.52429 1-52332 1.52235 1.52139 45 44 43 42 41 20 21 22 23 24 53484 .53509 53534 .53558 .53583 .84495 .84480 .84464 . 84448 .84433 .63299 .63340 .63380 .63421 .63462 1.57981 1.57879 1.57778 1.57676 1.57575 54951 .54975 54999 55024 .55048 .83549 .83533 .83517 .83501 .83485 -65771 -65813 -65854 -65896 -65938 1.52043 1.51946 1.51850 1.51754 1-51658 40 39 38 37 36 25 26 27 28 29 .53607 .53632 .53656 -53681 .53705 .84417 .84402 84386 .84370 .84355 63503 63544 .63584 63625 .63666 1.57474 1.57372 1.57271 1.57170 1.57069 .55072 .55097 .55121 55145 .55169 .83469 83453 .83437 -83421 .83405 .65980 -66021 -66063 -66105 -66147 1-51562 1.51466 1.51370 1.51275 1.51179 35 34 33 32 31 30 81 82 33 34 .53730 .53754 53779 .53804 .53828 84339 .84324 .84308 .84292 .84277 .63707 .63748 .63789 .63830 .63871 1.56969 1.56868 1.56767 1.56667 1.56566 .55194 .55218 .55242 .55266 .55291 .83389 83373 83356 83340 .83324 -66189 .66230 .66272 .66314 66356 1.51084 1.50988 1.50893 1-50797 1 . 50702 30 29 28 27 26 35 36 37 38 39 .53853 .53877 .53902 .53926 .53951 .84261 .84245 .84230 .84214 .84198 .63912 .63953 .63994 .64035 .64076 1.56466 1.56366 1.56265 1.56165 1.56065 55315 .55339 .55363 .55388 .55412 .83308 .83292 .83276 .83260 83244 66398 66440 .66482 .66524 66566 1-50607 1.50512 1-50417 1.50322 1-50228 25 24 23 22 21 40 41 42 43 44 .53975 .54000 . 54024 . 54049 .54073 .84182 .84167 .84151 84135 .84120 .64117 .64158 64199 . 64240 64281 1.55966 1.55866 1.55766 1.55666 1.55567 .55436 .55460 55484 .55509 .55533 83228 .83212 .83195 .83179 -83163 .66608 .66650 .66692 .66734 .66776 1.50133 1.50038 1.49944 1.49849 1.49755 20 19 18 17 16 45 46 47 48 49 54097 .54122 54146 .54171 .54195 84104 84038 84072 84057 . 84041 .64322 64363 . 64404 . 64446 .64487 1.55467 1.55368 .55269 .55170 .55071 55557 .55581 55605 55630 -55654 -83147 -83131 -83115 83098 83082 .66818 .66860 .66902 .66944 .66986 1.49661 1.49566 1-49472 1-49378 1-49284 15 14 13 12 11 50 51 59, 53 54 .54220 . 54244 .54269 .54293 .54317 84025 .84009 83994 .83978 83962 .64528 64569 64610 .64652 . 64693 54972 .54873 .54774 1-54675 1-54676 .55678 .55702 .55726 55750 .55775 83066 .83050 .83034 83017 .83001 .67028 .67071 .67113 .67155 .67197 1-49190 1-49097 1-49003 1-48909 1-48816 10 9 8 7 6 55 56 57 58 59 .54342 54366 54391 .54415 . 54440 .83946 .83930 83915 .83899 83883 .64734 .64775 .64817 64858 64899 1-54478 1-54379 1.54281 1.54183 1 54085 55799 .55823 .55847 55871 .55895 82985 82969 82953 82936 82920 .67239 .67282 .67324 .67366 .67409 1-48722 1.48629 1-48536 1-48442 1.48349 5 4 3 2 60 54464 .83867 64941 1-53986 55919 82904 .67451 1.48256 O ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 57 222 66 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS, 34 35 Sin. Cos. Tan. Cot, Sin. Cos. Tan. Cot. 9 2 8 55919 55943 55968 55992 56016 82904 82887 82871 82855 82839 67451 67493 67536 67578 67620 1.48256 1-48163 1.48070 1.47977 1.47885 57358 57381 57405 57429 57453 81915 81899 81882 81865 81848 70021 70064 .70107 70151 70194 1.42815 1.42726 1.42638 .42550 .42462 60 59 58 57 56 5 6 7 8 9 56040 56064 56088 56112 56136 82822 82806 82790 82773 82757 67663 67705 67748 67790 67832 1-47792 1-47699 1.47607 1.47514 1.47422 57477 57501 57524 57548 .57572 81832 81815 81798 .81782 81765 70238 70281 70325 70368 -70412 .42374 .42286 42198 .42110 1-42022 53 54 53 52 51 10 11 12 13 14 56160 56184 56208 56232 56256 82741 82724 82708 .82692 .82675 67875 67917 67960 68002 68045 1.47330 1.47238 1.47146 1-47053 1-46962 57596 .57619 57643 .57667 .57691 81748 81731 81714 81698 .81681 70455 70499 - 70542 -70586 -70629 1-41934 1.41847 1.41759 1.41672 1.41584 5O 49 48 47 46 15 16 17 18 19 56280 56305 56329 56353 56377 .82659 .82643 .82626 .82610 .82593 68088 68130 .68173 .68215 .68258 1.46870 1.46778 1.46686 1.46595 1.46503 -57715 .57738 -57762 57786 .57810 -81664 .81647 .81631 .81614 81597 -70673 -70717 -70760 -70804 - 70848 1.41497 1.41409 1-41322 1.41235 1.41148 45 44 43 42 41 20 21 22 23 24 56401 .56425 . 56449 . 56473 .56497 .82577 .82561 .82544 .82528 .82511 .68301 .68343 .68386 .68429 .68471 1.46411 1-46320 1-46229 1.4*6137 1.46046 .57833 57857 57881 57904 .57928 -81580 .81563 .81546 .81530 -81513 70891 70935 -70979 71023 71066 1.41061 1.40974 1.40887 1.40800 1-40714 4O 39 38 37 36 25 26 27 28 29 .56521 56545 .56569 .56593 .56617 82495 .82478 .82462 .82446 .82429 .68514 .68557 .68600 .68642 .68685 1.45955 1.45864 1.45773 1.45682 1.45592 .57952 -57976 .57999 -58023 .58047 -81496 81479 .81462 .81445 .81428 .71110 71154 -71198 .71242 -71285 40627 .40540 .40454 .40367 .40281 35 34 33 32 31 30 31 32 33 34 .56641 .56665 .56689 .56713 56736 .82413 .82396 .82380 .82363 .82347 .68728 .68771 .68814 .68857 .68900 1.45501 1.45410 1.45320 1.45229 1.45139 .58070 .58094 .58118 .58141 .58165 .81412 .81395 .81378 .81361 .81344 71329 71373 .71417 .71461 .71505 .40195 .40109 1-40022 1.39936 1-39850 30 29 28 27 26 35 36 37 38 39 .56760 . 5678< c .56803 .56832 -568'i6_ .82330 .82314 .82297 .82281 .82264 .68942 68985 .69028 .69071 -69114 1-45049 1-44958 1.44868 1-44778 1-44688 .58189 .58212 .58236 .58260 -58283 81327 .81310 .81293 .81276 .81259 .71549 .71593 .71637 .71681 71725 1-39764 1.39679 1.39503 1-39507 1-39421 25 24 23 22 -21 40 41 42 43 44 .56880 .56904 .56928 56952 .56976 .82248 82231 .82214 .82198 .82181 .69157 .69200 .69243 .69286 -69329 1-44598 1.44508 1.44418 1-44329 1.44239 -58307 -58330 -58354 -58378 .58401 81242 81225 81208 81191 81174 -71769 -71813 .71857 .71901 .71946 1-39336 1-39250 1-39165 1.39079 1-38994 20 19 18 17 16 45 46 47 48 49 .57000 .57024 57047 .57071 .57095 .82165 .82148 82132 .82115 .82098 69372 .69416 69459 .69502 .69545 1.44149 1.44060 1-43970 1.43881 1-43792 .58425 . 58449 .58472 .58496 .58519 .81157 81140 .81123 81106 .81089 -71990 -72034 .72078 .72122 72167 1-38909 1-38824 1.38738 1-38653 1.38568 15 14 13 12 Ui 50 51 52 53 54 .57119 .57143 .57167 57191 .57215 82082 82065 82048 .82032 .82015 69588 69631 69675 69718 69761 1.43703 1-43614 1.43525 1-43436 1.43347 .58543 .58567 .58590 .58614 58637 81072 81055 .81038 -81021 .81004 72211 72255 .72299 . 72344 72388 1.38484 1.38399 1-38314 1-38229 1.38145 10 9 8 7 _6 55 56 57 58 59 57238 57262 57286 57310 57334 .81999 81982 81965 81949 81932 69804 69847 69891 69934 69977 1-43258 1.43169 1-43080 1.42992 1.42903 .58661 .58684 58708 .58731 .58755 -80987 .80970 .80953 .80936 .80919 .72432 72477 .72521 .72565 -72610 1.38060 1.37976 1.37891 1.37807 1-37722 9 3 2 60 .57358 81915 .70021 1-42815 .58779 -80902 -72654 1-37638 0. "^ Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ;s 2 23 fi 4 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS 36 37 Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. , 2 3 58779 58802 58826 .58849 .58873 80902 80835 80867 80350 80833 .72654 .72699 .72743 .72788 .72832 1-37638 1.37554 1.37470 1.37386 1-37302 60182 60205 60228 60251 . 60274 -79864 - 79846 -79829 .79811 -79793 -75355 . 75401 75447 -75492 -75538 1-32704 1-32624 1-32544 1-32464 1-32384 60 59 58 57 56 5 6 7 8 9 58896 58920 58943 58967 58990 80816 80799 80782 80765 80748 72877 .72921 72966 .73010 73055 1.37218 1.37134 1.37050 1.36967 1.36883 . 60298 .60321 .60344 -60367 .60390 .79776 .79758 .79741 .79723 .79706 -75584 -75629 -75675 .75721 .75767 1-32304 1-32224 1-32144 1-32064 1.31984 55 54 53 52 SI 10 11 12 13 14 59014 59037 59061 59084 59108 80730 80713 80696 80679 80662 73100 73144 73189 73234 73278 1.36800 1.36716 1.36633 1.36549 1.36466 . 60414 .60437 . 60460 .60483 -60506 .79688 .79671 .79653 .79635 .79618 .75812 .75858 -75904 75950 .75996 1.31904 1.31825 1-31745 1.31666 1-31586 50 49 48 47 46 15 16 17 18 19 59131 59154 59178 59201 59225 80644 80627 80610 80593 80576 73323 73368 73413 73457 73502 1.36383 1.36300 1.36217 1.36134 1.36053 60529 .60553 .60576 .60599 .60622 79600 79583 .79565 -79547 .79530 76042 .76088 .76134 76180 76226 1-31507 1.31427 1-31348 1-31269 1-31190 45 44 43 42 41 20 21 22 23 24 59248 59272 59295 59318 59342 80558 8054.1 80524 80507 80489 73547 73592 73637 73681 73726 1.35968 1.35885 1.35802 1.35719 1-35637 . 60645 .60668 60691 -60714 60738 79512 .79494 .79477 79459 79441 .76272 .76318 .76364 76410 .76456 1-31110 1.31031 1.30952 1.30873 1-30795 4O 39 38 37 36 25 26 27 28 29 59365 59389 59412 59436 59459 80472 80455 80438 80420 80403 73771 73816 73861 73906 73951 1.35554 1-35472 1.35389 1.35307 1.35224 .60761 6G784 60807 60830 60853 .79424 79406 79388 79371 79353 .76502 .76548 .76594 -76640 -76686 1-30716 1.30637 1.30558 1 30480 1-30401 35 34 33 32 31 30 81 32 33 34 59482 59506 59529 59552 59576 80386 80368 80351 80334 80316 73996 .74041 74086 74131 74176 1.35142 1.35060 1.34978 1.34896 1.34814 60876 60899 60922 60945 .60968 .79335 79318 .79300 .79282 .79264 .76733 .76779 -76825 -76871 .76918 1-30323 1.30244 1-30166 1-30087 1-30009 30 29 28 27 26 35 36 37 38 39_ 59599 59622 59646 59669 . 59693^ 80299 80282 80264 80247 80230 74221 74267 74312 74357 .74402 1.34732 1.34650 1-34568 1.34487 1-34405 .60991 .61015 61038 61061 61084 79247 .79229 -79211 .79193 .79176 .76964 .77010 77057 -77103 77149 1-29931 1-29853 1.29775 1-29696 1-29318 25 24 23 22 21 40 41 42 43 44 .59716 .59739 .59763 .59786 .59809 80212 80195 .80178 .80160 .80143 74447 .74492 .74538 .74583 .74628 1.34323 1-34242 1.34160 1.34079 1.33998 61107 .61130 61153 81176 61199 -79158 -79140 .79122 .79105 .79087 77196 -77242 77289 77335 .77382 1-29541 1-29463 1-29385 1.29307 1-29229 20 19 18 17 16 45 46 47 48 40 59832 .59856 .59879 .59902 .59926 .80125 80108 .80091 .80073 .80056 .74674 .74719 .74764 .74810 74855 1.33916 1.33835 1.33754 1.33673 1.33592 61222 61245 61268 .61291 61314 .79069 .79051 .79033 .79016 .78998 - 77428 -77475 -77521 -77568 -77615 29152 29074 28997 .28919 .28842 15 14 13 12 11 50 51 52 53 54 .59949 .59972 59995 60019 .60042 .80038 .80021 .80003 .79986 .79968 .74900 .74946 .74991 .75037 .75082 1.33511 1.33430 1.33349 1.33268 1.33187 61337 61360 161383 j. 61406 .61429 .78980 .78962 .78944 .78926 .78908 -77661 .77708 .77754 .77801 .77848 .28764 28687 .28610 .28533 .28456 10 9 8 7 6 55 56 57 58 59 60065 .60089 60112 60135 .60158 .79951 .79934 .79916 .79899 .79881 .75128 75173 .75219 .75264 .75310 1.33107 1.33026 1-32946 1.32865 1.32785 61451 .61474 .61497 61520 .61543 -78891 78873 .78855 -78837 78819 .77895 .77941 -77988 -78035 -78082 -28379 -28302 -28225 -28148 1.28071 5 4 3 2 60 .60182 .79864 .75355 1.32704 .61566 78801 -78129 1-27994 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. '- I a* 2'< VL i -NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 38 39 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. / 1 2 3 4 .61566 61589 61612 61635 61658 78801 78783 .78765 78747 78729 78129 .78175 78222 -78269 78316 1-27994 1.27917 1.27841 1.27764 1-27688 62932 .62955 .62977 63000 63022 77715 .77696 .77678 .77660 . 77641 .80978 81027 .81075 81123 81171 1-23490 1-23416 1-23343 1.23270 1.23196 6O 59 58 57 56 5 6 7 8 9 61681 .61704 61726 .61749 61772 .78711 .78694 78676 78658 . 78640 .78363 78410 . 78457 78504 78551 1.27611 1.27535 1.27458 1-27382 1.27306 . 63045 -63068 63090 63113 .63135 77623 -77605 77586 -77568 -77550 81220 .81268 .81316 .81364 81413 1.23123 1.23050 1-22977 1.22904 1-22831 55 54 53 52 51 10 11 12 13 14 61795 .61818 .61841 61864 .61887 .78622 78604 .78586 78568 -78550 .78598 78645 78692 .78739 78786 1.27230 1.27153 1.27077 1.27001 1.26925 .63158 63180 .63203 63225 63248 .77531 .77513 . 77494 77476 77458 81461 81510 .81558 81606 .81655 1.22758 1.22685 1-22612 1-22539 1-22467 50 49 48 47 46 15 16 17 18 19 61909 61932 .61955 61978 62001 .78532 78514 78496 78478 -78460 78834 .78881 .78928 -78975 -79022 1.26849 1.26774 1.26698 1.26622 1.26546 63271 63293 63316 -63338 -63361 77439 . 77421 . 77402 .77384 77366 .81703 .81752 81800 81849 -81898 1-22394 1.22321 1.22249 1-22176 1.22104 45 44 43 42 41 20 21 22 23 24 62024 .62046 62069 .62092 62115 . 78442 78424 78405 78387 .78369 .79070 .79117 79164 .79212 79259 1-26471 1-26395 1.26319 1-26244 1.26169 -63383 -63406 -63428 -63451 -63473 77347 .77329 .77310 .77292 77273 -81946 -81995 -82044 82092 .82141 1-22031 1-21959 1-21886 1-21814 1.21742 40 39 38 37 36 25 26 27 28 29 62138 62160 62183 62206 62229 78351 .78333 78315 .78297 78279 79306 .79354 . 79401 . 79449 .79496 1-26093 1-26018 1-25943 1-25867 1-25792 -63496 -63518 -63540 -63563 .63585 77255 77236 -77218 .77199 77181 82190 82238 82287 .82336 .82385 1-21670 1.21598 1.21526 1.21454 1-21382 35 34 33 32 31 30 31 32 33 34 62251 62274 .62297 .62320 .62342 78261 .78243 78225 78206 .78188 79544 .79591 .79639 79686 .79734 1.25717 1-25642 1.25567 1-25492 1.25417 63608 .63630 .63653 63675 -63698 -77162 . 77144 -77125 77107 77088 . 82434 82483 .82531 -82580 -82629 1.21310 1-21238 1-21166 1-21094 1-21023 30 29 28 27 26 35 36 37 38 39 62365 .62388 .62411 .62433 62456 78170 .78152 78134 .78116 .78098 .79781 .79829 .79877 -79924 79972 1.25343 1.25268 1.25193 1.25118 1.25044 .63720 .63742 -63765 -63787 -63810 77070 77051 77033 .77014 76996 -82678 82727 82776 82825 .82874 1-20951 1-20879 1-20808 1-20736 1-20665 25 24 23 22 21 40 41 42 43 44 62479 .62502 -62524 62547 .62570 78079 .78061 .78043 78025 -78007 .80020 80067 .80115 80163 .80211 1.24969 1.24895 1-24820 1.24746 1.24672 63832 .63854 63877 63899 .63922 76977 76959 . 76940 76921 76903 .82923 .82972 83022 .83071 .83120 1-20593 -20522 20451 -20379 20308 2O 19 18 17 16 45 46 47 48 49 .62592 62615 62638 62660 62683 .77988 .77970 .77952 .77934 .77916 80258 80306 80354 .80402 80450 1.24597 1.24523 1.24449 1.24375 1.24301 63944 63966 -63989 64011 .64033 76884 76866 76847 76828 76810 .83169 .83218 83268 83317 -83366 .20237 20166 20095 1-20024 1-19953 15 14 13 12 11 50 51 52 53 54 62706 62728 62751 62774 .62796 77897 .77879 .77861 . 77843 .77824 80498 . 80548 -80594 80642 .80690 1-24227 1.24153 1.24079 1.24005 1.23931 64056 64078 . 64100 .64123 64145 -76791 -76772 -76754 .76735 76717 -83415 -83465 -83514 -83564 -83613 1-19882 1-19811 1-19740 1-19669 1-19599 10 9 8 7 6 55 56 57 58 59 62819 62842 62864 62887 .62909 77806 -77788 .77769 77751 .77733 80738 .80786 .80834 80882 80930 1.23858 1-23784 1.23710 1-23637 1-23563 .64167 64190 .64212 .64234 64256 -76698 76679 76661 76642 76623 -83662 -83712 -83761 -83811 -83860 1-19528 1.19457 1.19387 1.19316 1-19246 5 4 3 2 1 60 62932 77715 80978 1-23490 -64279 -76604 -83910 1.19175 / Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 51 225 50 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 40 41 / Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t 2 3 4 64279 64301 64323 64346 64368 76604 76586 76567 76548 76530 83910 83960 84009 84059 84108 1.19175 1-19105 1-19035 1-18964 1-18894 65606 65628 65650 65672 65694 75471 75452 75433 75414 75395 86929 86980 87031 87082 87133 1.15037 1.14969 1-14902 1-14834 1-14767 60 69 58 57 56 5 6 7 8 9 64390 64412 64435 64457 64479 76511 76492 76473 76455 76436 84158 84208 84258 84307 84357 1.18824 1-18754 1.18684 1-18614 1.18544 65716 65738 65759 65781 65803 75375 75356 75337 75318 75299 87184 87236 87287 87338 87389 1-14699 1-14632 1-14565 1.14498 1.14430 55 54 53 52 51 10 11 12 13 14 64501 64524 64546 64568 64590 76417 76398 76380 76361 76342 84407 84457 84507 84556 84606 1.18474 1 . 18404 1.18334 1-18264 1.18194 65825 65847 .65869 .65891 65913 .75280 75261 . 75241 75222 .75203 87441 87492 87543 87595 87646 1-14363 1-14296 1-14229 1.14162 1.14095 50 49 48 47 46 15 16 17 18 19 , 64612 64635 64657 64679 64701 76323 76304 76286 .76267 76248 84656 84706 84756 84806 84856 1.18125 1.18055 1-17986 1-17916 1-17846 .65935 .65956 65978 - 66000 -66022 75184 75165 75146 75126 75107 .87698 87749 87801 87852 .87904 1-14028 1-13961 1.13894 1-13828 1-13761 45 44 43 42 41 2O 21 22 23 24 64723 . 64746 .64768 64790 64812 .76229 76210 76192 76173 76154 84906 ii4956 85006 85057 85107 1.17777 1-17708 1-17638 1-17569 1.17500 . 66044 -66066 -66088 -66109 66131 .75088 75069 -75050 75030 75011 .87955 .88007 88059 88110 88162 1-13694 1-13627 1-13561 1-13494 1-13428 40 39 38 37 36 25 26 27 28 29 .64834 .64856 64878 .64901 .64923 -76135 76116 .76097 .76078 .76059 85157 85207 85257 85308 85358 1-17430 1.17361 1-17292 1.17223 1-17154 66053 66175 66197 66218 . 66240 .74992 74973 74953 . 74934 74915 88204 88265 88317 88369 88421 1.13361 1-13295 1-13228 1-13162 1-13096 35 34 33 32 31 30 31 32 33 34 . 64945 64967 64989 .65011 .65033 76041 .76022 .76003 75984 75965 85408 85458 85509 85559 85609 1-17085 1-17016 1-16947 1.16878 1.16809 66262 .66284 66306 66327 66349 74896 -74876 -74857 74838 74818 88473 88524 88576 -88628 -88680 1.13029 1.12963 1.12897 1-12831 1-12765 30 29 28 27 26 35 36 37 38 39 .65055 65077 .65100 65122 65144 .75946 75927 75908 75889 75870 85660 85710 85761 85811 -85862 1-16741 1.16672 1-16603 1-16535 1-16466 66371 66393 66414 66436 -66458 74799 74780 . 74760 . 74741 -74722 88732 -88784 -88836 -88888 -88940 1-12699 1-12633 1-12567 1-12501 1-12435 25 24 23 22 21 40 41 42 43 44 65166 65188 65210 65232 65254 75851 75832 -75813 75794 75775 -85912 -85963 86014 86064 -86115 1.16398 1.16329 1-16261 1-16192 1.16124 . 66480 66501 .66523 66545 66566 - 74703 -74683 . 74664 74644 -74625 .88992 89045 -89097 89149 .89201 1.12369 1.12303 1.12238 1-12172 1.12106 20 19 18 17 16 45 46 47 48 49 65276 65298 65320 65342 65364 .75756 75738 75719 75700 75680 86166 86216 .86267 86318 -86368 1.16056 1-15987 1-15919 1-15851 1-15783 66588 66610 66632 66653 66675 . 74606 74586 74567 74548 74528 89253 .89306 89358 -89410 .89463 1.12041 1-11975 1.11909 1-11844 1-11778 15 14 13 12 11 50 51 52 53 54 .65386 .65408 .65430 65452 .65474 75661 .75642 75623 75604 75585 86419 86470 86521 86572 86623 1-15715 1.15647 1-15579 1-15511 1-15443 66697 66718 66740 66762 66783 . 