■>:■■' W ■ <■ ;■■/«! ■ ■ ■ \ u ^ "^ 8 I A > ,#• $> * *4-> > -71 .., ' ' ^ >S -71 '" c» > V s ^ 1 v V ^ HANDBOOK FOR HIGHWAY ENGINEERS g^^^^'K ll ! ^■;;" l l ^■^ l M^ ^: l: ! ; L ;;v l l l ■ ^ -x. ;"^,^,r: ' ^ ,- ; l :-,■ ■■■:■ , ■ !| -: l " ^ '-^-^ l ';. i; v 1 .-.' :^-:' ' '- ■'' ■ | : i ,' i 1 ,. ■ ' -'-..';■':;■' ■ ^^..'i 1 - :;i; i :; ;i - y.'-'^ 1 : ' • McGraw-Hill BookCompany Pu/>fis/iers qf3oo£s/br Electrical World The Engineering andMining Journal Engineering Record , Engineering News Railway Age G aze tte American Machinist Signal ELnginoer American Engineer Electric Railway Journal Coal Age Metallurgical and Chemical Engineering Power iillHlllllllllllillllll HANDBOOK FOR HIGHWAY ENGINEERS CONTAINING INFORMATION ORDIN\RILY USED IN THE DESIGN AND CONSTRUC- TION OF ROADS WARRANTING AN EXPENDITURE OF $5,000 TO $30,000 PER MILE PART I. Principles of Design. PART II. Practice of Design and Construction BY WILSON G. HARGER, C.E. First Assistant Engineer, New York State Department of High- ways; Associate Member American Society of Civil Engineers AND EDMUND A. BONNEY Supervising Engineer, New York State Department of Highways SECOND EDITION Entirely Revised and Enlarged First Impression McGRAW-HILL BOOK COMPANY, Inc. 239 WEST 39TH STREET, NEW YORK LONDON: HILL PUBLISHING CO., Ltd. 6 & 8 BOUVERIE ST., E.C I916 1"^! ^ H^ «.* Copyright, 1912, 1916, by the McGraw-Hill Book Company, Inc. F/iOT EDITION First Impression, August, 191 2 Second Impression, September, 191 3 SECOND EDITION First Impression, June, 1916 JUL 21 1916 CI.A4311M2 PREFACE TO SECOND EDITION Since the publication of the first edition of this book four years ago, considerable progress has been made in the practice of road design and construction. To meet this advance, this handbook has been revised by bringing the material on top courses up-to-date, and by adding considerable data on tests, designs, costs, mainte- nance and specifications. Not only has much of the old material been revised, but new material, totaling approximately ioo pages, has been added. The criticisms and suggestions of many who have used the book in the field and office have aided the authors in this revision. A more complete and systematic index has been prepared by Mr. Percy Waller. The general arrangement of the book remains untouched. W. G. H. E. A. B. Rochester, N.Y., May, 1916. PREFACE TO FIRST EDITION The purpose of this book is to collect, in a compact and conven- ient form, information ordinarily required in the field and office practice of road design and construction. The book is designed to meet the requirements of both experienced and inexperienced road men. The material on the relative impor- tance of the different parts of the design, and the possibilities of economy, without impairing the efficiency of the road, are primarily for the inexperienced engineer. The collection of cost data and the tables will be useful to any one engaged in road work. As it is difficult to avoid clerical errors and mistakes in proof- reading in first editions, we shall appreciate the cooperation of read- ers in calling our attention to any errors. W. G. H. E. A. B. Rochester, N.Y., April, 19 12. TABLE OF CONTENTS PART I PAGE Introductory i General 1-3 Chapter I. Grades and Alignment 4-18 Maximum Grades Relative importance of automobile and horse traffic in the selection of grades 4 Difficulty of ascent and ease and safety of descent ... 6 The theoretical grades that fulfil certain traffic require- ments and the practical considerations which govern the selection n The construction of ruling grades 12 Minimum Grades On score of drainage 12 Level grades 12 Intermediate Grades Economy of earthwork 13 Short grades 13 Reverse vertical curves 13 Conditions intermediate grades must fulfil 13 Common mistakes in designing 13 Alignment Minimum radius of curvature 18 Utilization of old roadbed 18 Sight distance 18 Chapter II. Sections 19-32 Conditions they must fulfil 19 Premises of Design Crowns 19 Shoulder slope 19 Width of metaling 20 Width of shoulder 21 Depth of ditch 21 Economies effected by a judicious selection 22 Examples of sections in current use 23 Discussion of widths 28 Shoulder treatment 29 viii CONTENTS Chapter III. Drainage 33-60 Culverts Kinds used 33 Self-cleaning velocities 33 Determination of size 34 From existing structures 34 By maximum run-off formulae 34 Discharge capacity of small culverts 38 Side culverts 38 Village culverts . " 39 Small-span Bridges Determination of span . 39 Scour 40 Stream velocities 40 Examples of small-span bridges 41 Examples of small culverts . . . . 41 Under Drainage Side and center drains 52 Styles of construction 52 Design of outlet 53 Tables of weights of cast-iron pipe 54 Mesh reinforcement 55 Reinforcing bars 56 Cost of small culverts 58 Properties of I-beams 59 Chapter IV. Foundations for Broken Stone Roads. 61-73 The bearing power of different soils 61 Concentrated wheel loads on improved roads .... 62 The distributing action of foundation courses and the depth required for different soils 63-64 Examples of styles of construction in use 65 The distribution of stone in foundations 70 Special cases 71 Chapter V. Top Courses and their Maintenance . 74-108 Waterbound macadam 74 Waterbound macadam with surface treatment .... 76 Bituminous macadam 78 Rock asphalt 82 Amiesite 82 Brick pavements S3 Asphalt block 85 Stone block pavements 85 Concrete pavements 87 Small stone cube pavements 91 McClintock cube pavements 92 Rocmac 94 CONTENTS ix Maintenance General Discussion 96 Summarized costs 103 Typical detail costs 105 Summary of yearly costs 108 Chapter VI. Minor Points 109-117 Guard-rail Wooden . 109 Concrete . no Retaining Walls • Plain .....in Reinforced in Toe walls 113 Curbs Concrete 113 Stone 113 Guide signs 114 Danger signs 114-115 Cobble gutters • 115-116 Brick gutters, etc 116 Catch basins 116 Grates 116 Riprap 117 Dykes 117 Repointing old masonry 117 Facing old abutments 117 Chapter VII. Materials 1 18-143 Top course, macadam stone 118 Screenings 128 Bottom course, macadam stone 129 Fillers 129 Brick 129 Bituminous binders 130 Concrete materials 141 PART II Chapter VIII. The Survey 144-205 Center line 144 Levels and cross-sections 146 Drainage 149 Topography 150 Traffic reports 151 Foundation soils 151 Location and character of materials 152 X CONTENTS Right of way 154 Stadia reduction tables 156 Diversion lines 164 Adjustment of instruments 164 Curve tables and formulae 166 Examples of curve problems 199 Chapter IX. Office Practice 206-265 Mapping the Preliminary Survey Scales 206 Plotting center line 206 Table of sight distances 208 Plotting topography 208 Bench level computations 208 Cross-section levels, computations 209 Plotting cross-sections 209 Plotting profile 209 The Design Preliminary report 210 Shrinkage of earthwork 219 Templets 220 Economical grade line 221 Vertical curves 223 Formulae 223 Sight distance 225 Planimeter work ) . . . . 226 Methods 226 Accuracy 226 Computation of earthwork 227 Overhaul 250 Mass diagram 253 Macadam 255 Concrete * 255 Minor features 255 Final design report 255 Construction plans 257 Miscellaneous Points Grade line over railroad grade crossings 258 Clearances for railroad grade crossing eliminations . . 258 Computation of right-of-way areas 262 Parabolic crowns 262 Summary of economical design 263 Chapter X. Cost Data and Estimates 266-338 Macadam Roads 266 Earth excavation 266 Rock excavation 266 Unloading broken stone 268 CONTENTS xi Hauling 269 Loading fence stone 271 Spreading crushed stone 271 Placing boulder stone 272 Ratio of loose to rolled depths 272 Amounts of filler and binder 272 Loading filler sand 273 Spreading filler and binder 273 Rolling 273 Crushing Cost of 274 Proportions of different sizes in output 275 Sledging boulders for crusher 278 Dustless screenings 280 Stone fill, bottom course 281 Sub-base, bottom course 281 Applying residuum bituminous binder 281 Kentucky rock asphalt 283 Puddling waterbound roads . 283 McClintock cube surfacing 284 Amiesite 285 Hassam concrete pavement 288 Mixed concrete pavement 288-303 Asphalt block 293-291 Concrete culvert work 302-305 Guard-rail 306 Wooden 306 Concrete 306 Cobblestone gutter 307 Vitrified pipe 307 Speed of work 308 Plant and pay-roll 309 Forms of estimate 312 Sample estimate, macadam construction 318 Unit price, minor items 320 Brick Pavement on Country Roads ........ 327 Excavation 327 Concrete base 327 Preparing sand cushion 328 Laying brick 329 Grouting brick 329 Expansion joints 329 Edging^ . 330 Unloading brick 330 Hauling brick 330 Form of estimate 331 Sample estimate 331 Maintenance and repair 332 Cold oiling 333 Calcium chloride 334 xii CONTENTS Recapping 335 Scarifying and reshaping 335 Patrol maintenance, New York State 337 Automobile maintenance truck 337 Distribution of maintenance costs 338 Chapter XL Notes on Construction 339-371 Staking out 339 Rough grading 341 Fine grading 343 Sub-base 345 Bottom stone 346 Top course 348 Hassam concrete 350 Mixed concrete 350 Sheet asphalt, etc 351 Brick roads 354 Culverts • . . . 358 Chapter XII. Specifications 372-458 Materials Cement 372 Concrete aggregate 374 Stone gravel, etc., for pavements •. 376 Bituminous materials 377-389 Brick . 390 . Stone block 396 Cast iron pipe 398 Reinforcements 398 Tiles 400 Timber . 400 Methods of Construction Clearing and grubbing 401 Excavation . 401 Overhaul 404 Tiles and under drains 405 Leaching basins 405 Catch basins 406 Cast iron pipe 407 Stone fill 408 Piles 408 Timber and lumber 409 Riprap 409 Concrete masonry 410 Stone masonry 414 Stone curbing 415 Concrete curbing 416 Concrete edging 417 CONTENTS xiii Cobble gutters 417 Concrete gutters 418 Brick gutters 418 Concrete reinforcement 418 Guard rail 420 Guide signs 421 Sign posts 422 Loose stone 423 Sub-base 423 Telford base 424 Bottom courses 425 Concrete foundation 427 Top courses macadam 428 Scarifying and reshaping 431 Bituminous surface treatments . 432 Bituminous macadams 433 Bitulithic 439 Amiesite 441 Hassam concrete pavement . . • 442 Mixed concrete pavement 443 Glutrin 445 Wood block 446 Asphalt block 449 Brick 451 Medina sandstone block 456 General tables and formulae 459-589 Appendix A Traffic rules and regulations of the State of Ohio . . . 591 Traffic regulations State of New York 602 LIST OF TABLES PAGE i. Ruling grades in present use 5 2. Tractive effort of a team of horses 6 3. Effect of tire width on tractive resistance 7 4. Effect of size of wheel on tractive resistance .... 8 5. Effect of tire width on tractive resistance 9 6. Rolling resistance on different surfaces 10 7. Maximum loads on improved and dirt roads .... 10 8. Amount of excavation on improved roads 14 9. Maximum and usual widths of traveled way .... 20 10. Maximum run-off, small watersheds 34 n. N. Y. C. &H. R. R. culvert sizes, small drainage areas 35 1 1 a. Iowa State Highway Commission culvert sizes, small aieas 35 12. Run-off small areas, village streets ^ 37 13. Discharge capacity, small culverts 38 14. Weights, cast-iron pipe 54 15. Mesh reinforcement 55 16. Reinforcing steel bars -. 56 17 Approximate cost small concrete and C. I. P. culverts . 58 18. Properties of Cambria I-beams 59 19. Sizes of stone, Telford foundations 70 20. Loss of crown on macadam roads 75 21. Properties of road rocks 122 22. Properties of road rocks 123 23. Properties of road rocks .... 123 24. Geological classification of road rocks 125 25. Composition of tar products 131 25 a. Analysis of crude coke-oven tars . . . . . . .132 26. Composition of crude petroleum and petroleum residuums 136 27. Effect of cross-section interval on quantities of excava- tion . 147 28. Stadia reduction table . . . . 156 29 Curve table, radii and deflections 167 30. Tangents and externals, one-degree curve 170 31. Sight distance on curves in cut 208 32. Shrinkage of excavation in fill 220 33. Radii of vertical curves 225 34. Sight distance, vertical curves 225 35. Table of volumes, 50' cross-sections . . * 228 xvi LIST OF TABLES 36. Conversion table, cubic feet to cubic yards 230 37. Table of volumes for preliminary estimates 238 38. Conversion table, feet to miles 251 39. Weights of crushed stone per 100' of road different widths and loose depths 252 40. Number of cu. yds. macadam per ioo' of road . . . . 255 41. Number of sq. yds. per 100' road, different widths . . . 263 42. No. gals, bitumen per 100' road, different rates per sq. yd. 264 43. Cost of earth excavation 267 44. Cost of hauling broken stone 270 45. Cost of spreading broken stone 272 46. Ratio of loose to rolled stone depths 272 47. Amounts of filler and binder per cu. yd 272 48. Proportion of sizes, crusher output 275 49. Amount of materials for concrete 304 50. Speed of work, value of plant, force account .... 308 51. Cost of cold oiling ^^^ 52. Amounts of filler (spacing of loads) 347 53. Steam temperatures for heating bituminous materials . 349 54. Amounts of materials for culverts 361 General Tables and Formulae 55. Conversion table of weights and measures 459 56. Decimal equivalent of inches 460 57. Areas and volumes 462 58. Table of squares, cubes, square roots, and cube roots . 464 59. Trigonometric functions and solution of triangles . . . 478 60. Tables of natural sines, cosines, tangents, etc 479 61. Logarithms of numbers 514 62. Logarithmic sines, cosines, tangents, and cotangents 540 63. Weights of materials 584^ 64. Strength of materials 585* 65. General flexure formulae, bending moments, etc. . . . 586 66. Centers of gravity for ordinary shapes . . . . . . 588 67. Moments of inertia of ordinary shapes 589 HANDBOOK FOR HIGHWAY ENGINEERS PART I. THEORY OF DESIGN INTRODUCTORY The necessity for the permanent improvement of the main country roads has been so well recognized by all the States that the work promises in a few years to equal in magnitude that of Railroad, Canal, and River transportation. Highway construction has increased so rapidly that there are not enough experienced engineers to handle it. Most of the de- partments have been forced to use untrained men and have tried to make their plans "fool-proof" by standardizing the designs in detail. Road work is peculiarly unfitted for such treatment, as an appropriate and economical design often calls for changes every ioo feet and too much Standardization has resulted in a waste of money and unsatisfactory plans. The general public still believes that the work requires only common- sense and that the money spent on engineering is wasted; even in the Road Departments, many of the men take this view, but it is a relic of the old "hit or miss" style of town-road construction. There is no doubt that money judiciously spent in engineering is justified by the resultant saving in total cost; there is also no doubt that much of the money spent in so-called engineering is absolutely wasted. In order to handle satisfactorily the work already in sight, we must have a larger force of technically trained roadmen, who realize the importance of the problem as an engineering problem, and who understand that a good design depends on them and not on a mechani- cal use of Standards. As soon as such a force is developed we can do justice to the roads. GENERAL Highways are improved to reduce the cost of hauling and to in- crease the safety and ease of light traffic. The parts of the design are more or less important in proportion to their necessity for the fulfilment of these purposes, and may be ranked as follows : i. Selection of Roads 2. Grades and Alignment , 3. Cross Sections 4. Drainage 5. Foundations 6. Top Courses 7. Minor Details The Selection of Roads to improve is a matter of broad policy „ it becomes an engineering question only when a number of roads 2 THEORY OF DESIGN serve the same district, in which case the considerations of grade and economy govern. Grades, Alignment, and Section are the most permanent features of an improvement. The ruling grade largely controls the maximum load that can be hauled; section and grade combined determine the convenience of the road and the economy of earthwork, while align- ment and section affect the safety and are also important factors in the appearance of the highway. For these reasons these three points can be ranked as equal and first in importance. Drainage, Foundation Stone, and the Top Course keep the section intact and firm under heavy traffic. The bearing power of the sub- grade and shoulders is increased by the surface and subsurface drainage; the concentrated wheel loads of heavily loaded vehicles are spread over a safe area of subgrade by the foundation stone; the top course provides a surface that will withstand the abrasive action of wheels and horse-shoes, that gives a good footing and offers slight rolling resistance. At the present time the problem of the top course is more troublesome than all the other points combined, and various new styles of construction are being tried to meet the de- mands of both automobile and horse traffic. There is so much discus- sion of this one feature that it is easy to give it too much weight, and there is a tendency to economize on the more permanent and impor- tant parts of the design in order to get a higher grade top course. In the writer's opinion this is a mistake. The different types of experimental top courses will be described in detail, but as yet no definite conclusions can be drawn. Minor Details Minor Details include guard-rail, danger signs, guide signs, and other points affecting the safety and general appearance of the road. The steps of the design will be taken up in the order of their im- portance as indicated on page i. ROAD BONDS Road improvements are usually paid for by long term bonds; 50 year bonds have been very generally used. This method has been justly criticised as too long a term considering the fact that a large amount of money will be required for construction renewals before the original bonds expire. Serial bonds are a more rational method of payment. The following tabulation is based on the average costs per mile for 200 miles of 16 ft. State roads in Western New York and shows the cost of the permanent features and temporary parts of the different forms of construction. ROAD BONDS * Excavation * Drainage Structures * Foundations and sub-base . Surfacing . Edging . . Minor points * Total Permanent features " Temporary " . Probable life " " Brick Cost per mile » 2 200 700 63OO I47OO 500 9200 15200 % Total Cost 9.0 2.8 25-9 60.1 2.2 37-7 62.3 10 to 25 years Bit. Mac. Cost per mile $1900 700 3300 5900 500 5900 6400 % Total Cost Water Mac. Cost per mile 15.9 5-3 27.0 47-5 4-3 48.2 51.8 5 to 10 years $1900 700 3300 4000 500 5900 4500 % Total Cost 18.3 6.7 3i-7 38.5 4.8 S6.7 43-3 5 to 10 years CHAPTER I GRADES AND ALIGNMENT Grades can be divided into Maximum, Minimum, and Intermediate. Maximum or Ruling Grades It is impossible to do justice to the question of ruling grades in the brief discussion called for by a book of this character, but the main points will be covered in the following order: i. The relative importance of automobile and horse traffic in the selection of grades. 2. The difficulty of ascent and the ease and safety of descent. 3. The theoretical grades that fulfil certain traffic requirements, and the practical considerations that govern the selection. 4. The construction of ruling grades. 1. Under favorable conditions, gasoline and electric trucks can haul for about $0.08 to $0.10 per ton mile, traveling empty one way, while the cost of team hauling cannot be reduced much below $0.16 to $0.18. This looks like a big advantage for the trucks, but they are helpless on a poor foundation and their use for general purposes in the country is limited by bad side-roads and snow, and for produce hauling is confined to the short period of the year in which the crops are marketed. Near cities they are coming into use for milk routes, gardening produce, and the rural delivery of merchandise, but only on improved roads and only by concerns that are able to use them continuously enough to warrant the investment. Farmers must keep horses for their ordinary work and, having them, will continue to draw with teams. Mechanical trucking is bound to increase, but there seems to be no reason to believe that in the immediate future it will become more important than team hauling in rural districts and as the machines in use have sufficient power to take them up any firm surfaced grade that has heretofore been considered suitable for horse traffic, it is evident that for heavy hauling, teams still govern the selection of grade. In Europe, mechanical tractors drawing trains of farm wagons have been used successfully. This style of hauling will probably be adopted here for limited areas, but its development into general use is a matter of conjecture. The number of wagons drawn by one machine is limited to seven or eight by the difficulties at turns and the danger of obstructing the road, rather than by the present grades. Reduced grades would lessen the fuel consumption and increase the speed slightly, but would not materially increase the train load. It would seem that such a small saving for a class of traffic that is to be developed in the future would not warrant any reduction of grade below current practice. Light automobiles are not handicapped as much by bad roads as the heavy trucks; on fair roads their ability to cover long distances quickly makes them adapted to many uses, but they are not now MAXIMUM OR RULING GRADES and probably never will be, as effective as horses for general use under all conditions. The least powerful of these light machines have no difficulty on firm surfaced 8% to 10% grades, which eliminates them as a factor in determining the maximum rate. From the preceding statements of the present and probable future conditions of both light and heavy traffic it is reasonable to conclude that the horse and not the machine should govern the design of the Ruling Grade. 