74509 . 74489 . 74470 . 74451 .74431 89515 89567 89620 89672 89725 1-11713 1.11648 1.11582 1.11517 1-11452 10 9 8 7 6 55 56 57 58 59 65496 .65518 65540 .65562 65584 75566 75547 75528 .75509 .75490 86674 86725 .86776 .86827 .86878 1.15375 1.15308 1-15240 1-15172 1-15104 -66805 66827 66848 66870 66891 .74412 74392 74373 74353 . 74334 .89777 .89830 89883 89935 89988 1-11387 1-11321 1-11256 1-11191 1.11126 5 3 2 60 .65606 .75471 -86929 1.15037 -66913 -74314 -90040 1-11061 ' Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. ' 49 226 48 -NATURAL SINES, COSINES, TANGENTS, AND COTANuENTS. 43 43 t Sin. Cos. Tan. Cot. Sin. Cos. Tan. Cot. t o 2 8 .66913 66935 .66956 .66978 .66999 74314 74295 74276 74256 74237 .90040 90093 90146 90199 .90251 1-11061 1 10996 1-10931 1.10867 1.10802 68200 68221 68242 68264 68285 73135 -73116 73096 -73076 -73056 93252 -93306 -93360 -93415 -93469 1-07237 1-07174 1-07112 1-07049 1-06987 60 59 58 57 56 5 6 7 8 9 .67021 67043 .67064 67086 .67107 74217 74198 74178 74159 74139 90304 90357 90410 90463 90516 1-10737 1.10672 1-10607 1-10543 1 10478 68306 68327 68349 68370 68391 73036 -73016 72996 72976 -72.957 -93524 -93578 93633 -93688 -93742 1-06925 1-06862 1-06800 1-06738 1-06676 55 54 53 52 51 10 11 12 13 14 .67129 .67151 67172 67194 67215 74120 74100 74080 74061 74041 90569 90621 90674 90727 90781 1-10414 1-10349 1-10285 1-10220 1.10156 68412 68434 68455 68476 68497 -72937 72917 72897 72877 72857 -93797 -93852 -93906 -93961 -94016 1-06613 1-06551 1-06489 1-06427 1 06365 5O 49 48 47 46 15 16 17 18 19 67237 67258 67280 67301 87333 74022 74002 73983 73963 73944 90834 90887 90940 90993 91046 1.10091 1.10027 1.09963 1.09899 1-09834 68518 68539 68561 68582 68603 72837 72817 72797 72777 72757 .94071 -94125 -94180 -94235 -94290 -06303 -06241 -06179 -06117 .06056 45 44 43 42 .41 20 21 22 23 24 67344 67366 67387 67409 67430 73924 73904 73885 73865 73846 91099 91153 91206 91259 91313 1.09770 1-09706 1.09642 1-09578 1-09514 68624 68645 68666 68688 68709 72737 72717 72697 72677 72657 -94345 -94400 94455 -94510 -94565 -05994 -05932 1-05870 1-05809 1-05747 40 39 38 37 36 25 26 27 28 29 67452 67473 67495 67516 67538 73826 73806 73787 73767 73747 91368 91419 91473 91526 91580 1-09450 1-09386 1-09322 1-09258 1-09195 68730 68751 68772 68793 68814 72637 72617 72597 72577 72557 .94620 .94676 .94731 94786 . 94841 1-05685 1-05624 1-05562 1-05501 1-05439 35 34 33 32 31 30 31 32 33 34 67559 67580 67602 67623 67645 73728 73703 73688 73669 73849 91633 91687 91740 91794 91847 1-09131 1-09067 1-09003 1-08940 1-08876 68835 68857 68378 68899 68920 72537 72517 72497 72477 .72457 94896 94952 95007 95062 .95118 1-05378 1-05317 1-05255 1-05194 1-05133 30 29 28 le 35 36 37 38 39 67666 67688 -67709 67730 67752 73629 73810 73590 73570 73551 91901 91955 92008 92062 92116 1-08813 1-08749 1.08686 1-08622 1-08559 68941 68962 68983 69004 69025 .72437 -72417 -72397 -72377 -72357 95173 95229 95284 95340 .95395 1.05072 1.05010 1 . 04949 1-04888 1.04827 25 24 23 22 21 40 41 42 43 44 .67773 67795 67816 67837 .67859 73531 73511 73491 73472 73452 92170 92224 92277 92331 92385 1-08496 1-08432 1.08369 1-08306 1-08243 69046 69067 69088 69109 .69130 -72337 -72317 72297 .72277 .72257 .95451 .95506 .95562 .95618 .95673 1-04766 1-04705 1.04644 1-04583 1-04522 20 19 18 17 _16 45 46 47 48 49 -67880 67901 .67923 67944 67965 73432 73413 73393 73373 73353 92439 92493 92547 92601 .92655 1.08179 1-08118 1.08053 1.07990 1.07927 -69151 -69172 -69193 -69214 .69235 .72236 .72216 72196 72176 .72156 .95729 .95785 -95841 .95897 .95952 1-04461 1-04401 1-04340 1 . 04279 1-04218 15 14 13 12 11 50 51 52 53 54_ .67987 68008 68029 68051 68072 73333 73314 73294 73274 73254 92709 92763 .92817 92872 .92926 1-07864 1-07801 1-07738 1-07676 1-07613 -69256 69277 -69298 .69319 - 69340 .72136 .72116 .72095 .72075 .72055 .96008 .96064 .96120 .96176 .98232 1.04158 1.04097 1-04038 1.03976 1-03915 10 9 ? _6 55 56 57 58 59 68093 68115 .68138 68157 .68179 73234 .73215 .73195 73175 73155 .92980 93034 .93088 93143 93197 1.07550 1.07487 1-07425 1-07362 1-07299 -69361 .69382 .69403 69424 . 69445 72035 .72015 .71995 .71974 .71954 .96288 -96344 -96400 96457 .96513 1-03855 1.03794 1.03734 1-03674 1.03613 5 3 2 60 .68200 .73135 93252 1-07237 69466 71934 -96569 1.03553 7 Cos. Sin. Cot. Tan. Cos. Sin. Cot. Tan. 47 227 46 TABLE 10. NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS 44 44 / 2 3 ,4 5 6 7 8 9 Sin. Cos. Tan. Cot. ' ; Sin. Cos. Tan. Cot. / .69466 69487 .69508 69529 69549 71934 .71914 71894 71873 71853 .96569 .96625 .96681 .96738 .96794 1.03553 1.03493 1.03433 1-03372 1-03312 60 59 58 57 56 3O 31 32 33 34 .70091 .70112 .70132 70153 .70174 71325 -71305 -71284 -71264 -71243 .98270 -98327 .98384 -98441 .98499 1-01761 1-01702 1-01642 1.01583 1-01524 30 29 28 27 26 25 24 23 22 21 20 19 18 17 -IS 15 14 13 12 11 .69570 69591 69612 69633 69654 71833 .71813 .71792 .71772 .71752 96850 96907 .96963 .97020 .97076 1.03252 1.03192 1-03132 1.03072 1.03012 55 54 53 52 51 35 36 37 38 39 70195 70215 70236 70257 .70277 71223 -71203 -71182 -71162 71141 .98556 .98613 98671 .98728 .98786 1-01465 1.01406 1.01347 1.01288 1-01229 10 11 12 13 ii_ 15 16 17 18 i- 20 21 22 23 24_ 25 26 27 28 29_ &L 69675 69696 .69717 69737 .69758 69779 .69800 .69821 .69842 .69862 71732 .71711 .71691 71671 .71650 .97133 .97189 .97246 97302 97359 1-02952 1.02892 1.02832 1.02772 1.02713 50 49 48 47 46 40 41 42 43 44 -70298 -70319 . 70339 .70360 .70381 -71121 .71100 . 71080 .71059 .71039 98843 .98901 98958 .99016 -99073 1-01170 1.01112 1.01053 1-00994 1.00935 .71630 .71610 .71590 .71569 71549 97416 .97472 97529 .97586 .97643 1-02653 1.02593 1.02533 1.02474 1-02414 45 44 43 42 41 45 46 47 48 49 70401 - 70422 70443 -70463 - 70484 . 71019 70998 -70978 -70957 70937 -99131 -99189 -99247 .99304 -99362 1.00876 1.00818 1-00759 1.00701 1.00642 .69883 69904 .69925 .69946 .69966 .71529 .71508 71488 .71468 71447 97700 .97756 97813 .97870 .97927 1-02355 1.02295 1.02236 1.02176 1.02117 1-02057 1.01998 1-01939 1-01879 1.01820 40 39 38 37 36 50 51 52 53 54 70505 70525 70546 .70567 70587 70916 70896 70875 70855 70834 .99420 99478 -99536 .99594 .99652 1.00583 1.00525 1-00467 1-00408 1.00350 10 9 8 7 6 69987 70008 . 70029 . 70049 . 70070 .71427 . 71407 71386 71366 .71345 71325 97984 98041 .98098 98155 .98213 35 34 33 32 31 55 56 57 58 59 - 70608 -70628 - 70649 -70670 . 70690 . 70813 70793 - 70772 - 70752 -70731 .99710 .99768 .99826 .99884 .99942 1-00291 1-00233 1-00175 1-00116 1-00058 5 4 3 2 1 -P 70091 .98270 1-01761 30 6O .70711 70711 1 00000 1-00000 Cos. Sin. Cot. Tan. ' ' Cos. Sin. Cot. Tan. 45 228 45 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 1 2 3 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' 2 3 4 .00000 00000 00000 .00000 -00000 .00000 .00000 .00000 .00000 .00000 00015 .00016 .00016 .00017 00017 .00015 .00016 .00016 .00017 .00017 .00061 .00062 00063 00064 00065 .00061 .00062 .00063 .00064 .00065 00137 00139 00140 00142 00143 .00137 .00139 00140 00142 00143 1 2 3 5 6 7 8 9 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 . 00000 00018 .00018 .00019 .00020 .00020 .00018 .00018 .00019 .00020 .00020 .00066 .00067 .00068 .00069 .00070 .00066 00067 .00068 .00069 00070 00145 00146 00148 00150 00151 .00145 .00147 00148 .00150 00151 5 6 7 8 9 10 11 12 13 14 .00000 00001 .00001 .00001 .00001 .00000 .00001 .00001 .00001 .00001 00021 00021 00022 00023 00023 .00021 .00021 .00022 .00023 .00023 00071 00073 00074 00075 -00076 .00072 .00073 .00074 .00075 .00076 00153 00154 00156 .00158 .00159 .00153 .00155 .00156 .00158 .00159 1O 11 12 13 14 15 16 17 18 19 .00001 .00001 .00001 .00001 .00002 .00001 .00001 .00001 .00001 -00002 00024 00024 00025 00026 -00026 .00024 00024 .00025 .00026 .00026 00077 .00078 .00079 .00081 -00082 .00077 .00078 00079 .00081 .00082 .00161 00162 00164 00166 00168 .00161 .00163 .00164 00166 .00168 15 16 17 18 19 20 21 22 23 24 .00002 -00002 00002 -00002 00002 .00002 .00002 .00002 .00002 .00002 -00027 00028 .00028 .00029 -00030 00027 .00028 .00028 .00029 .00030 00083 -00084 00085 -00087 00088 .00083 00084 .00085 00087 .00088 00169 00171 00173 00174 00176 .00169 .00171 00173 00175 00176 20 21 22 23 24 25 26 27 28 29 -00003 00003 -00003 -00003 - 00004 .00003 .00003 .00003 .00003 00004 00031 .00031 00032 .00033 00034 .00031 .00031 .00032 00033 00034 -00089 00090 00091 00093 00094 00089 .00090 .00091 00093 00094 00178 00179 00181 00183 00185 00178 .00180 .00182 .00183 .00185 25 26 27 28 29 30 31 32 33 34 .00004 .00004 . 00004 - 00005 -00005 .00004 .00004 .00004 00005 .00005 -00034 00035 00036 00037 .00037 00034 .00035 00036 .00037 00037 00095 .00096 -00098 .00099 .00100 00095 -00097 00098 00099 00100 00187 00188 00190 00192 00194 .00187 00189 .00190 00192 .00194 3O 31 32 33 34 35 36 37 38 39 00005 .00005 .00006 .00006 00006 .00005 .00005 00006 00006 .00006 -00038 .00039 00040 -00041 -00041 00038 .00039 00040 00041 -00041 00102 00103 00104 00106 -00107 .00102 00103 00104 00106 .00107 00196 00197 00199 00201 00203 .00196 .00198 00200 00201 -00203 35 36 37 38 39 40 41 42 43 44 -00007 00007 -00007 00008 -00008 00007 00007 .00007 00008 00008 . 00042 00043 00044 00045 . 00046 00042 00043 00044 .00045 00046 -00108 00110 00111 00112 00114 00108 00110 .00111 00113 00114 00205 00207 00208 00210 00212 00205 00207 .00209 .00211 .00213 40 41 42 43 44 45 46 47 47 49 -00009 00009 00009 -00010 00010 00009 .00009 .00009 .00010 .00010 00047 00048 -00048 00049 -00050 .00047 .00048 00048 00049 .00050 00115 00117 00118 00119 -00121 .00115 00117 00118 00120 00121 00214 00216 .00218 00220 00222 00215 .'0216 .00218 .00220 .00222 45 46 47 48 49 50 51 52 53 54 -00011 00011 -00011 00012 00012 .00011 00011 .00011 00012 .00012 00051 00052 00053 00054 00055 .00051 .00052 .00053 00054 .00055 00122 00124 00125 00127 00128 00122 00124 00125 00127 00128 00224 00226 00228 -00230 00232 .00224 00226 .00228 .00230 00232 5O 51 52 53 54 55 56 57 58 59 -00013 -00013 -00014 -00014 .00015 .00013 00013 00014 .00014 .00015 00056 00057 .00058 00059 00060 .00056 00057 .00058 .00059 00060 00130 00131 00133 00134 -00136 .00130 00131 00133 .00134 .00136 00234 00236 -00238 00240 .00242 00234 .00236 .00238 .00240 -00242 55 56 57 58 59 60 -00015 00015 00061 00061 00137 .00137 00244 .00244 60 229 Inserted by permission from Webb's Railroad Construction. TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 4 5 6 7 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' o 1 2 8 4 00244 00246 .00248 .00250 00252 00244 .00246 .00248 .00250 .00252 .00381 00383 00386 -00388 .00391 .00382 .00385 .00387 .00390 00392 -00548 00551 .00554 00557 -00560 00551 00554 00557 00560 00563 .00745 00749 00752 .00756 .00760 00751 00755 00758 00762 00765 1 2 3 4 5 6 7 8 9 .00254 .00256 .00258 00260 .00262 .00254 .00257 00259 .00261 00263 .00393 00396 .00398 .00401 .00404 .00395 00397 00400 00403 00405 00563 .00566 00569 00572 .00576 00566 .00569 .00573 .00576 00579 00763 -00767 00770 -00774 .00778 00769 00773 00776 00780 -00784 6 7 8 9 1O 11 12 13 1.4 00264 -00266 .00269 .00271 00273 00265 .00267 .00269 .00271 00274 00406 .00409 00412 .00114 00417 00408 00411 .00413 00416 00419 00579 .00582 00585 00588 00591 .00582 .00585 00588 00592 00595 00781 00785 -00789 00792 00796 00787 -00791 00795 00799 00802 10 11 12 13 14 15 16 17 18 19 -00275 .00277 .00279 .00281 -00284 -00276 .00278 00280 00282 00284 .00420 .00422 .00425 .00428 -00430 00421 .00424 .00427 00429 -00432 00594 -00598 -00601 .00604 -00607 00598 00601 00604 .00608 00611 . 00800 . 00803 00807 00811 -00814 .00806 .00810 00813 00817 .00821 15 16 17 18 19 20 21 22 23 24 -00286 -00288 OC290 .00293 -00295 00287 00289 .00291 00293 00296 .00433 00436 00438 - 00441 00444 .00435 00438 '00440 .00443 .00446 -00610 00614 00617 00620 -00623 00614 00617 00621 -00624 00627 00818 00822 00825 00829 -00833 .00825 00828 00832 .00836 .00840 20 21 22 23 24 25 26 27 28 29 00297 00299 C0301 00304 -00306 .00298 .00300 .00302 .00305 .00307 00447 . 00449 00452 00455 00458 .00449 .00451 .00454 .00457 .00460 00626 00630 -00633 00636 -00640 .00630 .00634 .00637 00640 00644 00837 00840 .00844 00848 00852 .00844 00848 00851 .00855 .00859 25 26 27 28 29 30 31 32 33 34 00308 .00311 C0313 -00315 -00317 .00309 00312 .00314 00316 .00318 00460 00463 00466 00469 00472 004b3 .00465 .00468 .00471 00474 00643 -00646 00649 -00653 -00656 00647 .00650 00654 00657 00660 .00856 00859 -00863 00867 00871 -00863 .00867 -00871 -00875 .00878 30 31 32 33 34 35 36 37 38 39 00320 00322 00324 -00327 -00329 .00321 .00323 .00328 .00328 .00330 .00474 .00477 00480 -00483 00486 00477 .00480 -00482 .00485 -00488 00659 00663 -00666 00669 -00673 .00664 -00667 -00671 -00674 00677 00875 -OC878 -00882 -00886 -00890 -00882 -00886 00890 -00894 00898 35 36 37 38 39 40 41 42 43 44 00332 00334 -00336 -00339 00341 .00333 00335 00337 00340 .00342 00489 00492 00494 00497 00500 00491 .00494 -00497 .00500 .00503 -00676 00680 00683 -00686 -00690 -00681 .00684 .00688 -00691 00695 00894 00898 00902 00908 00909 00902 -00906 -00910 -00914 00918 40 41 42 43 44 45 46 47 48 49 00343 00346 00348 00351 00353 .00345 00347 00350 .00352 00354 00503 00506 00509 00512 00515 -00506 -00509 -00512 00515 .00518 -00693 00607 00700 00703 .00707 -00698 -00701 -00705 .00708 .00712 00913 00917 00921 00925 00929 -00922 -00926 .00930 -00934 00938 45 46 47 4& 49 50 5i 52 53 54 00356 .00358 00361 0036d -00365 00357 .00359 .00362 00364 .00367 00518 00521 00524 00527 .00530 -00521 -00524 -00527 .00530 .00533 007.10 00714 00717 00721 .00724 -00715 .00719 -00722 -00726 -00730 .00933 00937 00941 .00945 .00949 -00942 -00946 .00950 .00954 00958 50 51 52 53 54 55 56 57 58 59. 00368 .00370 00373 .00375 -00378 .00369 .00372 00374 00377 00379 .00533 .00536 00539 .00542 00545 -00536 -00539 00542 -00545 00548 00728 -00731 00735 00738 00742 -00733 .00737 -00740 -00744 00747 00953 00957 00961 .00965 00969 .00962 -00966 -00970 .00975 -00979 55 56 57 58 ,59 60 00381 .00382 00548 -00551 -00745 .00751 00973 -00983 60 230 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 8 9 10 11 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. / 2 3 4 .00973 00977 .00981 .00985 -00989 .00983 .00987 00991 00995 .00999 01231 .01236 01240 .01245 .01249 .01247 .01251 .01256 .01261 01265 -01519 .01524 .01529 .01534 .01540 01543 .01548 .01553 .01558 .01564 -01837 -01843 01843 .01854 .01860 01872 .01877 01883 .01889 01895 O 2 3 5 6 7 9 00994 00998 .01002 .01006 .01010 -01004 01008 .01012 .01016 .01020 01254 01259 01263 -01268 -01272 01270 .01275 .01279 -01284 .01289 01545 01550 .01555 01560 .01565 .01569 .01574 .01579 01585 .01590 01865 01871 01876 .01382 -01888 .01901 .01906 .01912 .01918 01924 5 6 7 8 9 10 11 12 13 14 .01014 01018 .01022 01027 .01031 01024 .01029 .01033 -01037 .01041 01277 .01282 01286 .01291 .01296 .01294 01298 01303 01308 01313 01570 .01575 01580 .01586 .01591 .01595 .01601 .01606 .01611 .01616 01893 01899 01904 01910 .01916 .01930 01936 .01941 01947 .01953 10 11 12 13 14 15 16 17 18 19 .01035 -01039 01043 -01047 .01052 .01046 .01050 -01054 01059 .01063 .01300 .01305 01310 01314 01319 01318 01322 01327 01332 01337 01596 -01601 .01606 01612 -01617 01622 .01627 .01633 .01638 .01643 01921 .01927 .01933 01939 01944 01959 01965 01971 .01977 01983 15 16 17 18 -19 20 21 22 23- 24 -01056 01060 01064 -01069 01073 .01067 .01071 .01076 .01080 .01084 01324 01329 -01333 01338 .01343 .01342 01346 01351^ .01356 .01361 .01622 01627 .01632 01638 01643 01649 01654 .01659 .01665 .01670 -01950 01956 01961 01967 01973 .01989 01995 02001 02007 02013 2O 21 22 23 24 25 26 27 28 29 .01077 .01081 .01086 -01090 .01094 .01089 .01093 .01097 .01102 .01106 .01348 01352 .01357 01362 .01367 .01366 .01371 .01376 .01381 .01386 01648 -01653 01659 01664 01669 .01676 .01681 .01687 .01692 .01698 -01979 01984 01990 -01996 02002 02019 02025 .02031 .02037 .02043 25 28 27 28 29 30 31 32 33 34 01098 .01103 -01107 .01111 .01116 .01111 .01115 .01119 .01124 .01128 01371 .01376 01381 .01386 .01391 .01391 01395 01400 .01405 .01410 .01675 01680 01685 01690 .01696 .01703 .01709 .01714 .01720 .01725 .02008 .02013 02019 .02025 -02031 .02049 .02055 .02061 02067 .02073 30 31 32 33 34 35 36 37 38 39 01120 .01124 .01129 01133 -01137 .01133 .01137 .01142 .01146 -01151 01396 -01400 .01405 01410 01415 .01415 .01420 01425 .01430 OUS'i 01701 01706 01712 -01717 -01723 .01731 -01736 .01742 .01747 .01753 02037 .02042 02048 02054 .02060 .02079 .02085 .02091 02097 02103 35 36 37 38 39 40 41 42 43 44 01142 .01148 01151 -01155 -01159 01155 .01160 01164 .01169 01173 -01420 .01425 01430 -01435 .01439 01440 .01445 01450 -01455 .01461 01728 01733 .01739 .01744 .01750 01758 .01764 .01769 .01775 .01781 .02066 .02072 02078 02084 -02090 02110 02116 02122 .02128 02134 40 41 42 43 44 45 46 47 48 49 .01164 -01168 01173 -01177 -01182 .01178 -01182 .01187 .01191 01196 01444 01449 01454 01459 .01464 .01466 01471 .01476 01481 01486 .01755 01760 01766 01771 .01777 01786 .01792 .01793 .01803 .01809 02095 02101 02107 .02113 -02119 02140 02146 02153 02159 02165 45 48 47 48 49 50 51 R2 53 54 01186 -01191 .01195 -01200 -01204 .01200 01205 .01209 .01214 .01219 01469 01474 01479 01484 .01489 01491 01496 01501 01506 .01512 01782 01788 .01793 .0179^ -01804 .01815 .01820 01826 01832 01837 .02125 02131 .02137 .02143 02149 .02171 .02178 02184 02190 02196 50 51 52 53 54 55 56 57 58 59, .01209 .01213 -01218 -01222 -01227 .01223 01228 .01233 .01237 .01242 01494 01499 01504 01509 -01514 .01517 .01522 .01527 .01532 .01537 .01810 01815 01821 .01826 .01832 01843 01849 .01854 01860 01866 .02155 02161 .02167 02173 .02179 02203 02209 02215 -02221 .02228 55 58 57 58 59 6O 01231 .01247 .01519 .01543 .01837 .01872 02185 .02234 60 231 TABLE 11 NATURAL VERSED SINES AND EXTERNAL SECANTS. 