2. Various grades on country roads have been under observation for so many years that it is safer to be guided by present practice which 13 the result of such observation, than to trust too much to a theoretical discussion. The adoption of the ruling grades given in Table i has depended partly on the ease of maintenance as well as the traffic considerations; the maximum grades on which different top courses can be safely used, either on account of foothold for horses or the maintenance of the surface, properly come under a discussion of such courses, and will be included in chapter V. Table i Ruling Grades in Foreign Countries Prussia , Hanover Baden Brunswick Holyrod Road in England Mountainous Districts Hilly Districts 5 % 4 % 2\% 4 % 3§% 2*% 8 % 6 % s % S*% 4 % 3 % 6 % 3 i% Military Highway over the Alps Italian side National Roads Level Districts \\% Swiss side. .6 % France 3/0 Departmental Roads 4%.... Subordinate Roads 6% Ruling Grades in the United States State New York . . . Massachusetts Connecticut . New Jersey . . Michigan .... Missouri .... Washington . Illinois Main Roads Side Roads Unusual Cases 5% 7&8% H% 5% 7% 5% — 5% 6-7% 9% 6% 5-6% 5% 5% 6% 9% 6 GRADES AND ALIGNMENT European observers claim that on a stone road, 5% is the maxi- mum grade that can be descended safely by a trotting team without the application of brakes, and that 12% is the maximum that can be descended safely with brakes. Safe descent with brakes need not be considered, as it would result in a grade far beyond ordinary practice; safe and easy descent without brakes very evidently plays a part in the selection of the ruling grade, but is more important for light teams with a small load, traveling at a comparatively rapid rate, than for heavy hauling teams which rarely trot. The writer knows of no records of actual maximum loads that can be drawn up different grades by an ordinary team; it is probably better to discuss this point theoretically, as any experiments would be affected by too many variable local conditions to be worth much as a basis for comparison. A summary of Prof. I. O. Baker's discussion of maximum team loads is given below, and through his courtesy we are enabled to include a collection of tables taken from his work " Roads and Pavements." Various trials have determined that the normal tractive power of a horse traveling three miles per hour for ten hours a day is approxi- mately one tenth of its weight; that when hauling up a steep grade it can exert one fourth of its weight for a short time; that for a con- tinuous exertion of one fourth, the grade should not be over 1200 feet long, and if over that, resting places must be provided every 600 to 800 feet; that in starting and for a distance of 50 to 100 feet, one half of its weight can be used; and that the net tractive power exerted by a horse on a grade equals (| its weight) — (the effort required to lift itself) , or approximately W/4 — W X per cent of grade expressed in hundredths, i.e. (W/4 — 0.04W) for a 4% grade. Table 2 shows the effective tractive power of a team of 1 200-pound horses on different grades. Table 2 W = Weight of team, 2400 lbs P = Per cent of grade in hundredths Grade Level 2i% 4 % S % 6 % 7 % 8 % 9 % 10 % Theoretical Tractive Effort tVW w/4 -PW w/4 -PW w/4 -PW w/4 -PW w/4 -PW w/4 -PW w/4 -PW w/4 -PW T Tractive Effort in Pounds 240 540 504 480 456 432 408 384 360 This power is used in overcoming axle friction, gravity resistance, and rolling resistance. MAXIMUM OR RULING GRADES 7 The axle friction is small, amounting to three or four pounds per ton for American farm wagons. Grade resistance (gravity) equals (Load x per cent of grade ex- pressed in hundredths) and expressed in pounds per ton of load equals (2000 x P). The rolling resistance varies for different surfaces and for each surface depends on the diameter of wheel, width of tire, speed of travel, and the presence or absence of springs on the wagon. The best diameter of wheels, best width of tires, and the use of springs as they affect the ease of hauling for both farm and road use are prob- lems for the wagon manufacturers. Morin, a French engineer, concluded, from a series of careful ex- periments, that the harder the surface of the road the less effect the width of tire had on rolling resistance. We are dealing with com- paratively hard surfacing only and with small differences in wheel diameters and can disregard these factors. As a matter of interest Tables 3, 4, and 5 are included to show the results of experiments on different soils and roads. The question of wide tires is necessary to road engineers only as it affects the distribution of wheel loads over a safe area and will be taken up under Foundations. Table 3. — Effect of Width of Tire upon Tractive Power 1 Resistances in Pounds per Ton Ref. No. Description of the Road Surface Sod Earth road (hard) . . " (muddy) Sand " (hard) . . . (deep) .... Gravel road (good) . Wood Block (round) Diameters of the Front & Rear Wheels respectively 3-6" & 3-10" 199 37i 5i 4" 108 243 162 351 49 3-6" & 3-IO" Width I*" 268 171 98 61 304 164 117 70 3-8" & 4-6" of if 236 141 83 35 4" 254 168 80 46 3-6" & 3'-io" Tires 1* 283 152 239 152 54 3-8" & 4-6" I*" 3" 189 228 114 265 114 228 66 28 76 38 Pamphlet by Studebaker Brothers Manufacturing Company, 1892. s GRADES AND ALIGNMENT Table 4. — Effect of Size of Wheels on Tractive Resistance 1 Pounds per ton Ref. No. Description of Road Surface Mean Diameter of Front & Rear Wheels 50" 38" 26" 1 2 3 4 5 6 7 8 9 10 Macadam, slightly worn, fair condition Gravel road, sand i" deep, loose stones " upgrade 2.2%, one-half inch wet sand, frozen below 57 84 123 69 IOI 132 173 178 252 130 61 90 132 75 119 145 203 201 303 148 70 no 173 79 139 179 281 265 374 186 Earth road. Dry and hard |* sticky mud, frozen below Timothy & blue grass sod, dry grass cut " " " " wet & spongy Cornfield; flat culture across rows, dry Plowed ground; not harrowed, dry & cloddy . . Average Value of Tractive Power Experiments of Mr. T. I. Mairs at the Missouri Agricultural Experiment Station. MAXIMUM OR RULING GRADES CSMHHCSCNCOMHHCSIOHM WCNHCSHCSHIHI o H o -H> • I— I 00 "t t^ O "tO O ^ lOvO oO O O cn O 0v0 O C>NH\0 CSOO co co t O ". co CD rQ co~ M as CD Hi 73 cvJ P rt o ft~2 >^ cd - co ^ ,£> T3^^^ O £ uhn., a a M - cu" fe bo bo °aa ^^ CO CD Si2 o • *» o rf - -M O FS~h ^ >~ O d^ Hi »-l PQO . H o a'g ^ « ©12 O C C cs _r co >, >^ta g > So 13 *U co co O - . ~ CO - - ^ +-> O CO cu o P4 >, CO a ^ pj a 1* - ~hOhO. 2 b a a a Cr u BBS &. a c rG 4-> bo a 5 a "as a s a g -<-> a 2*° »-H M fl) g ^ % ^0 Xfl CO o> o So i o E^ en -O "X3 to C3 ^ ^ ^ u o ^ ^ £/ en ccj S 1-4 ^ cu HI T3 bO a o (D Hi a CO "§ -H> OO o s ^►2 h (N rot lOO l>-00 O O m .O0 O O M CS CO t MMMHtHIHIHIHI -r ~l 7-12 Width of traveled way on 160 roads in Massachusetts, measured during the years 1896, 1897, 1898, and 1899, an d printed in the report the Massachusetts Highway Commission for 1900. The width of stone on these roads is given as 15' wide on 130, 12' wide on 3, and 10' wide on 2. It should be remembered that the stone is put on very much thicker in the middle than at the edges. The maximum width of traveled way as measured was as follows: 9 ft wide on 2 roads l8 ft. wide on 23 roads 10 tt n " 6 n 19 " n 1 tt 11 a it " 2 a 20 " tt 10 a 12 u n " 28 it 21 " a 10 a x 3 tt tt " 8 a 22 " tt 1 a 14 16 u it n n it u " 23 " 30 " 8 tt a (C 24 " 25 " 26 " it a it 2 4 1 a a a 17 it n " 1 a 33 " u 1 a The width of commonly traveled way as measured was as follows : 7 ft. wide on 12 roads 14 ft. wide on 8 roads 8 q a u 1 " 17 ' " 25 a a 15 ' 16 ' t it 13 2 a a [O n ' " 3 2 a 18 ' t 11 4 a li n 1 " 10 it 20 ' I It 2 11 12 tt ' " 3° a 22 ' c tt 1 n - C3 a ( " 3 it 25 ' ( a 1 11 The author has measured a number of the New York State im- proved roads and found that the width of heavy travel checked the Massachusetts results but that the maximum widths were more, averaging from 18 to 21 ft.; this probably can be explained by the INDIAN FALLS — CORFU ROAD 21 increase in automobile traffic since 1900, which, because of its higher speed, requires more room in passing. . Briefly stated, the widths subjected to hard wear on unimportant roads ranged from 8' to io'; on well traveled roads io' to 14', and in unusual cases 14/ to 16'. The maximum widths used varied from 12' to 14/ on the side roads, to 17' and 18' on the main thoroughfares, and as mentioned above have increased to 18-21' in the last few years. From this data, it seems that the best practice at present requires a driving width for "turn out" traffic of about 22', with a variable width of strong metaling determined by the traffic require- ments and ranging from io' to 20'. We have now practically developed a standard for the 22' of driving width; the metaling that is to carry the heavy traffic has a specified crown for each variety and from the edge of the metaling to the limits of the 22', the earth shoulder must have a slope of 1" to i' or possibly §" to i'. / "to I 'Crown^J^^^ Fig. 1 The flexibility of the section depends on the portion outside of this 22'. The function of the extra width is to keep the longitudinal drainage of surface water beyond the portion used for driving. To do this we are limited to a minimum slope of 1" to i' to insure trans- verse drainage and a maximum of 3" to i' on the score of safety. It is by the good judgment of the designer in using various slopes be- tween these limits and various widths and depths of ditches, combined with the possibilities of different grades, that the economies in earth- work are effected and at the same time the design is made appropriate to the local conditions. Two examples are given to illustrate this point. 1. INDIAN FALLS— CORFU ROAD IN NEW YORK STATE Original Djesign Revised Design Length 1.85 Miles. NO CHANGE IN PROFILE No Change in Ratio of Cut to Fill Width of Macadam 14' Width of Macadam 14' " Section 30' " " Section 24' Depth of Ditch 18" Depth of Ditch 14" Original estimated Revised estimated excavation 7500 Cu. Yds. excavation 5200 Cu. Yds. This change is section alone resulted in a saving of 2300 cu. yds. excavation or at the rate of 1 240 cu. yds. per mile, or in money about $600.00 per mile. 22 SECTIONS 2. PITTSFORD — NORTH HENRIETTA ROAD IN NEW YORK STATE Length 2.67 Miles Original Design Revised Design " 14" Width of Section 24' Depth of Ditch i2 y Ratio of cut to fill 1.25% Maximum Grade 5.0% Profile : — Rolling grades and reverse vertical curves used. Revised estimated excavation 6,620 Cu. Yds. Width of Section 30' Depth of Ditch 18" Ratio of cut to fill 1.35% Maximum Grade 5.0% Profile: — Designed with straight instead of rolling grades and tangents of ioo' between vertical curves. Original estimated excavation 11^450 Cu. Yds. A saving of 4,820 cu. yds; 1,800 cu. yds. per mile, or, in money, approximately $900.00 per mile. The revised design on this road is a good example of what can be saved by the use of a section that fits the conditions, a rolling grade, and a ratio of cut to fill that we have found from experience to be sufficient. The author's experience has indicated that an open ditch does not have much effect on ground water; that its part in the design is to drain the surface water, thus preventing seepage into the road-bed with a resulting softening of the surface; and consequently, whenever ground water is encountered under drains should be used. Deep ditches are not only useless but dangerous, and the best practice calls for the least depth that will handle the surface water. The following section is, therefore, suitable where there is no probability of much surface water; it is the writer's idea of the minimum width section that will be satisfactory, and where it can be adopted will give the most economical grading design for light cuts and fills. El. Theoretical Grade^<> x ^^nKj^I* f'fo/' ^°" , l< 22' _ ->| k - 24-' -26' —-- H Fig. 2 SECTIONS 23 New York State 19 15 Standards Cement Concrete .,-Efev. of Theoreffcai Grade i \y £ ' I \Q1> --»/W?k— -, -„ --24'-0"fo 32- '0 ->^K- 1 I Crowns per Ff\, ^Elev. of Theoreticaf Grade II >. \%* 1 5 ub -bo ttom Course if Required \ p ft*' 14-0-16-0- B ■--H ^ Waterbound Steep Grades -w?j«— —24'-0" to32'-0- — ->vrf ^c I Croyvn^perFty * Elevat ion of Theorefica Grade '//j Tj \5ub-bottom Course if Required \ Contracted Section I ! ^ . Y'Elev. of T heoretical Gr x ade ^ ^ Bituminous Macadam ..^ .. 24 L 0" to 32-0"- >j/f|«- CrovynJperEf-, ^-Elev. of Theoretical Gfade \ I I JJ riia /«?][<<. .4/^^ I "per Ft \ -IS'-O" Expansion Joint -- Expansion joints, if ot poured type shall be /"thick. For premou/ded type they shall be fforl6'widfh J §"for20'to24'width and $" for 32 '-width. Half Section Half Section f-i F-2 7b/? Course same as A.BorE as Specified'-^ 7777777777777A Sub- base as Bottom Course G /777777777777\ Top Course same as A, B or E as Spec if fed T777777777, 777/////"'" ' ' / / /////// ^7////7/// / s When lei ford replaces Sub -base, if is made 8" thick (bottom course to have the same thickness at center and edges where Sub -base or Telford is used) Sub-base or Telford H 1,4 -Asphalt Class Concrete .1 ~" ~ a iy->\ \<6-8->i Asphalt-^ i-d Kv ?" Sand— -:-->'• 7T M \<2 Brick *ft*&* 4 k— /*-*—->! ^2^ Class Concrete TYPICAL SECTIONS 25 ■r. ~ .2 "-S to •ij jad „| uavojo 5 CN JS 00 o> *b fc ~£ CO 5: in IH J3 "* vO 00 d .0 v> 00 < c "3 U '5, "U I9d #/! UA\OI3 "3 3 cd O to NO s M M 5: 00 M ~d "* V) UO 5: NO 'X a ml-* CO 10 10 c. "*-+ 10 japinoqs VO vO M-H lo 00 2^ ■73" "" 3£ JCC5 ,d- ^ in cj- ^* m-i ■- ^ V. C w D fi ■ O X a ori rt^ > d +J -»_> 0) C be d r u- O O O CN CO CD~ J3 iO Ed r> XJ- - T3 l) II II u fr d 0- - c ff ■j tin - d w 0) W H O H O ^ Q 26 SECTIONS Plate 2 MS I5'0". 2I'0 25'0 Trap : 46 tons per 100 it Local -4/ it >r u 15' 0:- 2/'0„ 25' Trap .; 38 tons per 100^ it. Local - 34 » » < 15 0" - 2I'Q" 25'0 Trap : 55 tons per 100 jh_ Local: 50 if » tt » I5'0"- 21' 25 '0 Trap : 4-3 tons per WO it. Local: 39 n ( New 5toneNo.2 J (New Stone No. I.) [<- - iS'O"—'- >f (Original Width of Macadam) Section for Resurfacing «Q .dr' '/' 72&2I&&A& 7ZZZ7ZZZZZ%Z&ZZZZ7Z72ZZZZZ7zzz?, .-IS' 0- > Trap • 64 tons per 100 it_ .^ Local: 58 " » » For Village Streets i "to I v^ — 7ZZZZZZ?ZZ%ZZ7Z?ZZZZZZ7Z. IS'O Trap_ ■ 55*onsper/00_ it_ Local : 50 » i> v For Village Streets. MASSACHUSETTS STANDARD FOR ROAD SECTIONS 27 Plate 2 — continued Note: The Backs of 'Guard- Pail Posts fobe set- one foot from Edge of Embankment for all Widths. m" A " V- Underdrain Cobble Filled - f Large Stone at Bottom, Small Stone and Gravel at Top. v Halve together over Posts "ft- 8'0" ft— 8'0" ----- ffi^ 8'0" ff U 1 I ' ! U •to I 2 - 2-~" I5 > >. ___„,! ^ Condition No. i. — See note below. Trap Rock — Lower course, No. 1 stone, 24 tons; screenings for binder, 4 tons. Upper course, No. 2 stone, 16 tons. Local Stone — Lower course, No. 1 stone, 22 tons; screenings for binder, 4 tons. Upper course, No. 2 stone, 14 tons. Total tonnage per 100': Trap, 44; Local, 40. Condition No. 2 — See note below. Trap Rock — Lower course, No. 1 stone, No. 2 stone, 16 tons; screenings for binder, Local Stone — Lower course, No. 1 stone. No. 2 stone, 14 tons; screenings for binder, 24 tons. 7 tons. 22 tons. 7 tons. Upper course, Upper course, Total tonnage per ioo': Trap, 475 Local, 43. Note.— For both penetration methods — grouting or the modi- fied Glad well method — there should be two applications of asphaltic oil, each f gal. per sq. yd. There may be also a third application of I gal. per sq. yd. for a surface finish. For surface treatment there should be one application of § gal. of oil per sq. yd. or two applica- tions of \ gal. each per sq. yd. on the finished surface of the roadway. Plate 2 — continued 2". .*" i'tol' ?ee: 2' j<- IS'O ,/; i Z Condition No. i. Trap Rock — Lower course, No. 1 stone, 19 tons; screenings for binder, 3 tons. Upper course, No. 2 stone, 17 tons. 28 SECTIONS Local stone — Lower course, No. i stone, 17 tons; screenings for binder, 3 tons. Upper course, No. 2 stone, 15 tons. Total tonnage per ioo': Trap, 39; Local, 35. Condition No. 2. Trap Rock — Lower course, No. 1 stone, 19 tons. Upper course, No. 2 stone, 17 tons; screenings for binder, 6 tons. Local Stone — Lower course, No. 1 stone, 17 tons. Upper course, No. 2 stone, 15 tons; screenings for binder, 6 tons. Total tonnage per 100': Trap, 42; Local, 38. Note. — Condition No. 1 : Bituminous Treatment — Penetration — lower course bound with stone screenings or sand. Condition No. 2: Bituminous Treatment — Surface Spraying — screenings of sand binder in upper course. Plate 3 Shoulder Drain I wide at Intervals of 200. 3 ^ini'Jlm£tC^l .4— ->«- q' s4<-3:. Laid *\I6& IZinCut >k— - inBox ■ I ^j I! 2 ml 13 ml - ■12 In Fill State of Washington Standard Section 80- ■* 8.0'- 1? S 1 Si This Section is the Arc of a Circle drawn through the Points a.birc Crown for Waterbound Macadam %tot! rt n Bituminous ** jfiot. New Jersey Standard Section Plates Nos. 1,2, and 3 show some of the Standard Sections in use at the present time. Widths of metaling can be discussed at this point, leaving depths for the chapter on "Foundations." There are two sets of widths in general use, 12 ft., 15 ft., 20 ft. and 14 ft., 16 ft., 20 ft. 20 ft. widths are not often required and it is evident that the use of 12 ft. instead of 14 ft. or 15 ft. instead of 16 ft. means a large saving STANDARD SECTIONS 29 (see footnote) ! and is good policy provided the narrower width serves the purpose. There are two ways of approaching this problem. The first is to build the strong metaling just wide enough to com- fortably take the heavy traffic, and if the natural shoulder material is not suitable, treat the shoulders to a width of i4 / -2o / with gravel, waste #2 stone, or S3 stone filled and rolled but not puddled or tarred, making them suitable and wide enough for the light "turn out traffic"; this method results in the 12' and 15' widths. The second way is to make the full depth of the macadam just wide enough to allow two vehicles to pass with a minimum safe clearance, not giving the shoulders any special treatment. This method results in the 14' width on unimportant roads. The 16' width is harder to justify, as on the main traveled roads it is wider than necessary for the heavy travel and too narrow for the automobile "turn out traffic." In the writer's opinion 12' should be used in preference to 14' on the side roads where the shoulder material is good or where gravel is cheap or local crushed stone is used, making it possible to get cheap #2 or #3 stone; the 14' width should be used in preference to 12' where the shoulder material is bad and gravel or stone are imported. On the main roads 15' is as satisfactory as 16' and is cheaper under all conditions, because the 16' width does not overcome the necessity for a good shoulder. The importance of shoulder treatment on the side roads should not, however, be overestimated. One of the New York State Highway engineers made a trip from Albany to Binghamton (130 miles) in the Fall of 19 10 and counted the rigs he passed; they averaged one every four miles outside of the villages; from this it would seem that for roads of this class shoulder treatment is not worth while unless fine shifting sand or heavy clay is encountered. The sketches given below show a number of variations of section for bituminous macadam which are applicable to special conditions. Figure 3 shows the distribution of stone on unimportant road sections. Figure 4 gives a good typical section for ordinary conditions on a main road. .it > Gravel- #2or#'55tone Filled and /"to/ kj. Rolled, butnot Puddled or Tarred. ' . k /4'-l8 - " -*1 K- 24-30 H Fig. 3. — Bituminous Macadam Gravel, #2 or #3 Stone ^ | About 3"deep f loose "~~ v 1 K IZ'-I5'-- — H *< —— -70'-20'- 31 Fig. 3 a. — Shoulder Treatment 1 The amount saved per mile, 'assuming a depth of macadam of 6" and an average price of stone at $3.50 per cu. yd. in place would be approximately $700.00 for use of 12' in place of 14/ and $350.00 for use of 15' in place of 16'. 3° SECTIONS ,- Dirt Shoulders No Shoulder Treatment CM E[. Theoretica l Grade ^-C ra ca ms' Ann' .\\\\v^ .V A\vfr |<- is 1 h -19'- k- — - 22'- K-- Z4'~ze' wnztol nXnnnNNVT Fig. 4. — Bituminous Macadam El. Theoretical Grade J^sOvwnf:/! %'.f. -^ S *777777777^ / //////\///A /S77~h7?r, ' Ur-r—JT— -* N. — \£ *— - Any Width which keeps the Top of Slope insidefHeR.oFW.—'-* Fig. 5 Figure 5 shows a section adapted to the top of hills where a small amount of surface water is expected. If for any reason it is not practicable to cut into the hill beyond a certain depth and more dirt is needed for fill than is given by the 26' section at this depth, the shoulders can be widened, provided the tops of the slopes keep within the right-of-way. It is always best to use as shallow a ditch as possible, as it simplifies the construction and maintenance of entrances to the abutting properties. 'Concrete Guardrail & Wooden § Guard Rail- k-~ -IO'-20'- J 24' Figure 6 gives a section showing the variations in fill. A slope of 1" to i' beyond the 22' width is used on shallow fills; a side slope of 1 on 4 is used for all ordinary fills up to a 7' depth; beyond a 7' depth it is cheaper to erect and maintain guard-rail, using a 1 on ij slope. The cost of guard-rail is taken up under "Minor Points." The section shown in Figure 7 is used for unusually heavy cuts to keep the excavation down as much as possible; it should never be used on a sharp curve because of the difficulty in seeing ahead. (See Alignment, page 18, and Office Practice, page 208.) VARIATIONS OF SECTIONS 31 Fig. 7 Fig. 8. — Banked Section in Excavation figure 8 shows a section well suited for sharp curves on steep grades; the slope of j" to i' is not objectionable for slow traffic up the hill and makes easier riding for vehicles traveling rapidly down grade; this section has also been used successfully on sharp curves on level grades and is becoming a standard feature of the New York State work. hi I "to I' I "to I Cr own 2 1 to I ' 15-20' M 3Z-40' : " : -"- :: -~-^- Fig. 9 Figure 9 is a satisfactory village section and by the use of a variable width can be made to fit conditions on most streets. S^ Elevation, Theoretical Grade k 8.0'- -* 15.0 F -16.0'- -+- 17.5 Fig. 10. — Bituminous Macadam Tracks on Side Expansion Joint \ ~ Curbing I S S Class i Tie fi"c„ jr ,l- Concrete- z Izoand Cushion^ * T i W Jf 5 -2"-* Class Cone. Base .