12 13 14 15 t Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' o 1 2 3 4 02185 02191 02197 .02203 02210 .02234 .02240 .02247 .02253 .02259 .02563 .02570 .02576 02583 .02589 02630 .02637 02644 .02651 02658 02970 02977 02985 02992 02999 .03081 03069 .03076 03084 03091 03407 03415 -03422 03430 03438 03528 03536 03544 03552 03560 O 2 3 4 5 6 7 8 9 .02216 .02222 .02228 02234 02240 .02266 .02272 .02279 .02285 -02291 02596 02602 .02609 -02616 .02622 02665 .02672 .02679 .02686 .02693 03006 03013 03020 03027 03034 03099 .03106 03114 .03121 .03129 -03445 03453 03460 03468 .03476 .03568 .03576 -03584 -03592 03601 5 6 7 8 9 10 11 12 13 14 .02246 02252 02258 .02265 02271 .02298 -02304 .02311 .02317 02323 02629 -02635 02642 .02649 .02655 .02700 .02707 02714 .02721 .02728 .03041 03048 03055 03063 03070 03137 .03144 .03152 .03159 03167 03483 03491 -03498 -03506 -03514 -03609 -03617 -03625 -03633 -03642 10 11 12 13 14 15 16 17 18 19 02277 .02283 02289 .02295 02302 02330 .02336 .02343 .02349 .02356 .02662 02669 02675 02682 .02689 .02735 .02742 .02749 .02756 02763 .03077 .03084 .03091 .03098 03106 03175 .03182 03190 -03198 -03205 03521 03529 03537 03544 03552 -03650 -03658 03666 .03674 03683 15 16 17 18 19 20 21 22 23 24 02308 02314 02320 02327 . 02333 .02362 02369 02375 .02382 02388 02696 02702 02709 .02716 02722 .02770 .02777 .02784 .02791 .02799 03113 03120 03127 031S4 03142 C3213 03221 03228 .03236 .03244 .03560 03567 03575 .03583 03590 -03691 -03699 .03708 03716 03724 20 21 22 23 24 25 26 27 28 29 .02339 02345 02352 -02358 02364 .02395 02402 02408 02415 .02421 02729 02736 02743 .02749 .02756 .02806 .02813 .02820 .02827 .02834 .03149 .03156 03163 -03171 03178 .03251 .03259 .03267 .03275 03282 .03598 03606 .03614 03621 03629 03732 03741 03749 03758 .03766 25 26 27 28 29 30 31 32 33 34 02370 02377 -02383 02389 -02396 02428 .02435 .02441 .02448 02454 .02763 02770 02777 02783 02790 .02842 02C49 .02856 .02863 .02870 -03185 03193 03200 03207 03214 03290 03298 03306 03313 03321 03637 03645 03653 03660 03668 03774 03783 03791 03799 03808 30 31 32 33 34 3o 36 37 38 39 02402 -02408 02415 02421 -02427 .02461 02468 02474 .02481 .02488 02797 02804 02811 02818 -02824 .02878 .02885 .02892 .02899 02907 03222 03229 03236 03244 03251 .03329 -03337 .03345 03353 03360 03676 03684 .03692 03699 03707 03816 03825 -03833 -03842 03850 35 36 37 38 39 40 41 42 43 44 -02434 -02440 02447 02453 02459 02494 .02501 02508 02515 02521 02831 02838 02845 02852 02859 .02914 .02921 .02928 .02936 02943 03258 03266 03273 03281 03288 -03368 03376 03384 03392 .03400 03715 -03723 03731 03739 03747 03858 .03867 -03875 .03884 .03892 40 41 42 43 44 45 46 47 48 49 02466 02472 02479 02485 02492 02528 02535 02542 02548 02555 02866 02873 02880 02887 02894 .02950 .02958 .02965 .02972 .02980 03295 03303 03310 03318 03325 03408 03416 .03424 .03432 03439 03754 03762 03770 03778 03786 .03901 .03909 .03918 .03927 .03935 45 46 47 48 49 50 51 52 53 54 02498 -02504 02511 02517 -02524 02562 02569 02576 02582 02589 02900 02907 02914 02921 .02928 .02987 02994 03002 03009 .03017 03333 03340 -03347 .03355 03362 03447 03455 .03463 03471 03479 03794 .03802 .03810 03818 03826 .03944 -03952 .03961 .03969 03978 50 51 52 53 54 55 56 57 58 59 -02530 02537 02543 02550 -02556 02596 02603 02610 02617 -02624 09935 -02942 02949 02956 02963 .03024 03032 .03039 03046 03054 .03370 .03377 03385 03392 .03400 03487 -03495 .03503 03512 03520 .03834 .03842 03850 03858 03866 .03987 -03995 04004 .04013 .04021 55 56 57 58 59 60 02563 02630 02970 .03061 03407 .03528 03874 .04030 60 232 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 16 17 18 19 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec I o 1 2 3 03874 03882 03890 03898 03906 .04030 .04039 .04047 .04056 .04065 04370 04378 04387 04395 04404 04569 04578 -04588 .04597 -04606 04894 04903 04912 04921 04930 .05146 -05156 .05166 05176 .05186 05448 05458 05467 05477 05486 .05762 -05773 05783 05794 .05805 1 2 3 4 5 6 7 8 9 03914 J3922 03930 03938 03946 04073 .04082 .04091 .04100 -04108 04412 04421 04429 04438 04446 .04616 04625 04635 .04644 .04653 04939 04948 04957 04967 04976 .05196 05206 .05216 .05226 05'236 05496 05505 05515 05524 05534 05815 05826 -05836 05847 .05858 5 6 7 8 9 10 11 12 13 14 03954 03963 03971 03979 03987 .04117 04126 .04135 04144 .04152 04455 04464 04472 04481 04489 .04663 .04672 04682 .04691 .04700 04985 04904 05003 05012 05021 .05246 .05256 .05266 -05276 .05286 .05543 05553 05562 05572 .05582 05869 -05879 05890 05901 .05911 10 11 12 13 14 15 16 17 18 19 03995 04003 04011 04019 04028 .04161 04170 04179 04188 -04197 04498 04507 04515 04524 04533 .04710 .04719 .04729 .04738 -04748 05030 05039 05048 05057 05067 .05297 .05307 .05317 .05327 .05337 05591 .05601 05610 05620 .05630 .05922 .05933 .05944 .05955 .05965 15 16 17 18 19 20 21 22 23 24 04036 04044 04052 04060 04069 04206 .04214 .04223 .04232 .04241 04541 04550 0455S 04567 04576 .04757 .04767 .04776 .04786 .04795 05076 05085 .05094 .05103 .05112 -05347 .05357 .05367 .05378 -05388 .05639 .05649 .05688 .05668 .05678 05976 .05987 .05998 .06009 .06020 20 21 22 23 24 25 26 27 28 29 04077 04085 04093 04102 04110 04250 04259 .04268 .04277 04286 04585 04593 .04602 .04611 .04620 .04805 .04815 .04824 04834 04843 .05122 .05131 .05140 05149 .05158 05398 -05408 .05418 .05429 -05439 .05687 05697 05707 05716 .05726 .06030 .06041 .06052 .06063 .06074 25 26 27 28 29 30 31 32 33 34 04118 04126 04135 04143 .04151 04295 .04304 .04313 .04322 .04331 -04628 04637 -04646 04655 .04663 .04853 .04863 04872 .04882 .04891 .05168 .05177 .05186 .05195 .05205 .05449 .05460 .05470 .05480 .05490 .05736 .05746 .05755 .05765 .05775 .06085 .06096 .06107 -06118 .06129 30 31 32 33 34 35 36 37 38 39 04159 .04168 .04176 .04184 .04193 04340 04349 04358 04367 .04376 .04672 04681 .04690 -04699 -04707 .04901 04911 04920 04930 -04940 -05214 .05223 .05232 .05242 .05251 .05501 05511 .05521 05532 .05542 .05785 .05794 .05804 .05814 .05824 06140 .06151 06162 06173 .06184 35 - 36 37 38 -39 40 41 42 43 44 -04201 .04209 .04218 .04226 .04234 .04385 .04394 .04403 .04413 .04422 -04716 .04725 .04734 .04743 -04752 -04950 04959 04969 -04979 .04989 .05260 .05270 .05279 .05288 .05298 .05552 .05563 .05573 .05584 .05594 .05833 .05843 .05853 .05863 .05873 .06195 .06206 06217 -06228 .06239 40 41 42 43 44 45 46 47 48 49 .04243 .04251 04260 04268 04276 .04431 04440 . 04449 04458 .04468 -04760 04769 .04778 .04787 .04796 .04998 05008 .05018 .05028 .05038 .05307 .05316 .05326 .05335 .05344 .05604 .05615 .05625 .05636 .05646 .05882 .05892 -05902 .05912 .05922 .06250 .06261 -06272 .06283 .06295 45 46 47 48 49 50 51 52 53 54 04285 .04293 04302 .04310 -04319 04477 04486 .04495 .04504 .04514 04805 .04814 -04323 .04832 -04841 05047 .05057 .05067 05077 05087 05354 .05363 .05373 .05382 -05391 .05657 .05667 .05678 .05688 -05699 .05932 .05942 .05951 .05961 .05971 .06306 .06317 .06328 .06339 .06350 5O 51 52 53 54 55 56 57 58 59 .04327 .04336 04344 .04353 .043fil 04523 04532 .04541 .04551 -04560 -04850 04858 -04867 -04876 04885 05097 05107 .05116 .05126 .05136 .05401 .05410 .05420 .05429 .05439 .05709 .05720 .05730 .05741 .05751 .05981 -05991 -06001 .06011 .06021 .06362 .06373 .06384 .06395 -06407 55 56 57 58 59 60 04370 -04569 -04894 05146 .05448 .05762 .06031 .06418 60 2 33 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 30 21 22 23 t Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. f 2 3 4 -06031 .06041 06051 06061 .06071 06418 .06429 .06440 .06452 06463 06642 06652 06663 .06673 06684 .07115 .07126 .07138 .07150 .07162 .07282 .07293 .07303 .07314 .07325 .07853 07866 .07879 07892 07904 07950 07961 07972 07984 -07995 .08636 08649 08663 .08676 .08690 O 2 3 4 5 6 7 8 9 06081 06091 .06101 06111 06121 .06474 .06486 06497 .06508 .06520 06694 06705 06715 .06726 06736 .07174 .07186 07199 .07211 .07223 07336 07347 -07358 07369 -07380 .07917 07930 07943 07955 .07968 -C8006 -C8018 -08029 C8041 .08052 .08703 08717 08730 08744 .08757 5 6 7 8 9 10 11 12 13 14 06131 06141 06151 .06161 .06171 .06531 .06542 .06554 .06565 06577 .06747 06757 06768 06778 .06789 .07235 07247 .07259 .07271 .07283 07391 07402 07413 07424 .07485 .07981 .07994 .08006 .08019 08032 08064 08075 08086 08098 08109 08771 08784 .08798 .08811 .08825 10 11 12 IS 14 15 16 17 18 19 06181 06191 06201 06211 .06221 06588 .06600 .06611 .06622 .06634 06799 06810 06820 06831 .06841 07295 .07307 .07320 .07332 07344 .07446 .07457 07468 .07479 .07490 .08045 .08058 .08071 -08084 .08097 .08321 .08132 .08144 .08155 08167 .08839 .08852 .08866 .08880 .08893 15 13 17 18 19 2O 21 22 23 24 06231 06241 06252 06262 .06272 .06645 .06657 06668 .06680 .06691 06852 .06863 06873 06884 06894 -07356 .07368 .07380 .07393 .07405 .07501 .07512 .07523 .07534 .07545 08109 .08122 .08135 .08148 .08161 .08178 08190 08201 08213 08225 .08907 .08921 08934 .08948 .08962 20 21 22 23 24 25 26 27 28 29 -06282 .06292 06302 06312 06323 .06703 .06715 .06726 .06738 .06749 06905 .06916 .06926 .06937 .06948 .07417 .07429 .07442 .07454 .07486 .07556 .07568 .07579 .07590 07601 .08174 .08087 .08200 .08213 08226 08236 08248 08259 08271 08282 .08975 .08989 .09003 09017 .09030 25 26 27 28 29 30 31 32 33 34 08333 06343 06353 06363 06374 .06761 .06773 .06784 .06796 .06807 .06958 .06969 06980 .06990 .07001 .07479 .07491 .07503 .07516 .07528 .07612 07623 .07634 .07645 07657 .08239 .08252 08265 .08278 .08291 08294 08306 08317 .'08329 . 08340 .09044 .09058 .09072 .09086 .09099 30 31 32 .33 34 35 36 37 38 39 06384 06394 06404 06415 06425 .06819 .06831 06843 06854 06866 .07012 .07022 .07033 07044 07055 .07540 .07553 .07565 .07578 07590 07668 .07679 .07690 -07701 07713 08305 .08318 .08331 08344 .08357 08352 08364 -08375 08387 08399 09113 09127 .09141 .09155 .09169 35 36 37 38 39 40 41 42 43 44 06435 06445 06456 06466 06476 .06878 .06889 06901 .06913 .06925 07065 07076 07087 07098 07108 .07J02 .07615 .07627 07U40 .07652 .07724 .07735 07746 .07757 07769 .08370 .08383 08397 .08410 .08423 08410 08422 08434 08445 08457 09183 .09197 .09211 09224 .09238 40 41 42 48 44- 45 46 47 48 49 06486 06497 06507 06517 06528 .06936 -06948 .06960 .06972 06984 07119 07130 07141 07151 07162 .07665 .07677 .07690 .07702 .07715 07780 07791 07802 07814 07825 .08436 .08449 .08463 08476 .08489 08469 08481 .08492 .08504 .08516 .09252 .09266 09280 09294 09308 45 46 47 48 49 5O 51 52 53 54 06538 06548 06559 06569 06580 .06995 .07007 .07019 -07031 07043 07173 07184 07195 07206 07216 .07727 .07740 .07752 .07765 .07778 07836 07848 07859 07870 07881 .08503 .08516 .08529 .08542 .08556 08528 08539 08551 08563 08575 .09323 -09337 .09351 09365 09379 50 51 52 53 54 55 56 57 58 59 06590 06600 06611 06621 06632 .07055 .07067 .07079 .07091 .07103 07227 07238 07249 07260 07271 .07790 07803 .07816 07828 -07841 07893 07904 07915 07927 07938 .08569 08582 .08596 .08069 08623 08586 08598 08610 08622 08634 09393 09407 09421 09435 09449 55 56 57 58 59 60 06642 .07115 07282 07853 07950 .08636 08645 09464 60 234 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 24 25 26 27 2 3 ~5~ 6 7 8 _9_ 10 11 12 13 14 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. t ~ 2 3 4 5 6 7 8 9 08645 .08657 08669 08681 .08693 .09464 .09478 .09492 .09506 .09520 .09369 .09382 .09394 09406 -09418 10338 .10353 .10368 .10383 10398 10121 10133 .10146 .10159 -10172 11260 11276 11292 11308 11323 10899 10913 10926 10939 -10952 12233 -1224b 12266 -12283 12299 .08705 .08717 .08728 .08740 .08752 09535 09549 .09563 .09577 .09592 09431 09443 09455 09468 .09480 -10413 . 10428 .10443 .10458 .10473 10184 10197 .10210 10223 10236 11339 11355 11371 11387 11403 10965 10979 -10992 -11005 .11019 -12316 -12333 - 12349 .12366 -12383 .08764 08776 .08788 .088CO .08812 .09608 09620 .09635 .09649 .09663 .09493 .09505 09517 09530 .09542 .10488 10503 10518 10533 .10549 .10248 10261 10274 .10287 .10300 11419 11435 .11451 -11467 .11483 11032 11045 .11058 11072 .11085 12400 12416 12433 12450 .12467 10 11 12 13 14 15 16 17 18 19 .08824 08836 .08848 .08860 .08872 .09678 .09692 .09707 .09721 -09735 09554 09567 09579 09592 .09604 .10564 .10579 .10594 10609 .10625 10313 10326 .10338 10351 10364 11499 11515 11531 11547 .11563 11098 11112 11125 -11138 11152 .12484 .12501 .12518 -125b4 -12551 15 16 17 18 19 20 21 22 23 24 .08884 08896 08908 .08920 .08932 .09750 .09764 .09779 .09793 .09808 -09617 09629 09642 09654 .09666 10640 10655 10670 .10686 -10701 10377 10390 10403 10416 10429 11579 11595 -11611 11627 -11643 11165 11178 11192 11205 11218 12568 -12585 .12602 .12619 12636 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 25 26 27 28 29_ 30 31 32 33 34 35 36 37 38 39 .08944 .08956 .08968 .08980 .08992 .09822 09837 .09851 .09866 09880 09679 09691 .09704 09716 09729 .10716 .10731 10747 10762 10777 . 10442 10455 .10468 10481 .10494 11659 -11675 11691 -11708 -11724 11232 11245 11259 11272 .11285 -12653 -12670 -12687 .12704 12721 .09004 .09016 .09028 -.09040 .09052 .09895 09909 .09924 -09939 09953 09741 .09754 09767 .09779 .09792 .10793 10808 10824 -10839 -10854 10507 10520 .10533 10546 .10559 .11740 11756 -11772 -11789 11805 11299 11312 11326 11339 .11353 .12738 12755 12772 12789 . 12807 .09064 .09076 .09089 .09101 .09113 .09968 09982 .09997 .10012 .10026 .09804 09817 09829 09842 .09854 .10870 .10885 -10901 10916 .10932 10572 -10585 .10598 10611 10624 11821 .11838 11854 11870 -11886 11366 11380 11393 11407 11420 12824 12841 -12858 .12875 -12892 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 09125 09137 .09149 .09161 .09174 .10041 .10055 .10071 .10085 .10100 09867 09880 09892 09905, .09918 . 10947 10963 .10978 .10994 11009 10637 .10650 .10663 10676 .10689 11903 11919 11936 11952 .11968 11434 11447 11461 11474 .11488 12910 -12927 12944 12961 12979 4O 41 42 43 44 45 46 47 48 49 50 51 52 53 54 .09186 09198 09210 .09222 -09234 .10115 .10130 .10144 .10159 .10174 09930 09943 09955 09968 -09981 -11025 11041 -11056 11072 11087 10702 10715 -10728 10741 10755 11985 12001 12018 12034 12051 11501 11515 -11528 -11542 -11555 12996 13013 -13031 13048 13065 .09247 09259 09271 -09283 .09296 10189 .10204 .10218 .10233 .10248 09993 10006 10019 10032 . 10044 11103 -11119 11134 .11150 11166 10768 10781 10794 -10807 10820 12067 12084 12100 12117 12133 -11569 11583 -11596 11610 -11623 . 13083 13100 13117 13135 13152 55 56 57 58 59 09308 09320 .09332 -09345 .09357 .10263 10278 .10293 .10308 .10323 10057 10070 10082 10095 10108 .11181 .11197 .11213 .11229 11244 .10833 10847 10860 10873 10886 -12150 12166 12183 .12199 12216 11637 11651 11664 11678 .11692 13170 13187 .13205 13222 .13240 55 56 57 58 59 60 .09369 .10338 10121 .11260 .10899 .12233 11705 13257 60 235 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS, 28 29 30 31 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' 2 3 4 11705 11719 .11733 11746 11760 .13257 .13275 13292 13310 13327 .12538 .12552 12566 12580 12595 14335 -14354 .14372 .14391 14409 13397 13412 13427 13441 -13456 15470 .15489 15509 .15528 15548 14283 .14298 14313 14328 . 14343 16663 16684 16704 .16725 .16745 O 1 2 3 4 5 6 7 8 9 11774 11787 -11801 11815 11828 .13345 .13362 13380 .13398 13415 12609 12623 12637 12651 12665 .14428 -14446 14465 .14483 14502 13470 13485 -13499 -13514 -13529 .15567 15587 15606 -15626 -15645 14358 14373 14388 14403 -14418 .16766 .16786 .16806 16827 16848 5 6 7 8 9 10 11 12 13 14 11842 11856 11870 11883 11897 13433 13451 13468 13486 -13504 12679 12694 .12708 12722 12736 14521 .14539 .14558 .14576 14595 .13543 13558 .13573 -13587 -13602 -15665 -15684 15704 15724 15743 . 14433 14449 . 14464 14479 . 14494 16868 16889 -16909 .16930 -16950 10 11 12 13 14 15 16 17 18 19 .11911 11925 11938 11952 .11966 13521 13539 13557 .13575 13593 12750 12765 .12779 .12793 12807 14614 .14632 .14651 14670 -14689 -13616 -13631 -13646 -13660 13675 15763 15782 .15802 15822 -15841 14509 14524 14539 14554 .14569 -16971 16992 17012 -17033 -17054 15 16 17 18 19 20 21 22 23 24 .11980 .11994 12007 12021 .12035 13610 .13628 .13646 13664 13682 .12822 12836 12850 12864 12879 -14707 .14726 .14745 14764 .14782 -13690 -13705 -13719 -13734 - 13749 .15861 .15881 .35901 .15920 .15940 -14584 -14599 -14615 14630 14645 17075 17095 17116 -17137 17158 20 21 22 23 24 25 26 27 28 29 . 12049 12063 -12077 12091 12104 -13700 13718 .13735 .13753 .13771 .12893 12907 12921 .12936 .12950 .14801 .14820 .14839 .14858 .14877 13763 13778 13793 13808 .13822 15960 15980 .16000 16019 -16039 14660 14675 14690 14706 .14721 -17178 .17199 .17220 17241 17262 25 28 27 28 29 30 31 32 33 34 12118 .12132 12146 12160 12174 .13789 .13807 .13825 .13843 .13861 12964 .12979 .12993 .13007 .13022 .14896 .14914 .14933 -14952 .14971 33837 13852 13867 13881 -13896 16059 .16079 .16099 .16119 .16139 14736 14751 14766 14782 14797 17283 17304 17325 17346 17367 30 31 32 33 34 35 36 37 38 39 .12188 12202 12216 12230 12244 13879 .13897 .13916 13934 13952 .13036 .13051 13065 13079 .13094 .14990 .15009 15028 .15047 .15066 -13911 -13926 -13941 -13955 -18970 .16159 .16179 16199 16219 -16239 -14812 .14827 14843 14858 -14873 17388 17409 37430 17451 -17472 35 36 37 38 39 40 41 42 43 44 .12257 12271 12285 .12299 .12313 13970 .13988 14006 14024 14042 .13108 13122 13137 13151 .13166 15085 -1510u .15124 -15143 15162 13985 - 14000 -14015 - 14030 -14044 .16259 .16279 .16299 9. 