^'Concrete underground Tie i"°St.Ba\ -*- Fig. 10 A. — Village Street, Brick Pavement. Tracks in center, "T"- Rail Special Grooved Brick The preceding discussion attempts to show only the main points to be considered, for every road presents local conditions peculiar to 32 SECTIONS itself that require special solutions. However, if the Engineer keeps these points in mind, he will make an economical and appropriate design. Macadam Hitching Post / Plain Curb Brick \ Crown 2 to f. k- io'o"-- >j '^Concrete Base j| Fig. ii. — Village Section. Combined Brick and Macadam Section in Front of Stores, where Horses will be Hitched Close to the Curb. Prevents Pawing up the Macadam It may be said in closing that many of the road widths, as actually built, do not represent the engineering judgment of the Highway Departments. On a road where it is evident that a 12' or 14/ width of metaling would amply serve the traffic there is often a strong senti- ment that this locality is being defrauded because some other road is 16' wide, and if political influence can be successfully used a 16' width is constructed. This is mentioned to show one of the practical diffi- culties in attempting to build an economical road that meets the actual traffic requirements. The general complaint that roads are becoming more expensive overlooks this contributory cause, and, while it is true that more expensive constructions are necessary on account of changed traffic conditions, it can be safely asserted that in most cases where political expediency overrules engineering judgment, either in regard to widths or materials, an unnecessary expense is incurred. This condition is, however, sometimes due to defects in the Highway laws which allow too much interference by local officials who are not qualified to judge in such matters, and it has been demon- strated that better results are obtained by centralizing the control of the design, particularly in regard to widths, alignment, and materials, in some executive or commission, which is as independent as possible of such local pressure. CHAPTER III CULVERTS — SMALL SPAN BRIDGES — UNDER DRAINS This chapter deals with the smaller drainage structures only. For the theory and practice of reinforced concrete long-span structures, masonry arches, or steel bridges, the reader is referred to the standard works on those subjects. ^ The conditions for transverse surface drainage to the ditches were given in chapter II and the minimum ditch grades that insure the longitudinal drainage were mentioned under the heading of "Mini- mum Grades," page n. Ditches on steep grades must be protected from wash by cobble paving, cement gutters, or loose stone, and these designs are considered under " Minor Points," page 115. I. Culverts Engineers do not differ much in the design of these structures. They should be permanent; should be large enough to take the maxi- mum flood flow; should, if possible, be self-cleaning; must admit of being cleaned easily, when necessary, and must be long enough to include the normal width of section between parapets. There is nothing more unsightly and dangerous than to have the width of roadway narrowed at a culvert. Cast-iron pipe or reinforced concrete boxes are generally used Cast-iron pipe culverts larger than 18" are rarely designed, as they are not economical. (See Table No. 17, page 58.) Vitrified pipe should never be placed under the roadbed proper unless encased in concrete; even then cast-iron pipe is preferable and probably cheaper. Where the head room is small, usual practice calls for cast-iron pipe, and if the flow is large, a double or triple line of pipe may be constructed. For small drainage areas the size of the culvert is determined by the con- venience of cleaning, rather than by the discharge capacity. Where sufficient fall can be obtained to make it self-cleaning, a 12" pipe is feasible, but where the flow is sluggish, nothing less than a 16" or 18" pipe will serve satisfactorily. The self-cleaning velocity of flow for sand and earth particles is about one foot per second; for coarse gravel about three feet per second (Ogden's Sewer Design, page 134). A pipe laid on a slope that gives a velocity of five feet per second when flowing one-quarter full should keep clean; this requires a fall of approximately two feet in one hundred for a 12" pipe, and is the minimum grade at which the 12 size should be used. For the smaller concrete culverts the shape of the opening should be designed to allow the use of collapsible forms. The desired size of a culvert is usually determined in the field by notmgthe dimensions of the old culvert, if any, and by inquiries of the neighboring residents and the road commissioner as to how the 33 34 CULVERTS existing structure has handled the water in the past; any such con- clusion should be checked by computing the probable maximum run- off from the area tributary to the culvert. For the convenience of designers, Table No. 10 is given, showing the approximate maximum run-off for small watersheds in flat, rolling, and hilly country. Of course, it is understood that such a table is to be used simply as a guide for judgment. Table io. Maximum Run-off for Small Watersheds Using Dickens' Formula D = C^/]VP. Run-off Expressed in Second Feet Area in Square Miles Flat Country C 200 Rolling Country C 250 Hilly Country C 300 o.i =64 acres 36 45 54 0.2 60 75 90 0.3 81 IOI 121 0.4 IOO 125 I50 0.5 119 149 l8o 0.6 136 170 204 0.7 153 191 229 0.8 169 211 253 0.9 185 231 277 1.0 200 250 300 2.0 . 334 4i7 50I 3-o 456 57o 684 4.0 564 705 846 5-o 668 835 I002 6.0 764 955 1 146 7.0 860 1075 I29O 8.0 95o 1188 I426 9.0 1038 1297 1556 10.0 1122 1402 l682 20.0 1890 2362 2834 30.0 2560 3200 384O 40.0 3180 3975 4770 50.0 3760 4700 564O 60.0 43io 5400 6480 70.0 4840 6050 7260 80.0 536o 6700 804O 90.0 5840 7300 8760 100.0 6320 7900 9480 For areas under 0.1 square mile, see Table 12. CULVERT DESIGN 35 Dickens' formula takes into consideration the rate of rainfall and character of the catchment basin by the coefficient "C" and is as reliable as any of the maximum run-off formulae. Wilson in his " Irrigation Engineering," page 19, gives the following values of "C": Rainfall 3.5 to 4 inches in 24 hours. Flat country C 200 Mixed " C Hilly " C 200 250 300 These values are safe for the Northern and Eastern Atlantic States. Rainfall 6 inches in 24 hours. Flat country C 300 Mixed " C 325 Hilly " C 350 Table ii. New York Central and Hudson River R.R. Cul- verts for Small Drainage Areas Steep, Rocky Ground. Acres Flat Cultivation, Long Vallej r . Acres Size. Diameter in Inches Equivalent Capacity. Pipes 5 10 10" 10 20 12" 20 40 16* 25 50 18" two 16" pipes 30 60 20" two 16" pipes 45 90 24 " two 18" pipes 70 140 30" two 24" pipes no 220 36 " two 30 " pipes 150 300 42" two 30 " pipes 1 So 360 48 " two 36" pipes 280 560 60" Note. — To be used only in the absence 1 of more reliable infor- mation, particularly existing culverts over t he same stream. Table iia. Culvert Design. Iowa State Highway Com- mission l Size of Culvert Opening Maximum Acres Minimum Acres 2'X 2' 4 r X 4 ' 6'X 6' 8' X 8' 10' X 10' 70 376 1300 2700 5000 28 140 520 1 1 20 2000 36 CULVERTS Types of Structures Used l i. Box culverts and slab bridges 2' to 20' span. Not economical over 20' span. 2. Reinforced concrete arches 8' to 100'. Constant tendency to destroy by temperature strains and settlement. 3. Pony truss steel bridges. 30' to 8o' span with reinforced con- crete floor. Adapted to districts where concrete materials are scarce. 4. Reinforced concrete girders, 20' to 50' span. Very economical, but require careful design. Not economical for spans over 50'. Where the road runs through a village, a closer computation may be obtained by using a sewer run-off formula. The Burkle-Ziegler formula for such approximations is as follows: Cubic ft. per sec. ] ( £v. cu- f t of ] 4 /Av slope of ground per acre reaching \ = C X \ ^^LltL \ X \l m ft. per 100 No. of acres Svert ^ b r " e! r a ^ t u ,y ing I heaviest fall J T No. of acres drained C = °-75 f° r paved streets and built up business blocks. C = 0.625 for ordinary city streets. C = 0.30 for villages with gardens, lawns, and macadamized streets. Trautwine states that 1" of rainfall per hour equals 1 cu. ft. per second per acre approximately. For drainage areas of under 1 square mile, it is probably better to use the Burkle-Ziegler formula even for farming country, using the coefficient C = 0.25. Table 12 shows the amount of run-off computed by this formula assuming a maximum rainfall rate of 4" per hour for the constants C = 0.30 and C = 0.25 for areas up to 1 cquare mile. Note: — Quantities in Tables 10, 12, and 13 computed and checked by slide-rule; sufficiently accurate for the purpose for which these tables are intended. Table 13 gives the velocity of flow and the discharge capacity of pipe and box culverts for different rates of fall per 100 feet. Examples of the use of tables 10 to 13 dn checking culvert sizes. 1. Determine the character and area of watershed tributary to culvert; say rolling country, one square mile. 2. Determine flood flow for this area of rolling country from Table No. 10; equals 250 second feet. 3. From the profile of the stream where it crosses the road de- termine the fall in feet per 100; say 1.0 ft. 4. In Table 13 opposite 1.0 ft. in the "Rate of Fall" column, pick out the size that has a discharge capacity of 250 second feet; equals 4/X4/ culvert. Where the road runs through a depression which has no outlet, a culvert should be placed at the lowest point to keep the water at the same elevation on both sides of the road, and the grade line raised above high-water level. It is our opinion that a culvert should have the same slope as the stream bed. If given a greater slope the outlet end tends to clog, and if a lesser the inlet end will plug. It is unusual for culverts to fill badly, except when placed at the foot of a steep hill where the 1 See Table at foot of p. 35. TYPES OF STRUCTURES USED 37 stream velocity is naturally reduced. At such points an extra large structure should be designed with the idea of providing sufficient waterway even after the contraction caused by this settlement has occurred. Such a culvert should be cleaned after each freshet. More trouble is experienced from culverts becoming filled with ice due to alternate freezing and thawing weather; this is particularly true of small culverts draining springs. Culverts as large as 2^2' have frozen solid in this manner, and if this condition is anticipated the size should be regulated accordingly or trouble will be experienced during the Spring break-up. In designing culverts under side roads, the length must be great enough to provide an easy turn; many times a saving in length can be made by placing the culvert a short distance down the side road, as shown in figure No. 12, page 39. Table 12. Run-off for Small Areas Discharge in cu. ft. per second for a maximum rainfall rate of 4 inches per hour. Fall of 5 ' in 1000 Fall of 20' in 1000 Fall of 50' in 1000 Area in Acres C = 0.30 C = 0.25 C = 0.30 C = 0.25 C = 0.30 C = 0.25 1 1.8 i-S 2.5 2.1 3-i 2.7 2 3-o 2-5 4.2 3-5 5-4 4-5 3 4.1 3-4 5-7 4.8 7-2 6.0 4 5-o 4.2 7-2. 6.0 9.0 7-5 5 6.0 5.o 8-5 7-i 10.7 8.9 6 6.8 5-7 9-7 8.1 12.2 10.2 7 7-7 6.4 10.9 9.1 13-7 11.4 8 8.5 7-i 12.0 10.0 I5-I 12.6 9 9-3 7-8 13-2 11. 16.5 13-8 10 10. 1 8.4 14-3 11.9 18.0 15.0 20 16.9 14.1 24.0 20.0 30.2 25.2 30 23.0 19.2 32.5 27.1 40.7 33-9 40 28.5 23.8 40.3 33-6 50.9 42.4 50 33-6 28.0 47-7 39-8 60.0 50.0 60 38.6 32.2 54-6 45-5 68.7 57-3 70 43-3 36.1 61.4 51.2 77-3 64.4 80 48.0 40.0 67.9 56.6 85.2 71.0 90 5 2 -4 43-7 73-9 61.6 93-i 77.6 100 56.7 47-3 80.2 66.8 100.8 84.0 200 95-4 79-5 134-6 112. 2 169.7 I4I-4 300 129.0 107.7 182.9 152.4 229.7 191.4 400 160.0 133-6 227.0 189.2 285.6 238.0 500 190.0 158.0 268.0 223-5 336.6 280.5 600 640 ) 1 sq. mile J 216.0 180.0 307.0 256.0 387.3 322.8 230.0 1 I92.0 323-0 269.0 406.3 338.6 1 200 second feet by Dickens' formula, Table 10. 38 CULVERTS •a B u 9 in I p 93 G >> *o _c u > U & U u pH u u y 0> O* OC "+ O r~» H M ro^f lovO vO (n rooo n q co Q\ oi 4nOh M M M CS iO« O N rOOO <0 M 04 tf> *0 ^t *t ^ OO 04 io oi io O O tooo h 4oco d ■**• r>- qv <* co co q »0 t^ O fO>ONOl O O O 04 04 04 IO ionO w 4-vo r-~ co rooo rovq oooh to4 \o r^co o> o -t (Nvqoo q NtJO H fO 9 £ O o a & u p.. r> C/2 co O y5 s vO O^ • IO00 N N HVO M ■4vO d H co 4"d H H M H On co CO co w d v> o o o q o p ci h ci to4 tod !^P4 cr o <> CO w X o pq M H w erf o o U IT) X IO C/) s O ioO ^ t^ o rO ^ 1-- »o t-i 04 >• oq q q ro dod M M 01 OOfO <5 NO) 04 CO *0 X *»o 5 O io d h 04 13 O to oi cood d M M 04 fO X "Vo 5 O ^ r^ co H N CO"t q co io w a»oo *o h\6 rod M M 04 04 "Vt- X 5 vo r^oo "* 00 v£> t^vO H N fO't q co vd h M is >• VO l>."0 O HVO rod M M 04 04 X "V* 5 O O ^00\0 cooo O m O M IH 04 CO CO O 04 04 M H 13 > oo O O to io O to oi d O H M 04 04 CO X s 00 O 04 OO VO 00 *>. H lOOO M ^J" M H M 04 04 o q 00 H 1> > t>. r^ q »oq q (jtodfONH M M 04 04 CO X "Vo 5 Nf^H 04 00 0» 00 N NH rt- r>. M M OJ 04 04 q q > t^ t>. o to io q O* CO O co *■» w H M 04 04 CO 04 X CO s 04 ioioO 0> io O ionO 'O ^tO 00 VO Ooo vd d "o vo to tovo co vo O 00 ci N H 4 N 6 M M 04 04 04 CO 04 X 5 M 04 0> fO Tt Tj- tJ- co tJ- io t-00 On O vo o vo 4o r| »>• ^00 co O to O t^- O ^00 H COO M M M 04 04 04 IO X 04 c/5 5 M O 04 04 O00 04 04 co ^t iovo vo J>- q vo fO O 13 > q q q oj q vo o *>■ d -4- r^. d oi -4- M M M 04 (N 04 4 ( < u I c 1 i a D h ^p p p p q o d m oi co 4- iovo cr'o cd 3 fl) — <> SMALL SPAN, SOLID FLOOR BRIDGES 39 Main Road Macadam Side Culv. in Ditch Line Side Culvert Set Back on Side Road Fig. 12 The following section shows a form of culvert often used in village streets where deep ditches at the culvert site would be objectionable: Sidewalk IN Drop 6 rate 1 kV^\^^^^\^^^^^^^^^^^^^\^^^^^^x^^j^^^^^^^^^^^.^^^^^^^^^^\^,^ l v:^^?^g ^rw^W^'XwXw^ Inlet Sidewalk -4- ^>>^>>>^ Fig. i3 For the small-sized structures required to carry ditch drainage under driveways vitrified tile well laid is as suitable as any style of construction; the wooden boxes built by some Departments are not economical, which is shown in the following estimate of relative cost of small culverts, given by A. R. Hirsch in Wisconsin Road Pamphlet No. 4 : Kind Size of Opening Length First Cost and Maintenance for 100 Years 3 " Hemlock box Concrete box 15 in. sq. 15 in. sq. 18 in. 18 in. 18 in. 18 in. 18 in. 24' 2o' 2o' 30' 2S r 24' 26' $252.00 40.OO 35-°° 41.00 42.00 166.00 196.00 Concrete pipe Single strength V. T. P. . Double strength V. T. P. . Cast-iron pipe . Corrugated steel SMALL SPAN, SOLID FLOOR BRIDGES Under this head are included spans of 5 to 25 feet; they are gener- ally designed from one of three types: reinforced concrete slabs, 4Q CULVERTS Plate 4.— New York State Slab Bridges -Rods 0.25°" Net Area-, Spaced lE'X.toC. W\{Spans5'to25!) Plan. (Heights 5 'to 15.') , z DowehO.25^ ii'xtrs ;;, Pile Foundations tobe %_'. Used in Lightand Shift- ing Soils. Pave when Ordered byDiv.Engineer. Note All rods to have a deformed cross-section. All rib metal to be of medium steel. 2d class concrete in all slabs and parapets. 3d class concrete in wings invert and abutments. Wing walls on the outlet end of all square culverts with concrete floors to be built parallel to the center line of the culvert. Round all exposed edges to 1^ inch radius. -Rods in Slab to be Exten- \ ded 24 Diams. beyond Neat Lines of \ Abutment Elevation. Bottom Width of the Abutments not less than j~ of Total Height from Bottom of Abutment to lop of Slab. Section onCenterLine For Typical Section "F" Where culvert covers become a part of con- crete base for brick pavement, transverse reinforcement should be extended 12" beyond back of abutment intc concrete base. Rods in Slab to be Extended 24 Diams. beyond Neat Lines A of the Abutments. "Rods 0.25°' NetArea-, , Spaced IZ C.toC. NEW YORK STATE SLAB BRIDGES 41 ! Span Thickness of Slab Net Area of Rods Rod Spacing C-C Length of Dowels 5 8" o.25sq." 4*" 12* 6 9" it 4" u 7 10" o^gsq/' - 3'/ 54 u 8 IO" 11 si" a 9 II" it 5" u 10 12" a 4-r a 11 12" o.s6sq." er a 12 13" u 6" l8* 13 13" It - 3// 54 a 14 14" It -3// 58 11 15 14" a 5" it 16 15" it .2 If 44 11 17 i5' r a 44 ti 18 16* u A 1 " 42 ti 19 17' u 44 a 20 18" o.77sq." ,-10 54 a 21 18" a si" a 22 19" u 5" 24" 23 19" it S" 11 24 20" a .5// 4s 11 25 21* i.oosq." r-1" 58 11 For Spans 5' to 19' W = 18" For Clear Height 10' or less 5' to 19' W = 24* " " " ii' to is' " " 2o' tO 25' W = 24" " " For Clear Height 7' or less E = 3'- o" " 8' to 10' E = 4'- o" above 10' E = 5'- o" 15' or less 42 CULVERTS steel I-beam stringers supporting thin reinforced concrete floor slabs, or plain and reinforced concrete arches. Central piers will often reduce the cost of culverts having a long span with small height. For structures of this class more care must be taken in determining the span and height. On streams requiring spans of more than 10 feet there are generally existing structures above and below the proposed bridge site which will afford the best basis for judgment. While it is usually good policy not to reduce the span of an existing structure it is often found that the present bridge, particularly if it is a steel bridge that has been sold to the town by an enterprising bridge company, has a needlessly long span. If the freshet velocity of the stream is high the stream bed and the abutment foundations may be protected from scour by riprap. However, it is not often necessary to take this precaution for small span bridges. According to Trautwine a velocity of eight miles an hour, or 12' per second, will not derange quarry rubble-stones exceeding half a cu. ft. deposited around piers or abutments. A rough approximation of small stream velocities can be made by assuming a value of 60 for the constant C in the formula V = C \/RS where V = velocity of flow in feet per second; constant C = 60. „ ,* , .. ,. Cross sectional area of flow R = Hydraulic radius = ^ r~^ — = Wetted Perimeter. S = slope of stream. Example. To approximate the freshet velocity of the stream shown having a fall of i.o' per ioo', or 53' per mile 2S.0' v = cvrs C = 60 IOO R 25 S = = 0.01 100 V = 60^/4 X 0.01 = 6o\/^°4 = 60 X .2 = 12 ft. per second. Plates No. 4 to No. 6 c show the standards for culverts and small bridges as used by various State Departments. Under Drainage The purpose of under drains is to intercept the ground water before it reaches and softens the subgrade. On a side hill road the drain is usually placed under the ditch on the up-hill side (see Figure No. 14, position No. 1, page 52), where the greatest depth can be obtained NEW YORK STATE I-BEAM BRIDGES 43 Plate 4 a. — New York State I-Beam Bridges Expanded Metal Embedded in 6"Concrete( '2* 'Class) \ Length of Culvert — - d V-1'6"-^ Exp.Metal- Steel Bars I- Beams Spaced as/Shown In .A\ Table/. ' " Conor Cover && ' j : ? , [VWc • no more than 4. l-'-oj r 3 « Cross Section of Parapet Showing Reinforcing. '-■'■■■■'ff.-iWSIlSTO 2 Steel Bars- ^ ? ^ 55 ^ .',' 2 -Class Concrete Parapet 3 Paint all Beams p\ 2 Coats Lead and Oil. Round all Exposed Edqesto2£Rad. Exp. Metal to be 3 6' "or -6* 12' 'Mesh, weiqhinq not less than 1. 2 lbs. persq. ft. - — . __: — -L_____ _' _J£! Longitudinal Section. PartEnd Elevation. I- Beams Spaced as Shown on Table I. Expanded Metal Embedded in 6" ?d r//-rcc rnnrre+t h- -x 6 'tolZ'high, r=24+(lix2{") I3'to20'" ,x=24+(Hx3 n ) Spansl8to30Feet. 6-2?aa c k---x~>| 6 'tol2'high, x=ldt(tlx2fj I3'to20'f f x=ffl(Hx3") Spans 6 to 17 Feet Length of Culvert 'L enqth of Culvert is taken as the 'Distance from Outs, to Outs ide of Para- pets or from 0. toO.ofOuardRailand Meas- ured on a Line at Right Angles toC.L.of Road. Span is Taken as me Distance bet. Abutm'rs Measured on a Line parallel to the C.L.ofRoad. - 'at Top of I- Beams. Angle of Skew is the Angle bet. the C.L.of Culvert and a Line atR.Angles to C.L.ofRoad. ti= Total Height of Abutment A and B = Deflection of Wings in Degrees. L and M= Length of Winqs. P = Length • of Abutment Measured along Face of lop. Skew Culvert. 44 CULVERTS ii^h ^\d 'u " u n O 00 O CN ^tO 00 O cn "^-vo 00 O N *tO CO O N "tO 00 O cn rj- M M CN CN CN CN t-~ g.3 q^2u9-j }00J I9J O fO^O O CN tOOO H ttNO fO>0 O CN IO00 H ^NQ COO O CN tO'OrOcO , t , t , t 1 O l O iOO OOO M>. t^OO 00 OO 0*00*00 M qiSugq; ;ooj £s 10 CN cn ro-rj-iOM rOfONOO CNOO oO r^.00 O O m cn co »oO r^-00 OO w f^t MHWMMMHMMMCNCNCNCNCNCNCNCNCNCOrOCOCO Cu. Yds. 2d Class Concrete Cover and Parapets }OOJ I9J lONOO N ^O OOOwcOvoOmpO -to 00 O cn tJ-O r-» O w M M M CN CN CN CN CN CO' / )fOfOtO^ ,, t'tt , t>01010IJHO U")0 6 6 6 d 6 d 6 6 d 6 6 6 6 d 6 d 6 d o d 6 6 6 6 6 }OOJ Ss ^00 CO cOOO 00 GO O CO O Ow O h m -rj-O cocO cn O O O cn 00 1J"> O O O CNOO "t M N^IHOO O ^0 ^ W t-^'tO NtO^HOO T}- «t io voO t» r^-oo OOO m m cn co -t 10 10O r-- r^-oo O O O m MMMHMMMI-1MMMMMCNCNCN CO X qigug'j lOOJ I9J oO O O w cn co "3" ioO r>-00 O m cn ro "*t 10*0 t-^co O O w cn co hwhhmmmmihmCncncncncncncncncncocococo q;§n9i ;ooj Ss OioO^OioOoio0 1 oOioO'oOioO^Oio OnionO«")nOw' / )N(nionOnionO«ionOn CN CN CN CN tOfOfOtO^ - ^" T t^" l 01^ U-JO 0*0 t*» f-» t-» t^OO CO M o 'too hionhooo O m cn t O r^oo OOO HMCNCNCNCNCOCOCOCO»OVO 100 t^OO O O O O 9ZIg »h H=o-c!oort|a>HeOi-<("*'Hh!'«:-*^ i^i«|N-!t)H|fiH|NHM r/J a a a> PQ i i— i -00 M COO CO O CN rto 00 O V)0>^ O toioOiOf^ONO co*£> O^O O^O O i^OfONH ioOO m •}j i9d *1A\ CN lO IOOO MHMliOlOMMMMI-lCNCNCNCNCNCNCNlClDlOU-) M W M M CN CN CN CN CN COcOcOCOCO T t' : t't' ; t't' , t ,, t 1 O l Ol010 }93J q^Sug^j CO O O m cn co rt ir,\o t-»oO O m cn co ^ ^00 r-00 O »-* cn co MMMHMMMMMWCNCNCNCNCNCNCNCNCNCOCOCOCO Sup'Bds •"' "^' ^' ^' ^' ^' ^' ^' ^' ^' ^' ' ^' ^' ^' v 1 "J ' ^' ^' ^' ' CN rOcOCN cOCN cocOcocoCN rocOcOcOcOCN CN cOcocOfO s9qouj q;d9Q O t>i t^GO OOOO O CN CN CN CN N lo IO >0 V) m 10 IOOO 00 OO 00 t/3 0 O t^cO O O O m MMMMMMMHMMMMMMCNCN M9^S O O CO OO <0 OO CN O^O CN 0\\0 CN O^CNCO l^5CN00 IOMOO CN O O I-^O UO CO CN M OOO Nlfl'tfOH O O r^-O io CO CN M O lAo o t^-oo o d h cn cn co 4- too r^-oo t>t>6 h cn nt>nm HMMMMMMMHHHMCNCNCNCNCNCNCN AV9^S .Si O O CO 0*o t^oo O O m cn co "t io*0 t^co O O m cn co 't >o*0 r^oo OO MHHMMMMMMMCNCNCNCNCNCNCNCNCNCN Span, all Culverts O t^oo O O m cn co ■* mO r-^00 O O m cn co -^- too r^oo O O (HMMMMMMMMMCNCNCNCNCNCNCNCNCNCNCO QUANTITIES IN CULVERTS 45 o LO M II pq o CO II < w m o LO HI Cubic Yds. each ft. in length of Culvert more or less than 25 ft. Xjuosej^; M N ^t CO ^O^C O CN TJ-X-- - r t do" CN CO 't -t V)0 NOO O O HI CN CO^LO HI HI HI HI HI HI Cubic, Yards Third Class Concrete sSui yv * LO CN 00 i" Tj- TT H LO -^f* Tj- CN O T^- CN LO lo C> co O *0 «0 cn 10 OO WOO lo-3-O O HI HI cn O O ^ J>-l>.C>rj-H H H CN CO Tf tOO 00 O M CO tOOO M H t-4 H M CI s^nqy z 10 1000 to to cm 10 to coco OOOOO O OCO 0606 OO* h to O to 6 N co cn rf ^ tosO NOO O cn co TfvO OO O h CO MHMMMMHCNCN Cubic Yards Third Class Concrete s2u IAV ^ ION IO CO »00 O O h -f co to X^ O cn -^-O O0 O N ^t TJj-iOOCO OO h co^toO NOO h ^-Ood O* CN IONOH COtONOCN "4- MMfHHHClCNCMCNCNCOCO h1 ^■o^t-o^o^t-o^t-o^-o^-o^- co^rO t^OO °q cotoOOO O t-t cn t^- HHHHH HHCNCNO) }U9ui:mqv jo iqSpH ffl O t^oo O O h cn co ^r too i>-oo O " 1 CULVERTS Plate 4 a — continued 47 Table No. 6 Number I- Beams For Concrete Covers only P ^3 T pnrrf Vi r\( AKntmontc Spacing Length of Culvert 2'- 6" 2'- o" 3'-o" 15° Skew 30 Skew 45° Skew 18 5 5 4 18.64 20.79 25.46 19 5 5 5 19.67 21.94 26.97 20 6 5 5 20.71 23.09 28.28 21 6 6 5 21.74 24.25 29.70 22 6 6 5 22.78 25-40 31.II 23 7 6 6 23.81 26.66 32.53 24 7 6 6 24.85 27.71 33-94 25 7 7 6 25.88 28.87 35-36 26 8 8 7 26.92 30.02 36.77 27 8 8 7 27-95 31.18 38.18 28 9 8 7 28.99 32-33 39.60 29 9 8 8 30.02 33-49 41.01 30 9 9 8 31.06 34-64 42.43 31 10 9 9 32.09 35-So 43-84 32 10 9 9 33-13 36.95 45-26 33 11 10 9 34.16 38.10 46.67 1 Application of Tables Quantities for a 30 Skew Concrete Culvert, concrete top, length 30 feet, opening 13 feet high and 12 feet wide. From Table 1, an opening 12.12 ft. wide 30 Skew is a 14-ft. span requiring (see 30-ft. length, Table 6) 9 I-B earns spaced 2 r -9" c. to c. (9 X 400) = 3600 lbs. I-Beams; 218 lbs. Bars; 400 + (5X16) = 480 sq. ft. Ex'p'd Metal; 9-78 + (5 X 30) = 11.28 cu. yds. 2d class Concrete 32 lin. ft. Pipe Rail. An opening 13 ft. high will require Abutments, 16 ft. high (13' + 2' in ground + 10" I-Beam = 15'- 10"). From Table 4, Abutments = 118.0 cu. yds., Wings= 102.9 cu. yds. (5X4.79 = 23.95 cu. yds. 5 ft. extra length of Culvert) 118. o + 102.9 -f- 23.95 = 244.85 cu. yds. 3d Class Concrete. For Spans of more than 17 feet, use Masonry Tables for Con- crete Abutments and Wings. 4 8 CULVERTS Plate 5 k- W - ->j End Elevation. General Dimensions Semi-Circular Arch Culverts s Span Thickness at Springing Line Thickness of Ring Height of Haunch T Concrete K Masonry C Concrete R Masonry J Concrete V Masonry 6 8 10 12 14 16 18 20 2'-6" 2 , -6" 3'-o' / 3'-6" 3 , -9" 4 '-o" 4 '-6" ■>'-o" 2'-6" 2'-6" 3'-o" 3'-6" 3'-9" 4 '-o" 4 '-6" S'-o" 10" 11" 12" 14* 15" 16" 18" 18" 10" 12" 12" 15" 15* 15' 18" 18" iV 2'-6" 3'-°" 3'-6" 4 '-o" 4 '-8" S'-o" S'-o" 2'-0" 2'-6" 3'-°" 3'-9" 4 '-6" S'-o" S'-6" 6'-o" CULVERTS 49 Plate 6 Alternative Cross Section for Collapsible Forms. k- 6'6"or- ' 8'0" Longitudinal Section. ( All 1^2'and 2'x2' Culverts to be Bui It with EndWa/h.) End Wall. c a - t .ZOO .c x> 1- < 2|3 2'0" 6" 12" a 5 3'0" 6" 12" /.2 4'0" 6" 12" /2 S'O" 8" /2" /.2 Table of Dimensions. Reinforcement^ ^«9 |. , d:o'd'. , 'a:o"c, , ,r».i.Y' mt r*r ■ • • — ■ — ' Longitudinal Section. Slope m ^Dowels 0.25°" NetArea , I'Olon^lZC.ioC. Part Cross Section. New York State Small Box Culverts So CULVERTS Plate 6 a. — Massachusetts Standard for Concrete Arch Culverts |4--4'6"--->i f*\ !F\&y -* i (*!