16319 -16339 14888 14904 34819 14934 34949 17493 17514 -17535 17556 17577 40 41 42 43 44 45 46 47 48 49 -12327 .32341 .12355 .12369 12383 14061 14079 14097 14115 14134 13180 -13195 .13209 .13223 .13238 -15181 .15200 .15219 .15239 .15258 -14059 14074 -14089 14104 14119 16359 16380 .16400 16420 16440 14965 3 4980 14995 15011 15026 -17598 -17620 17641 -17662 176S3 45 46 47 48 49 50 51 52 53 54 .1239'/ 12411 .12425 . 12439 .12454 .14152 .14170 14188 .14207 14225 .13252 .13267 .13281 13296 13310 .15277 .15296 .15315 .15335 .15354 14134 14149 14164 14179 14194 .16460 16481 16501 .16521 .16541 15041 15057 15072 15087 35103 .17704 17726 17747 .17768 .17790 50 51 52 53 54 55 56 57 58 59 .12468 12482 12496 12510 .12524 .14243 14262 14280 .14299 .14317 .13325 .13339 .13354 .13368 .13383 .15373 .15393 .15412 .15431 -15451 14208 14223 14238 14253 14268 16562 16582 16602 16623 -16643 15118 15134 15149 15164 15180 17811 -17832 17854 17875 .17896 55 56 57 58 59 60 .12538 .14335 13397 .15470 -14283 16663 15195 17918 60 236 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 32 33 34 35 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. t ~ 2 3 a 6 7 8 9 10 11 12 13 14 15 16 17 18 19 o 2 3 ~5~ 7 9 10 11 12 13 14 15 16 17 18 19- 20 21 22 23 24 25 26 27 28 29 15195 .15211 .15226 .15241 15257 17918 17939 .17961 .17982 .18004 16133 .16149 .16165 -16181 -16196 .19236 19259 19281 .19304 -19327 .17096 .17113 17129 .17145 .17161 .20622 .20645 20669 20693 -20717 -18085 -18101 -18118 .18135 -18152 -22077 22102 22127 22152 -22177 15272 .15288 .15303 .15319 .15334 .18025 18047 18068 18090 .18111 .16212 .16228 .16244 .16260 .16276 19349 19372 19394 -19417 - 19440 17178 17194 .17210 17227 .17243 .20740 20764 20788 20812 -20836 18168 18185 -18202 .18218 -18235 . 22202 22227 22252 .22277 -22302 .15350 -15365 .15381 -15396 -15412 -18133 .18155 .18176 18198 .18220 16292 16308 -16324 .16340 -16355 .19463 .19485 .19508 -19531 -19554 .17259 .17276 -17292 .17308 17325 .20859 20883 -20907 .20931 .20955 -18252 -18269 -18286 -18302 -18319 22327 .22352 22377 . 22402 - 22428 .15427 15443 .15458 .15474 .15489 .18241 18263 18285 .18307 .18328 -16371 -16387 -16403 -16419 -16435 .19576 19599 .19622 -19645 -19668 .17341 17357 17374 -17390 -17407 .20979 -21003 21027 -21051 -21075 -18336 -18353 -18369 -18386 -18403 .22453 .22478 .22503 22528 22554 .15505 -15520 -15536 -15552 -15567 -18350 -18372 .18394 18416 .18437 .16451 -16467 -16483 .16499 .16515 -19691 .19713 .19736 .19759 .19782 -17423 -17439 -17456 17472 .17489 -21099 21123 -21147 -21171 -21195 18420 18437 -18454 -18470 -18487 -22579 22604 22629 -22655 -22680 20 21 22 23 24 25 26 27 28 29 -15583 -15598 -15614 -15630 .15645 .18459 .18481 18503 .18525 .18547 - 16531 16547 16563 16579 -16595 .19805 .19828 19851 .19874 -19897 17505 -17522 -17538 .17554 .17571 21220 21244 21268 21292 -21316 18504 -18521 -18538 -18555 .18572 . 22706 -22731 .22756 -22782 22807 30 31 32 33 34 .15661 15676 15692 15708 15723 .18569 .18591 .18613 .18635 -18657 .16611 -16627 16644 16660 -16676 .19920 19944 19967 19990 -20013 17587 17604 -17620 17637 -17653 .21341 21365 -21389 -21414 -21438 -18588 18605 -18622 -18639 -18656 22833 -22858 22884 . 22909 .22935 30 31 32 33 34 35 36 87 38 39 .15739 -15755 -15770 15786 .15802 -18679 .18701 .18723 .18745 18767 -16692 -16708 -16724 -16740 -16756 20036 20059 20083 20106 20129 -17670 -17686 -17703 -17719 -17736 21462 .21487 .21511 -21535 -21560 18673 -18690 -18707 -18724 .18741 -22960 -22986 23012 -23037 .23063 35 36 37 38 _39 4O 41 42 43 44 40 41 42 43 44 45 46 47 48 49 .15818 -15833 15849 -15865 .15880 .18790 .18812 18834 -18856 .18878 16772 -16788 -16805 -16821 -16837 -20152 -20176 -20199 .20222 -20246 17752 .17769 17786 .17802 .17819 -21584 -21609 -21633 -21658 -21682 -18758 -18775 18792 18809 -18826 .23089 23114 .23140 -23166 23192 15896 .15912 15928 -15943 15959 -18901 -18923 -18945 -18967 -18990 -16853 -16869 -16885 -16902 -16918 -20269 -20292 -20316 -20339 -20363 -17835 -17852 -17868 -17885 -17902 21707 .21731 21756 21781 -21805 -18843 18860 -18877 18894 18911 .23217 .23243 -23269 .23295 -23321 45 46 47 48 49 5O 51 52 53 _5J 55 56 57 58 59 50 51 52 53 54 55 56 57 58 59 15975 15991 .16006 -16022 18038 -19012 .19034 .19057 19079 -19102 -16934 -16950 -16966 -16983 -16999 .20386 20410 20433 .20457 - 20480 -17918 .17935 -17952 17968 -17985 21830 21855 21879 21904 21929 -18928 .18945 18962 18979 18996 .23347 .23373 .23399 . 23424 -23450 16054 -16070 16085 .16101 .16117 .19124 .19146 .19169 .19191 -19214 .17015 .17031 17047 17064 17080 -20504 -20527 20551 -20575 -20598 -18001 18018 18035 .18051 -18068 .21953 .21978 .22003 .22028 -22053 19013 19030 19047 19064 19081 .23476 .23502 -23529 -23555 23581 60 16133 .19236 -17096 .20622 -18085 .22077 19098 23607 60 237 TABLE 11 NATURAL VERSED SINES AND EXTERNAL SECANTS 36 37 38 39 / o 2 8 ~6~ 6 8 G 1O 11 12 13 li- 15 16 17 18 -L9_ 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39_ 40 41 42 43 44 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ~0 2 _! 5 8 9 10 11 12 13 -I* 15 16 17 18 19 19098 19115 19133 .19150 -19167 .23607 23633 .23659 .23685 23711 .20136 -20154 .20171 20189 20207 .25214 25241 .25269 .25296 25324 .21199 .21217 .21235 21253 .21271 .26902 .26931 26960 .26988 .27017 .22285 22304 22322 .22340 22359 .28676 -28706 -28737 -28767 28797 .19184 .19201 .19218 .19235 .19252 .23738 .23764 .23790 .23816 .23843 .20224 .20242 .20259 .20277 .20294 .25351 25379 25406 .25434 .25462 21289 21307 21324 .21342 .21360 27046 .27075 27104 27133 27162 22377 22395 22414 22432 . 22450 28828 .28858 .28889 .28919 28950 .19270 .19287 19304 .19321 19338 .23869 .23895 .23922 .23948 .23975 20312 .20329 .20347 20365 .20382 .25489 .25517 .25545 .25572 .25600 21378 .21396 .21414 .21432 21450 .27191 27221 .27250 .27279 27308 .22469 -22487 -22506 22524 .22542 .28980 29011 .29042 29072 29103 .19356 .19373 .19390 19407 -19424 .24001 .24028 .24054 .24081 .24107 20400 .20417 .20435 .20453 .20470 .25628 .25656 .25683 .25711 .25739 21468 21486 .21504 .21522 .21540 .27337 .27366 .27396 .27425 -27454 22561 .22579 -22598 22616 22634 .29133 .29164 .29195 .29226 29256 19442 19459 19476 19493 .19511 .24134 -24160 .24187 .24213 .24240 .20488 .20506 20523 20541 .20559 25767 .25795 .25823 .25851 .25879 .21558 .21578 21595 21613 .21631 27483 .27513 .27542 .27572 .27601 .22653 22671 22690 22708 .22727 29287 29318 -29349 .29380 .29411 20 21 22 23 24 25 26 27 28 _29 30 31 32 33 34 .19528 19545 19562 .19580 19597 .24267 .24293 . 24320 .24347 24373 20576 20594 .20612 .20629 .20647 .25907 .25935 25963 .25991 .26019 21649 .21667 .21685 21703 .21721 .27630 .27660 27689 .27719 .27748 22745 .22764 .22782 .22801 -22819 -29442 -29473 .29504 .29535 29566 .19614 .19632 .19649 .19666 -19684 . 24400 .24427 .24454 .24481 .24508 20665 .20682 .20700 .20718 20736 .26047 .26075 .26104 .26132 .26160 .21739 21757 21775 .21794 21812 -27778 -27807 -27837 .27867 .27896 .22838 .22856 22875 .22893 22912 .29597 .29628 .29659 .29690 29721 .19701 .19718 19736 19753 .19770 .24534 .24561 .24588 .24615 . 24642 20753 20771 20789 .20807 .20824 .26188 .26216 .26245 .26273 26301 21830 .21848 .21866 21884 -21902 -27926 .27956 .27985 28015 .28045 22930 22949 22967 22986 . 23004 .29752 -29784 -29815 -29846 -29877 35 38 37 38 -39 40 41 42 43 _44 45 46 47 48 49 19788 .19805 19822 .19840 .19857 .24669 .24696 .24723 .24750 24777 20842 20860 20878 20895 .20913 .26330 .26358 .26387 26415 . 26443 .21921 21939 -21957 21975 .21993 28075 .28105 -28134 .28164 28194 23023 .23041 23060 .23079 .23097 - 29909 -29940 -29971 .30003 30034 45 46 47 48 49 50 51 52 53 54 19875 19892 19909 19927 .19944 .24804 .24832 24859 .24886 24913 .20931 20949 20967 20985 .21002 26472 .26500 .26529 .26557 26586 22012 .22030 22048 22066 . 22084 22103 22121 22139 22157 22176 -28224 -28254 28284 .28314 28344 23116 23134 -23153 23172 -23190 .30066 .30097 .30129 .30160 30192 19962 .19979 19997 20014 20032 . 24940 .24967 .24995 .25022 - 25049 21020 .21038 .21056 21074 .21092 .26615 .26643 .26672 .26701 .26729 .28374 - 28404 -28434 28464 28495 23209 -23228 -23246 23265 -23283 .30223 .30255 30287 .30318 .30350 50 51 52 53 54 55 56 57 58 -fifl 60 55 56 57 58 M. 60 .20049 20066 20084 20101 -20119 .25077 .25104 .25131 .25159 .25186 21109 21127 21145 21163 -21181 .26758 .26787 .26815 .26844 .26873 22194 22212 22231 22249 .22267 -28525 28555 .28585 .28615 .28646 23302 23321 -23339 23358 -23377 .30382 .30413 .30445 .30477 -30509 20136 25214 .21199 .26902 22285 28676 23396 .30541 238 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 40 41 43 43 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. O 1 2 3 4 5 6 7 9 23396 .23414 23433 .23452 .23470 .30541 .30573 .30605 .30636 .30668 .24529 .24548 24567 24586 .24605 ,32501 .32535 .32568 .32602 .32636 .25686 25705 .25724 .25744 .25763 .34563 .34599 .34634 .34669 .34704 .26865 26884 26904 .26924 .26944 36733 .36770 .36807 .36844 .36881 2 3 _1 5 6 7 8 9 1O 11 12 13 14 "15 16 17 18 19 23489 .23508 .23527 .23545 23564 .30700 30732 .30764 .30796 .30829 24625 .24644 24663 24682 .24701 .32669 .32703 .32737 .32770 .32804 .25783 25802 25822 .25841 25861 .34740 .34775 .34811 34846 34882 26964 26984 27004 27024 27043 36919 .36956 .36993 .37030 37068 10 11 12 13 14 15 16 17 18 19 .23583 .23602 .23620 .23639 23658 .30801 .30393 .30925 .30957 .30989 . 24720 24739 .24759 .24778 .24797 .32838 .32872 .32905 .32939 .32973 .25880 25900 .25920 .25939 25959 .34917 34953 .34988 .35024 .35060 27063 27083 27103 .27123 27143 37105 .37143 37180 .37218 .37255 .23677 23696 23714 23733 -23752 31C22 .31054 .31086 31119 .31151 24816 24835 .24854 .24874 .24893 .33007 .33041 .33075 .33109 33143 .25978 .25998 .26017 .26037 .26056 .35095 .35131 .35167 .35203 .35238 .27163 .27183 .27203 .27223 . 27243 .37293 .37330 .37368 .37406 37443 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 23771 23790 .23808 .23827 23846 31183 .31216 .31248 .31281 .31313 24912 24931 24950 24970 24989 .33177 .33211 .33245 .33279 .33314 .26076 .26096 .26115 .26135 .26154 .35274 .35310 .35346 .35382 .35418 .27263 .27283 .27303 27323 .27343 .37481 .37519 37556 .37594 .37632 2O 21 22 23 _24 25 28 27 28 29 30 31 32 33 34 23865 23884 23903 23922 23941 .31346 .31378 .31411 .31443 .31476 25008 25027 25047 25066 25085 .33348 .33382 .33416 .33451 .33485 .26174 .26194 .26213 26233 .26253 .35454 .35490 .35526 .35562 .35598 .27363 27383 27403 27423 .27443 37670 .37708 .37746 .37784 .37822 .23959 .23978 23997 24016 24035 .31509 31541 .31574 .31607 .31640 25104 25124 25143 25162 25182 .33519 .33554 .33588 .33622 .33657 26272 26292 26312 26331 26351 .35634 .35670 35707 .35743 35779 27463 27483 27503 27523 27543 .37860 .37898 .37936 .37974 38012 35 36 37 38 39 24054 24073 24092 24111 24130 .31672 .31705 .31738 .31771 .31804 25201 25220 25240 25259 25278 .33691 .33726 .33760 .33795 .33830 26371 26390 26410 26430 26449 .35815 35852 .35888 .35924 .35961 27563 27583 27603 27623 27643 .38051 .38089 .38127 .38165 .38204 35 36 37 38 -39 40 41 42 43 44 40 41 42 43 44 24149 24168 24187 24206 24225 .31837 .31870 .31903 .31936 31969 25297 25317 25336 25356 25375 .33864 .33899 .33934 .33968 .34003 26469 26489 26509 26528 26548 .35997 .36034 .36070 .36107 .36143 27663 27683 27703 27723 27743 38242 .38280 .38319 .38357 .38396 , 45 46 : 47 48 49 24244 24262 24281 24300 24320 .32002 .32035 .32068 .32101 .32134 25394 25414 25433 25452 25472 .34038 .34073 .34108 .34142 .34177 26568 26588 26607 26627 26647 .36180 .36217 .36253 .36290 36327 27764 27784 27804 27824 27844 .38434 .38473 .38512 .38550 .38589 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 5O 51 52 53 54 24339 24358 24377 24396 24415 .32168 .32201 32234 .32267 .32301 25491 25511 25530 25549 25569 .34212 .34247 34282 34317 34352 26667 26686 26706 26726 26746 .36363 .36400 .36437 .36474 .36511 27864 27884 27905 27925 27945 .38628 .38666 .38705 .38744 .38783 55 56 57 58 59 24434 24453 24472 24491 24510 .32334 .32368 .32401 32434 .32468 25588 25608 25627 25647 25666 34387 .34423 .34458 .34493 .34528 26766 26785 26805 26825 26845 .36548 36585 .36622 .36669 .36696 27965 27985 28005 28026 28046 .38822 .38860 .38899 .38938 .38977 60 24529 .32501 25686 .34563 26865 .36733 28066 .39016 60 239 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 44 45 46 47 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' 1 2 3 4 .28066 .28086 .28106 .28127 .28147 .39016 .39055 .39095 .39134 .39173 29289 -29310 29330 29351 29372 .41421 .41463 .41504 .41545 .41586 .30534 .30555 .30576 30597 .30618 43956 .43999 .44042 .44086 -44129 31SOO 31821 31843 31864 311*5 46628 46674 .46719 46765 -46811 1 2 3 4 5 6 7 8 9 .28167 .28187 28208 .28228 .28248 .39212 .39251 .39291 .39330 39369 29392 .29413 .29433 29454 .29475 .41627 .41669 .41710 41752 41793 30639 .30660 30681 .30702 30723 .44173 .44217 44260 44304 . 44347 31907 31928 31949 31971 31992 46857 .46903 46949 46995 47041 5 6 7 8 9 10 11 12 13 14 .28268 .28289 28309 28329 .28350 .39409 .39448 .39487 .39527 .39566 .29495 .29516 .29537 29557 29578 .41835 .41876 .41918 .41959 42001 .30744 .30765 .30786 .30807 .30828 44391 .44435 .44479 .44523 44567 32013 32035 32C56 -32077 32099 .47087 .47134 47180 .47226 47272 10 11 12 13 14 15 16 17 18 19 .28370 .28390 .28410 .28431 .28451 .39606 39646 .39685 .39725 39764 .29599 .29619 .29640 .29661 29681 .42042 .42084 .42126 .42168 .42210 .30849 .30870 .30891 .30912 -30933 44610 .44654 .44698 .44742 -44787 32120 -32141 -32163 -32184 32205 .47319 .47365 .47411 47458 -47504 15 16 17 18 19 20 21 22 23 24 .28471 .28492 .28512 .28532 .28553 39804 .39844 .39884 .39924 .39963 29702 .29723 .29743 29764 -29785 .42251 .42293 .42335 .42377 .42419 -30954 .30975 -30996 -31017 -31038 .44831 .44875 .44919 .44963 .45007 -32227 32248 32270 32291 32312 .47551 47598 .47644 .47691 -47738 20 21 22 23 24 25 26 27 28 29 28573 28593 .28614 .28634 .28655 40003 .40043 .40083 .40123 .40163 ,29805 .29826 .29847 29868 .29888 .42461 .42503 .42545 .42587 .42630 -31059 .31080 .31101 31122 -31143 .45052 .45096 .45141 .45185 .45229 32334 -32355 32377 32398 -32420 47784 .47831 47878 47925 -47972 25 26 27 28 29 30 31 32 33 34 .28675 .28695 28716 28736 28757 .40203 .40243 .40283 .40324 .40364 29909 .29930 29951 29971 .29992 .42672 .42714 .42756 .42799 .42841 11165 31186 -31207 -31228 .31249 .45274 .45319 .45363 .45408 -45452 32441 -32462 32484 32505 32527 48019 48066 .48113 .48160 48207 30 31 32 33 34 35 36 37 38 39 .28777 28797 28818 .28838 .28859 .40404 .40444 40485 .40525 405P5 .30013 30034 30054 30075 30096 42883 .42926 .42968 .43011 .43053 31270 -31291 -31312 -31334 -31355 .45497 .45542 .45587 45631 -45676 -32548 -32570 -32591 -32613 32634 48254 48301 -48349 48396 48443 35 36 37 38 39 40 41 42 43 44 .28879 28900 .28920 .28941 .28961 .40606 . 40646 .40687 .40727 .40768 30117 30138 30158 .30179 .30200 .43096 .43139 .43181 .43224 .43267 -31376 -31397 31418 31439 .31461 45721 45766 .45811 .45856 .45901 32656 -32677 -32699 -32720 -32742 48491 -48538 48586 48633 48681 40 41 42 43 44 45 46 47 48 49 28981 29002 29022 29043 .29063 .40808 .40849 .40890 .40930 .40971 30221 30242 30263 30283 .30304 .43310 .43352 .43395 .43438 -43481 31482 31503 31524 31545 -31567 .45946 .45992 .46037 -46082 46127 32763 -32785 -32806 -32828 - 32849 48728 .48776 48824 48871 -48919 45 46 47 48 49 50 51 52 53 54 29084 .29104 ..'39125 .29145 29166 .41012 .41053 .41093 .41134 .41175 30325 30346 -30367 30388 . 30409 .43524 .43567 .43610 .43653 .43696 31588 31609 31630 31651 31673 46173 46218 46263 46309 46354 -32871 32893 32914 32936 3?957 48967 49015 49063 .49111 .49159 50 51 52 53 54 55 56 57 58 59 .29187 29207 29228 29248 29269 .41216 .41257 .41298 .41339 .41380 30430 -30451 30471 .30492 -30513 .43739 .43783 .43826 .43869 .43912 31694 31715 31736 31758 31779 .46400 .46445 46491 .46537 -46582 32979 33001 33022 33044 330P5 .49207 49255 .49303 .49351 .4P399 55 56 57 58 59 60 29289 .41421 -30534 .43956 31800 .46628 33087 49448 60 240 TABLE 11 NATURAL VERSED SINES AND EXTERNAL SECANT3. 48* 49 50 51 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' o 1 2 3 4 33087 33109 33130 33152 33173 .49448 .49496 .49544 .49593 .49641 34394 34416 34438 34400 34482 52425 .52476 .52527 .52579 -52630 35721 35744 35766 35788 35810 55572 55626 .55680 .55734 .55789 37068 37091 37113 37136 37158 58902 58959 59016 .59073 -59130 1 2 3 4 5 6 7 8 9 10 11 12 13 14 33195 33217 33238 33260 33282 49690 .49738 .49787 .49835 .49884 34504 34526 34548 34570 34592 .52681 .52732 .52784 .52835 .52886 35833 35855 35877 35900 35922 .55843 .55897 .55951 .56005 . 56.060 37181 37204 37226 37249 37272 -59188 -59245 59302 .59360 -59418 5 6 7 8 9 33303 33325 33347 33368 33390 .49933 .49981 .50030 .50079 .50128 34614 34636 34658 34680 34702 .52938 .52989 .53041 .53092 .53144 35944 35967 35989 36011 36034 .56114 .56169 .56223 .56278 .56332 37294 37317 37340 37362 37385 .59475 .59533 59590 .59648 .59706 10 11 12 13 14 15 16 17 18 19 33412 33434 33455 33477 33499 .50177 .50226 .50275 .50324 .50373 34724 34746 34768 34790 34815? .53196 .53247 .53299 .53351 .53403 36056 36078 36101 36123 36146 .56387 .56442 .56497 .56551 .56606 37408 37430 37453 37476 37498 .59764 .59822 .59880 .59938 .59996 15 16 17 18 19 20 21 22 23 24 . 25 26 27 28 29 33520 33542 33564 33586 33607 50422 .50471 .50521 .50570 .50619 34834 34856 34878 34900 34923 .53455 .53507 .53559 .53611 .53663 36168 36190 36213 36235 36258 56661 .56716 .56771 .56826 .56881 37521 37544 37567 37589 37612 .60054 .60112 .60171 .60229 .60287 20 21 22 23 24 25 26 27 28 29 33629 33651 33673 33694 33716 50669 .50718 50767 .50817 .50866 34945 34967 34989 35011 35033 .53715 .53768 .53820 .53872 .53924 36280 36302 36325 36347 36370 .56937 .56992 .57047 .57103 .57158 37635 37658 .37680 .37703 .37726 .60346 .60404 .60463 .60521 -60580 30 31 32 33 ! 34 33738 33760 33782 33803 33825 50916 50966 51015 .51065 .51115 35055 35077 35099 35122 35144 .53977 . 54029 54082 .54134 .54187 36392 36415 36437 36460 36482 .57213 -57269 .57324 .57380 .57436 .37749 37771 -37794 .37817 .37840 .60639 .60698 .60756 .60815 .60874 3O 31 32 33 34 35 36 37 38 -39 4O 41 42 43 44 45 46 47 48 49 5O 51 52 53 54 55 56 57 58 59 60 35 36 37 38 39_ 40 41 42 43 44 45 46 47 48 49 33847 33869 33891 33912 33934 .51165 .51215 .51265 .51314 .51364 35166 35188 35210 35232 35254 .54240 .54292 . 54345 .54398 .54451 36504 36527 .36549 36572 .36594 -57491 .57547 .57603 .57659 .57715 -37862 -37885 -37908 -37931 .37954 .60933 .60992 .61061 .61111 -61170 33958 33978 34000 34022 34044 .51415 .51465 .51515 .51565 .51615 35277 35299 35321 35343 35365 .54504 .54557 .54610 .54663 .54716 .36617 36639 36662 36684 .36707 57771 .57827 .57C83 .57939 .57995 .37976 .37999 .38022 -38045 -38068 .61229 . 61288 .61348 61407 .61467 34065 -34087 34109 34131 -34153 .51665 .51716 .51766 .51817 .51867 35388 35410 35432 .35454 35476 .54769 .54822 .54876 .54929 .54982 36729 36752 -36775 36797 -36820 .58051 .58108 .58164 .58221 .58277 -38091 -38113 38136 38159 .38182 .61526 .61586 .61646 .61705 .61765 50 51 52 53 54 -34175 34197 -34219 34241 34262 .51918 .51968 .52019 .52069 .52120 -35499 35521 35543 .35565 .35588 .55036 .55089 .55143 .55196 .55250 .36842 36865 36887 .36910 .36932 .58333 .58390 . 58447 .58503 .58560 .38205 38228 38251 -38274 .38296 - 61825 .61885 .61945 .62005 .62065 '55 56 57 58 59 60 34284 34306 34328 34350 -34372 .52171 .52222 .52273 .52323 .52374 .35610 35632 35654 35677 35699 .55303 .55357 .55411 .55465 .55518 .36955 -36978 .37000 -37023 -37045 .58617 .58674 .58731 .58788 .58845 -38319 .38342 .38365 .38388 .38411 .62125 .62185 .62246 .62306 -62366 34394 .52425 .35721 .55572 -37068 .58902 38434 .62427 241 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANT& 53 53 54 55 ' Vers. Ex. sec Vers. Ex. sec Vers. Ex. sec Vers. Ex. sec I . 38434 .38457 .38480 .38503 38526 .62427 . 62487 .62548 .62609 62669 .39819 39842 39865 39888 39911 .66164 .66228 .66292 .66357 66421 41221 41245 41269 41292 41316 .70130 .70198 .70267 70335 . 70403 42642 .42666 42690 .42714 42738 . 74345 74417 . 74490 74562 74635 2 3 4 1 e 1 8 9 38549 38571 38594 38617 .38640 .62730 .62791 .62852 62913 .62974 39935 39958 .39981 40005 .40028 66486 66550 .66615 .66679 66744 41339 41363 41386 41410 41433 . 70472 . 70540 . 70609 70677 70746 42762 42785 42809 42833 42857 74708 . 74781 .74854 .74927 75000 5 6 7 8 g 10 11 12 13 14 38663 .38686 .38709 38732 38755 .63035 .63096 .63157 .63218 .63279 40051 40074 .40098 40121 .40144 .66809 .66873 66938 .67003 67068 .41457 41481 41504 41528 -41551 70815 .70884 .70953 . 71022 71091 42881 .42905 42929 42953 42976 .75073 .75146 .75219 .75293 75366 10 11 12 13 14 15 16 17 18 -L2L .38778 .38801 .38824 38847 38870 .63341 .63402 .63464 .63525 .63587 40168 40191 40214 40237 40261 .67133 .67199 .67264 .67329 .67394 41575 41599 .41622 41646 .41670 .71160 71229 .71298 .71368 71437 .43000 .43024 43048 43072 . 75440 .75513 75587 .75661 75734 15 16 17 18 19 20 21 22 23 24_ 38893 .38916 .38939 .38962 .38985 .63648 .63710 .63772 .63834 .63895 .40284 40307 .40331 40354 .40378 .67460 .67525 .67591 .67656 .67722 41693 41717 41740 -41764 .41788 71506 .71576 .71646 .71715 71785 43120 43144 43168 43192 43216 .75808 .75882 .75956 .76031 76105 20 21 22 23 24 25 26 27 28 39009 39032 39055 39078 39101 .63957 .64019 .64081 64144 64206 .40401 .40424 .40448 40471 40494 .67788 .67853 67919 67985 68051 41811 41835 41859 41882 .41906 71855 .71925 .71995 .72065 72135 43240 43264 43287 43311 43335 76179 .76253 .76328 .76402 76477 25 26 27 28 29 30 31 32 33 34 39124 39147 39170 39193 39216 .64268 .64330 .64393 .64455 .64518 -40518 40541 40565 40588 40611 .68117 .68183 68250 .68316 .68382 41930 .41953 .41977 42001 .42024 .72205 . 72275 .72346 72416 .72487 43359 43383 43407 43431 43455 76552 .76626 .76701 .76776 76851 ~3O 31 32 33 34 (CO CO CO CO CO kOCO-JCDOl 39239 39262 39286 39309 39332 .64580 .64643 .64705 .64768 .64831 40635 40658 40682 40705 40728 . 68449 68515 .68582 68648 68715 42048 42072 42096 42119 42143 .72557 .72628 .72698 .72769 . 72840 43479 43503 43527 43551 .43575 76926 .77001 .77077 .77152 35 36 37 38 39 40 41 42 43 44 39355 39378 39401 39424 39447 64894 64957 .65020 .65083 65146 40752 40775 40799 40822 40846 68782 .68848 .68915 .68982 69049 42167 42191 42214 42238 42262 .72911 72982 .73053 .73124 .73195 43599 43623 43647 43671 43695 .77303 .77378 .77454 77530 77606 40 41 42 43 44 45 46 47 48 49 39471 39494 39517 39540 39563 .65209 .65272 .65336 .65399 65462 40869 40893 40916 40939 40963 .69116 .69183 .69250 .69318 .69385 42285 42309 42333 42357 42381 .73267 .73338 . 73409 .73481 .73552 43720 43744 43768 43792 43816 77681 77757 .77833 .77910 77986 45 46 47 48 49 50 51 52 53 54 39586 39610 39633 9656 9679 65526 65589 65653 65717 65780 40986 41010 41033 41057 41080 .69452 .69520 .69587 .69655 .69723 42404 42428 42452 42476 42499 .73624 .73696 .73768 .73840 .73911 43840 43864 43888 43912 43936 .78062 .78138 .78215 .78291 78368 50 51 52 53 54 55 56 57 58 -fiL. 9702 9726 9749 9772 9795 65844 65908 65972 66036 66100 41104 41127 41151 41174 41198 .69790 69858 .69926 .69994 70062 42523 42547 42571 42595 42619 .73983 .74056 .74128 . 74200 74272 43960 43984 44008 44032 44057 .78445 .78521 .78598 .78675 78752 55 56 57 58 59 60 9819 66164 41221 70130 42642 74345 44081 78829 60 242 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 56 57 58 59 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 44081 .44105 44129 44153 44177 .78829 .78906 .78984 .79061 79138 .45536 45560 .45585 -45609 .45634 .83608 .83690 .83773 .83855 .83938 47008 47033 47057 .47082 .47107 .88708 .88796 .88884 .88972 .89060 48496 48521 48546 48571 -48596 .94160 .94254 . 94349 .94443 .94537 2 3 4 5 6 7 9 44201 .44225 44250 -44274 -44298 .79216 .79293 .79371 . 79449 .79527 -45658 45683 -45707 45731 .45756 .84020 .84103 .84186 .84269 .84352 .47131 .47156 .47181 47206 .47230 .89148 89237 .89325 .89414 .89503 48621 -48646 48671 .48696 -48721 .94632 .94726 .94821 .94916 .95011 44322 44346 .44370 44395 .44439 .79604 .79682 .79761 .79839 .79917 .45780 -45805 45829 .45854 .45878 .84435 84518 84601 .84685 -84768 47255 .47280 .47304 .47329 .47354 .89591 .89680 .89769 .89858 .89948 48746 -48771 .48796 .48821 .48846 .95106 95201 .95296 .95392 .95487 10 11 12 13 14 15 16 17 18 19 44443 44467 44491 .44516 .4454.0 .79995 .80074 80152 .80231 .80309 .45903 45927 .45951 45976 .46000 .84852 .84935 .85019 85103 85187 .47379 47403 .47428 47453 47478 .90037 .90126 90216 .90305 .90395 48871 48896 48921 .48946 .48971 .95583 -95678 .95774 .95870 95966 20 21 22 23 24 25 28 27 28 29 44564 .44588 44612 .44637 44661 .80388 .80467 .80546 .80625 .80704 .46025 -46049 46074 46098 .46123 85271 .85355 .85439 85523 .85608 .47502 .47527 .47552 .47577 .47601 .90485 .90575 .90665 .90755 .90845 .48996 49021 49046 49071 49096 .96062 .96158 .96255 .96351 .96448 20 21 22 23 24 25 26 27 28 29 30 3] 32 33 34 35 36 37 38 -39 40 41 42 43 44 45 46 47 48 49 5O 51 52 53 54 55 56 57 58 59 44685 .44709 44734 44758 44782 .80783 .80862 .80942 .81021 .81101 46147 46172 -46196 -46221 46246 .85692 .85777 .85861 .85946 .86031 47626 47651 47676 .47701 47725 .90935 .91026 .91116 .91207 .91297 49121 49146 -49171 -49196 .49221 .96544 .96641 .96738 .96835 .96932 30 31 32 33 84 35 36 37 38 39 44806 .44831 .44855 44879 .44903 .81180 81260 81340 .81419 .81499 -46270 46295 46319 46344 -46368 .86116 .86201 .86286 .86371 .86457 47750 .47775 -47800 47825 .47849 .91388 .91479 .91570 .91661 91752 .49246 .49271 49296 .49321 .49346 .97029 .97127 .97224 .97322 .97420 44928 44952 44976 45001 .45025 81579 .81659 .81740 .81820 .81900 -46393 46417 - 46442 46466 -46491 86542 .86627 86713 .86799 86885 47874 .47899 .47924 47949 -47974 .91844 .91935 .92027 .92118 .92210 49372 49397 49422 -49447 -49472 .97517 .97615 .97713 .97811 .97910 40 41 42 43 44 45 46 47 48 49 45049 -45073 45098 45122 -45146 81981 .82061 82142 82222 82303 46516 -46540 46565 46589 .46614 86970 .87056 .87142 87229 87315 47998 .48023 48048 48073 .48098 .92302 .92394 .92486 .92578 .92670 49497 49522 .49547 .49572 .49597 .98008 .98107 .98205 .98304 .98403 -45171 45195 -45219 -45244 -45268 82384 82465 82546 .82627 .82709 46639 46663 46688 46712 .46737 .87401 87488 87574 .87661 .87748 48123 .48148 48172 .48197 -48222 .92762 92855 .92947 .93040 .93133 49623 49648 49673 49698 .49723 .98502 .98601 .98700 .98799 .98899 50 51 52 53 54 55 56 57 58 59 45292 45317 45341 -45365 -45390 82790 82871 82953 .83034 .83116 46762 46786 46811 46836 46860 .87834 .87921 .88008 .88095 88183 .48247 -48272 .48297 48322 -48347 .93226 .93319 .93412 .93505 .93598 -49748 -49773 -49799 49824 49849 .98998 .99098 99198 .99298 .99398 45414 45439 -45463 45487 45512 .83198 .83280 .83362 .83444 .83526 46885 46909 46934 -46959 46983 .88270 88357 88445 .88532 88620 -48372 48396 48421 -48446 48471 .93692 .93785 .93879 .93973 .94066 49874 49899 49924 49950 49975 .99498 .99598 .99698 99799 -99899 60 45536 .83608 47008 .88708 .48496 .94160 50000 1.00000 6O 243 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 60 61 63 63 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. t 1 2 3 4 50000 50025 50050 50076 50101 1.00000 1-00101 1-00202 1-00303 1 - 00404 51519 51544 51570 51595 51621 1-06267 1-06375 1.06483 1-06592 1-06701 53053 53079 53104 53130 -53156 1.13005 1.13122 1-13239 1-13356 1-13473 54601 54627 54653 54679 54705 1-20269 1-20395 1-20521 1-20647 1-20773 1 2 3 4 ~~5~ 6 7 8 g 50126 50151 50176 50202 50227 1-00505 1-00607 1-00708 1-00810 1-00912 51646 51672 51697 51723 -51748 1-06809 1-06918 1-07027 1-07137 1-07246 -53181 53207 -53233 .53258 .53284 1-13590 1.13707 1-13825 1-13942 1.14060 54731 .54757 - 54782 -54808 . 54834 1-20900 1-21026 1.21153 1.21280 1-21407 5 6 7 8 9 ib~ 11 12 13 14 50252 50277 50303 50328 50353 1-01014 1-01116 1-01218 1-01320 1-01422 -51774 -51799 -51825 -51850 -51876 1-07356 1.07465 1.07575 1-07685 1-07795 -53310 53336 .53361 .53387 -53413 1-14178 1.14296 1-14414 1-14533 1-14651 - 54860 .54886 -54912 54938 -54964 1-21535 1.21662 1.21790 1-21918 1.22045 10 11 12 13 14 15 16 17 18 19 50378 50404 50429 50454 50479 1.01525 1-01628 1-01730 1-01833 1.01936 -51901 51927 .51952 .51978 .52003 1-07905 1.08015 1.08126 1.08236 1-08347 -53439 -53464 -53490 -53516 -53542 1.14770 1.14889 1.150C8 1.15127 1-15246 54990 -55016 . 55042 .55068 - 55094 1-22174 1-22302 1-22430 1-22559 1.22688 15 16 17 18 19 20 21 22 23 24 50505 -50530 .50555 .50581 .50606 1-02039 1-02143 1-02246 1.02349 1.02453 -52029 -52054 -52080 -52105 -52131 1-08458 1.08569 1.08680 1.08791 1-08903 -53567 -53593 -53619 -53645 -53670 1-15366 1-15485 1-15605 1-15725 1-15845 -55120 -55146 55172 55198 .55224 1-22817 1-22946 1-23075 1-23205 ] .23334 20 21 22 23 24 25 26 27 28 29 50631 .50656 .50682 .50707 .50732 1-02557 1-02661 1-02765 1-02869 1-02973 -52156 -52182 .52207 -52233 .52259 1-09014 1.09126 1-09238 1.09350 1-09462 .53696 -53722 -53748 -53774 .53799 1-15965 1-16085 1.16206 1.16326 1-16447 55250 55276 55302 55328 .55354 1-23464 1-23594 1-23724 1-23855 1-23985 25 26 27 28 29 30 81 32 33 34 .50758 .50783 -50808 .50834 -50859 1-03077 1-03182 1-03286 1-03391 1-03496 .52284 -52310 .52335 -52361 .52386 1-09574 1-09686 1-09799 1-09911 1-10024 -53825 -53851 -53877 -53903 .53928 1-16568 1.16689 1.16810 1.16932 1-17053 55380 -55406 55432 55458 -55484 1-24116 1-24247 1-24378 1-24509 1-24640 30 31 32 33 34 35 36 37 38 39 .50884 -50910 .50935 -50960 . 50986 1-03601 1-03706 1-03811 1-03916 1-04022 .52412 -52438 . 52463 .52489 .52514 1-10137 1-10250 1-10363 1-10477 1-10590 53954 -53980 - 54006 -54032 . RAO 58 1-17175 1.17297 1-17419 1.17541 1.17663 55510 55536 55563 55589 -FF615 1-24772 1-24903 1-25035 1-25167 1 .95300 35 36 37 38 39 40 41 42 43 44 51011 -51036 -51062 .51087 .51113 1-04128 1-04233 1-04339 1 - 04445 1-04551 -52540 52566 -52591 52617 .52642 1-10704 1.10817 1-10931 1.11045 1-11159 -54083 -54109 .54135 -54161 -54187 1-17786 1-17909 1.18031 1-18154 1-18277 55641 55667 -55693 -55719 -55745 1-25432 1-25565 1-25697 1-25830 1.25963 40 41 42 43 44 45 46 47 48 49 51138 51163 51189 .51214 -51239 1-04658 1-04764 1.04870 1-04977 1-05084 52668 52694 52719 52745 .52771 1-11274 1-11388 1-11503 1.11617 1-11732 -54213 -54238 -54264 -54290 .54316 1-18401 1-18524 1-18648 1-18772 1-18895 55771 55797 -55823 -55849 -55876 1-26097 1-26230 1-26364 1-26498 1-26632 45 46 47 48 49 50 51 52 53 54 51265 51290 51316 51341 .51366 1-05191 1-05298 1-05405 1-05512 1-05619 .52796 52822 .52848 .52873 -52899 1.11847 1.11963 1.12078 1-12193 1-12309 -54342 54368 -54394 - 54420 . 54446 1-19019 1-19144 1.19268 1-19393 1-19517 .55902 -55928 -55954 -55980 -56006 1-26766 1-26900 1-27035 1-27169 1-27304 50 51 52 53 54 55 66 57 58 59 51392 51417 .51443 51468 -51494 1-05727 1-05835 1-05942 1-06050 1-06158 52924 52950 -52976 .53001 .53027 1-12425 1.12540 1.12657 1-12773 1-12889 - 54471 54497 -54523 . 54549 .54575 1-19642 1-19767 1-19892 1-20018 1-20143 -56032 .56058 .56084 -56111 56137 1-27439 1-27574 1-27710 1.27845 1-27981 bb 56 57 58 -C 60 51519 1-06267 .53053 1-13005 -54601 1-20269 -56163 1-28117 60 244 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 64 65 66 67 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. r o 1 2 3 4 56163 .56189 .56215 .56241 .56267 1.28117 1-28253 1-28390 1.28526 1-28663 .57738 .57765 .57791 .57817 . 57844 1-36620 1-36768 1.36916 1-37064 1.37212 -59326 .59353 .59379 .59406 .59433 .45859 .46020 .46181 .46342 -46504 -60927 60954 .60980 61007 .61034 55930 56106 .56282 .56458 .56634 O 3 5 6 7 8 9 .56294 .56320 .56346 .56372 .56398 1.28800 1-28937 1-29074 1-29211 1.29349 .57870 .57896 57923 .57949 .57976 1-37361 1.37509 1.37658 1-37808 1.37957 . 59459 .59486 59512 59539 .59566 .46665 46827 1.46989 1-47152 1-47314 -61061 .61088 -61114 61141 -61168 56811 -56988 57165 -57342 .57520 5 6 8 9 1O 11 12 13 14 .56425 .56451 56477 56503 .56529 1.29487 1-29625 .29763 .29901 . 30040 .58002 .58028 .58055 .58081 .58108 1-38107 1-38256 1.38406 1.38556 1-38707 .59592 -59619 -59645 -59672 -59699 1-47477 1-47640 1-47804 1-47967 1.48131 -61195 -61222 -61248 -61275 -61302 1-57698 1.57876 1-58054 1-58233 1.58412 1O 11 12 13 14 15 16 17 18 19 56555 .56582 .56608 .56634 .56660 30179 .30318 -30457 -30596 -30735 .58134 -58160 -58187 .58213 -58240 1-38857 1-39008 1-39159 1.39311 1-39462 -59725 59752 -59779 -59805 .59832 1.48295 1.48459 1.48624 1.48789 1-48954 .61329 -61356 -61383 -61409 .61436 1.58591 1.58771 1-58950 1-59130 1-59311 15 16 17 18 19 . 20 21 22 23 24 .56687 56713 56739 .56765 56791 30875 31015 31155 31295 -31436 -58266 58293 .58319 .58345 .58372 1.39614 1.39766 1-39918 1.40070 1.40222 -59859 -59885 .59912 -59938 .59965 1-49119 1-49284 1-49450 1-49616 1-49782 . 61463 .61490 .61517 .61544 -61570 1-59491 1-59672 1-59853 1.60035 1-60217 20 21 22 23 24 25 2G 27 28 29 56818 56844 56870 56896 56923 31576 31717 .31858 .31999 .32140 -58398 .58425 .58451 58478 58504 1.40375 1.40528 1.40681 1-40835 1.40988 -59992 -60018 -60045 60072 .60098 1 - 49948 1-50115 1-50282 1 . 50449 1.50617 61597 61624 .61651 .61678 .61705 1-60399 1-60581 1-60763 1-60946 1-61129 25 26 27 28 29 30 31 32 33 34 56949 56975 57001 57028 57054 32282 32424 .32566 1-32708 1-32850 58531 58557 58584 58610 58637 1-41142 1.41296 1.41450 1.41605 1.41760 60125 60152 60178 -60205 .60232 1.50784 1.50952 1.51120 1.51289 1-51457 .61732 .61759 -61785 61812 .61839 1-61313 1-61496 1.61680 1-61864 1 62049 30 S 33 34 35 36 37 38 39_ 57080 57106 57133 57159 57185 1-32993 1.33135 1-33278 1-33422 1-33565 58663 58690 58716 58743 58769 1-41914 1.42070 1.42225 1.42380 1-42536 60259 -60285 60312 60339 -60365 1.51626 1-51795 1.51965 1.52134 1 . 52304 61866 61893 -61920 -61947 -61974 1-62234 1.62419 1-62604 1.62790 1-62976 35 36 37 38 39 40 41 42 43 44 57212 57238 57264 .57291 .57317 1-33708 1-33852 1-33996 1-34140 1-34284 -58796 58822 58849 -58875 .58902 1.42692 1-42848 1.43005 1.43162 1-43318 60392 60419 - 60445 60472 . 60499 1.52474 1.52645 1.52815 1.52986 1.53157 -62001 -62027 -62054 -62081 -62108 1-63162 1.63348 1-63535 1-63722 1-63909 40 41 42 43 44 45 46 47 48 49 57343 57369 -57396 . 57422 . 57448 1 . 34429 1.34573 1-34718 1-34863 1-35009 .58928 58955 -58981 59008 -59034 1.43476 1-43633 1.43790 1.43948 1.44106 .60526 60552 .60579 .60606 60633 1.53329 1.53500 1.53672 1.53845 1-54017 62135 -62162 62189 -62216 - 62243 1.64097 1-64285 1-64473 64662 64851 45 46 47 48 49 50 51 52 53 54 .57475 -57501 .57527 .57554 -57580 1.35154 1.35300 1.35446 1-35592 1-35738 59061 -59087 59114 59140 .59167 1.44264 1 44423 1.44582 1-44741 1-44900 .60659 60886 .60713 . 60740 60766 1.54190 1-54363 1-54536 1 - 54709 1 - 54883 - 62270 -62297 62324 .62351 -62378 .65040 65229 65419 1-65609 1.65799 5O 51 52 53 54 55 56 57 58 .59. .57606 57633 57659 .57685 .57712 1-35885 1-36031 1-36178 1-36325 1-36473 59194 59220 -59247 -59273 -59300 1-45059 1-45219 1.45378 1.45539 1.45699 60793 60820 60847 -60873 -60900 1-55057 1-55231 1.55405 1.55580 1.55755 . 