% * — Si f, <-/8> ^.1 this requires more excavation for the same depth, and, in the writer s opinion, it is more likely to be broken. The usual depth for drains is three feet below the surface. Where the road is on a descending grade, the water will flow out ot : CULVERTS Si Plate .6 b See Note below. __1 ,# Concrete k— 5'— >l Assumption for Live Load. -J xt i Longitudinal Rods 20"to27'C.toC. 5 of k sEvery o/fte/- Rod bent J i v •ir / ■'t r * 71 Long. Rods not *■ Spans less than 8 ft Corners not less than 6. Long. Rods Same SizeasSides and Bottom t. Cross Section of Culvert. -fr-'No Corners ' ; or Corner Rods • or Reinforcing n for 5 ides and \ Bottom t on2' \ and 3 'Spans. U» Q d — ■* •r T d <3 u ri CO c ccj rt a a CO CO a CO a CO C a CO a CO OOl^t^NrOtOWMOOO»fO T tOMO-< t^ Ol 't t^ tO f) t^ ^t"00 O* f) O ^ >H g m to toO vo rf io^O CO O O co O 00 "*H00 m O NO O O 00 "O io^- coiohvO c* t^ O* m iooO m O fOO r^-O fONH tJ- Q\ co ^ t^. O tJ-00 O ^t O t 3" | - ,v O m \0 O *0 m t-.cs Ohhhnnnnn to (O to 't 't ■+ io^O *0 t^ t^ r^ O O O O O O O O n w to fO ^ lo'C d y o o 3 OOoOMoOoOooOOOOOooOOONNMNWNMN't^^Tj-^ v' *0x*0» M M *M *M M M *M 00*00 CO'oO CO 00 CO CO CO u a CO VOMDVO "3-Tj-Tt-C* (N NOMDOO tOtOtOtOtOtOtO^ N OS CN o 0> Oi Oi Oi o i ii i i i T o O W N *N C* fOtO^oWWNN t-~ i j^^O O^OvO^O H . r^O vOOcX)C00000O^OOOOOO v 0\0O'O<5vOO ( O^0 Tj-Ttioioiovoioioiovo io^o voO'OO t^t^r^.t^ t^oo oooooooooooooO 00*0*0*0* ^ % H|^H|nH|n ^ * ^ H|n«l«H|NHt1H|NHlNH|NH|Nrt|t• r* t^00 OOOOOOmmmmmmm MHNtOHNtONtO^'MtO'tNtO't'ONtO* lOO X>-00 -M U -1-1 0) O M 28 cu T3 ^ CCj .y «« &^ ^u 'o »-• en ^ { • w ^y/ ^^'^ - ■■ ■ Bcrckf/H^KM^ 'mMm^-Packfill No. 4 Stone-; No.2 Stone or Gravel*^ Farm Tile-. Joints \ , S ~ Wrapped in Burlap* 1 - * **P» Fig. 16 Open Throat There are two kinds of drain in general use : No. 1 is built entirely of stone with an open throat roughly laid as shown; it is satisfactory in a water-bearing strata of gravelly loam or clay, but does not work so well in quicksand, which is liable to fill it up. It is generally cheaper, however, than No. 2. No. 2 is built of porous farm tile or vitrified tile of a suitable size (usually 3" to 6") with open joints, wrapped with a double or triple layer of burlap; the pipe is surrounded and covered with clean gravel or f " crushed stone to a depth of 6", the remaining depth of the trench being filled with large stone. If this drain has a good fall and the outlet is kept free, it will rarely clog even in bad quicksand. The author has successfully used the following method to prevent UNDER DRAINAGE S3 the outlet from clogging: after being brought out from under the macadam, the drain is continued under and across the ditch line, then keeping outside the ditch line, and using a slightly smaller 1 mt v , f ,/,>>'. ^ n An i n , !rr=:j . I! Fig. 17 gradient than that of the open ditch, the tile is continued down the hill until it reaches a point eight or nine inches above the ditch grade. Here it is turned into the open ditch through a small concrete head- wall and what little material it tends to deposit is washed down the ditch by the surface water. (See figure 17.) In planning the drainage for a road improvement, it is well to make as few changes as possible from the existing scheme. New culverts or a change of direction and amount of water discharged through farm land is almost certain to result in some friction with the owners of the properties affected. 54 CULVERTS OO o o 3 & M n Ph H to < o U c^ O t-H rfl H ro CO < < CO Ph O m P4 O H r* W H o H w ££ CO M M o w H CO a H !^ < < w « H M O VO t^ CN () io O O lo 00 o o o o PI ^ a M -»-> f4 o O COOO N o M fO N Ol ^ o o o o £ 4-) OO LOOO lovO o- 0\00 H H COOO H 00 j>- r-oo O CO CN OOO t«- CO o o !>. CO LO CN O M VO CO LOOO ON o LOVO 00 H CN CO LO t"^ ON H "* J>- O ^ o *fr H o H H M CN CN ^ LO *>. ON O in LOOO H VO CN Tt Tt LO LOVO SO J>-O0 OO ON O CN co lo i>- S3 p. •4-* rd Xi -t-> to d (U ^- O O O lo OnvO O J>-vo lo O O O O OO co O O O ON CN LOOO M M H M CN o o o o o O O lo O O oq^ O^ ^ w^ q^ CN -^- LO t^ ON o o CN vd H t^- CO LOOO H CON to CN CO "^J-vO h io q q o CN CN LO O LO OO O CN LO f^- H H H M CO CO CN t^ O CO co ^f M 6 CO co lo ON lo CN co tJ- lo n H tn en (D CN LOOO M t^ rt ^t LO LO cn vO O io O vo vo t^- t^OO ON CO LOOO CN CO O W CN tJ- i-i i-i i-i i-i o i>-00 O CN LO (N NNION tJ- h m cn vd O ON ON M VO > O ^ J>- lo lovO vO vO vO OO ON O vO N.00 ON M CN saqouj l^uiuio]yj; co "tvo OO O CN rtvO OO tj- OvO NOO CN co co ^ Tf WEIGHTS OF EXPANDED METAL 55 o Ph Weight in Pounds per Sq. Ft. OvO rj- o Tf O n^h r»vo -*tO vOOOO cn "- 66666666666 1 O -d § < CO vo vo vo vo O lOifllOcOfO COO «o CO cn CO ^tOOOoOOOOOoOt^-CN TtO t^-OO f>. VO CN VOO Ol CN CO O "^"OO CN M M CN " H s < u h Q < Section Area per Foot of Width O vo r^.vo cooo 00 r>-0 O O "1 vooo O M OOoo >t t^O vo O O O "^-vO CNCNCNMOWMCNCO 'tO O CN CO 66666666666666 CO -<-> M CO o Standard < < Light Standard Heavy Ex " Standard Heavy Old Style Standard Heavy o°3 cX)cocNCNvOOOOO v OOvO' : t ,, =t MMMMMMMMM M ,-fi e* cn m CN cococococococo ^1"nO O 56 CULVERTS Table 16. Table of Round and Square Bar Weights Round Bars Plain Sq uare Bars and Twisted Square Bars Diameter Area Weight Dimension Area Weight 1 4 .0491 .167 1 4 .0625 .212 _5_ 16 .0767 .261 A .0977 •332 3 8 .1104 .376 3 8 .1406 .478 A .1503 •511 7 T6~ .1914 .651 i "2 .1963 .668 1 2 .2500 .850 A .2485 .845 9 16 .3164 I.O76 5 8 .3068 1.043 5 8 .3906 I.328 \i •3712 1.262 11 T6 •4727 I.607 3 4 .4418 1.502 3 4 .5625 I-9I3 ^4 16 .5185 1-763 13 16 .6602 2.245 7 8 .6013 2.044 7 8 .7656 2.603 ti- .6903 2-347 15 16 .8789 2.988 i •7854 2.670 I 1 .0000 3.4OO ii ..9940 3-38o i* 1.2656 4.303 ii 1.2272 4.172 i| 1.5625 5-3I3 if I.4849 5-049 if 1.8906 6.428 ii 1. 7671 6.008 if 2.2500 7.650 Diameters expressed in inches. Areas expressed in square inches. Weights expressed in pounds per foot of length. The twisted square bar is known as the Ransome Bar. RcmsomeBar. Kahn Cup Bar. Corrugated Bars, Diamond bar. ThacherBar. ROUND AND SQUARE BAR RATES 57 •8 O ■Q N < PQ c B B I 1 & Q fa o w < W < o P u u CD Q < « O w w n pq CJ O B s X! cn ^t-oo co q\0 ^ co 6 6 O H H « Wj IT) < CN H O M COO O CO O "t^ OMD O Q O O w cn co to ^ O to 000000* m M c3 pq M D rC (J CJ Xi H X! n coO O cooo ^t" O t*- dddoHHNfOfo M < ^ o ^f O CO 00 m u->m O O O M M (N ^ lO N O^ H dddddddoH a w H M 1-4 O U -t-> X! vO to»T)iO O OO OnO «*■ CO O M CN CO *0 V < to O r^ O O CN ION O lO d O O H M Ih PR M en '£ H x cncnOOOOOOO cn omnio^(-t|-ioion CN -rfOO CO OO ^ CO CO OOO^'-iCNcO'^'^O tOO O o too o o to cnCOOcnvoO^ocn O "3- to oo o o o o Omcncoi-O^Ocnlo OOOOOOmmm H w PQ $ XI IS 1 '53 cn co tJ-oO co O CO CO O OOO tONH t^ to CO coO NupWiO OOwcNcncO'sJ-io < OOO too O O to OOOCNtOO^OCN rf to o^^ o o o o M CN co>-Ot^O cn to OOOOOmmm Nominal Size of Bar iHl-*»lQCH|NiO|00e9i^H°0 iH|0CiH|tJ( H M M 58 CULVERTS CO H > •J u w PL, 1— I Ph o 1— 1 1 H CO < O 0* rfvOOO O 0 w Ph r~» m vo O «o O ro io OO O M H M H CN O 4 *5 n Ph CO M O r-~ VO OO O m to >0 H M H M 4& VO q co oo 9* m Ph cn co ^ O »o t^> co O vn co vN vO m rt r^ o t)- vo !>. 00 O m M M S9= CO CN CO •"+ <+ *0 iO VO t^ CO O 4£= o oq Tt- CN M O OO Tf to vo vo *>. M CO H t4 W > u w H w u o u O w O o H N CO X »o O m co -^-vo t^OO O m co rj- io t^OO O M t^CO O O m ovo r-»CO Ov O m co ^f o to X r^oo 00>0 O h m cn c\» coco^Tfioio Tf tovo NdO H W (O* u->vo x^co O O HMHHHNNNNMMNNOin^ ©9= CS to "Vo X t*. t>»vo vOvOioio^l-Tt'^-cococococvi oi CN CO "^ lOO t^-00 ©O H N to^t IOVO r^ MHHMMMMHCviCNCNvNCNCNMCvj CO 00 4 6^ X t^VO lOrJ-fON M O OCO f^vo *0 rf co CN 0 m m CN co^J-Tf loO vo i^OO O O* #^ MMMHMMMWMMMMMM o VO CO X "co n i-Hoovo coOoo low o\ r^. •<+ w owo OOMMC^ro^t^- »OVO vo t^OO O O O S^MWHHMHHMMMMMMMvN co vq co X "co OvvO vn OiiOHOO •^J-M N WiOvO coO»»0 r^CO O O* O m m OlOOHHWN(*)W)'tTt-V5 4^ MHMHMMMMMMM o to X CM COCO MNH^O Hl^O TJ-0> ^tO0 fONN O vo r~- r^co cooO\000>HM(NiNro €^ MMMMMMMM- o CO ■*- 6 c a 4> : > R-a 3 u Cd "5 3 be 8 a** ""CO X co 4) 4-> +-> vJ rr 2^ OB -.6 X -1 n en T3 n L bC cd ac3. £ffi M u o H s o a I: r" o c •> t- — k §8 PQ«» CAMBRIA STANDARD I-BEAMS 59 Table 18. Properties of Cambria Standard I-Beams Depth of Weight per Area of Thick- ness of Web Width of For Fiber Stress of 12,500 lbs. per Sq. In. for Bridges Beam Foot Section Flange Inches Pounds Sq. Inches Inch Inches Coefficient of Strength 3 5.50 I.63 •17 2-33 13,790 3 6.50 I.QI .26 2.42 14,950 3 7-50 2.21 .36 2-52* 16,180 4 7-50 2.21 .19 2.66 24,850 4 8.50 2.50 .26 2-73 26,480 4 9-50 2-79 •34 2.81 28,110 4 IO.50 3-09 .41 2.88 29,750 5 9-75 2.87 .21 3.00 40,300 5 12.25 3.60 ,36 3-*5 45,390 5 14-75 4-34 •50 3.29 50,490 6 12.25 3.6l .23 3-33 60,520 6 14.75 4-34 ■35 3-45 66,6lO 6 17-25 5-07 •47 3.57 72,740 7 ' 15.00 4.42 •25 3-66 86,260 7 17.50 5-15 ■35 3.76 93,290 7 20.00 5-88 .46 3-S7 100,430 . 8 18.00 5-33 .27 4.00 118,490 8 20.25 5-96 •35 4.08 125,400 8 22.75 6.69 •44 4.17 133,57° 8 25.25 7-43 •53 4.26 141,740 9 21.00 6.31 .29 4-33 157,260 9 25.00 7-35 .41 4-45 170,260 9 30.00 8.82 •57 4.61 188,640 9 35-oo 10.29 •73 4-77 207,020 10 25.00 7-37 •3i 4.66 203,500 10 30.00 8.82 •45 4.80 223,630 10 35-oo 10.29 .60 4-95 244,050 10 40.00 11.76 •75 5.10 264,480 12 31.50 9.26 •35 5.00 299,740 12 35-oo 10.29 •44 5-09 317,030 12 40.00 11.76 .56 5-21 341,540 6o CAMBRIA STANDARD I-BEAMS Table 18 . Continued Depth of Beam Weight per Foot Area of Section Thick- ness of Web Width of Flange For Fiber Stress of 12,500 lbs. per Sq. In. for Bridges Inches Pounds Sq. Inches Inch Inches Coefficient of Strength 15 42.00 12.48 .41 5-50 490,840 15 45.OO 13.24 .46 5-55 506,490 15 50.00 14.71 .56 5-65 537,130 " 15 55-oo 16.18 .66 5-75 567,770 15 60.00 I7-65 •75 5.84 598,410 18 55-oo 15-93 .46 6.00 736,620 18 60.00 17-65 .56 6.10 779,440 18 65.00 19.12 .64 6.18 816,200 18 70.00 20.59 .72 6.26 852,970 20 65.00 19.08 •50 6.25 974,6oo 20 70.00 20.59 .58 6-33 1,016,490 20 75.00 22.06 ^5 6.40 1,057,340 24 80.00 23-32 •5o 7.00 l,449,46o 24 85.00 25.00 -57 7.07 1,505,430 24 90.00 26.47 .63 7-i3 1,554,450 24 95.00 27.94 .69 7.19 1,603,470 24 100.00 29.41 •75 7-25 1,652,490 Explanation of the coefficient of strength in the above table and examples showing use in practice. The coefficient of strength for each sized beam represents the maximum uniformly distributed load, in pounds, that will produce a fiber stress not exceeding 12,500 lbs. per sq. inch multiplied by the span in feet. If the load to be investigated is a concentrated load it must be changed to an equivalent uniform load in order to use the values given. This is done by multiplying the concentrated load by 2. Example: Suppose that it is required to determine the size I-beam that will carry a 40,000 lb. load in the center of a 15' span and a uniformly distributed load of 20,000 lbs. The coefficient of resistance for the concentrated load will be 2 (40,000) X 15 = 1200000 Uniform load 20,000 X 15 = 300000 1500000 The required beam must have a coefficient of resistance of 1500000 plus the coefficient due to its own weight. A 24" beam weighing 90 lbs. per foot has a coefficient of 1,554,450. The beam weighs 90 X 15 = 1,350. The coefficient for the beam weight is 1,350X15 = 20,250, which deducted from 1,554,450 gives a co- efficient of 1,534,200, which is slightly greater than required and is safe. CHAPTER IV FOUNDATIONS FOR BROKEN STONE ROADS Concrete foundations are considered under Brick Pavements in chapter V. The real foundation of a road is the earth subgrade; generally, however, the term foundation is used in speaking of the lower course of stone, gravel, etc., used to distribute the concentrated wheel loads. A discussion can 'be developed under the following heads: i. The bearing power of different soils. 2. The concentrated wheel loads on improved roads. 3. The distributing action of foundation courses and the depth required for different soils. 4. The different kinds of foundation courses. 5. The distribution of the stone in the foundations. 6. Special cases. 1. Bearing Power of Soils The subgrade develops its greatest bearing power when dryo In the following discussion we assume that the soils are protected by a well-designed drainage system. Mr. W. E. McClintock, Mem. Amer. Soc. C. E., Chairman of the Massachusetts Highway Commission, published in the 1901 report of that Commission a valuable statement of the results of their investigations on the bearing power of soils and the distribution of wheel loads by the macadam. The conclusions have been well tested in practice and found to be satisfactory. "The Commission has estimated that non-porous soils drained of ground water, at their worst will support a load of about 4 lb. per square inch; and having in mind these figures the thickness of broken stone has been adjusted to the traffic. "On a road built of fragments of broken stone the downward pressure takes a line at an angle of 45 degrees from the horizontal and is distributed over an area equal to the square of twice the depth of the broken stone. If a division of the load in pounds at any one point by the square of twice the depth of the stone in inches gives a quotient of four or less, then will the road foundation be safe at all seasons of the year. On sand or gravel the pressure can be safely put at twenty pounds per square inch. . . . "Acting on this theory the thickness of the stone varies from four inches to sixteen inches, the lesser thickness being placed over good gravel or sand, the greater over heavy clay, and varying thicknesses on other soils. In cases where the surfacing of broken stone exceeds six inches in thickness, the excess in the base may be broken stone, stony gravel or ledge stone; the material used for the excess depending entirely upon the cost, either being equally effective." 61 62 FOUNDATIONS FOR BROKEN STONE ROADS 2. Concentrated Wheel Loads There should be some limit placed by law to the maximum load per lineal inch of tire for vehicle using improved roads. The roads can then be designed for this load with no danger of failure from unreason- able pressures. Road work is handicapped in this country by the lack of wide tire statutes and the regulation of traction engines using sharp lugs on the wheels. At present it is necessary to assume a loading that will probably not be exceeded by the unregulated traffic. Many engineers favor a law limiting the load on improved roads to 700 to 800 lb., to the lineal inch, which is a reasonable limit; with a six inch thread this would mean a load of nine tons for a four wheel truck provided the load was uniformly distributed. This is beyond the limits of team hauling. Most of the mechanical trucks in present use have tires wide enough to reduce the pressure below this limit. Near some of the large cities, however, mechanical trucking has increased to proportions that amount to a regular freight line and excessive loads are carried; the load and speed for such trucks must be regulated, for no road can stand abuse of this character. The following regulations governing the control of motor trucks and traction-engines were prepared by the New York State Highway Commissioner to go into effect in 19 14. Regulations for State and County Highways Adopted by the Commissioner of Highways of the State of New York Sec. 1. No traction-engine, road-engine, hauling-engine, trailer, steam-roller, automobile truck, motor or other power vehicle shall be operated upon or over the state or county highways, the face of the wheels of which vehicle are fitted with flanges, ribs, clamps, cleats, lugs or spikes. This regulation applies to all rings or flanges upon guiding or steering wheels of any such vehicle. In case of traction-engines, road-engines or hauling-engines which are equipped or provided with flanges, ribs, clamps, cleats, rings or lugs, such vehicles shall be permitted to pass over said highways provided that cleats are fastened upon all the wheels of such vehicles, and are not less than i\ in. wide and not more than if in. high, and so placed that not less than two cleats on each wheel shall touch the ground at all times, and the weight shall be the same on all parts of said cleats. Sec. 2. No traction-engine, trailer, steam-roller, automobile truck, motor or other power vehicle shall be operated upon or over the state or county highways; nor shall any object be moved over or upon any such highways upon wheels, rollers or otherwise, in excess of a total weight of 14 tons, including the vehicle, object or contrivance and load, without first obtaining the permission of the State Commission of Highways as hereinafter provided. No weight in excess of 8 tons shall be carried on any one axle of [any such vehicle. Sec. 3. The tire of each wheel of a traction-engine, road-engine, hauling-engine, trailer, steam-roller, automobile truck, motor or other power vehicle (except traction-engines, road-engines, and hauling-engines) shall be smooth, and the weight of such vehicle, including load, shall not exceed 800 lb. upon any inch in width of the CONCENTRATED WHEEL LOADS 63 tire, wheel, roller or other object, and any weight in excess of 800 lb. upon an inch of tire is prohibited unless permission is obtained from the State Commissioner of Highways as hereinafter provided. Sec. 4. No motor or other power vehicle operated upon any state or county highway shall be of a greater width than 90 in., except traction-engines which may have a width of 1 10 in. Sec. 5. No traction-engine, road-engine, hauling-engine, trailer, steam-roller, automobile truck, motor or other power vehicle, carry- ing a weight in excess of 4 tons, including the vehicle, shall be operated upon any state or county highway at a speed greater than 15 mi. per hr.; and no such vehicle carrying a weight in excess of 6 tons, including the vehicle, shall be operated upon any such highway at a speed greater than 6 mi. per hr. when such vehicle is equipped with iron or steel tires, nor, a speed greater than 12 mi per hr. when the vehicle is equipped with tires of hard rubber or other similar substance. Sec. 6. The State Commisssioner of Highways, upon proper application in writing, may grant permission for the moving of heavy vehicles, loads, objects or structures in excess of a total weight of 14 tons over state and county highways, upon proper application in writing being made therefor, and under such restrictions as the Com- missioner may prescribe. Sec. 7. The owner, driver, operator or mover of any vehicle over any state or county highway shall be responsible for all damages which said highway may sustain as a result of a violation of any of the provisions of the foregoing Rules and Regulations, and the amount thereof may be recovered in an action of tort by the State Commis- sioner of Highways or by any County Superintendent of Highways of any county or by any Town Superintendent of Highways of any town in which said violation occurs. 1 Sec. 8. These regulations take effect October 20, 1913. "Section 24 of Chapter 25 of the Consolidated Laws entitled 'The Highway Law' provides that any disobedience of any of the foregoing rules and regulations shall be punishable by a fine of not less than $10 and not more than $100 to be prosecuted by the Town, County or District Superintendent, and paid to the County Treasurer to the credit of the fund for the maintenance of such highways in the town where such fine is collected." Under these regulations properly enforced any of the ordinary foundation courses can be successfully used except on the heaviest traffic roads (see Classification page 74) provided the depth is varied to meet the soil conditions. Heavily loaded farm wagons exert a pressure of about 350 lb., per lineal inch of tire width as determined from the records of produce dealers in Western New York, and the author believes that a road designed to distribute a 4,200-pound wheel load on a six-inch tire would be safe. Note : — The length of wheel bearing on a well-constructed mac- adam road is about 1". The use of this loading and the application of the rules for distribu- tion of pressure given by Mr. McClintock in the preceding quota- 1 This provision for enforcement does not work well; enforcement should be in the hands of the State authorities. 64 FOUNDATIONS FOR BROKEN STONE ROADS tion results in a depth of 15" for heavy clay or a fine sandy loam, and a depth of 5" for gravel, which check his results. The thickness to be used in the intermediate cases must depend on the judgment of the engineer. The following examples are intended only as a guide for the more common cases. The amount for special cases often depends on trial. Sand and gravel require from 4" to 6"; New York State uses 7" as a minimum where motor trucking is expected; Massachusetts uses the following section on good gravel. A ZZZZZZZ2EZZZIZZZ ■ ' ■ ■'■■'■'■ • • 1 > m — is! ^4"to5' Fig. 18 Wherever the total depth is less than 5" the stone should be laid in one course and classified as top stone. For a light clay loam an average depth of 7" to 9" is sufficient in cut; for fills over 2' deep 7" is enough; high fills even of clay after having once settled rarely give trouble with 7" of stone. Heavy clay requires at least 12" in cut; if the soil is springy and especially hard to drain, 15" to 18 " is advisable. For shallow fills (see figure 19) : Old Surface Good Materia!^ Fig. 19 In shallow or "pancake" fills, clay or fine sandy loam should never be used where the natural surface at this point is of a better variety, as they are almost certain to become saturated with water and will either squeeze or heave out of shape; long, shallow fills are to be avoided, which is considered in placing the grade fine, but where unavoidable, the best available material should be obtained and the original surface well broken up to form a bond with the new fill. Where clay is used, it should be treated as in cut. For fills of inter- mediate depths [i' to 2'] 8" to 9" is satisfactory. A fine sandy loam is difficult to drain because of its strong capillary action. Mr. Charles Mills, Chief Engineer of the Massachusetts Highway Commission, in the report for the year 1902 states that a loam of which 30% or more will pass a 100 sieve will require from 10" to 15" of stone. To illustrate the different stone depths that may be used in a short distance, an extract follows from the construction report on founda- tions for " Clover Street, Section 1," a road near Rochester, N.Y. This was built in 1907-1908 and has held satisfactorily under farm traffic. KINDS OF FOUNDATION COURSES 65 Clover Street Road, Section i The normal depth of stone on this road was 7" > ** ^°P Station to Station Character of Subgrade Total Depth of Stone 180 183 + 25 183 + 25 186 -f 25 186 + 25 187 187 190 190 191 191 193 193 200 Cut in sand and gravel Clay fill Clay cut Sand, gravel and clay Clay cut Clay loam fill Sand and gravel 6" 8* n" 7" 12" 7" 6" PREPARATION OF SUBGRADE It is evident from the pressures to which a road is subjected that the subgrade must be well consolidated before placing the foundation stone. This is usually effected by rolling with a 10 or 15 ton steam roller, exerting a pressure of 350 to 500 pounds per linear inch of wheel width, and is continued until the grade is firm and compact. The difficulties of consolidation in different soils and the methods of overcoming them will be included in chapter XL KINDS OF FOUNDATION COURSES The foundation courses in ordinary use are as follows : 1. Crushed stone 2. Screened gravel 3. Field stone sub-base 4. Pit gravel sub-base 5. Field stone-sub base bottom course 6. Pit gravel sub base-bottom course 7. Quarry stone base or Telford. 