62405 .62431 -62458 -62485 -62512 1-65989 1.66180 1 66371 1-66563 1-66755 55 56 57 58 59 60 57738 1-36620 -59326 1-45859 .60927 1.55930 .62539 1-66947 60 245 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 68 69 70 71 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' 2 3 .62539 .62566 .62593 62620 62647 1-66947 1.67139 1-67332 1.67525 1-67718 64163 -64190 .64218 -64245 - 64272 1-79043 1.79254 1.79466 1-79679 1-79891 65798 65825 65853 .65880 .65907 1-92380 1-92614 1-92849 1-93083 1.93318 - 67443 67471 - 67498 -67526 -67553 2.07155 2-07415 '2-07675 2-07936 2-08197 2 3 4 5 6 7 8 . 9 .62674 .62701 .62728 62755 .62782 1-67911 1.68105 1-68299 1.68494 1-68689 -64299 64326 64353 .64381 . 64408 1.80104 1.80318 1-80531 1-80746 1-80960 65935 -65962 -65989 -66017 66044 1-93554 1-93790 1.94026 1-94263 1-94500 -67581 -67608 -67636 -67663 -67691 2-08459 2-08721 2-08983 2-09246 2-09510 5 6 7 8 9 10 11 12 13 14 62809 62836 62863 .62890 62917 1.68884 1-69079 1.69275 1.69471 1-69667 . 64435 - 64462 . 64489 -64517 - 64544 1-81175 1.81390 1-81605 1-81821 1-82037 -66071 -66099 -66126 -66154 -66181 1-94737 1-94975 1-95213 1-95452 1.95691 -67718 -67746 -67773 -67801 -67829 2.09774 2-10038 2-10303 2.10568 2-10834 10 11 12 13 34 15 16 17 18 19 62944 62971 62998 63025 .63052 1.69864 1-70061 1.70258 1.70455 1.70653 -64571 .64598 64625 64653 . 64680 1-82254 1-82471 1-82688 1-82906 1-83124 -66208 -66236 -66263 -66290 66318 1-95931 1.96171 1-96411 1-96652 1-96893 -67856 -67884 -67911 -67939 -67966 2-11101 2-11367 2-11635 2.11903 2-12171 15 16 17 18 19 20 21 22 23 24 .63079 63106 63133 63161 63188 1.70851 1.71050 1.71249 1 71448 1.71647 . 64707 . 64734 .64761 64789 .64816 1-83342 1-83561 1.83780 1-83999 1-84219 66345 -66373 66400 66427 66455 1 97135 1-97377 1-97619 1-97862 1.98106 -67994 -68021 -68049 -68077 -68104 2-12440 2-12709 2-12979 2-13249 2-13520 20 21 22 23 24 25 26 27 28 29 63215 -63242 63269 63296 63323 1-71847 1 . 72047 1.72247 1 . 72448 1.72649 . 64843 . 64870 .64898 . 64925 64952 1.84439 1.84659 .84880 .85102 85323 66482 66510 66537 66564 .66592 1-98349 1-98594 1-98838 1-99083 1-99329 68132 68159 68187 68214 - 68242 2-13791 2-14063 2-14335 2-14608 2-14881 25 26 27 28 29 30 31 32 33 34 63350 63377 63404 63431 63458 1-72850 1-73052 1-73254 1-73456 1-73659 -64979 65007 65034 65061 65088 -85545 -85767 .85990 -86213 1.86437 66619 66647 66674 66702 66729 1-99574 1-99821 2-00067 2-00315 2-00562 68270 -68297 68325 68352 -68380 2-15155 2-15429 2.15704 2.15979 2-16255 30 sa 32 33 34 35 36 37 38 39 63485 63512 63539 63566 63594 1.73862 1-74065 1.74269 1 - 74473 1-74677 65116 65143 65170 65197 65225 1.86661 1-86885 1-87109 1-87334 1-87560 66756 66784 66811 66839 -6R866 2-00810 2-01059 2-01308 2-01557 2-01807 68408 68435 -68463 68490 .68518 2-16531 2-16808 2-17085 2-17363 2-17641 35 36 37 38 39 40 41 42 43 44 63621 63648 63675 63702 63729 1.74881 1.75086 1.75292 1.75497 1-75703 65252 65279 65306 65334 65361 1-87785 1-88011 1-88238 1-88465 1-88692 66894 66921 66949 66976 . 67003 2-02057 2-02308 2-02559 2-02810 2-03062 -68546 68573 68601 .68628 -68656 2-17920 2-13199 2-18479 2-18759 2-19040 40 41 42 43 44 45 46 47 48 49 63756 63783 63810 63838 63865 1.75909 1-76116 1.76323 1.76530 1.76737 65388 65416 65443 65470 65497 1-88920 1-89148 1.89376 1-89605 1.89834 -67031 67058 67086 67113 67141 2-03315 2-03568 2-03821 2-04075 2-04329 .68684 68711 68739 68767 .68794 2-19322 2-19604 2-19886 2-20169 2-20453 45 46 47 48 49 50 51 52 53 54 63892 63919 63946 63973 64000 1.76945 1-77154 1-77362 1.77571 1.77780 65525 65552 65579 65607 65634 .90063 .90293 .90524 90754 -90986 67168 67196 67223 67251 67278 2-04584 2-04839 2-05094 2-05350 2-05607 -68822 . 68849 68877 68905 68932 2-20737 2-21021 2-21306 2-21592 2-21878 50 51 52 53 54 55 56 57 58 1L. 64027 64055 64082 64109 64136 1.77990 1-78200 1-78410 1.78621 1-78832 65661 65689 65716 65743 65771 -91217 -91449 1-91681 1.91914 1-92147 67306 67333 67361 67388 67416 2-05864 2-06121 2-06379 2-06637 2-06896 68960 68988 69015 69043 69071 2-22165 2-22452 2-22740 2-23028 2.23317 55 56 57 58 59 60 64U63 1.79043 657.98 1-92380 67443 2-07155 69098 2-23607 60 246 TABLE 11. NATURAL VERSED STNES AND EXTERNAL SECANTS. 72 73 74 75 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' o 1 2 3 | -69098 .69126 .69154 69181 .69209 2-23607 2.23897 2-24187 2-24478 2-24770 -70763 -70791 -70818 -70846 -70874 2-42030 2-42356 2-42683 2-43010 2-43337 -72436 -72464 -72492 .72520 . 72548 2-62796 2.63164 2-63533 2-63903 2-64274 -74118 -74146 -74174 -74202 -74231 2-86370 2-86790 2-87211 2-87633 2-88056 O 1 2 3 5 6 7 8 9 .69237 .69264 .69292 .69320 .69347 2-25062 2-25355 2-25648 2-25942 2-26237 .70902 .70930 .70958 -70985 .71013 2-43666 2-43995 2-44324 2-44655 2.44986 72576 -72604 72632 -72660 .72688 2.64645 2.65018 2-65391 2-65765 2.66140 -74259 .74287 .74315 .74343 .74371 2-88479 2-88904 2-89330 2-89756 2-90184 5 6 7 8 9 10 11 12 13 14 -69375 - 69403 -69430 .69458 .69486 2-26531 2-26827 2-27123 2-27420 2-27717 -71041 -71069 -71097 -71125 -71153 2-45317 2-45650 2-45983 2-46316 2-46651 .72716 - 72744 -72772 -72800 .72828 2-66515 2-66892 2-67269 2-67647 2-68025 74399 . 74427 74455 - 74484 .74512 2.90613 2-91042 2-91473 2.91904 2-92337 10 11 12 13 14 15 16 17 18 19 69514 .69541 -69569 -69597 -69624 2-28015 2-28313 2-28612 2-28912 2-29212 -71180 .71208 -71236 -71264 -71292 2-46986 2-47321 2-47658 2-47995 2 48333 .72856 -72884 -72912 -72940 .72988 2-68405 2-68785 2-69167 2-69549 2-69931 . 74540 -74568 -74596 -74624 -74652 2-92770 2-93204 2-93640 2-94076 2-94514 15 16 17 18 19 20 21 22 23 24 -69652 -69680 .69708 .69735 .69763 2-29512 2-29814 2-30115 2-30418 2.30721 -71320 -71348 -71375 71403 .71431 2-48671 2-49010 2-49350 2-49691 2-50032 -72996 -73024 -73052 73080 -73108 2-70315 2-70700 2-71085 2.71471 2-71858 - 74680 - 74709 -74737 -74765 . 74793 2.94952 2-95392 2-95832 2-96274 2-96716 20 21 22 23 24 25 26 27 28 29 .69791 .69818 -69846 -69874 -69902 2.31024 2.31328 2-31633 2-31939 2-32244 -71459 -71487 -71515 .71543 .71571 2-50374 2-50716 2-51060 2-51404 2-51748 -73136 -73164 -73192 .73220 -73248 2.72246 2-72635 2-73024 2-73414 2-73806 . 74821 . 74849 .74878 - 74906 - 74934 2.97160 2-97604 2.98050 2.98497 2-98944 23 26 27 28 29 30 31 32 33 34 -69929 -69957 -69985 -70013 - 70040 2.32551 2-32858 2-33166 2-33474 2-33783 71598 -71626 -71654 -71682 -71710 2.52094 2-52440 2-52787 2-53134 2-53482 -73276 -73304 73332 -73360 -73388 2-74198 2-74591 2-74984 2-75379 2-75775 -74962 -74990 -75018 .75047 .75075 2-99393 2-99843 3-00293 3-00745 3-01198 3O 31 32 33 34 35 36 37 38 19- 70068 70096 -70124 70151 -70179 2-34092 2-34403 2-34713 2-35025 2-35336 -71738 -71766 -71794 -71822 .71850 2-53831 2-54181 2-54531 2-54883 2-55235 -73416 - 73444 - 73472 -73500 -73529 2-76171 2-76568 2-76966 2-77365 2-777R5 -75103 -75131 -75159 -75187 .75216 3-01652 3.02107 3-02563 3-03020 3-03479 35 36 37 38 39 40 41 42 43 44 -70207 -70235 - 70263 - 70290 -70318 2-35649 2-35962 2-36276 2.36590 2-36905 -71877 -71905 -71933 -71961 -71989 2-55587 2-55940 2.56294 2-56649 2.57005 -73557 -73585 -73613 -73641 -73669 2-78166 2-78568 2-78970 2-79374 2-79778 - 75244 -75272 -75300 -75328 -75356 3-03938 3-04398 3-04860 3-05322 3-05786 40 41 42 43 44 45 46 47 48 49 - 70346 -70374 - 70401 . 70429 -70457 2-37221 2-37537 2-37854 2.38171 2-38489 -72017 - 72045 -72073 -72101 .72129 2-57361 2-57718 2-58076 2-58434 2-58794 -73697 -73725 -73753 -73781 -73809 2-80183 2-80589 2-80996 2-81404 2-81813 -75385 -75413 - 75441 .75469 -75497 3-06251 3-06717 3-07184 3-07652 3-08121 45 46 47 48 49 50 51 52 53 54 -70485 -70513 -70540 70568 .70596 2-38808 2-39128 2-39448 2.39768 2.40089 .72157 72185 .72213 72241 72269 2.59154 2-59514 2-59876 2-60238 2.60601 -73837 -73865 -73893 -73921 -73950 2-82223 2-82633 2-83045 2-83457 2-83871 -75526 -75554 -75582 -75610 -75639 3-08591 3-09063 3-09535 3-10009 3-10484 5O 51 52 53 54 55 56 57 58 59. .70624 70652 70679 70707 -70735 2-40411 2-40734 2-41057 2-41381 2-41705 72296 -72324 -72352 72380 -72408 2-60965 2-61330 2-61695 2-62061 2-62428 -73978 -74006 - 74034 -74062 -74090 2-84285 2-84700 2-85116 2-85533 2-85951 -75667 -75695 -75723 -75751 -75780 3-10960 3.11437 3-11915 3.12394 3-12875 55 56 57 58 59 60 70763 2-42030 .72436 2-62796 -74118 2-86370 -75808 3-13357 60 247 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 76 77 78 79 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' 2 3 4 -75808 75836 75864 75892 75921 3-13357 3-13839 3.14323 3-14809 3-15295 77505 77533 77562 77590 77618 3-44541 3-45102 3-45664 3-46228 3-46793 .79209 79237 79266 79294 -79323 3-80973 3-81633 3-82294 3-82956 3-83621 -80919 80948 -80976 -81005 81033 4-24084 4-24870 4-25658 4-26448 4-27241 O 1 2 3 4 5 6 7 8 9 75949 75977 76005 76034 76062 3-15782 3.16271 3-16761 3-17252 3-17744 77647 77675 77703 77732 77760 3-47360 3-47928 3-48498 3-49069 3-49642 79351 .79380 79408 -79437 -79465 3.84288 3-84956 3.85627 3-86299 3-86973 -81062 81090 81119 81148 81176 4.28036 4-28833 4-29634 4-30436 4-31241 5 6 7 8 9 10 11 12 13 14 76090 76118 76147 76175 76203 3-18238 3.18733 3-19228 3.19725 3.20224 .77788 .77817 .77845 .77874 77902 3-50216 3-50791 3-51368 3-51947 3-52527 - 79493 -79522 -79550 -79579 - 79607 3-87649 3-88327 3-89007 3-89689 3-90373 .81205 81233 81262 -81290 -81319 4-32049 4.32859 4.33671 4.34486 4-35304 10 11 12 13 14 15 16 17 18 19 76231 76260 76288 76316 76344 3-20723 3-21224 3-21726 3-22229 3-22734 .77930 .77959 -77987 78015 -78044 3-53109 3-53692 3.54277 3 - 54863 3-55451 -79636 -79664 -79693 .79721 -79750 3.91058 3-91746 3.92436 3-93128 3-93821 81348 81376 81405 -81433 .81462 4-36124 4-36947 4-37772 4-38600 4-39430 15 16 17 18 19 20 21 22 23 24 76373 . 76401 76429 76458 76486 3-23239 3-23746 3-24255 3-24764 3-25275 . 78072 .78101 .78129 78157 78186 3 . 56041 3-56632 3.57224 3.57819 3-58414 -79778 -79807 -79835 79864 79892 3-94517 3-95215 3-95914 3-96616 3-97320 81491 -81519 .81548 .81576 .81605 4-40263 4-41099 4-41937 4-42778 4-43622 20 2. 2i. 23 24 25 26 27 28 29 76514 76542 76571 76599 .76627 3.25787 3-26300 3-26814 3-27330 3-27847 . 78214 . 78242 . 78271 . 78299 -78328 3-59012 3-59611 3-60211 3.60813 3-61417 79921 79949 79978 8C006 80035 3-98025 3-98733 3-99443 4-00155 4-00869 81633 81662 81691 81719 -81748 4-44468 4-45317 4-46169 4-47023 4-47881 25 26 27 28 29 30 31 32 33 34 76655 76684 .76712 76740 76769 3-28366 3-28885 3 - 29406 3-29929 3-30452 78356 78384 78413 . 78441 . 78470 3 - 62023 3-62630 3-63238 3 63849 3-64461 -80063 -80092 -80120 -80149 -80177 4-01585 4-02303 4-03024 4-03746 4-04471 81776 81805 -81834 31862 81891 4-48740 4-49603 4-50468 4-51337 4-52208 30 31 32 33 34 35 36 37 38 2* 76797 76825 76854 76882 76910 3-30977 3-31503 3-32031 3-32560 3-33090 . 78498 78526 78555 .78583 78612 3-65074 3-65690 3-66307 3-66925 3-67545 -80206 -80234 -80263 -80291 .80320 4-05197 4-05926 4-06657 4-07390 4.08125 81919 81948 81977 82005 -82034 4-53081 4-53958 4-54837 4-55720 4.56605 35 36 37 38 -29 40 41 42 43 44 76938 76967 76995 77023 77052 3-33622 3-34154 3-34689 3-35224 3-35761 78640 78669 78697 78725 78754 3-68167 3-68791 3.69417 3 - 70044 3 - 70673 80348 80377 -80405 80434 80462 4-08863 4-09602 4-10344 4-11088 4-11835 82063 82091 82120 82148 -82177 4-57493 4-58383 4-59277 4-60174 4-61073 40 41 42 43 44 45 46 47 48 49 77080 77108 77137 77165 77193 3-36299 3-36839 3-37380 3-37923 3-38466 78782 78811 78839 78868 78896 3-71303 3-71935 3-72569 3-73205 3-73843 -80491 80520 80548 80577 80605 4-12583 4-13334 4-14087 4-14842 4-15599 82206 82234 -82263 82292 82320 4-61976 4-62881 4-63790 4-64701 4-65616 45 46 47 48 49 50 51 52 53 54 77222 77250 77278 77307 77335 3-39012 3-39558 3-40106 3-40656 3-41206 78924 78953 78981 79010 79038 3 74482 3-75123 3-75766 3-76411 3-77057 80634 80662 80691 80719 80748 4-16359 4-17121 4-17886 4-18652 4-19421 82349 82377 82406 82435 82463 4-66533 4-67454 4-68377 4-69304 4.70234 50 51 52 53 54 55 56 57 58 19 77363 77392 77420 77448 77477 3-41759 3-42312 3-42867 3-43424 3.43982 79067 79095 79123 79152 79180 -77705 -78355 -79007 -79661 -80316 80776 80805 80833 80862 80891 4.20193 4-20966 4-21742 4-22521 4-23301 82492 82521 82549 82578 82607 4-71166 4-72102 4-73041 4-73983 4-74929 55 56 57 58 59 60 77505 8-44541 79209 .80973 80919 4-24084 82635 4-75877 60 248 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 80 81 82 83 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. t 2 3 4 -82635 82664 .82692 .82721 -82750 4-75877 4-76829 4.77784 4-78742 4.79703 .84357 .84385 .84414 .84443 . 84471 5-39245 5-40422 5.41602 5.42787 5-43977 -86083 -86112 -86140 -86169 -86198 6-18530 6-20020 6.21517 6.23019 6-24529 .87813 .87842 -87871 -87900 .87929 7.20551 7.22500 7.24457 7-26425 7-28402 O 2 3 5 6 7 8 9 82778 -82807 82836 82864 -82893 4.80667 4-81635 4-82606 4-83581 4.84558 .84500 -84529 -84558 .84586 .84615 5-45171 5-46369 5.47572 5.48779 5-49991 -86227 -86256 -86284 86313 .86342 6-26044 6-27566 6-29095 6-30630 6-32171 87957 .87986 88015 . 88044 88073 7-30388 7-32384 7-34390 7-36405 7-38431 5 6 7 8 9 10 11 12 13 14 82922 82950 -82979 .83008 83036 4.85539 4-86524 4-87511 4-88502 4-89497 .84644 .84673 .84701 .84730 .84759 5.51208 5-52429 5-53655 5.54886 5-56121 -86371 86400 -86428 -86457 .86486 6-33719 6-35274 6-36835 6-38403 6-39978 .88102 88131 -88160 .88188 .88217 7-40466 7-42511 7-44566 7-46632 7-48707 10 11 12 13 14 15 16 17 18 19 83065 -83094 -83122 -83151 .83180 4.90495 4-91496 4-92501 4-93509 4-94521 .84788 .84816 .84845 .84874 . 84903 5.57361 5-58606 5.59855 5-61110 5-62369 -86515 -86544 -86573 -86601 -86630 6-41560 6-43148 6-44743 6-46346 6-47955 -88246 88275 -88304 -88333 88362 7-50793 7-52889 7-54996 7-57113 7-59241 15 16 17 18 19 20 21 22 23 24 83208 83237 -83266 83294 83323 4-95536 4-96555 4-97577 4-98603 4-99633 -84931 -84960 -84989 .85018 .85046 5.63633 5-64902 5-66176 5-67454 5-68738 -86659 -86688 -86717 86746 .86774 6-49571 6-51194 6-52825 6-54462 6-56107 -88391 -88420 88448 -88477 .88506 7-61379 7-63528 7-65688 7-67859 7-70041 20 21 22 23 24 25 26 27 28 29 83352 83380 83409 83438 83467 5-00666 5-01703 5.02743 5-03787 5-04834 .85075 .85104 .85133 85162 .85190 5-70027 5-71321 5-72620 5-73924 5-75233 86803 .86832 86861 .86890 .86919 6-57759 6-59418 6-61085 6-62759 6 - 64441 88535 88564 -88593 -88622 -88651 7-72234 7.74438 7.76653 7.78880 7.81118 25 26 27 28 29 30 31 32 33 34 83495 83524 83553 83581 83610 5-05886 5-06941 5-08000 5-09062 5-10129 .85219 -85248 -85277 -85305 -85334 5-76547 5-77866 5-79191 5-80521 5-81856 -86947 -86976 -87005 -87034 -87063 6.66130 6-67826 6-69530 6-71242 6-72962 -88680 -88709 88737 .88766 .88795 7.83367 7-85628 7-87901 7.90186 7-92482 30 31 32 33 34 35 36 37 38 39- 83639 83667 83696 -83725 83754 5-11199 5.12273 5-13350 5-14432 5-15517 -85363 .85392 .85420 .85449 .85478 5-83196 5-84542 5-85893 5-87250 5-88612 .87092 .87120 -87149 -87178 -87207 6-74689 6-76424 6-78167 6-79918 6-81677 -88824 -88853 .88882 .88911 - 88940 7,94791 7-97111 7-99444 8-01788 8.04146 35 36 37 38 -39 40 41 42 43 44 -83782 -83811 83840 -83868 -83897 5-16607 5-17700 5-18797 5.19898 5-21004 -85507 -85536 -85564 -85593 -85622 5-89979 5.91352 5.92731 5.94115 5-S5505 -87236 -87265 -87294 87322 -87351 6.83443 6-85218 6-87001 6-88792 6-90592 -88969 -88998 89027 .89055 -89084 8-06515 8-08897 8-11292 8-13699 8-16120 40 41 42 43 44 45 46 47 48 49 .83926 .83954 .83983 -84012 . 84041 5.22113 5-23226 5.24343 5.25464 5.26590 -85651 -85680 -85708 85737 .85766 5.96900 5.98301 5-99708 6-01120 6-02538 -87380 -87409 87438 87467 -87496 6-92400 6.94216 6-96040 6-97873 6-99714 89113 89142 .89171 -89200 .89229 8-18553 8-20999 8-23459 8-25931 8-28417 45 46 47 48 49 50 51 52 53 54 -84069 84098 .84127 .84155 .84184 5-27719 5.28853 5-29991 5-31133 5-32279 .85795 .85823 .85852 -85881 -85910 6-03962 6-05392 6-06828 6-08269 6.09717 -87524 -87553 -87582 -87611 -87640 7-01565 7-03423 7-05291 7-07167 7.09052 -89258 -89287 .89316 .89345 .89374 8-30917 8-33430 8-35957 8-38497 8-41052 5O 51 52 53 54 55 56 57 58 59 .84213 . 84242 .84270 84299 -84328 5.33429 5-34584 5-35743 5-36906 5-38073 .85939 -85967 -85996 .86025 .86054 6.11171 6.12630 6.14096 6.15568 8-17046 87669 .87698 -87726 .87755 .87784 7.10946 7.12849 7.14760 7-16681 7-18612 -89403 .89431 89460 .89489 .89518 8-43620 8-46203 8.48800 8-51411 8-54037 55 56 57 58 59 60 84357 5-39245 .86083 6.18530 -87813 7-20551 -89547 8-56677 60 249 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 84 85 86 / Vers. Ex. sec. Vers. JSx. sec. Vers. Ex. sec. t 2 3 4 89547 89576 .89605 .89634 .89663 8.56677 8.59332 8.62002 8-64687 8.67387 .91284 .91313 .91342 .91371 91400 10.47371 10.51199 10.55052 10.58932 10-62837 .93024 .93053 .93082 .93111 93140 13.33559 13.39547 13-45586 13-51676 13.57817 O I 2 3 4 5 6 7 8 9 5 6 7 8 JL 10 11 12 13 14 .89692 .89721 .89750 .89779 .89808 8.70103 8-72833 8.75579 8-78341 8.81119 .91429 .91458 .91487 -91516 .91545 10.66769 10.70728 10.74714 10.78727 10-82768 .93169 .93198 .93227 .93257 93286 13-64011 13-70258 13-76558 13-82913 13.89323 .89836 89865 .89894 89923 89952 8.83912 8-86722 8.89547 8.92389 8-95248 91574 .91603 .91632 .91661 .91690 10.86837 10-90934 10.95060 10.99214 11.03397 .93315 .93344 .93373 93402 93431 13-95788 14-02310 14-08890 14-15527 14-22223 1O 11 12 13 14 15 16 17 18 2L. 20 21 22 23 24 89981 90010 90039 90068 90097 8-98123 9.01015 9-03923 9.06849 9.09792 91719 .91748 .91777 .91806 -91835 11.07610 11.11852 11.16125 11.20427 11.24761 .93460 .93489 .93518 .93547 -93576 14-28979 14-35795 14-42672 14-49611 14-56614 15 16 17 18 19 90126 .90155 90184 90213 .90242 9.12752 9.15730 9.18725 9.21739 9-24770 91864 .91893 .91922 91951 .91980 11.29125 11.33521 11.37948 11.42408 11.46900 93605 -93634 .93663 .93692 .93721 14-63679 14-70810 14-78005 14-85268 14.92597 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 25 26 27 28 29 90271 -90300 .90329 .90358 .90386 9.27819 9.30887 9.33973 9.37077 9.40201 .92009 .92038 92067 .92096 -92125 11.51424 11.55982 11.60572 11.65197 11.69856 .93750 93779 .93808 93837 93866 14-99995 15-07462 15-14999 15-22607 15-30287 30 31 32 33 34 90415 90444 90473 90502 .90531 9.43343 9.46505 9.49685 9.52886 9-56106 .92154 .92183 92212 .92241 .92270 11.74550 11.79278 11.84042 11.88841 11-93677 .93895 -93924 .93953 -93982 .94011 15-38041 15-45869 15-53772 15-61751 15.69808 35 36 37 38 39 .90560 .90589 .90618 .90647 .90676 9.59346 9-62605 9.65885 9.69186 9-72507 .92299 .92328 .92357 92386 .92415 11.98549 12.03458 12.08040 12.13388 12.18411 - 94040 .94069 -94098 .94127 94156 15-77944 15-86159 15-94456 16-02835 16.11297 35 36 37 38 39 40 41 42 43 44 .90705 .90734 .90763 .90792 .90821 9.75849 9.79212 9.82596 9.86001 9-89428 .92444 .92473 .92502 -92531 -92560 12-23472 12.28572 12.33712 12.38891 12.44112 .94186 .94215 . 