1. Broken Stone Bottom Course. This style of construction is the one in most general use. Where local stone is abundant and well distributed, such a course will cost from S2.00 to $2.50 per cubic yard rolled in place; where imported stone is necessary, the cost depends largely upon the freight rate and the length of haul and may run as high as S5.00. Bottom of this kind is generally used where the total depth of stone metaling does not exceed 6" to 8" after rolling. Beyond these depths it is often cheaper to substitute sub-base or sub-base bottom course for a part or the whole of the broken stone course. The method of construction by the New York State Highway Commission is shown in the following extract from their 191 1 specifi- cations : 66 FOUNDATIONS FOR BROKEN STONE ROADS Stone Macadam Bottom Course " After the subgrade has been prepared and has been accepted by the Engineer, a layer of broken stone of the approved size and quality for bottom course shall be spread evenly over it to such a depth that it shall have, when rolled, the required thickness. The depth of the loose stone shall be gauged by laying upon the subgrade cubical blocks of wood of the proper size and spreading the stone evenly to conform to them." "The roller shall be run along the edge of the stone backward and forward several times on each side before rolling the center. Before putting on the filler the course shall be rolled until the stone does not creep or weave ahead of the roller. In no case shall the screenings or sand for filler be dumped in mass upon the crushed stone, but they shall be spread uniformly over the surface from wagons or from piles that have been placed on the shoulders. It shall then be swept in with rattan or steel brooms and rolled dry. This process shall be continued until no more will go in dry, when the surface shall, if required by the Engineer, be sprinkled to more effectually fill the voids. No filler shall be left on the surface, and surface of bottom course stone shall be swept clean before covering with top course. Only such teaming as is necessary for distributing the materials will be allowed on the bottom course. Any irregularities or depres- sions, the result of settlement, rolling or teaming, if slight, shall be made good with broken stone of the same size used in the bottom course, otherwise the stone shall be removed and the subgrade re- graded and rolled. Such removal and restoring of the surface shall be made at the expense of the Contractor. Screenings shall not be used in leveling up irregularities or depressions." Massachusetts uses no filler; otherwise their construction is substantially the same as New York. Where imported stone is specified or the local stone is suitable for both top and bottom courses, the size used for bottom course is known commercially as "No. 4 stone" and ranges from 2f " to 3!" in its greatest dimension; the smaller sizes are used for the top course, for concrete and for filler; where the local material is only fit for bottom, the course is made up of stone ranging from i n to 3!" in order to use up the total output of the crusher. The stone smaller than 1" is used for filler, on the shoulders, and sometimes for the cheaper grades of concrete. In specifying the sized stone for a particular job, economy is considered. Stone sized from 1" to 3%" is perfectly satisfactory. The only reason for limiting the usual size from 2f " to 3!" is that it leaves the 1" to 2f " stone for the top course; a uniform grade is important for the top and the size men- tioned gives a smooth finish. The ratio of loose depth to rolled depth is given on page 272. Where filler is not used in the construction of the bottom course more binder is required for the top; it is our opinion that the use of filler is the better construction. The clause concerning teaming in the quoted specifications is a dead letter; teaming helps to consolidate the bottom provided it is KINDS OF FOUNDATION COURSES 67 distributed over the full width and care is taken in watching the course to prevent loss of shape when the traffic is first turned on or after a long continued rainfall. 2. Gravel Bottom Course. Screened gravel 1" to $%" in size is used in place of crushed stone; the course is constructed in the same manner as described above, except that a filler containing some clay or clay loam is preferable to a coarse sand, and it is often necessary to wet the course in order to consolidate it satisfactorily. A gravel bottom should be made somewhat thicker than a crushed stone bottom as the fragments do not interlock as firmly as crushed stone. The choice between a screened gravel or crushed stone bottom depends entirely on the relative cost. Under favorable conditions a screened gravel bottom course will cost from $1.30 to $2.00 per cubic yard, rolled in place. 3. Field Stone Sub-base. Field stone sub-base is constructed, as shown in the cut, of field boulders roughly placed and rilled with gravel, waste No. 2 stone or stone chips; no attempt is made to finish the top of the course exactly to line and grade, as any small inequalities can be filled with bottom stone. The depth varies from 5" to 12" depending on the soil encountered in the size of the available field stone. In designing a bottom course of this kind, care must be taken to have accurate data as to the average size of stone available. If the de- mands of a foundation were fully satisfied by a 5" sub-base course, it might still be more economical to use a 7" course if the stone aver- aged seven inches, because the extra work of sorting and sledging to a 5" size would result in a higher cost per square yard than for a 7" depth. Top Course,^ ^!o , : Bottom Course ia&jkmrf es ■Sub-Base Fig. 20 ^ The amount of stone and filler required per cubic yard in place is given on page 272. Under favorable conditions this sub-base can be constructed for $1.00 to $1.50 per cubic yard. 4. Pit Gravel or Creek Gravel Sub-base. Stony gravel is a satisfactory material for sub-base; it can be readily constructed for any depth from 2" to 24" if required, and where a pit or creek bar is near, the cost of such a course should run from $0.80 to $e. 2 5 per cubic yd. 68 FOUNDATIONS FOR BROKEN STONE ROADS The ratio of loose to consolidated gravel for such a course is given on page 272. 5. Field Stone Sub-base Bottom Course. Sub-base bottom course is essentially the same construction as sub-base, except that, as the top course is placed directly upon it, the stone must be more carefully assorted as to size, more carefully placed as to line and grade, and a better grade of filler must be used. Crushed stone (crusher run) or coarse gravel make a satisfactory filler. /Top Course 1 ^Sub-Baw Bottom Course Fig. 21 The course can be of any depth from 5" up, depending, as for sub- base, on the soil and average size of stone; it is practically impossible to make a large stone bottom of this kind conform exactly to line and grade; a variation of 1" either above or below grade is usually allowed and the inequalities taken out with the top stone; this requires that the top course must be at least 3" deep after rolling. Sub-base bottom is especially applicable for long stretches of road requiring a depth of 9" to 12"; it usually costs from $1.30 to $i.;o per cubic yard in place where fence stone is available, and by its use the item of higher-priced bottom stone is reduced. However, on a hard foundation it is generally better to use 4" to 5" of ordinary broken stone bottom course instead of the sub-base bottoin course even if more expensive, because the small stone construction is more uniform in its resistance to heavy loads and the top course will wear more evenly and longer. . . An extract from the 1915 New York State Specifications is given below. „ Sub-base Bottom Course When field or quarry stone is used for constructing the foundation course it shall be of a hard, sound and durable quality, acceptable to the Engineer; the stones shall be placed by hand so as to bring them in as close contact as possible. When quarry stones are used they shall be placed on edge. The depth of the stone shall m no ca^e be greater than the depth specified for the course, the width shall not be greater than the depth, nor more than six inches; and the length shall not be greater than one and one-half times the depth, nor more than 12 inches. The distribution of the stone shall be ol a uniformity satisfactory to the Engineer. The long dimension sha always be placed crosswise the road. After laying^ this course shall be thoroughly rolled with an approved roller weighing not less than 10 tons, and shall then be filled with stone or coarse gravel as directed and again rolled until the stones are bound together and thoroughly KINDS OF FOUNDATION COURSES 69 compacted; but no gravel shall be used for filling except under written permission of the Engineer. .All holes or depressions found in rolling shall be filled with material of the same quality and the surface shall be rerolled until it conforms to the lines and grades shown on the plans. When field stone is used approved tailings may be used for filling. In all cases a sufficient amount of fine material shall be used to fill all voids. In limited areas where the use of a roller is impracticable heavy tampers may be used to con- solidate the material. 6. Pit Gravel Bottom or Sub^Base Bottom. A stony gravel containing not over 1$% of loam makes a satis- factory course; the depths vary from 4" to 18"; pit or creek gravel even when unusually coarse has from 40 to 60 c ~ c of fine material; a suitable gravel for pit run bottom should not contain more fine material passing a j" screen than coarse material retained on a J" screen. If there is a large excess of fine the gravel should be screened and remixed at the bin in proper proportions. The great difficulty in this construction is to get proper consolida- tion without too much delay. It is advisable to lay a course of this kind at least two weeks ahead of the top stone in order to give traffic and rains a chance to help consolidate the course. The addition of 10 % of loam to clean gravel will quicken the consolidation. This can be done either at the pit by leaving a thin layer of loam when stripping which runs down with the gravel in loading or by placing from \" to 1" of loam on top of the gravel as spread on the road; the author has succeeded in getting rapid consolidation by snatching loaded teams over the loose course with the road roller; the roller continually smooths out the gravel and eases the haul for the teams; the horses' hoofs and wagon wheels punch into the gravel and pack it down rapidly. Sprinkling helps. A gravel bottom consolidates unevenly and it is always necessary to reshape it somewhat after consolidation; about So. 05 per cu. yd. should be allowed for this re- shaping of crown and elimination of humps and hollows. A prop- erly consolidated gravel bottom will permit a 4-ton load on 3 V tires passing over it without making a wheel mark over \" deep; this is a simple available construction test. We have gone into some detail covering this construction as it is the most economical type of bottom in a large number of cases but is not generally favored because it is harder to consolidate than, the other types of bottom. With a 3" or preferably a 4" madacam top it has proved perfectly satisfactory on all but the heaviest traffic roads. The cost of a gravel bottom ranges from $0.80 to $1.50 per cu. yd. in place provided the hauls are short. The depths of gravel is gauged by blocks or lines and the ratio of loose to rolled depth is approx. 1.2 (see page 272). 7. Telford Base. Telford base is rapidly going out of use in the United States because of the difficulty of maintaining a top course laid upon it. It seems 70 FOUNDATIONS FOR BROKEN STONE ROADS to be too rigid and is. more expensive than sub-base or sub-base bottom course, costing about $1.80 to $2.00 per cubic yard under favorable conditions. A good description of a telford construction is given by Mr. William Pierson Judson in " Roads and Pavements." The following quota- tion is an extract from his book: "On this subgrade are then placed by hand the stones forming the telford foundation, which may vary in size as shown below: each stone must be set vertically upon its broadest edge, lengthwise across the road and forming courses and breaking joints with the next course, so as to form a close and firm pavement. The stones are then bound by inserting and driving stones of proper size and shape to wedge the stones in their proper position. All projecting points are then broken with a sledge or hammer so that no projections shall be within four inches of the finished grade fine. "The telford foundation is then rolled with a steam roller of ten or more tons weight, until all stones are firmly bedded and none move under the roller. All depressions are then rilled with stone chips not larger than two and one-half inches, and the whole left true and even and four inches below the fine of finished grade and cross-section. "A good workman will average about twenty minutes in setting a square yard of this telford foundation, which may be formed of any kind of quarried rock which is most available: cobble-stones are not suitable. "The practice in 1901 in the states named is here shown:" Table 19. Sizes of Stone for Telford Foundation, in Inches State Depth, as set on Edge Width, as set Length set across Road Remarks Max. Min. Max. Min. Max. Min. New Jersey . Mass Conn New York . . 8 6 8 8 8 5 8 6 4 10 10 10 4 6 4 10 15 18 15 6 8 6 Alternate end-stones double length. Two inches gravel rolled on subgrade as base. Macadam covering formed in one layer. Used only on unstable ground as foundation for macadam. Distribution of Stone in Foundations. In the discussion of sections, Table 9 shows that most of the traffic keeps to the middle 10 or 12 feet; to make a consistent design the foundation should therefore be thicker in the middle than on the sides for the ordinary crushed stone bottom, and where sub-base is required it is often unnecessary to place it the full width of the metaling. KINDS OF FOUNDATION COURSES 71 Figure 22 is an example of such a foundation course for ordi- nary soils as used by the- New York State Highway Commission in 1910. 4 to 8 Fig. 22 Figure 23 is an example of an economical sub-base for a light traffic road as used by the Illinois Highway Commission in 19 10. Fig. 23 On a heavy traffic road, however, the writer does not believe that the width of sub-base should be less than the width of metaling. Special Cases. Long stretches of comparatively level ledge rock, peat, muck, and vegetable loam may be placed under this head. Where a road is on the surface of ledge rock for any distance, the usual cross-section of part cut and part fill cannot be used because of the high cost of shallow rock excavation for ditches; the grade should be lifted to make the normal section fill and the best avail- able material (not clay) used in its construction. Where conditions of this kind prevail, dirt is usually hard to obtain and often a stone fill is cheaper and also more satisfactory. The construction shown below was used for a stretch of two and one-half miles on the Leroy-Caledonia State Highway in New York, where ledge rock was encountered as described. The price for the stone fill was $1.23 per cubic yard in place con- structed as shown; the road was built in 19 10 and has given satisfac- tion; such a base, however, is very rigid, which will probably cause a more rapid deterioration of the top course than if earth were used; the minimum thickness of top for such a fill is 3" as it is impossible to construct it exactly to line and grade; it was found that by allow- ing a variation of 1" either above or below the grade elevation, the fill could be readily constructed, and these small inequalities were taken out with the top stone. A top course having such a variable thickness should be paid for by weight and not by volume in place. (See page 268, Cost Data.) 72 FOUNDATIONS FOR BROKEN STONE ROADS Screened Gravel or \' ''^E^in£f Top j ^rusher Run abovej Broken Stone 'j-^ ^^jL,^ ' »^__ -Lj£ ^_ ,- Best Available ■ uravetoroKen '^\ ' Ledge Rock- Stone, Fence or / J Quarry Stone Fill./ ^Method "A*! j ' Fig. 24 Fill can be made of fence stone, gravel, quarry spalls, stone chips, or run of crusher stone over f " in size. Method A. — Boulders up to 2 cu. ft. can be used, placing the largest in the bottom of the fill; the top layer must be fairly uniform and not over 8" in size and must be roughly placed by hand to reduce the voids as much as possible, provided this layer of large stone is within 4" of the bottom of the top course. The top 8" to be filled with stone chips or gravel and a cushion of at least 2" of screened gravel, stone chips or crusher run of broken stone over f " in size to be placed on top to bring the fill to the correct grade and crown for the top course. Method B. — Same materials and manipulation as Method A, except that provided the top of the boulder fill is more than 4" from the bottom of the top course the top layer of the boulder fill need not be placed by hand. (See sketch, Method B.) Peat, Muck, Vegetable Loam, or Silt. Where the material is semifluid the only solution is a pile and grillage foundation. Swamps, as ordinarily encountered, can be treated successfully by using a corduroy or mattress foundation covered with a deep fill of gravel or large stone. In some cases where the muck is com- paratively stiff, a gravel or boulder fill alone will give a satisfactory foundation. Where swamps are crossed by improved roads, the location usually follows the old road which has often been corduroyed in the past; in such a case the old foundation should not be disturbed; a sufficient additional depth of stone can be added to keep the shape of the section intact. As an example, the Scottsville-Mumford New York State im- provement crossed a 1000 ft. stretch of muck on the old road location; it was found that the original cedar corduroy was in good shape; an 18" depth of large boulders was placed on the old foundation and surfaced with 6" of broken stone macadam. This stretch of road has kept its shape and has not settled; it affords a good example of the statement made on page 61, that in many special cases the depth of the stone is determined by trial; the boulders were KINDS OF FOUNDATION COURSES 73 put on in successive layers of 6" each until there was no material movement under the roller and then surfaced with the broken stone macadam. Under a heavy load the whole road-bed will vibrate for 100 feet, but the shape remains intact. Swamp <■'!-'■ ■■u^:^0^^':.^o/a/ Corduroy?. : '" r - 'Black Muck' K Fig. 25 Economical Foundation Design Macadam Roads The economical design of foundation courses may be summarized as follows. For moderate traffic use pit run coarse local gravel if available varying the depths to suit the soil. If gravel is not available use a macadam bottom for ordinary soils and field stone sub-base or sub- base bottom for bad foundations. The economy in the design of macadam roads is greatly increased by utilizing local material, pref- erably uncrushed, to its fullest extent. We wish to emphasize this point (see design report page 210). If the supply of local material is limited it should be used for as much of the road as possible and advantage should be taken of the different local supplies by changing the design to allow their use with short hauls. Uniform designs which disregard limited amounts of local materials often raise the cost from $500 to $1,000 per mile. Conclusions. In the design of a road, the amount of material required for the foundation courses can be only approximated. This is the only item in the preliminary estimate that cannot be figured within definite limits. It can be closely estimated if careful data on the soils is obtained from local people and from the preliminary survey (see page 151), but a certain leeway must be given the constructing engineer so that he may vary the estimated depths to meet the construction conditions and build a consistent road. CHAPTER V TOP COURSES AND THEIR MAINTENANCE The selection of the most economical top course that is suitable for a given road is the hardest problem of Highway Engineering. The relative economy of the different constructions is theoretically expressed by the sum of the first cost and the capitalized cost of maintenance and renewal. The first can be readily estimated, but the cost of maintenance and renewal cannot be figured with any degree of accuracy for single special cases, and even on large systems it can only be approximated because of the new factor of motor vehicle traffic. The life of any surfacing is comparatively short, a fact generally overlooked in most of the popular literature on Good Roads. On any road the amount and class of traffic will fluctuate, and roads that are designed for light travel will often fail under temporary heavy traffic which, for some reason, is diverted from its normal course. The first improved roads built in any locality will for a time carry more than their share of the traffic, which is naturally reduced by the subsequent construction of adjacent improvements. It can be readily seen that it is difficult to judge the amount of traffic a road will handle and that a short-time traffic estimate is valueless as a basis for a definite conclusion. The design of the top course is usually based on a comparison of the actions of different kinds of previously improved roads that serve districts similar to that under consideration. Roads may be divided into four general traffic classes. Class I. Main trunk roads, for 5 to 20 miles out of cities of say 5o,coo and upwards and in the business sections of villages, which carry the concentrated farm and truck garden traffic of a large area and are subjected to continuous auto truck and touring car traffic. Class II. Main through automobile routes, at greater distances from the cities, which have a large touring car traffic and medium heavy farm produce traffic. Class III. Secondary or feeder roads and cross roads having a medium heavy farm produce traffic and light auto traffic. Class IV. Pleasure roads or scenic routes that have a heavy touring car and light steel tire traffic. This chapter describes the advantages and disadvantages of the various types and in the discussion of maintenance indicates in a general way their economic limitations. Waterbound Macadam Waterbound macadam is constructed of crushed fragments of suitable rock, filled with rock dust and sprinkled and rolled until firm and hard. The cost varies from about $3.50 per cubic yard 74 TOP COURSES 75 where local materials are available to $6.00 where the stone is im- ported and the haul is long. A fair average price for roads in Western New York would be $4.30 per cubic yard, or 35^ per square yard for a three-inch depth. Depth of Course. As the top stone is relatively more expensive than the bottom course a good design calls for the least thickness of top which can be successfully constructed and maintained. In 1901 the thickness used for top-course macadam in Massachu- setts, New York, Connecticut, and New Jersey was 2", and the size of the top-course stone fragments ranged from \" to \\" in Massa- chusetts to 1" to 2" in New York. Experience demonstrated that with a course as thin as 2", the larger stone fragments tended to "kick out" under traffic and that the top wore out by raveling rather than by the abrasive action of the teaming. For this reason the best practice at present calls for a 3* depth of finished top course, using stone ranging in size from ij" to 2! "; this depth makes it possible for the large stone fragments to interlock more firmly than in a 2" course. Crowns. The crowns used on plain macadam are \" to i' to f " to 1'; while V to 1 is satisfactory when first built, the gradual loss of crown due to traffic and weather action soon makes it too flat to shed the water. Mr. Charles Mills, Chief Engineer of the Massachusetts Highway Commission, reports the following loss of crown on State roads in Massachusetts and concludes that an original crown of f " to i' is advisable, except in villages where the traffic is in two lines. A f " to 1/ crown has proved satisfactory in Xew York State. Table 20. Tests Made in December, 1901 Date of Original Construction Number of Tests Original Crown (Inches per Foot) Present Crown (Inches per Foot) 1895 1896 1897 1898 1899 7 9 12 7 2 O.694 O.583 O.645 O.625 O.688 O.500 0.5I4 O.500 O.500 O.625 From the Massachusetts Highway Report for 1901. Maximum Grades. Waterbound macadam gives a good footing for horses on the steepest grades that are ever constructed; the limit of grade for this construction is determined by the cost of maintenance; on steep grades macadam washes badly and the cost of maintenance is high. Good pracjtice limits its use to grades of 5% or under, although 76 TOP COURSES it has been used and maintained successfully on grades as high as 12%. Advantages and Disadvantages. Waterbound macadam does not require particularly rigid inspection during construction and can be built under almost any weather conditions except freezing. By its method of construction the voids between the large stone fragments are completely filled with solid material and there is no tendency to squeeze or creep as in some of the asphaltic macadams. If carefully built it maintains its longi- tudinal and transverse shape and is an easy riding road for both team and motor traffic. Plain waterbound roads generally loosen up during the spring thaw and if subjected to much traffic at this time are liable to ravel. This trouble is not experienced with the bituminous macadams. Under heavy automobile traffic a plain waterbound macadam is not satis- factory as the machines remove the fine dust particles between the larger stones, leaving a rough surface which "kicks out " under team traffic. For this reason waterbound roads which are receiving much motor traffic are generally being treated with some kind of a dust layer or a bituminous protecting coat, that will better resist the wear of automobile travel. Waterbound Roads Treated with Dust Layers or Protected by Flush Coats. If waterbound macadam is kept moist by sprinkling with water, rapid disintegration under light machine traffic, traveling at medium speeds, is prevented. For light traffic, city or village streets, this is feasible, but the cost of sprinkling long stretches of country roads is prohibitive, and where the speed is high, as usually occurs on the main improved country roads, sprinkling alone will not satisfactorily protect a plain macadam. The application of calcium chloride to a road surface keeps the dust down for a longer period than sprinkling with water, as this salt has the property of absorbing moisture from the atmosphere and condensing it on the road surface; on side roads two applica- tions a season have kept the surface in good condition. The salt is applied with an ordinary agricultural drill, using about i§ pounds per square yard for the first application and less for the succeeding applications. In Western New York the cost of the first applica- tion i2 r wide has been approximately $100 per mile. Complaints have been made that the application of too much calcium chloride has caused soreness to horses' feet, but using the quantities given above, no trouble has been experienced, to the writer's knowledge. 1 The application of calcium chloride does not build up the road or form a wearing cushion that protects the stone; it merely prevents the fine surface dust from being blown away or removed by the machines. 1 We are indebted to Mr. Frank Bristow, Superintendent of Repairs, New York State Department of Highways, for much of the data on Calcium Chlorine, Glutrin and Cold Oiling. WATERBOUND MACADAM 77 Glutrin. Glutrin is a trade name for the liquid which is run out of sul- phide tanks in the manufacture of pulp; it is distilled and the acids neutralized. It resembles molasses in color and consistency, is soluble in water, and is applied by sprinkling the surface of the road with one part glutrin dissolved in one or more parts of water, using from 0.3 to 0.5 gallons of the glutrin mixture per square yard treated. The road surface need not be swept if the dust is not more than \" deep. It hardens the surface to a certain extent, and, apparently, prevents raveling if applied twice during a season on roads receiving a moderately heavy traffic. According to Hubbard an addition of 5% to 15% of semiasphaltic oil to the glutrin prolongs its efficiency, but such an addition tends to produce an oily mud in continued wet weather; glutrin alone does not produce this objectionable condition. Glutrin has been laid in New York State under an agreement with the Robeson Process Company of Ausable Forks, at a cost of $0.04^ to $o.o6| per square yard of surface actually treated. Cold Oiling. Macadam surfaces treated with light refined tar or asphaltic oil give a nearly ideal surface after the slippery, sticky condition has disappeared. The road to be treated is swept clean of dust and the oil is applied by pressure sprinklers, using from 0.2 to 0.3 gallons per square yard. The surface may be dry or slightly moist when the oil is applied. It is then covered with a good quality of pea gravel, stone or slag screenings or a sharp, coarse sand. In Western Xew York the cost has ranged from $0.02 to $0.04 per square yard, including sweeping, materials (oil and cover) and the labor of placing. To derive a season's benefit from the application of light oil or tar, the surface of the macadam must be thoroughly impregnated with the bitumen. Some of the lighter oils will evaporate. The cover will absorb some more. To get the greatest degree of saturation of road surface therefore, with a resultant freedom from dust and dis- integration, the cover should be the smallest amount of stone that will smooth out or eradicate that "toothy" or " mosaic" effect of small shallow voids between the firmly locked top stone. (See page 100.) On medium traffic roads, one application a season is sufficient and on light traffic roads one application will sometimes last for two seasons. Hot Tar and and Asphaltic Residuum Flush Coats. Bituminous flush coats are applied by sweeping the macadam carefully to remove all surface dirt as well as the stone or sand filler to a depth of about \" below the top of the larger stone fragments. On this rough, clean, dry surface a heavy refined tar or a bituminous residuum of the binder grade is spread hot, using from 0.2 to 0.8 gallons per square yard. The binder is applied at temperatures ranging from 250 to 400 F., and is spread either by hand-sprinkling 78 TOP COURSES pots or is sprayed on by specially devised pressure sprinklers. It is then covered with a layer of clean No. 2 stone (J"), or dustless screenings and thoroughly rolled. A well constructed surface of this kind resembles asphalt. It protects the macadam from raveling, is ^waterproof, forms a surface which takes the wear of the traliic from the large stone fragments, and gives a pleasing appearance. However, it cannot be laid in wet or cold weather; like asphalt, it is slippery and will not give satisfactory footing for horses on grades over 4%, and, unless laid evenly, will develop short, sharp waves or humps, which are very disagreeable for fast-moving automobile traffic. Some engineers advance the argument that by successive applications of such a flush coat a road can be maintained indefinitely without recaping, but as far as the writer has been able to observe, the roads become so humpy from continued treatment of this kind that recapping will be necessary to even up the surface on the score of comfort alone. The use of a hot tar application on a concrete road will be dis- cussed on Page 87. For use on an existing macadam road as repair, the authors believe that there is just one condition where a hot application should be specified; where an old road has begun to disintegrate unexpectedly, has passed the stage where cold oiling would rejuvenate it and funds are not available in the current year for resurfacing, then the hot oil or tar treatment may be used as a stop-gap to save it from complete disintegration for another year. The cost of flush coats exclusive of covering ranges from $0.12 to $0.16 per gallon, or about So. 09 per square yard. If applied to a macadam road during construction the cost of the plain macadam is increased approximately So. 10 per square yard, making So. 45 per square yard a fair comparative figure for flush coat and waterbound macadam construction. The crown ordinarily used on flush coat roads is \" to 1'. All bituminous binders have the following practical disadvantages whether applied as surface coats or as binders in bituminous mac- adams. The composition of residuum products is so complex and so easily varied that, to get uniform results, each shipment must be sampled and analyzed to insure certain required properties. In heating, care must be taken not to char the binder, as this destroys its life and effectiveness. They cannot be applied in wet or cold weather, which reduces the length of the construction season, and unless evenly spread a rough, humpy road results. Bituminous Macadam. Bituminous macadams are constructed in two ways, by the pene- tration method and by the mixing method. Penetration Method. Most of the bituminous roads in Xew York State have been built by this method. The larger stone fragments, ranging in size from 1" to 2" ', to 1" to 25*, depending on the depth of the course, are spread and rolled; BITUMINOUS MACADAMS 79 a heavy grade of refined tar, residuum bituminous material, or fluxed natural asphalt, is then poured hot, either by hand or machines, 1 into the voids of the stone so that the stone fragments are covered with a thin coat of bituminous material; No. 2 stone, or dustless screenings are spread over the surface and broomed and rolled until the voids are filled; if a flush coat is to be used the excess filler is broomed off and the surface coat applied in the same manner as described for plain macadam. Where the flush coat is not applied, a wearing coat of clean screenings is spread over the surface. The amount of bituminous material used as binder varies from 1.25 gallons to 1.75 gallons per square yard, depending on the depth of the course. The amount used for flush coats ranges from 0.2 to 0.5 gallon per square yard. The cost of a one-coat 2" bituminous top, using 1.25 gallons per square yard, will range from $0.35 to $0.45, and a 3" one- coat top, using 1.75 gallons per square yard, from $0.50 to $0.60 a square yard. The flush coat using 0.4 gallon per square yard will add about $0.06 to the above costs. For the purpose of comparison with madacam a fair set of prices is, 2" Bituminous top, one coat of bitumen . .$0.40 per square yard 2" " " flush coat $0.45 " 3" " " one coat of bitumen .$0.55 " " " 3" " " flush coat $0.60 " Depth of Top Courses for Bituminous Macadams. In 1910 New York State adopted a depth of 2" using 1.25 gallons as binder and 0.5 gallon as flush coat per square yard. In 191 1 a 3" depth was used with 1.25 gallons per square yard as binder and 0.4 gallon as flush coat. In 19 1 5 a 3" depth was used with 1.75 gallons as binder and 0.5 gallon as flush coat. A2" bituminous top will not fail by raveling, the defect mentioned for a 2" waterbound macadam course, but it has certain construc- tional difficulties. To construct a 2" course no stone should be over 2" in its largest dimension. Because of the tendency to crack under concentrated wheel loads, none of the stone forming the main body of the course should be less than one inch in size. These limits of size are so narrow that difficulty has been experienced in procuring sufficient stone for top when crushing local material, and even when the stone is obtained from a commercial plant the same difficulty is often encountered. Also in spreading such a depth with stone ranging in size from 1" to 2", there will be places where the metaling is only one stone deep and the fragments do not fit as closely together nor have the same chance to interlock as in a deeper course. The spaces between these stones are filled with the No. 2 (f ") size, which 1 The author has had better success with hand pouring for the first coat than with machine work. For thin flush coats, however, a pressure machine is absolutely neces- sary. If bitumen is poured by hand it must be poured across the road (never along the road) as this method of work largely eliminates humps formed by overlap. It is much easier to control the hand spread than the machine spread as to amounts and the stone spread is not disturbed or rutted up during the pouring. 80 TOP COURSES wears more rapidly under traffic than the larger pieces and the road tends to become rougher than would occur if the ij" stone fitted closer together. This last argument does not apply to flush coat roads. The argument is often made that a 3" top will last one and one- half times as long as a 2" top because it has one and one-half times as much material, but the life of a top course rarely depends on its total thickness, as it will become so badly out of shape before the general elevation has worn down an inch that it will need recapping. In attempting to meet these difficulties, %\ w and 3" courses have been built; as far as the author has been able to judge, the 2\" depth remedies the defects. When pouring bitumen in the penetration method, a pocket of fine stone, dirt, etc., will sometimes hold the binder near the top in too great quantities; during hot weather the bitumen swells and, as the voids are full in these spots, it rises to the surface and forms a hump or wave. This trouble is not so frequent on either 2 \" or 3" courses as on the 2" depth. The writer's present opinion is that a 2\" depth, using about 1.4 gallons bitumen per square yard in one coat, will give satisfaction. Crowns. The crowns used on bituminous macadams range from \" to 1' to |" to i'; \" to i' is generally used and is apparently satisfactory. Footing. A single coat road affords good footing on any grade that will be adopted as suitable for heavy hauling; such a top course will not wash, which makes it easy to maintain on hills. A flush coat, however, cannot be used to advantage On grades over 4%. Advantages and Disadvantages. Bituminous macadam without a flush coat provides good footing for horses; it will not ravel, is easy to repair for small depressions and ruts, is comparatively dustless and keeps its longitudinal and transverse shape well, making a comfortable riding road for fast travel. On the other hand, it will probably wear more rapidly than the flush coat construction as the traffic comes directly on the stone; it is subject to the practical disadvantages of construction of all roads where bituminous materials are used; it is not waterproof when first constructed; this last defect, however, is remedied by the traffic which grinds up the surface wearing coat and forces it into the voids. As a matter of fact, the combined action of traffic and weather puddles the road, and after about six weeks' use we can say that the road has a bituminous bond and a water-puddle finish. Flush coat bituminous macadams are more dustless than the single coat, are more nearly waterproof when first built, look smoother at first, and will probably cost less to maintain. However, they do not BITUMINOUS MACADAMS 81 give as good a footing as the single coat and are liable to develop waves and humps disagreeable to fast traffic. If a flush coat is used there seems to be no advantage in a bitumi- nous binder, as the flush coat alone prevents raveling, and, if such is the case, the binder used throughout the depth of the course is a waste of money; a waterbound bituminous flush coat course might better be used. In choosing between a flush coat construction or a single coat bituminous madacam, the author believes that a single coat bituminous macadam is the better design; although it will probably cost more to maintain, the increased safety and comfort to the traveling public is worth the expenditure. Mixing Method. — Open Mix. Type I. The stone and bitumen are mixed hot in specially designed machine mixers. The mixture is then spread in the same way as sheet asphalt. A flush coat can be used if desired. The 191 5 New York State specifications call for No. 2 stone (f to 1})" — when finished thickness is to be two inches or less and a mixture of No. 2 and No. 3 stone (ij to 2\)"\ when finished top course is greater than 2", the stone to be proportioned as directed by the Engineer. Approximately 18 gallons of bituminous material to each cubic yard of loose stone. In this "open" mix, it is unavoidable that pockets of mixed top material will be placed which have a greater percentage of voids than the average. Whether or not a seal coat is used, these pockets will wear more rapidly than the surrounding pavement. In a similar manner, variations in the size of the stone will cause uneven wear. Both conditions tend to produce a humpy pavement after some use. Mixing Method. — " Tight Mix " or " Topeka." Type II. The stone, sand and bitumen are mixed hot in specially designed machine mixers. The mixture is then spread in the same way as sheet asphalt. The thickness varies according to the foundation. It is generally a consolidated depth of 2" on a concrete foundation and 2^" on a firm macadam foundation. The various sizes of the mineral aggregate and the percentages of each are specified within certain limits varying slightly to meet gradations peculiar to the material available. Because of the fine aggregate used in work of this type, there is not sufficient stability to withstand a mixed traffic and the surface ultimately forms in disagreeable waves. Attempts have been made to prevent this waving by using a high penetration asphaltic cement which will permit the pavement to iron itself out. However, if a heavy slow-moving traffic be carried on this type of road, the surface will rut. Apparently, the best results in mixed Bituminous Macadam have been secured when the coarse aggregate was used — stone between three-quarter inch and one and one-half inches in size, which were filled with a matrix of fine material of sand and bituminous material. Such pavements have sufficient u body" to materially decrease the "creeping" under use and take a more even wear than the open mixed type. 82 TOP COURSES The prices for this type of top course run from Type I, 6ii^ . $1.10^ , -JJL — > lr to - ? per sq. yd. Type II, 66^ $1.25 ^ H 'Natural Rock Asphalts. Sandstones and limestones containing a certain percentage of bitumen are known as rock asphalts. The most common source of supply for the Eastern States is Kentucky, and the product is known as "Kentucky Rock Asphalt." It is a sandstone containing about 7% to 10% of maltha. It is pulverized at the mine and is shipped and applied cold in the following manner: 2" to 2i"of stone, ranging in size from f" to i|", are spread and rolled slightly. The rock asphalt is run through a shredding machine and spread over the stone, using approximately forty pounds per square yard. The whole mass is then thoroughly rolled, preferably with a six or eight ton tandem roller; forty pounds per square yard of pure rock asphalt is then spread as a wearing coat and well rolled; the rolling is continued intermittently for a number of days after the traffic is turned on the road. The cost of such a course has been about $0.70 per square yard in Western New York. The crown ordinarily used is \" to i'. Advantages and Disadvantages. The road is pleasing in appearance, is not as slippery as sheet asphalt, and will not ravel under motor traffic. However, it is hard to construct in cold weather, is not uniform, and will ravel in spots. It has defects in common with sheet asphalt of showing wear by de- veloping short humps and hollows disagreeable to fast traffic. The steepest grade on which it can be used advantageously is about 5%, as it becomes slippery in cold weather, and in warm weather it some- times softens enough to make hard pulling for heavy loads. Amiesite. Amiesite, a patented material made of crushed stone coated with asphaltic cement, has been used on many miles of road with good results. It is shipped cold in a friable and granulated state, spread on either macadam or concrete base and well rolled. Amiesite screenings are then spread and rolled, forming the surface. This construction costs about $1.00 per sq. yd., 3" thick. It resembles asphalt in appearance and has the advantages and disadvantages of all roads of this class. It is particularly adapted for small jobs where it would not pay to set up an asphalt plant or where suitable asphalt materials are not locally available. For further information see chapter on Cost Data and Specifica- tions. Other Surfaces of a Bituminous Nature. There are any number of patented pavements that can be classed under this head to which we cannot give space. BRICK PAVEMENTS 83 Sheet Asphalt and Warren Brothers' Bitulithic are used in unusual cases, but constitute such a small percentage of the mileage that for information concerning them we refer the readers to books by Richardson, Hubbard, Tillotson, etc. We includes some notes on inspection of construction, page 351. Brick Pavements. The ordinary brick pavement construction is probably familiar to most readers. On a concrete foundation 5" to 7" in thickness a sand cushion varying in depth from 1" to 2" is spread and the paving brick are laid on this sand bed so as to break joints; the brick are well rolled and the joints are rilled with sand, cement grout or pav- ing pitch. Longitudinal expansion joints of bituminous material are provided next to the curbs or edgings; transverse expansion joints, space 30' to 50' apart, are used by' some designers. The latest practice tends to make the sand cushion as thin as possible, acting merely as an evener of the concrete surface. It is also rare to find any material but cement grout used for filler. The use for traverse • J Jg Pitch Expansion Joint Concrete ;M^^^2:C6ncr e^Bwe:°5' t o d.-o.-r 3 .