94244 - 94273 .94302 16-19843 16-28476 16.37196 16-46005 16-54903 40 41 42 43 44 45 46 47 48 49 45 46 47 48 49 .90850 .90879 .90908 .90937 .90966 9-92877 9.96348 9.99841 10-03356 10.06894 92589 -92618 .92647 .92676 -92705 12.49373 12.54676 12.60021 12.65408 12.70838 -94331 -94360 -94389 -94418 . 94447 16-63893 16-72975 16-82152 16-91424 17-00794 50 51 52 53 54 .90995 91024 .91053 .91082 .91111 10-10455 10-14039 10.17646 10.21277 10-24932 .92734 .92763 .92792 .92821 -92850 12.76312 12-81829 12-87391 12-92999 12-98651 -94476 .94505 -94534 .94563 .94592 17.10262 17.19830 17.29501 17-39274 17-49153 5O 51 52 53 54 55 56 57 58 59 .91140 .91169 .91197 .91226 91255 10-28610 10.32313 10-36040 10.39792 10-43569 .92879 .92908 .92937 .92966 .92995 13.04350 13.10096 13.15889 13.21730 13-27620 -94621 -94650 .94679 .94708 -94737 17-59139 17.69233 17.79438 17-89755 18-00185 55 56 57 58 59 60 .91284 10.47371 .93024 13-33559 .94766 18-10732 60 250 TABLE 11. NATURAL VERSED SINES AND EXTERNAL SECANTS. 87 88 89 1 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ' o 2 3 4 94766 .94795 .94825 .94854 .94883 18-10732 18-21397 18-32182 18.43088 18-54119 .96510 .96539 .96568 96597 .96626 27.65371 27.89440 28-13917 28-38812 28-64137 .98255 .98284 98313 .98342 -98371 F.3- 29869 57.26976 58-27431 59-31411 60.39105 O 1 2 3 4 5 6 7 8 9 .94912 .94941 .94970 .94999 .95028 18.65275 18.76560 18-87976 18-99524 19.11208 .96655 .96684 96714 .96743 .96772 28-89903 29-16120 29.42802 29.69960 29.97607 98400 .98429 -98458 -98487 -98517 61-50715 62.66460 63-86572 65-11304 66-40927 5 6 7 8 9 10 11 12 13 14 .95057 .95086 .95115 .95144 .95173 19.23028 19.34989 19-47093 19.59341 19-71737 .96801 .96830 .96859 .96888 .96917 30.25758 30-54425 30.83623 31-13366 31-43671 .98546 98575 .98604 .98633 -98662 67-75736 69-16047 70-62285 72-14583 73-73586 10 11 12 13 14 15 16 17 18 J9 .95202 .95231 .95260 .95289 .95318 19-84283 19-96982 20-09838 20.22852 20.36027 .96946 .96975 .97004 .97033 970R2 31-74554 32.06030 32-38118 32-70835 33-04199 -98691 .98720 .98749 .98778 .98807 75-39655 77.13274 78.94968 80-85315 82-84947 15 16 17 18 19 20 21 22 23 24 .95347 .95377 .95406 95435 .95464 20.49368 20-62876 20.76555 20.90409 21-04440 .97092 .97121 .97150 .97179 .97208 33-38232 33.72952 34-08380 34-44539 34-81452 -98836 -98866 .98895 98924 .98953 84-94561 87-14924 89-46886 91-91387 94-49471 20 21 22 23 24 25 26 27 28 29 .95493 .95522 .95551 .95580 .95609 21.18653 21.33050 21-47635 21-62413 21-77386 .97237 .97266 .97295 .97324 .97353 35-19141 35-57633 35-96953 36-37127 36.78185 .98982 .99011 .99040 .99069 .99098 97-22303 100-1119 103-1757 106-4311 109-8966 25 26 27 28 29 30 31 32 33 34 .95638 95667 .95696 .95725 .95754 21-92559 22-07935 22-23520 22-39316 22-55328 .97382 .97411 .97440 .97470 -97499 37.20155 37-63068 38-06957 38.51855 38-97797 .99127 .99156 .99186 .99215 -99244 113-5930 117-5444 121-7780 126-3253 131-2223 30 31 32 33 34 35 36 37 38 39 .95783 .95812 .95842 95871 .95900 22-71563 22-88022 23-04712 23-21637 23-38802 .97528 .97557 97586 .97615 97644 39-44820 39-92963 40-42266 40-92772 41-44525 .99278 .99302 99331 .99360 .99389 136-5111 142-2406 148-4684 155-2623 162-7033 35 36 37 38 39 40 41 42 43 44 .95929 .95958 .95987 .96016 .96045 23-56212 23-73873 23-91790 24-09969 24-28414 97673 .97702 .97731 .97760 .97789 41-97571 42-51961 43-07746 43-64980 44-23720 .99418 99447 .99476 .99505 -99535 170-8883 179-9350 189-9868 201-2212 213-8600 40 41 42 43 44 45 46 47 48 49 96074 .96103 .96132 .96161 .96190 24-47134 24-66132 24-85417 25-04994 25-24869 .97819 .97848 .97877 .97906 .97935 44-84026 45-45963 46-09596 46-74997 47-42241 .99564 .99593 .99622 .99651 .99680 228-1839 244-5540 263-4427 285-4795 311-5230 45 46 47 48 49 50 51 52 53 54 .96219 .96248 .96277 .96307 .96336 25-45051 25.65546 25.86360 26.07503 26-28981 .97964 .97993 .98022 98051 -98080 48.11406 48-82576 49-55840 50.31290 51-09027 .99709 .99738 99767 .99796 .99825 342-7752 380.9723 428.7187 490-1070 571-9581 50 51 52 53 54 55 56 57 58 59 .96365 .96394 .96423 .96452 .96481 26-50804 26-72978 26-95513 27-18417 27-41700 .98109 .98138 .98168 .98197 .98226 51.89156 52.71790 53-57046 54.45053 55-35946 .99855 .99884 .99913 .99942 .99971 686.5496 858-4369 1144-916 1717-874 3436-747 55 56 57 58 59 60 96510 27-65371 .98255 56-29869 1.00000 Infinite 60 251 252 12. CUBIC YARDS PER 100 FEET. SLOPES i : 1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 22 24 26 28 1 45 53 60 68 82 90 i 97 105 2 93 107 122 137 167 181 196 211 3 142 163 186 208 253 275 297 319 4 193 222 252 281 341 370 400 430 5 245 282 319 356 431 468 605 542 6 300 344 389 433 522 567 611 656 7 356 408 460 512 616 668 719 771 8 415 474 533 593 711 770 830 889 9 475 542 608 675 808 875 942 1008 10 537 611 685 759 907 981 1056 1130 11 601 682 764 845 1008 1090 1171 1253 12 667 756 844 933 1111 1200 1289 1378 13 734 831 926 1023 1216 1312 1408 1505 14 804 907 1010 1115 1322 1426 1530 1633 15 8<3 986 1096 1208 1431 1542 1653 1764 16 948 1067 1184 1304 1541 1659 1778 1896 17 1023 1149 1274 1401 1653 1779 1905 2031 18 1100 1233 1366 1500 1767 1900 2033 2167 19 1179 1319 1460 1601 1882 2023 2164 2305 20 1259 1407 1555 1704 2000 2148 2296 2444 21 1342 1497 1653 1808 2119 2275 2431 2586 22 1426 1589 1752 1915 2241 2404 2567 *730 23 1512 1682 1853 2023 2364 2534 2705 2875 24 1600 1778 1955 2133 2489 2667 2844 3022 25 1690 1875 2060 2245 2616 J2801 2986 3171 26 1781 1974 2166 2359 2744 2937 3130 3322 27 1875 2075 2274 2475 2875 3075 3275 3475 28 1970 2178 2384 2593 3007 3215 8422 3630 29 2068 2282 2496 2712 3142 3356 3571 3786 30 2167 2389 2610 2833 3278 3500 3722 3944 31 2288 2497 2726 2956 3416 3645 3875 4105 32 2370 2607 2844 3081 3556 3793 4030 4267 33 2475 2719 2964 3208 3697 3942 4186 4431 34 2581 2833 3085 3337 3841 4093 4344 4596 35 2690 2949 3208 3468 3986 4245 4505 4764 36 2800 3067 3333 3600 4133 4400 4667 4933 37 2912 3186 3460 3734 4282 4556 4831 5105 38 3026 3307 3589 3870 4433 4715 4996 5278 39 3142 3431 3719 4008 4586 4875 6164 5453 40 3259 3556 3852 4148 4741 5037 5333 6630 41 3379 3682 3986 4290 4897 5201 5505 5808 42 3500 3811 4122 4433 5056 5367 6678 5989 43 3623 3942 4260 4579 5216 5534 5853 6171 44 3748 4074 4400 4726 5378 5704 6030 6356 45 3875 4208 4541 4875 5542 5875 6208 6542 46 4004 4344 4684 5026 5707 6048 6389 6730 47 4134 4482 4830 5179 5875 6223 6571 6919 48 4267 4622 4978 5333 6044 6400 6756 7111 49 4401 4764 5127 5490 6216 6579 6943 7305 50 4537 4907 5278 5648 6389 6759 7130 7500 51 4675 5053 5430 5808 6564 6942 7319 7697 52 4815 5200 5584 597'0 6741 7126 7511 7896 53 4956 5349 5741 6134 i 6919 7312 7705 8097 54 5100 5500 5900 6300 ! 7100 7500 7900 &300 55 5245 5653 6060 6468 7282 7690 8097 8505 56 5393 5807 6222 6637 7467 7881 8296 8711 57 5542 5964 6386 6808 7653 8075 8497 8919 58 5693 6122 6552 6981 7841 8270 8700 9130 59 5845 6282 6719 7156 8031 3468 8905 9342 60 6000 6444 6889 7333 8222 8667 9111 9556 12. CUBIC YARDS PER 100 FEET. SLOPES * : 1. 253 Depth Base 12 Base 14 Base 1G Base 13 Base 22 Base 24 Base 26 Base 28 1 46 54 Cl CD 83 91 98 106 2 96 111 126 141 170 185 200 215 3 150 172 194 217 261 283 306 328 4 207 237 267 296 356 385 415 444 5 269 306 343 380 454 491 528 565 6 333 378 422 467 556 600 644 689 7 402 454 506 557 661 713 765 817 8 474 533 593 652 770 830 889 948 9 550 617 683 750 883 950 1017 1083 10 630 704 778 852 1000 1074 1148 1222 11 713 794 876 957 1120 1202 * 1283 1365 12 800 889 978 1067 1244 1333 1422 1511 13 891 987 1083 1180 1372 1469 1565 1661 14 985 1089 1193 1296 1504 1607 1711 1815 15 1083 1194 1306 1417 1639 1750 1861 1972 16 1185 1304 1422 1541 1779 1896 2015 2133 17 1291 1417 1543 1669 1920 2046 2172 2298 18 1400 1533 1667 1800 2067 2200 2333 2467 19 1513 1654 1794 1935 2217 2357 2498 2639 20 1630 1778 1926 2074 ' 2370 2519 2667 2815 21 1750 1906 2061 2217 2528 2683 2839 2994 22 1874 2037 2200 2363 2689 2852 3015 3178 23 2002 2172 2343 2513 2854 3024 3194 8365 24 2133 2311 2489 2667 3022 3200 3378 3556 25 2269 2454 2639 2824 3194 3380 3565 3750 26 2407 2600 2793 2985 3370 35C3 3756 3948 27 2550 2750 2950 3150 3550 3750 3950 4151 28 2696 2904 3111 3319 3733 3941 4148 4356 29 2846 3061 3276 3491 3920 4135 4350 4565 30 3000 3222 3444 3667 4111 4333 4556 4778 31 3157 3387 3617 3846 4306 4535 4765 4994 32 3319 3556 3793 4030 4504 4741 4978 5215 33 3483 3728 3972 4217 4706 4950 5194 5439 34 3652 3904 4156 4407 4911 5163 5415 5667 35 3824 4083 4343 4602 5120 5380 5639 5898 36 4000 4267 4533 4800 5333 5600 5867 6133 . 37 4180 4454 4728 5002 5550 5824 6098 6372 38 4363 4644 4926 5207 5770 6052 6333 6615 39 4550 4839 5128 5417 5994 6283 6572 6861 40 4741 5037 5333 5630 6222 6519 6815 7111 41 4935 5239 5543 5846 6454 6757 7061 7365 42 5133 5444 5756 6067 6689 7000 7311 7623 43 5335 5654 5972 6291 6928 7246 7565 7883 44 5541 5867 6193 6519 7170 7496 7822 8148 45 5750 60&3 6417 6750 7417 7750 8083 8417 46 5963 6304 6644 6985 7667 8007 8348 8689 47 6180 6528 6876 7224 7920 8269 8617 8965 48 49 6400 6624 6756 6987 7111 7350 m m 8889 9165 50 6852 7222 7593 7963 87C4 9074 9444 9815 51 7083 7461 7839 8217 8972 9350 9728 10106 52. 7319 7704 8089 8474 9244 9630 10015 10400 53 7557 7950 8343 8735 9520 9913 10306 10698 54 7800 8200 8600 9000 9800 10200 10600 11000 55 8046 8454 8861 9269 10083 10491 10898 11306 56 8296 8711 9126 9541 10370 10785 11200 11615 57 8550 8972 9394 9817 10661 11083 11506 11928 58 8807 9237 9667 10096 10956 11385 11815 12244 59 9069 9506 9943 10380 11254 11691 12128 12565 60 9333 9778 10222 10667 11556 12000 12444 12889 254 12. CUBIC YARDS PER 100 FEET. SLOPES 1 : 1. Depth Base 12 Base 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 1 48 56 63 70 78 107 115 122 104 119 133 148 163 222 237 252 3 167 189 211 233 256 344 367 389 237 267 296 326 356 474 504 533 5 315 352 389 426 463 611 648 685 6 400 444 489 533 578 756 800 844 7 493 544 596 648 700 907 959 1011 8 693 652 711 77'0 830 1067 1126 1185 9 700 767 833 900 967 1233 1300 1367 10 815 889 963 1037 1111 1407 1481 1556 11 937 1019 1100 1181 1263 1589 1670 1752 12 1067 1156 1244 1333 1422 1778 1867 1956 13 1204 1300 1396 1493 1589 1974 2070 2167 14 1348 1452 1556 1659 1763 2178 2281 2385 15 1500 1611 1722 1833 1944 2389 2500 2611 16 1659 1778 1896 2015 2133 2607 2726 2844 17 1826 1952 2078 2204 2330 2833 2959 3085 18 2000 2133 2267 2400 2533 3067 3200 3333 19 2181 2322 2463 2604 2744 3307 3448 3589 20 2370 2519 2667 2815 2963 3556 3704 3852 21 2567 272 2878 3033 3189 3811 3967 4122 22 2770 2933 3096 3259 3422 4074 4237 4444 23 2981 8152 3322 3493 3663 4344 4515 4685 24 3200 3378 3556 3733 3911 4622 4800 4978 25 3426 3611 3796 3981 4167 4907 5093 5278 26 8659 3852 4044 4237 4430 5200 5393 5585 27 3900 4100 4300 4500 4700 5500 5700 5900 28 4148 4356 4563 4770 4978 5807 6015 6222 29 4404 4619 4833 5048 5263 6122 6337 6552 30 4667 4889 5111 5333 5556 6444 6667 6889 31 4937 5167 5396 5626 5856 6774 7004 7233 32 5215 5452 5689 5926 6163 7111 7348 7585 33 5500 5744 5989 6233 6478 7456 7700 7944 34 6793 6044 6296 6548 6800 7807 8059 8311 35 6093 6352 6611 6870 7130 8167 8426 8685 , 36 6400 6667 6933 7200 7467 8533 8800 9067 37 6715 6989 7263 7537 7811 8907 9181 9456 38 7037 7319 7600 7881 8163 9289 9570 9852 39 7367 7656 7944 8233 8522 9678 9967 10256 40 7704 8000 8296 8593 8889 10074 10370 10667 41 8048 8352 8656 8959 9263 10478 10781 11085 42 8400 8711 9022 9333 9644 10889 11200 11511 43 8759 9078 9396 9715 10033 11307 11626 11944 44 9126 9452 9778 10104 10430 11733 12059 12385 45 9500 9833 10167 10500 10833 12167 12500 12833 46 9881 10222 10563 10904 11244 12607 12948 13289 47 10270 10619 10967 11315 11663 13056 13404 13752 48 10667 11022 11378 11733 12089 13511 13867 14222 49 11070 11433 11796 12159 12522 13974 14337 14700 60 11481 11852 12222 12593 12963 14444 14815 15185 51 11900 12278 12656 13033 13411 14922 15300 15678 62 12326 12711 13096 13481 13867 15407 15793 16178 63 12759 13152 13544 13937 14330 15900 16293 16G85 64 13200 13600 14000 14400 14800 16400 16800 17200 65 13648 14056 14463 14870 15278 16907 17315 17722 66 14104 14519 14933 15348 15763 17422 17837 18252 67 14567 14989 15411 15833 16256 17944 18367 18789 68 15037 15467 15896 16326 16756 18474 18904 19333 69 15515 15952 16389 16826 17263 19011 19448 19885 60 16000 16444 16889 17333 17778 19556 20000 20444 12. CUBIC YARDS PER 100 FEET. SLOPES H : 1. 255 Depth Base 12 Base 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 1 50 57 65 72 80 109 117 124 2 111 126 141 156 170 230 244 259 3 183 206 228 250 272 361 383 406 4 267 296 326 356 385 504 533 563 6 361 398 435 472 509 657 694 731 6 467 511 556 600 644 822 867 911 7 583 635 687 739 791 998 1050 1102 8 71i 770 830 889 948 1185 1244 1304 9 850 917 983 1050 1116 1383 1450 1517 10 1000 1074 1148 1222 1296 1593 * 1667 1741 11 1161 1243 1324 1406 1487 1813 1894 1976 12 1333 1422 1511 1600 1689 2044 2133 2222 13 1517 1613 1709 1806 1902 2287 2383 2480 14 1711 1815 1919 2022 2126 2541 2644 2748 15 1917 2028 2139 2250 2361 2806 2917 3028 16 2133 2252 2370 2489 2607 3081 3200 3319 1? 2361 2487 2613 2739 2865 3369 3494 3620 18 2600 2733 2867 3000 3133 3667 3800 3933 19 2850 2991 3131 3272 3413 3976 4117 4257 20 3111 3259 3407 3556 3704 4296 4444 4592 21 3383 3539 3694 3850 4005 4628 4783 4939 22 3667 3830 3993 4156 4318 4970 5133 5296 23 3961 4131 4302 4472 4642 5324 5494 5665 24 4267 4444 4622 4800 4978 5689 5867 6044 25 4583 4769 4954 5139 5324 6065 6250 6435 26 4911 5104 5296 5489 5681 6452 6644 6837 2? 5250 5450 5650 5850 6050 6850 7050 7250 28 5600 5807 6015 6222 6430 7259 7467 7674 2ft 5961 6176 6391 6606 6820 7680 7894 8109 30 6333 6556 6778 7000 7222 8111 8333 8555 31 6717 6946 7176 7406 7635 8554 8783 9013 32 7111 7348 7585 7822 8059 9007 9244 9482 33 7517 7761 8006 8250 8494 9472 9717 9962 34 7933 8185 8437 8689 8941 9948 10200 10452 35 8361 8620 8880 9139 9398 10435 10694 10954 36 8800 9067 9333 9600 9867 10933 11200 11467 37 9250 9524 9798 10072 10346 11443 11717 11991 38 9711 9993 10274 10556 10837 11963 12244 12526 39 10183 10472 10761 11050 11339 12494 12783 13072 40 10667 10963 11259 11556 11852 13037 13333 13630 41 11161 11465 11769 12072 12376 13591 13894 14198 42 11667 11978 12289 12600 12911 14156 14467 14778 43 12183 12502 12820 13139 13457 14731 15050 15369 44 12711 13037 13363 13689 14015 15319 15644 15970 45 13250 13583 13917 14250 14583 15917 16250 16583 46 13800 14141 14481 14822 15163 16526 16867 17207 47 14361 14709 15057 15406 15754 17146 17494 17843 48 14933 15289 15644 16000 16356 17778 18133 18489 49 15517 15880 16243 16606 16968 18420 18783 19146 50 . 16111 16481 16852 17222 17592 19074 19444 19815 51 16717 17094 17472 17850 18228 19739 20117 20494 52 17333 17719 18104 18489 18874 20415 20800 21185 53 17961 18354 18746 19139 19531 21102 21494 21887 54 18600 19000 19400 19800 20200 21800 22200 22600 55 19250 19657 20065 20472 20880 22509 22917 23324 56 19911 20326 20741 21156 21570 23230 23644 24059 57 20583 , 21006 21428 21850 22272 23961 24383 24805 58 21267 21696 22126 22556 22985 24704 25133 25563 59 21961 22398 22835 23272 23709 25457 25894 26332 60 22667 23111 23556 24000 24444 26222 26667 27111 256 12. CUBIC YARDS PER 100 FEET. SLOPES 2 : 1. Depth Base 12 Base 14 Base 16 Base 18 Base 20 Bae 28 Base 30 Base 32 1 52 59 67 74 81 111 119 126 2 119 133 143 1C3 178 237 252 267 3 200 222 244 267 289 878 400 422 4 296 326 856 385 415 5-33 563 693 5 407 444 481 519 556 V04 741 778 6 533 578 622 637 711 889 933 979 7 674 726 778 830 881 1089 1141 1193 8 830 889 948 1007 1067 1304 1363 1422 9 1000 1067 1133 1200 1267 1533 1600 1667 10 1185 1259 1333 1407 1481 1778 1852 1926 11 1385 1467 1548 1630 1711 2037 2119 2200 12 1600 1689 1778 1867 1956 2311 2400 2489 13 1830 1926 2022 2119 2215 2600 2696 2793 14 2074 2178 2281 2385 2489 2904 3007 3111 15 2333 2444 2356 2667 2778 3222 3333 3444 16 2607 2726 2844 2963 3081 &55(3 3674 3793 17 2896 3022 8148 3274 3400 3904 4030 4156 18 3200 8333 3457 3600 3733 4207 4400 4533 19 3519 3059 3800 3941 4081 4644 4785 4926 20 3852 4000 4148 4296 4444 5037 5185 5333 21 4200 4356 4511 4667 4822 5444 5600 5756 22 4563 4730 4889 5053 5215 5867 6030 6193 23 4941 5111 5281 5452 5622 6304 6474 6644 24 5333 5511 5689 5867 6044 6756 6933 7111 25 5741 5926 6111 6296 6481 7222 7407 7593 26 6163 6356 6548 6741 6933 7704 7896 8089 27 6600 6800 7000 7200 7400 8200 8400 8600 28 7052 7259 7467 7674 7881 8711 8919 9126 29 7519 7733 7948 8163 8378 9237 9452 9667 30 8000 8222 8444 8667 8889 9778 10000 10222 31 8496 8726 8956 9185 9415 10333 10563 10793 32 9007 9244 9481 9719 9956 10904 11141 11378 33 9533 9778 10022 10267 10511 11489 11733 11978 34 10074 10326 10378 10830 11081 12089 12341 12593 35 10630 10889 11148 11407 11667 12704 12963 13222 36 11200 11467 11733 12000 12267 13333 13600 13867 37 11785 12059 12333 12607 12881 13978 14252 14526 38 12385 12667 12948 13230 13511 14637 14919 15200 39 13000 13289 13578 13867 14156 15311 15GOO 15889 40 13630 13926 14223 14519 14815 16000 16296 16593 41 14274 14578 14881 15185 15489 16704 17007 17311 42 14'.)33 15244 15556 15867 16178 17422 17733 18044 43 15607 15926 16224 16563 16881 18156 18474 18793 44 16296 16622 16948 17274 17600 16904 19230 19556 45 17000 17333 17667 18000 18333 19667 20000 20333 46 17719 18059 18400 18741 19081 20444 20785 21126 47 18452 18800 19148 19496 19844 21237 21585 21933 48 19200 19556 19911 20267 20622 22044 22400 22756 49 19963 20326 20689 21052 21415 22867 23230 23593 50 20741 20711 21481 21852 22222 23704 24074 24444 51 21533 21911 22289 22667 23044 24556 24933 25311 *2 22341 22726 23111 23496 23881 25422 25807 26193 53 23163 2J556 23948 24341 24733 26304 26696 2?089 54 24000 21400 24800 25200 25600 27200 27600 28000 55 24852 25259 25667 26074 26481 28111 28519 28926 56 25719 26133 26548 26963 27378 29037 29452 29867 57 26600 27022 27444 27867 28289 20978 30400 30822- 58 27496 27926 28356 28785 29215 30933 31363 31793 59 28407 28844 29281 29719 30156 31904 32341 32778 60 89333 29778 30222 30667 31111 32889 33333 33778 12. CUBIC YARDS PER 100 FEET. SLOPES 3 : 1. 