-^ Note t Transverse Expansion Joints Spaced K ton 30 to 50 ft may be used. Fig. 26. — Brick Pavement, Flush Edging expansion joints is also being relegated to the back ground. Pre- molded alsphaltic strips are designed for longitudinal expansion joints in most recent practice. In the last few years the former theory that the i\" sand cushion prevented crushing of the brick and gave the amount of resiliency necessary to a pavement of this type has been disputed, and apparently successfully so, by the increased use of a cement sand bed. Upon the prepared foundation a bed of cement and sand consisting of one part cement to four parts of approved sand is spread not more than 1" in depth. The sand and cement are thoroughly mixed dry until a uniform color is obtained, struck off with a template and rolled with a hand roller weighing about 300 lbs. After the brick are laid thereon and before grouting, the rolled bricks are thoroughly wet by sprinkling. It is important that the bricks be well wet so as to set up the cement sand bed. . It is probably true that the use of a mortar cushion reduces the tendency of the brick to loosen near cracks. In 19 1 5 several experimental brick pavements were constructed where the mortar cushion and brick were laid upon concrete which was still plastic. Concrete foundation was shaped by template and the brick laid, inspected and rolled with a hand roller before the cement had taken its initial set. This is immediately followed by grouting. It is too early to say whether or not this so-called " mono- 84 TOP COURSES lithic " construction will be successful. The expense and difficulty of manipulation are increased and it is doubtful if any material advan- tages are attained. Brick Pavement construction is essentially rigid, intended to with- stand heavy traffic. The cost, including foundation and surfacing, ranges from about $1.60 to $3.00 per square yard, the average price in Western New York being about $2.00. Brick pavements on heavy traffic roads have been extensively used in Ohio and New York. Macadam foundations for brick surfacing have not proved satisfactory in the Northern States, as the surface is too rigid and cracks under the heaving action of the frost. Even on a concrete foundation longitudinal cracks often develop from this same action. It is more difficult to prevent this on country roads than in cities where the sewers keep the earth sub-grade comparatively dry, and the necessity for a center drain under the concrete base is being recognized by many designers. Some engineers believe that the 1 to 1 cement grout in general use is too strong, and that if a weaker grout or a sand filler were adopted in its place the heaving frost action would merely separate the bricks slightly instead of breaking them and that as the road settled they would fall back into close contact. This is an attempt to make a theoretically rigid construction flexible and seems to be striving to adapt the construc- tion to conditions for which it is not fitted. Longitudinal Cracks. — These cracks have been carefully studied, as they seem to be the most discouraging feature of brick pavement construction on country roads. Mr. Wm. C. Perkins, Chief Engineer of the Dunn Wire Cut Lug Brick Co. states from a careful examination of a large mileage of brick roads built under his supervision, that longitudinal cracks have always occurred within 2' or 3' of the center of the road; that the cracks extend down through the concrete base and that less difficulty is experienced in preventing them as the crown of the pavement is reduced. From these observations he has been led to experiment with a concrete base having a perfectly flat bottom, as shown in figure 2 6 A, crowning the road by making the concrete thicker in the middle than on the edges. The claim is made that this style of construction is helping to prevent such cracks. Transverse Expansion Joints. — The use of transverse expansion joints has not been successful locally. Difficulty has been experi- enced with the brick loosening at these joints, and whenever a tem- perature heave has occurred it has appeared at the joint. Their use has been abandoned for rural roads in Western New York. .Sand Cushion .Longitudinal m V » ' ''^Concrete Foundation Fig. 26 a ASPHALT BLOCK 85 The crowns in use on brick pavements range from \" to 1', to f " to i'. For the methods of figuring ordinates for parabolic crowns see page 262. Brick pavement does not give a good foothold for horses on grades above 5 % unless some special form of brick is used. For steep grades, on heavy traffic roads, it is better practice to use some form of stone block. Stone block pavement, including concrete foundation, costs from $2.70 to $3.30 per sq. yd. It is suitable for the steepest grades that are constructed. .' Pitch Expansion Joint Slopelj«ULJ^ Concrete ^M^k^i^^^Co nc'rei^Baieis^to 6 'l?M **- 2"Sand w« S5J0W Lm"j' Note : If Pitch Filler is used between StoneBlock, ^ * no Special Expansion Joint is needed. Fig. 27. — Stone Block Pavement, Flush Edging Where stone blocks are used on hills it is better practice to use second quality blocks; these blocks are identical with the first quality blocks as to material but are not dressed as carefully and cost about fifty cents per square yard less; their rougher surfaces and wider joints afford better footing. For the difference in size and joints see specifications, Medina Block, page 396. The first cost of brick pavement for country roads restricts its use to roads where it can be conclusively proved that macadam will not be suitable. Asphalt Block The asphalt block pavement laid in New York State has been very satisfactory. The proportion of ingredients is about 70% crushed rock, usually trap, which has passed a \" ring, about 20% limestone dust to act as filler and approximately 10% of asphaltic cement, molded under a pressure of 2 tons per sq. inch of block having a 2" depth. This produces a dense asphalt much superior to the ordinary sheet. The asphalt used is Trinidad. This is refined and fluxed so that the resulting A. C. may be varied as to adhesiveness, penetration, etc. to meet varying conditions peculiar to different localities. The penetration is made high enough to give a certain amount of pliancy to the block, to avoid crumbling at the edges and to make the joints self-healing. The use of blocks containing steel anchors, laid across the road, approximately fifteen feet apart, has eliminated any movement of the block under traffic. These blocks are placed at more frequent intervals on curves. Block pavements have been laid using a longi- 86 TOP COURSES tudinal row of these anchor blocks in place of edging. The results appear satisfactory. Sfeel Reinforcement ■ /' \ %s . . . _J Plan ''Mortar Bed'. ""^V -p, * a; ^ > ,^>M >■> a; i;^^K- ^^v-^ bi- section Fig. 27 a ^Foundation After the base is prepared a mixture of 1 to 4 Portland cement mortar is spread \ inch thick. This mortar bed is carefully screeded and the block laid thereon, joints being broken at least 4 inches. Asphalt Block Data Highway No. County \ Mileage Bottom Per sq. yd. Top Per sq. yd. Per Mile 16'— 26' 5357 Westchester 0-95 $0.61 $1.49 $26,593 5375 n 1-34 Old Mac I.69 18,114 5388 Rockland 2.16 a a 1.70 1 27,025 0.59 I.70 1 32,525 ii53 Niagara 0.97 O.60 1-37 1 31,800 5482 Westchester 1. 16 O.66 1-50 29,270 1167 a 1.28 O.Ol 1.38 24,245 io53 (C 1-45 Old Mac I.60 21,205 5528 Warren 0.61 o.59 I.60 35,990 5356 Westchester 0-53 Old Mac I.63 1 26,960 536i ci 0.68 0.61 I.44 1 23,5 12 5302 Ci 0.25 0.67 i-37 25,569 53°4-A u 0.31 0.47 1.47 23,166 5373 tc 2.85 0.58 Av. 0.599 1-52 • Av. 1.533 1 Costs from preliminary estimate. All costs not marked with star from bid prices. CONCRETE PAVEMENTS 87 An interesting comparison with brick occurs in the " pinning in" at curbs. Instead of bats being broken by hand, a large mechanical shear is used. Each fractional block is measured and cut to fit exactly. After being laid, the pavement is given a light coat of sharp sand which is broomed into the joints. Traffic is permitted in four or five days. Advantages. — The pavement shows a smooth, uniform surface, dustless and practically noiseless. Its life has yet to be determined: Pavements that have been down ten or fifteen years are still in good shape. Within a reasonable freight radius from the point of manu- facture, it can be laid for approximately the cost of brick. Disadvantages. — A mist or light rain makes the pavement very slippery. It should not be used on grades over 4%. CONCRETE PAVEMENTS Introductory Inasmuch as there is some difference of opinion as to the value of this type each author has written his interpretation of the available facts. Concrete Pavements By W. G. Harger Many miles of these roads have been constructed in the last few years. The construction has varied from poor 1 to 6 pit run gravel con- crete to first-class 1: if: 3 stone concrete 6" to 9" thick. There is enough data to conclude that cheap concrete is a failure. An effort was made to protect the surface of such a mix with a thin bituminous surface coat of asphaltic oils or tars. These coats have not been successful as they peel off and produce an unsightly, rough riding and a high maintenance cost road. The 'type of concrete road now being built and which has many enthusiastic supporters is a first-class 1: ij: 3 stone or screened gravel concrete which takes the traffic directly on its surface. The concrete is carefully manipulated (see specifications, page 443). The ordinary section used is shown in figure 27B. Expansion joints of creosoted wood or patented steel plates with tarred paper filler are provided at intervals of approximately 30 ft.* The cost of this pavement has been from $1.10 to Si. 50 per sq. yd. They have the advantages and disadvantages of all rigid types of construction. They should not be used on grades over 5%. Pavements of this class have been built on roads having light, medium and heavy traffic and are advocated by Cement Manufac- turers as an economical road under all classes of traffic. The author * The author personally believes that better results will be obtained by eliminating these joints altogether. The artificial joints are sources of weakness in that they tend to localize the wear. Apparently less wear occurs at a natural crack and it is certain that a smoother riding road is obtained. / 88 TOP COURSES believes that while this type has its place that a great mileage is being constructed which from an engineering viewpoint is not justified. The roads have not been down long enough to obtain reliable data as to their length of life before resurfacing. Considering in a general way, however, what we know of the material and the action of the weather and traffic on rigid types of pavement, an allowance of 10 to 15 years would appear liberal. When they arrive at the point when they need resurfacing a large expense is involved. It has been demonstrated that cheap thin bituminous coats have not been suc- cessful; it is not possible to successfully resurface with a thin layer of concrete which means that probably Asphaltic Concrete, Asphalt Block, Brick or some other form of block or cube pavement will be used at a cost of from $9,000 to $16,000 per mile. The fact that resurfacing when it occurs requires such a large expenditure eliminates this type from use on any but the more important roads which con- stitute a small percentage of the mileage of any large system. 4"fo6' ^^mrn^^ZF'^ 8 Fig. 27B With the data at hand the indications are that this type is a good design for the heavier Class II traffic roads where it is desirable to keep the first cost as low as possible and where it is. expected that the necessary money will be available in later years for an expensive resurfacing. The fact that the first cost is moderate and the maintenance is low for a number of years blinds a great many highway officials to the final cost of upkeep, or if they are not blind many of them figure that the roads will outlast their tenure of office and what do they care about their successors. Concrete Bituminous Roads By E. A. Bonney Some four or five years ago, a tremendous wave of publicity swept concrete roads into the limelight. The construction at that time consisted of a second-class concrete base with a skin coat from \" to \" in depth, composed of screenings, mixed with hot oil or tar, and sometimes a combination of the two. The base was laid without joints and gravel or any kind of stone was used for aggregate. Under this type at least a dozen patented pavements were devel- oped practically none of which have to any degree borne out the extravagant claims made at that time. The bituminous skin coat has not been satisfactory. It is subject to all the disadvantages of other bituminous macadams and with few exceptions has not adhered to the concrete for any length of time. CONCRETE BITUMINOUS ROADS 89 There is a road known as the Bedford- Golden s Bridge State High- way in Westchester County, on which 2.67 miles of concrete base had been laid when the original contract was canceled. The un- finished portion was covered with an experimental skin-coat treat- ment which today is as sound and intact as when laid. Work was finished in the early summer of 191 5. The road was subjected to the enormous automobile traffic peculiar to W r estchester County all season. A brief description follows: The concrete was cleaned, all dust, dirt or caked material removed. It was then coated with a cold application of low carbon tar, very light grade, almost a creosote. This was spread about jq gallon per sq. yd. and allowed to dry for two hours. About f of a gallon per sq. yd. of Bit. Mat. U T" low carbon binder was then applied hot and covered with approximately 37 lbs. of No. 2 stone per sq. yd. A second coat of J gallon per sq. yd. was then applied and covered with about 32 lbs. per sq. yd. of No. 1 stone. This treatment so far looks extremely well and has not broken away from the concrete. It is still too early to classify as a success. The cost of the top course only was 17^ per sq. yd. Base cost 6Sc' making total of 85c. Concrete Pavements By E. A. Bonne y 1— 1 J— 3 Mix Concrete pavements are showing as each season passes by, that they are worthy of much more consideration than has been given them up to the present time. For roads subjected to heavily loaded and slow moving vehicular traffic or for roads so located or so traveled that any type of macadam road would be subjected to costly maintenance, the concrete pavement has come to stay. The wear seems to be inappreciable and because of the flat crown, traffic is spread over the entire width of metal. Great care must be exercised in the selection of aggregates. Many sands that are considered good enough for ordinary concrete work will not give satisfactory results in concrete pavement. Stone or gravel should be limited to those showing a high coefficient of wear. Considerable attention should be paid to the percentage of voids in the sand and stone. Experiments should be made to determine approximately the mixture giving the lowest percentage of voids. The authors do not believe in the blind adoption of a specified mix. It is often essential that the mix be varied to correspond to the grada- tion of available sand and voids in coarse aggregate. Several containers of uniform volume and a pair of scales are all the apparatus necessary to show whether or not the specified mix is the best mixture for the aggregates available. The approximate percentage of voids may be found by water. By making up several concrete cubes or cylinders of the same vol- ume, beginning with the specified mix and varying the others as 90 TOP COURSES indicated by the percentage of voids, the heaviest product will indi- cate the proper mixture. Any data given herewith is based upon a one-course road. The authors are not personally familiar with two-course roads. Bulletin No. 249 of the Office of Public Roads, U. S. Department of Agriculture, cites the advantages and disadvantages of concrete highways as follows: "Advantages: 1. As far as can be judged, they are durable under ordinary sub- urban and rural traffic conditions. While it is true that there are no very old concrete pavements in existence, the present condition of many of those which have undergone several years' service would seem to warrant the above statement. 2. They present a smooth, even surface, which offers very little resistance to traffic. In the past the surface of concrete pavements have sometimes been roughened in order to insure a good foothold for horses. This practice has now been abandoned, except on very steep grades, because it tends greatly to accelerate deterioration of the pavement, and because the smooth surface has been found to afford a fairly satisfactory foothold under all ordinary conditions. 3. They produce practically no dust and may be easily cleaned. 4. They can be maintained at comparatively small cost until renewals become necessary. 5. They may be made to serve as an excellent base for some other type of surface when resurfacing becomes desirable. 6. They present a pleasing appearance/ 5 "Disadvantages: 1. They are somewhat noisy under horse traffic. 2. There is no method of constructing necessary joints in the pavements which will entirely prevent excessive wear in their vicinity. Furthermore, joints do not altogether eliminate cracking and wherever a crack develops it must be given frequent attention in order to pre- vent rapid deterioration of the pavement. 3. They cannot be readily and effectively repaired as many other types of pavements." This summation of concrete roads in general seems eminently fair. We believe, however, that to the disadvantages should be added the inevitable rut which appears between the edge of the .concrete and the earth shoulder. These ruts are dangerous to fast-moving traffic and require constant maintenance for their elimination unless the shoulders are armored with crushed stone or gravel for two feet or more from the concrete. The questions of reinforcement and joints are still the subjects of much discussion among engineers. The item of reinforcement largely increases the cost of the roads and it is yet too early to say that the added expense is justified. The joint problem affords an unlimited field for a variance of opin- ions. Few engineers are satisfied with any of the existing armored joints, patented or otherwise. CONCRETE PAVEMENTS 91 The author believes that experience to date has divided the prob- lem of joints into two fields: i. e. on roads under continued mainte- nance a bituminous joint- will prove satisfactory and is renewable at small cost; on roads which receive spasmodic maintenance or none at all, some sort of steel joint should be used. On New York State work where maintenance is continuous the most satisfactory joint to date is of premolded asphalt, which is so placed that it projects from f " to \" above the surface of the concrete; thus Premolded Asphalt Joint as Laid A combination of hot weather and traffic spreads the asphalt out in this manner, leaving a bituminous mat over the joint. Premolded Asphalt Joint as it becomes under Traffic For concrete roads not under maintenance, the better joints are being made of soft steel tempered to the same relative hardness as the concrete. A hard steel joint simply transfers the point of wear from the joint-edge proper to the concrete back of the joint. The proper length of Concrete slabs between joints is another subject of speculation. Many roads are now being built with vary- ing distances between joints in an endeavor to determine how few can be used with success. The average cost of this type in New York State lor 6" depth of pavement is $1,121 per square yard of pavement only. Total average cost for mile of completed highway, including excavation, drainage structures and pavement, is $15,320. Small Stone Block Surfacing. In Germany, Hungary, Austria, and England a surfacing made of granite blocks, ranging in size from 2V to 4", has been used success- fully. This pavement is known as Kleinprlaster in Germany, and as "Durax" armoring in England. The stone cubes must be cut with considerable accuracy in order to give a smooth and durable surface. The blocks are laid on a thin sand cushion of about f " depth, on either a macadam or concrete foundation; they are thoroughly 9 2 TOP COURSES rammed to give a firm bearing and the joints filled either with clean sand flushed in, or a bituminous filler. The joints do not exceed \" in width. The courses of cubes are laid either diagonally to the direction of the traffic or in concentric rings. Where the stone is broken by hand the cost is high and it would be impossible to consider its use for rural roads in this country. A machine * has, however, been developed in Europe for breaking these cubes which is claimed to produce a satisfactory product at a reasonable rate. It is a belt-driven friction drop-hammer having a stone chisel mounted on the anvil; the hammer head is shaped like a stone-cutter's sledge. The power needed for each machine is about ij H. P. About 400 of these machines are in operation, and a plant in Sweden is turning out 700,000 square yards of pavement per year with 62 machines. Provided the pavement can be laid for $1.00 to $1.25 per square yard, it seems a type that must be seriously considered. A price as low as this, however, would necessitate the use of convict labor in the manufacture of the cubes. McClintock Cube Pavements, by W.G. Harger This is a patented pavement devised by J. Y. McClintock, County Engineer of Monroe County, N. Y. It is very similar to "Klein- pflaster" except that under his patent artificial cubes as well as stone cubes are proposed. It appears to be a very promising type. The construction is essentially as shown in figure 28 and consists of a top course of 2|" cubes placed on a thin sand cushion supported by either a macadam or concrete base. The cubes have been made of concrete, vitrified paving brick material and stone as in Conti- nental practice. Gravel or Stone Shoulders '-, ^Macadam Foundation FlG. 28 They are loaded, hauled and dumped like broken stone; laid in close contact by means of a pallet and rake 128 at a time on a sand cushion \ to \" thick, no care being taken to break joints. They are then rolled to bring to an even and firm bearing; the joints are filled with a sandy loam and the surface treated with a light coat of light road oil or cold tar if the foundation is macadam. The joints are grouted if the foundation is concrete. Temporary shoulders of 2" plank are put down during the laying of the cubes after which they are removed and replaced with broken stone or gravel as shown in figure 28. 