257 Depth Base 12 Base 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 1 56 63 70 78 85 115 122 130 2 133 148 163 178 193 252 267 281 3 233 256 278 300 322 411 433 456 4 356 385 415 444 474 593 622 652 5 500 537 574 611 648 796 833 870 6 667 711 756 800 &44 1022 1067 1111 7 856 907 959 1011 1063 1270 1322 1374 8 1067 1126 1185 1244 1304 1541 1600 1659 9 1300 1367 1433 1500 1567 1833 1900 1967 10 1556 1630 1704 1778 1852 2148 2222 2296 11 1833 1915 1996 2078 2159 2485 2567 2648 IS 2133 2222 2311 2400 2489 2844 2933 3022 13 2456 2552 2648 2744 2841 3226 3322 3419 14 2800 2904 3007 3111 3215 3630 3733 3837 15 3167 3278 3389 3500 3611 4056 4167 4278 16 3556 3674 3793 3911 4030 4504 4622 4741 17 3967 4093 4219 4344 4470 4974 5100 5226 18 4400 4533 4667 4800 4933 5467 5600 5733 19 4856 4996 5137 5278 5419 5981 6122 6263 20 5333 5481 5630 5778 5926 6519 6667 6815 21 5833 5989 6144 6300 6456 7078 7233 7389 22 6356 6519 6681 6844 7007 7659 7822 7985 23 6900 7070 7241 7411 7581 8263 8433 8504 24 7467 7644 7822 8000 8178 8889 9067 9144 25 8056 8241 8426 8611 8796 9537 9722 9807 26 8667 8859 9052 9244 9437 10207 10400 10593 27 9300 9500 9700 9900 10100 10900 11100 11300 28 9956 10163 10370 10578 10785 11615 11822 12030 29 10633 10848 11063 11278 11493 12352 12567 12781 30 11333 11556 11778 12000 12222 13111 13333 13556 31 12056 12285 12515 12744 12974 13893 14122 14352 32 12800 13037 13274 13511 13748 14696 14933 15170 as 13567 13811 14056 14300 14544 15522 15767 16011 34 14356 14607 14859 15111 15363 16370 16622 16874 35 15167 15426 15685 15944 16204 17241 17500 17759 36 16000 16267 16533 1G800 17067 18133 18400 18667 37 16856 17130 17404 17678 17959 19048 19322 19596 38 17733 18015 18296 18578 18859 19985 20267 20548 89 18633 18922 19211 19500 19789 20944 21233 21522 40 19556 19852 20148 20444 20741 21926 22222 22516 41 20500 20804 21107 21411 21715 22930 23233 23537 42 21467 21778 22089 22400 22711 23956 24267 24578 43 22456 22774 23093 23411 23730 25004 25322 25641 44 23467 23793 24119 24444 24770 26074 26400 26726 45 24500 24833 25167 25500 25833 27167 27500 27833 46 25556 25896 26237 26578 26919 28281 28622 28963 47 26633 26981 27330 27678 28026 29419 29767 30115 48 27733 28089 28800 29156 30578 30938 31289 49 28856 20219 21)581 29944 30307 31759 32122 32485 50 30000 30370 30741 31111 31481 32963 33333 33704 51 31167 31544 31922 32300 32678 34189 34567 34944 52 32356 32741 33126 33511 33396 35437 35822 36207 53 33567 33959 34352 34744 35137 36707 37100 87493 54 34800 35200 35600 36000 36400 38000 38400 38800 55 36056 36463 36870 37278 37685 39315 39722 4130 56 37333 37748 38163 38578 38993 40652 41067 41481 5? 38633 39056 39478 39900 40322 42011 42433 42856 58 39956 40385 40815 41244 41674 43393 43822 44252 59 41300 41737 42174 42611 43048 44796 45233 45670 60 42667 43111 43556 44000 44444 46222 46667 47111 258 TABLE 13. CUBIC YARDS IN 100 FEET LENGTH. Area. IS: Cubic Yards. Area. ft Cubic Yards. Area. ft Cubic Yards. Area. Sq, Ft. Cubic Yards. Area. Sq. Ft. Cubic Yards. 1 3.7 51 188.9 101 374.1 151 559.3 201 744.4 2 7.4 52 192.6 102 377.8 152 563.0 202 748.2 3 11.1 53 196.3 103 381.5 153 566.7 203 751.9 4 14.8 54 200.0 104 385.2 154 570.4 204 755.6 5 18.5 55 203.7 105 388.9 155 574.1 905 759.3 6 22.2 56 207.4 106 392.6 156 577.8 206 763.0 7 25.9 57 211.1 107 396.3 157 581.5 207. 766.7 8 29.6 58 214.8 108 400.0 158 585.2 208 770.4 9 33.3 59 218.5 109 403.7 159 588.9 209 774.1 10 37.0 60 222.2 110 407.4 160 592.6 210 777.8 11 40.7 61 225.9 111 411.1 161 596.3 211 781 5 12 44.4 62 229.6 112 414.8 162 600.0 212 785.2 13 48.1 63 233.3 113 418.5 163 603.7 213 788.9 14 51.9 64 237.0 114 422.2 164 607.4 214 792.6 15 55.6 65 240.7 115 425.9 165 611.1 215 796.3 16 59.3 66 244 .4 116 429.6 166 614.8 216 800.0 17 63.0 67 248.2 117 433.3 167 618.5 217 803.7 18 66.7 68 251.9 118 437.0 168 622.2 218 807.4 19 70.4 69 255.6 119 440.7 169 625.9 219 811.1 20 74.1 70 259.3 120 444.4 170 629.6 220 814.8 21 77.8 71 263.0 121 448.2 171 633.3 221 818.5 22 81.5 72 266.7 123 451.9 172 637.0 222 822.2 23 85.2 73 270.4 123 455 6 173 640.7 223 825.9 24 88.9 74 274.1 124 459.3 174 644.4 224 829.6 25 92.6 75 277.8 125 463.0 175 648.2 225 833.3 26 96.3 76 281.5 126 466.7 176 651.9 226 837.0 27 100.0 77 285 2 127 470.4 177 655.6 227 840.7 28 103.7 78 , 288.9 128 474 1 178 659.3 228 844.4 29 107.4 79 292.6 129 477.8 179 663.0 229 848.2 30 111.1 80 296.3 130 481.5 180 666.7 230 851.9 31 114.8 81 300.0 131 485.2 181 670.4 231 855.6 32 118.5 82 303.7 132 488 9 182 674.1 232 859.3 33 122.2 83 307.4 133 492.6 183 677.8 233 863.0 34 125.9 84 311.1 134 496.3 184 681.5 234 866.7 35 129 6 85 314.8 135 500.0 185 685.2 235 870.4 36 133.3 86 318.5 136 503.7 186 688.9 236 874.1 37 137.0 87 322.2 137 507.4 187 692.6 237 877.8 38 140.7 88 325.9 138 511.1 188 696.3 238 881.5 39 144 4 89 329.6 139 514.8 189 700.0 235 885.2 40 148.2 90 333.3 140 518.5 190 703.7 240 888.9 41 151.9 91 337.0 141 522.2 191 707.4 241 892.6 42 155.6 92 340.7 142 525.9 192 711.1 242 896.3 43 159 3 93 344.4 143 529.6 193 714.8 243 900.0 44 163.0 94 348.2 144 533.3 194 718.5 244 903.7 45 166.7 95 351.9 145 537.0 195 722.2 245 907.4 46 170.4 96 355.6 146 540.7 196 725.9 246 911.1 47 174.1 97 359.3 147 544.4 197 729.6 247 914.8 48 177.8 96 363.0 148 548.2 198 733.3 248 918.5 49 181.5 99 366.7 149 551.9 199 737.0 249 922.2 50 185.2 100 370.4 150 555.6 200 740.7 250 925.9 TABLE 13. CUBIC YARDS IN 100 FEET LENGTH. 259 Area * Cubic Yards. Area. ft Cubic Yards. Area ft Cubic Yards. Area ft Cubic Yards. Area ft Cubic Yards. 251 929.6 301 1114.8 351 1300.0 401 1485.2 451 1670.4 252 933.3 302 1118.5 352 1303.7 402 1488.9 152 1674.1 253 93?. 303 1122.2 353 1307.4 403 1492.6 453 1677.8 254 940.7 304 1125.9 a54 1311.1 404 1496.3 454 1681.5 255 944.4 305 1129.6 355 1314.8 405 1500.0 455 1685.2 256 948.2 306 1133.3 356 1318.5 406 1503.7 456 1688.9 1 257 951.9 307 1137.0 357 1322.2 407 1507.4 457 1692.6 258 955.6 308 1140.7 358 1325.9 408 1511.1 458 1696.3 259 959.3 309 1144.4 359 1329.6 409 1514.8 459 1700.0 260 963.0 310 1148.2 360 1333.3 410 1518.5 460 1703.7 261 966.7 311 1151.9 361 1337.0 411 1522.2. 461 1707.4 262 970.4 312 1155.6 362 1340.7 412 1525.9 462 1711.1 263 974.1 313 1159.3 363 1344.4 413 1529.6 463 1714.8 264 977.8 314 1163.0 364 1348.2 414 1533.3 -164 1718.5 265 981.5 315 1166.7 365 1351.9 415 1537.0 465 1722.2 266 985.2 316 1170.4 366 1355.6 416 1540.7 466 1725.9 267 988.9 317 1174.1 367 1359.3 417 1544.4 467 1729.6 268 992.6 318 1177.8 368 1363.0 418 1548.2 468 1733.3 269 996.3 319 1181.5 369 1366.7 419 1551.9 469 1737.0 270 1000.0 320 1185.2 370 1370.4 420 1555.6 470 1740.7 271 1003.7 321 1188.9 371 1374.1 421 1559.3 471 1744.4 272 1007.4 322 1192.6 372 1377.8 422 1563.0 472 1748.2 273 1011.1 323 1196.3 373 1381.5 423 1566.7 473 1751.9 274 1014.8 324 1200.0 374 1385.2 424 1570.4 474 1755.6 275 1018.5 325 1203.7 375 1388.9 425 1574.1 475 1759.3 276 1022.2 326 1207.4 376 1392.6 426 1577.8 476 1763.0 277 1025.9 327 1211.1 377 1396.3 427 1581.5 477 1766.7 278 1029.6 328 1214.8 378 1400.0 428 1585.2 478 1770.4 279 1033.3 329 1218.5 379 1403.7 429 1588.9 479 1774.1 280 1037.0 330 1222.2 380 1407.4 430 1592.6 480 1777.8 281 1040.7 331 1225.9 381 1411.1 431 1596.3 481 1781.5 282 1044.4 332 1229 6 382 1414.8 432 1600.0 482 1785.2 283 1048.2 333 1233.3 383 1418.5 433 1603.7 483 1788.9 284 1051.9 334 1237.0 384 1422.2 434 1607.4 484 1792.6 285 1055.6 335 1240.7 385 1425.9 435 1611.1 485 1796 3 286 1059.3 336 1244.4 386 1429.6 436 1614.8 486 1800.0 287 1063.0 337 1248.2 387 1433.3 437 1618.5 487 1803.7 288 1066.7 338 1251.9 388 1437.0 438 1622.2 488 1807.4 289 1070.4 339 1255.6 389 1440.7 439 1625.9 89 1811.1 290 1074.1 340 1259.3 390 1444.4 440 1629.6 490 1814.8 291 1077.8 341 1263.0 391 1448.2 441 1633.3 491 1818.5 292 1081.5 342 1266.7 392 1451.9 442 1637.0 492 1822.2 293 1085.2 : 343 1270.4 393 1455.6 443 1640.7 493 1825.9 294 1088.9 344 1274.1 394 1459.3 444 1644.4 494 1829.6 295 1092.6 345 1277.8 395 1463.0 445 1648.2 495 1833.3 296 1096.3 346 1281.5 396 1466.7 446 1651.9 496 1837.0 297 1100.0 347 1285.2 397 1470.4 447 1655.6 497 1840.7 298 1103.7 348 1288.9 398 1474.1 448 1659.3 498 1844.4 299 1107.4 349 1292.6 399 1477.8 449 1663.0 499 1848.2 300 1111.1 350 1296.3 400 1481.5 450 1G66.7 500 1851.9 260 TABLE 13. CUBIC YARDS IN 100 FEET LENGTH. Area. H: Cubic Yards. Area. 1?: Cubic Yards. Area. & Cubic Yards. Area. 1?: Cubic Yards. Area. 8t Cubic Yards. 501 1855.6 551 2040.7 601 2225.9 651 2411.1 701 2596.3 502 1859.3 552 2044.4 602 2229.6 652 2414.8 I 702 2000.0 503 1863.0 553 2048.2 603 2233.3 653 2418.5 i 703 2603.7 504 1866.7 554 2051.9 604 2237.0 654 2422.2 704 2607.4 505 1870.4 555 2055.6 605 2240.7 655 2425.9 705 2611.1 506 1874.1 556 2059.3 606 2244.4 656 2429.6 706 2614.8 507 1877.8 557 2063.0 607 2248.2 657 2433.3 707 2618.5 508 1881.5 558 2066.7 608 2251.9 658 2437.0 708 2622.2 509 1885.2 559 2070.4 609 2255.6 659 2440.7 709 2625.9 510 1888.9 560 2074.1 610 2259.3 660 2444 .4 710 2629.6 511 1892.6 561 2077.8 611 2263.0 661 2448.2 711 2633.3 512 1896.3 562 2081.5 612 2266.7 662 2451.9 712 2637.0 513 1900.0 563 20&>.2 613 2270.4 663 2455.6 713 2640.7 514 1903.7 564 2088.9 614 2274.1 664 2459.3 714 2644.4 515 1907.4 565 2092.6 615 2277.8 665 2463.0 715 2648.2 516 1911.1 566 2096.3 616 2281.5 666 2466.7 716 2651.9 517 1914.8 567 2100.0 617 2285.2 667 2470.4 717 2655.6 518 1918.5 568 2103.7 618 2288.9 668 2474.1 718 2659.3 519 1922.2 569 2107.4 619 2292.6 669 2477.8 719 2663.0 520 1925.9 570 2111.1 620 2296.3 670 2481.5 720 2666.7 521 1929 6 571 2114.8 621 2300.0 671 2485.2 721 2670.4 522 1933.3 572 2118.5 622 2303.7 672 2488.9 722 2674.1 523 1937.0 573 2122.2 623. 2307.4 673 2492 6 723 2677.8 524 1940.7 574 2125.9 624 2311.1 674 2496.3 724 2681.5 525 1944.4 575 2129.6 625 2314.8 675 2500.0 725 2685.2 526 1948.2 576 2133.3 626 2318.5 676 2503.7 726 2688.9 527 1951.9 577 2137.0 627 2322.2 677 2507.4 727 2692.6 528 1955.6 578 2140.7 628 2325.9 678 2511.1 728 2696.3 529 1959.3 579 2144.4 629 2329.6 679 2514.8 729 2700.0 530 1963.0 580 2148 2 630 2333.3 680 2518.5 730 2703.7 531 1966.7 581 2151.9 631 2337.0 681 2522.2 731 2707.4 532 1970.4 582 2155.6 632 2340.7 682 2525.9 732 2711.1 533 1974.1 583 2159.3 633 2344.4 683 2529.6 733 2714.8 534 1977.8 584 2163.0 634 2348.2 684 2533.3 734 2718.5 535 1981.5 585 2166.7 635 2351.9 685 2537.0 735 2722.2 536 1985.2 586 2170.4 636 2355.6 686 2540 7 736 2725.9 537 1988.9 587 2174.1 637 2359.3 687 2544.4 737 2729.6 538 1992.6 588 2177.8 638 2363 688 2548.2 738 2733.3 539 1996.3 589 2181.5 639 2366.7 689 2551.9 739 2737.0 540 2000.0 590 2185.2 640 2370.4 690 2555.6 740 2740.7 541 2003.7 591 2188.9 641 2374.1 691 2559.3 741 2744.4 542 2007.4 592 2192.6 642 2377.8 692 2563.0 742 2748.2 543 2011.1 593 2196.3 643 2381.5 693 2566.7 743 2751.9 544 2014.8 594 2200.0 644 2385.2 694 2570.4 744 2755.6 545 2018.5 595 2203.7 645 2388.9 695 2574.1 745 2759.3 546 2022.2 596 2207.4 646 2:592.6 696 2577.8 746 2763.0 547 2025.9 597 2211.1 ; 647 2396.3 697 2581.5 747 2766.7 548 2029.6 598 2214.8 648 2400.0 698 2585.2 748 2770.4 549 2033.3 599 2218.5 649 2403.7 699 2588.9 749 2774.1 550 2037.0 600 2222.2 650 2407.4 700 2592.6 750 2777.8 TABLE 13. CUBIC YARDS IN 100 FEET LENGTH. 261 Area. !?: Cubic Yards. Area. q- tt. Cubic Yards. Area. SSq. i F <- Cubic Yards. Area. ft Cubic Yards. Area. St Cubic Yards. 751 2781.5 801 2966.7 851 3151.9 901 3337.0 951 3522.2 752 2785.2 802 2970.4 852 3155.6 902 3340.7 952 3525.9 753 2788.9 803 2974 . 1 8o3 3159.3 9U3 3344.4 953 3529.6 754 2792.6 8o4 2977.8 854 3163.0 904 3348.2 954 3533.3 755 2796.3 805 2981.5 8f>5 3166.7 905 3351.9 955 3537.0 756 2800.0 806 2985.2 856 3170.4 906 3355.6 956 3540.7 757 2803.7 807 2988.9 857 3174.1 907 3359.3 957 3544.4 758 2807.4 808 2992.6 858 3177.8 908 3363.0 958 3548.2 759 2811.1 809 2996.3 859 3181.5 909 3366.7 959 3551.9 760 2814.8 810 3000.0 860 3185.2 910 3370.4 960 3555.6 761 2818.5 811 3003 7 861 3188.9 911 3374.1 961 3559.3 762 2822 2 812 3007.4 862 3192.6 912 3377.8 962 3563.0 763 2825.9 813 3011.1 863 3196.3 913 3381.5 963 3566.7 764 2829.6 814 3014.8 864 3200.0 914 3385.2 964 3570.4 765 2833 3 815 3018.5 865 3203.7 915 3388.9 965 3574.1 766 2837.0 816 3022.2 866 3207.4 916 3392.6 966 3577.8 767 2840.7 817 3025.9 867 3211.1 917 3396.3 967 3581.5 768 2844 4 818 3029.6 868 3214.8 918 3400.0 968 3585.2 769 2848.2 819 3033.3 869 3218.5 919 3403.7 969 3588.9 770 2851.9 820 3037.0 870 3222.2 920 3407.4 970 3592.6 771 2855.6 821 3040.7 871 3225.9 921 3411.1 971 3596.3 772 2859.3 822 3044.4 872 3229.6 922 3414.8 972 3600.0 773 2863.0 823 3048.2 873 3233.3 923 3418.5 973 3603.7 774 2866.7 824 3051.9 874 3237.0 924 3422.2 974 3607.4 775 2870.4 825 3055.6 875 3240.7 925 3425.9 975 3611.1 776 2874.1 826 3059.3 876 3244.4 926 3429.6 976 3614.8 777 2877.8 827" 3063.0 877 3248.2 927 3433.3 977 3618.5 778 2S81.5 828 3066.7 878 3251.9 928 3437.0 978 3622.2 779 2885.2 829 3070.4 879 3255.6 929 3440.7 979 3625.9 780 2888.9 830 3074.1 880 3259.3 930 3444.4 980 3629.6 781 2892.6 831 3077.8 881 3263.0 931 3448.2 981 3633.3 782 2896.3 832 3081.5 882 3266.7 932 3451.9 92 3637.0 783 2900.0 833 3085.2 883 3270.4 933 3455.6 983 3640.7 784 2903 7 834 3088.9 884 3274.1 934 3459.3 984 3644.4 785 2907.4 835 3092.6 885 3277.8 935 3463.0 985 3648.2 786 '^91 1 . 1 836 3096.3 886 3281.5 936 3466.7 9*6 3651.9 787 2914.8 837 3100.0 887 3285.2 937 3470.4 987 3655.6 788 2918.5 838 3103.7 888 3288.9 938 3474.1 988 3659.3 789 2922.2 839 3107.4 889 3292.6 939 3477.8 Q?9 3663.0 790 2925.9 840 3111.1 890 3296.3 940 3481.5 990 3666.7 791 2929.6 841 3114.8 891 3300.0 941 8485.2 991 3670.4 792 2933.3 842 3118.5 892 3303.7 942 3488.9 992 3674.1 793 2937.0 843 3122.2 893 3307.4 943 3492.6 993 3677.8 ?94 2940.7 844 3125.9 894 3311.1 9)4 3496.3 994 3681.5 795 2944.4 845 3129.6 895 3314.8 945 3MX).0 995 8685.2 796 2948.2 846 3133.3 896 3318.5 946 3503.7 996 3688.9 797 2051.9 847 3137.0 897 3322.2 947 3507.4 997 3692.6 798 2955.6 848 3140.7 808 3325.9 948 3511.1 998 3696.3 799 2959.3 849 3144.4 899 3329.6 949 3514.8 999 3700.0 800 2963.0 850 3148.2 900 3333.3 950 3518.5 1000 3703.7 262 TRIGONOMETRIC FORMULAS. TABLE 14. TRIGONOMETRIC FORMULAS. FIG. 98. RIGHT TRIANGLES. Z> FIG. 99, In Fig 99, let DCE be the arc of a quadrant, ABC a right triangle, the angle 13 A C subtended by the arc CE = A, and consider the radius A C = unity. Then BC = am AB = cosA. HE = tan A. DF =cotA. AH = sec A. ^ljP=cosec^.. J5 r = versinJ.. DI = co versing, CH= exsec-4. CF = coexsec^.. Using the small letters a, &, c, to represent the sides of a right triangle in Fig. 98 or 99, we may write sin^l = -; eosec.4 -; .-. sin>4 = ... 6 a cosec^l = -; sec A--, .'.cosA= 6 c sec A = -: cot A =-;.'. tan A = L . c a cot-4 OBLIQUE TRIANGLES 263 FORMULAS FOR LOGARITHMIC COMPUTATION. Given one side, a, and the three angles A, B, and C. The "sine proportion" states that each side of a triangle is proportional to the sine of the angle opposite it. , , a sin B . a sin C Then, b = -. -r- and C = Area . cauu \s - . 4 . sm A mnA a 2 sin B sin C 2 sin (B+ C) Given two sides a and 6 and the included angle C. Ftrs/ method: Determining the angles first. $(A + B) = J (180 - C) tan \(A - B) = ~-^ tan J (A + B). A = J (A + B) + | (A - B) B = i (A + B) - $(A - B). The third side, c, can now be determined by the sine proportion. Second method: Determining the third side first. 2 sin i C /-- tan x = - Va h. a b a - b c = COS X The angles A and B can now be found by the sine proportion Area = J a b sin C. Given the three sides a, 6, and c. Let s = >- (a + 6 -f c). \| (s a) (s b) (s c) Then *tan \A = *tan J -i- (s - a) ( - 6) (.s - c) - (s a) (s b) (s c) Area = -y/ s (s a) (s 6) (s c) * It will be noted that the numerator is the same in each case and that the denominators are quantities whose logarithms are known and hence two additional logarithmic operations over that required for one angle gives all three angles and a complete check on the entire compu- tation since A + B + c = 180. 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