1 A detailed description of this machine is given in Engineering News, March 27, 1912. CONCRETE PAVEMENTS 93 The experience of the past six years has shown that this form of construction using a sand-tarred joint is flexible under frost action which makes it suitable as a surfacing on a macadam base. It keeps its shape under traffic and shows no tendency to ravel or break down at the edges and can -be successfully held with a macadam or gravel shoulder without the formation of a rut along the edge which is a difficulty always encountered where a rigid edg- ing is designed. It gives a satisfactory surface in both wet and dry weather and can be laid late in the season. The cubes require com- paratively little inspection and can be successfully used as a patch in maintenance with simple manipulation. They reduce the tonnage and freight cost where imported materials are required. Concrete cubes have not served satisfactorily, failing in spots, but this is to be expected as it is not a reliable material for a road surfacing of this nature (that is for such small units). Vitrified shale cubes with wide sand joints laid on a macadam base have shown ability to stand medium traffic. Vitrified shale cubes with close tarred joints laid on a thick macadam base serve very satisfactorily under moderately heavy traffic, and the indications are that these cubes laid on a con- crete foundation and grouted will meet all but the heaviest traffic satisfactorily. Consider briefly the present tendencies in highway construction. There are two distinct types; the flexible form represented by the macadams and the rigids types, such as brick, asphalt, stone block, etc. having concrete foundations. Each has a distinct field and their relative economy depends largely on the traffic. It is sufficient for this discussion to note that macadams are suit- able for light and medium traffic (Classes II and III); that they are able to withstand climatic changes better than the rigid pave- ments and that with a moderate yearly expenditure they can be kept in good condition when used under the volume of traffic stipulated. They fail either under high velocity traffic or heavy hauling; the first being a surface failure and the second a foundation failure for most of the roads in this locality but a surface failure for some which have a thick well-consolidated base. That is, if some better flexible surface can be used on a first-class macadam foundation, this type of road will be able to handle a heavier volume of traffic than at present with a moderate maintenance charge. The indications are that the Brick cubes with sand-oiled joints will serve this purpose. The rigid roads develop defects due to temperature changes; frost heave and the settlement of fills. Subsequent movement is localized along these lines and eventually expensive repair and re- construction is necessary. Under heavy traffic, however, the cost is less than for the macadam type and the inconvenience of continual repairs is avoided. The first cost of Brick and Asphalt block which are generally con- sidered the best of the rigid types is so high that designers often hesitate to use them where they are actually needed. If it were possible to reduce the cost and yet obtain practically the same class of improvement a larger mileage could be used to advantage. 94 TOP COURSES The indications are that the Brick cubes on a concrete foundation will serve this purpose at a cost of about $0.40 per sq. yd. less than the present paving brick. Highway designers do not hesitate to use madacam for the light traffic roads or expensive rigid constructions for the extremely heavy traffic; the great mileage that lies on the verge of either form of construction offers the real difficulty. It is for this class of road that the cubes are particularly adapted by reducing the cost of brick and increasing the efficiency of macadam. This applies also to the resurfacing of concrete and macadam roads. The author believes that provided this type fulfills its present indications that it will meet a recognized need in highway construc- tion and for this reason has given more space than perhaps is justified to a method which has not been tested out by a large mileage of construction. A reasonable cost of the brick 2" cube surfacing is approximately $0.95 per sq. yd. in Western New York. This form of road material is adaptable to manufacture by convict labor. Rocmac. Rocmac is another patented pavement which deserves mention, as the roads which the author has seen built by this method compare favorably with other types of construction. The claim is made that, under favorable conditions, it will cost only fifteen cents per square yard more than plain macadam. The only available example of cost details given below is hardly a fair sample of what can be done. We quote an extract from the 19 10 report of the New York State Highway Commission: " Experimental pavement according to the Rocmac System as laid over the westerly portion of Buffalo Road, Section No. 2, County Highway No. 83, located in the Town of Gates, County of Monroe, New York. "The Rocmac system differs from ordinary macadam construction in that the aggregate of crushed stone is cemented together by a matrix composed of limestone dust (as rich as possible in carbonate of lime) mixed with a solution of silicate of soda and sugar, the silicate of soda combining with the carbonate of lime, an unstable compound, forming silicate of lime, which is a very stable compound. "The materials used in this experiment were Leroy limestone flour for the matrix, being the entire crusher product which would pass a screen of J-inch mesh, and Akron limestone No. 3 size with some No. 4 size mixed for the aggregate. The No. 3 size being retained on a screen of ij-inch mesh and passing a screen of 2-inch mesh, the No. 4 size being retained on a screen of 2-inch mesh and passing a screen of 3|-inch mesh. "The delivery point for material shipped by rail being Cold water Station, a dead haul of one mile to the beginning of the work. "The supervision given this work consisted of occasional inspections by the division superintendent of repairs and the inspector in charge of this section, neither of whom could devote much time to this particular work without interfering with other duties. Had the work been constantly directed by a competent foreman more progress ROCMAC 95 would have been made and the cost probably would have been decreased. "The method pursued during the laying of this surface was to scarify by hand the original foundation course, removing all loose material by brooming, upon this prepared foundation to spread the matrix composed of limestone dust and solution, to an average depth of about two inches, upon this spread the crushed limestone aggre- gate to such a depth as would give finished rolled thickness averaging about 3f inches when properly crowned, then rolling same until thoroughly consolidated and continuing rolling and sprinkling with water by hand until the matrix which flushed to the surface in the form of grout has nearly disappeared, when the pavement is covered with a light coat of screenings and considered complete. "The total length of this resurfacing extending from Station 237 to Station 275+ 76 is 3,876 lineal feet, aggregating an area of 6,890 square yards surface upon which was used 1,094 tons of No. 3 and No. 4 crushed limestone, 520 tons of limestone flour and 4,050 gallons of silicate of soda solution. "Deducting from total expenditure materials not used and expense of labor trimming shoulders and ditching would leave total cost of this resurfacing including all material and labor necessary to form pavement complete in place $6,400.82 or $0.9288 per square yard. This expense is itemized as follows: Item Total Per Sq. Yd. Cost of Stone f.o.b. cars delivery point. Cost of Rocmac solution $2,026.59 617.28 1,408.79 408.61 547-28 1,341.64 50.63 $0.2941 O.0896 0. 2044 O.0593 O.0794 O.1947 O.0074 Cost of teams hauling stone, solution, water and coal Freight and duty on solution Roller and coal Labor Tools, tank, blacksmith, oil and wood . . Total $6,400.82 $0.9288 "The average price paid per ton for all stone f.o.b. cars at delivery point is $1.25 J; price paid per hour for labor $0.22; for teams $0.56^ per hour; roller rent $10 per day. "During the progress of this resurfacing traffic was not interfered with at all, all traffic being permitted to go over the work in whatever stage of progress. This is an advantage worthy of consideration. "The finished surface after five months' traffic has the appearance of a well-constructed macadam road, being hard, smooth, well bound, and clean, no discoloration being apparent except immediately after a rain, when it shows light brown in spots, due to the solution, which being soluble in water comes to the surface. 96 TOP COURSES "No ravel developed during continued dry weather when freshly laid and under traffic; road is relatively dustless; this, however, depends upon the percentage of silica in the stone used. The theory being that whenever the pavement becomes wet the solution is brought to the surface, resulting in absorbing and hardening down any fine material which had been produced by the abrasion of tires. "It can be laid in all excepting freezing weather, and while smooth yet it is sufficiently rough to afford good footing for horses and rubber tires. There is nothing entering into the construction to soften under high temperature and nothing to form mud in wet weather. It is claimed to be self-healing, due to continual chemical reactions taking place whenever the road becomes wet." CONCLUSION In this chapter the authors have attempted to, show the approxi- mate costs of the different styles of construction in general use or such experimental tops which they have seen that promise well. The costs given can be considered as relative only, to be used in a comparison of the various constructions and are based on roads in Western New York. In selecting the kind of pavement for various traffic requirements, good present practice calls for Brick, Asphalt Block or Asphaltic Concrete for Class I traffic at a cost for a i6' road ranging from $20,000 to $28,000 per mile. For Class II, Bituminous Macadam, First-Class Concrete, Water- bound Macadam with Bituminous flush coats, Amiesite, and in the near future probably small blocks or stone cubes at a cost for a 16' road ranging from $11,000 to $18,000. For Class III, Waterbound Macadam treated with Calcium Chloride or light road oil or tar at a cost for 12' to 16' road of from $8,000 to $11,000. For Class IV, Bituminous Macadam, Concrete, Waterbound Macadam with Bituminous flush coats, Amiesite, Asphalt, etc. at a cost of $12,000 to $20,000 per mile. The type selections as given are based on satisfactory performance under traffic and moderate maintenance cost. MAINTENANCE In the development of any system of highways the methods and cost of maintenance become increasingly important. The rapid growth of motor traffic in the last few years has changed both methods and cost, making it necessary to give new figures which are reliable for present traffic conditions. We have therefore confined ourselves in the discussion to recent costs with which we are familiar in order that in stating general conclusions proper allowance is made for unusual conditions not shown in the reports of various State Highway Departments. The discussion will be based on the general maintenance costs and methods employed on 5600 miles of New York State Highways MAINTENANCE OF IMPROVED HIGHWAYS 97 of all types for the year 191 5 and detail costs on 600 miles of roads in Western New York for a term of years. We are indebted to Mr. Frank Bristow for the following discussion of general maintenance methods and summarized costs. It should be borne in mind that the discussion and costs apply to territory subjected to severe winters. Maintenance of Improved Highways By Frank W. Bristow, N. Y. S. Dept. of Highways, Division of Maintenance Maintenance comprises keeping the paved roadway surface in as nearly perfect condition as possible, keeping the earth shoulders smooth and safe for traffic; the drainage system free from obstruc- tions; all structures in good repair; removing obstacles to vision as brush or overhanging branches; and cutting tall weeds and grass. If the work of maintaining improved roadways is consistently performed through successive years it is certain that the efficient life of such roads will be lengthened. Maintenance should commence when construction leaves off, because in order to effectively and eco- nomically maintain improved roads it is necessary that the roadway be in a good state of repair at the time the maintenance work begins, and should the pavement be so worn as to be structurally weak it is not economy to postpone resurfacing. Maintenance work, including surface treatment with bituminous material and cover, should be distinguished from extensive repairs involving resurfacing or reconstruction. Maintenance of Macadam Roads It is especially desirable that all surface treatments be completed as early in the season as possible; say by mid-summer to permit traffic to enjoy the greatest benefit from such treatment, the season of heaviest motor traffic being from the middle of July to the middle of September. So far as practicable the correction of surface defects such as ruts and depressions should precede the surface treatments. While the elimination of dust on macadam roads is desirable as adding to the comfort of the traveling public, it is necessary from the maintenance point of view, inasmuch as dust means deterioration of the road which if permitted to continue results in a raveled con- dition and the macadam will disintegrate. Surface treatment with oil or tar also tends to seal or waterproof the pavement. Horse- drawn steel-tired traffic tends to destroy an oiled surface mat, while rubber-tired motor traffic is beneficial. It is good practice not to oil macadam roads upon which horse- drawn traffic greatly predominates, or new waterbound macadam which has not been under traffic at least two months, or extremely shady roads. The usual foundation defects which develop in gravel and macadam surfaces are ruts, due to a soft condition in the earth sub-grade, depressions due to settlement of fills which commonly develop at locations where new culverts were constructed and frost boils. 98 TOP COURSES Shallow ruts and surface depressions are corrected by being filled in with crushed stone of as large size as the depth of depressions will permit, the same being well tamped into place, and more lasting results are obtained if a proper grade of bituminous material is used to firmly bind the new stone; light asphaltic oils and tars have been used for this, purpose with unsatisfactory results, in that patches made by this method do not endure, the experience being that the material forming the patch is pushed ahead by traffic leaving the original depression exaggerated by the bunch of new patching material at the end. Heavier binder grade material has been used; a patch by this method is durable but does not wear away as rapidly as the adjacent surface resulting in a high spot in time. To date our expe- rience is that an asphaltic emulsion for cold patching is most satis- factory, being nearly fool proof and requiring no equipment but a broom and shovel. This material is not recommended for use with stone of greater size than will pass a one and a quarter inch ring. In using this material the depression to be repaired should be swept clean, so as to be free from mud or loose material, and tamped full of a mixture of the emulsion and broken stone. Such a patch will require an hour or two to set. The proportions of the mixture re- quired are, where the stone used are uniform in size about three- quarters of a gallon per cubic foot of stone, where the stone are graded about a gallon per cubic foot. This mixture may be made in moderate quantities as stock for use as required. Ruts in gravel surfaces may be eliminated by the use of a hone early in the season. Deep ruts indicate necessity of either subdrainage or reinforcement of the foundation; an inspection should determine which is the proper remedy. On side hill roads frequently a deep drain in the upper side ditch to intercept the ground water will be effective; where reinforce- ment is decided as necessary, usually sub-base construction about eight feet in width will be sufficient. Field stone, quarry spalls, broken stone, slag or gravel are proper materials for such reinforce- ment. Frost Boils so called are caused by wet spots in the earth founda- tion freezing and heaving, later when the frost leaves and the foun- dation soil is soft the thin macadam crust tends to break through under loaded wheels. These spots which usually occur where the road construction is in a cut, should be excavated, and drained if practicable; any wet clayey soil or silt removed and replaced by gravelly material, field stone, quarry spalls or other good material, the macadam is then replaced. Ravel is the term applied to describe the condition where the fragments of broken stone become loosened from the body of the road, due to the binding agent failing to perform its function. Bare toothy or a pitted condition of surface are the varying degrees of a slightly rough surface due to the interstices between the fragments of stone not being filled flush with the binding material or when the wearing surface has innumerable extremely slight depressions. Dust, which is self-explanatory. The remedy for raveled, pitted or dusty condition is a surface treatment of bituminous material and cover. MAINTENANCE OF IMPROVED HIGHWAYS 99 These treatments are generally made using a grade of asphaltic residuum oil or a refined tar product which can be applied cold, or which requires very little heating, and better and more uniform results are obtained where a pressure distributor is used. If a pressure machine is used not less than twenty pounds should be required. Asphaltic base oils, or tar products having a bituminous content of from 40 to 60 per cent may be applied by gravity sprinkler, but 60 to 75 per cent asphaltic oils or tars containing 60 to 70 per cent of pitch are preferably applied by pressure. Uniformity in application is important. As to the relative merits of asphaltic Residuum oils, cut back asphalts, high carbon, or low carbon tars there is a diversity of opinion. Relative cost and durability will naturally be the considerations controlling the selection. The material which is the cheaper at one delivery point may not be at some other. As to durability it is doubtful if there is any advantage as between the asphalt and tar- products. When applied, the tar material appears to take a set faster than the asphalt, which is a decided advantage, but more criticism is made as to slipperiness of the tarred surfaces during freezing weather. It is thought that the tars have the greater adhesive quality, but that the exposed surface due to evaporation of volatile constituents becomes crumbly or dead in a shorter time than a similar grade of Asphalt. Regarding rate of application per unit area, this will vary with the porosity of the surface to be treated; for the cold, or light hot appli- cation ranging between one-sixth and one- third gallon per square yard. Experience is that from one fifth to one quarter gallon will produce good results on the average surface. Preliminary to the applying of the bituminous material the surface to be treated should be swept clean if necessary, to free it from all loose and organic matter; after this has been done the application can proceed regardless of whether the surface is wet or dry, providing there are no pools of standing water on the surface, a slightly damp surface apparently gives better penetration than an absolutely dry surface, the object sought being to get the material into the texture of the road. The surface treatment should be confined to one side or half wddth of the road at a time, leaving the other side available for traffic. Some little time should be allowed for proper penetration, but within one hour after the application it should be lightly covered with suitable material. Traffic can now use this side and the treat- ment continued on the opposite side. The materials recommended for cover are crushed stone or slag which will pass a half inch mesh and are free from dust, ore tail- ings, fine screened gravel or coarse sharp sand. The toughness of the mineral aggregate used for oiling cover is an element in the durability of the mat formed by the treatment. Relative cost will determine the selection of material to be used for cover. The quantity of cover necessary will vary with the rate of application of the bitu- minous material and with the porosity of the surface treated. Where the rate of application of oil is from one-fifth to one-quarter gallon per square yard the range of cover may be stated as being ioo TOP COURSES between thirty-five and seventy cubic yards per mile of road sixteen feet wide, and generally forty to fifty cubic yards will be ample. This cover should be uniformly applied either by hand or by mechanical spreader; however, only sufficient to cover the oil lightly should be applied at one time. It will require two or three separate spreadings from time to time as the surface becomes shiny and sticky to produce a perfect mat. Any excess unused material delivered for cover should finally be gathered up and stored in neat piles back of the ditch line where possible. These treatments do not require rolling, although rolling tends to turn any coarse sharp fragments of cover material onto their broader sides, reducing danger of tire cuts to a minimum. 1 Thick mats formed of binder and three-quarter inch stones while durable are not generally satisfactory; they are expensive, costing from one to two thousand dollars per mile and frequently become rough under traffic, although they do serve at times to carry a road along for a few years which would otherwise be a resurfacing matter. This treatment also is used to restore a crown to a road worn flat. On gravel and new waterbound macadam and upon roads where there is little motor traffic, maintenance by calcium chloride is effective. Where this treatment is used the applications may be of the granulated crystals applied by hand or by a mechanical spreader, at the rate of one pound to one and a quarter pounds per square yard; preliminary sweeping is not necessary unless there is excessive dust, say a quarter inch depth or more upon the surface proposed to be treated. Should this treatment be made immediately preceding a rain, a considerable quantity of material would be lost. Two or three treatments as above should suffice for the average season, and the width treated may be confined to the width of the traveled way. This treatment has cost in New York State about one hundred fifty dollars a mile a year. Surfaces which have previously been oiled are not recommended for Calcium Chloride treatment. In cases where, continued surface treatments of Bituminous material through successive years has built up an excessive depth of mat, which has a tendency to be unstable and rut, it is suggested that such mat be removed and spread upon the shoulders, which will cost from fifty to one hundred fifty dollars a mile, and that sur- face treatments be again made upon the macadam itself. Should it be found that the macadam has become uneven, as to crown and grade, or is badly worn or has numerous holes, it is suggested that the road be scarified and thoroughly dragged with a heavy spike- tooth harrow, after which an agricultural weeder should be repeatedly hauled over the road, the object sought being to work all of the finer particles to the bottom of the scarified course, leaving fairly clean coarse stone at the surface; this should be shaped up by hand or scraper and rolled to develop any irregularities in the surface which should be corrected by the addition of new crushed stone. Any pockets of fine material should be removed and replaced by new top course stone, the weeder should again be used to loosen the stone, 1 The authors wish to emphasize the danger of using thick mats for ordinary maintenance. MAINTENANCE OF IMPROVED HIGHWAYS 101 which will then be ready for the first application of binder, which may be at the rate of three-quarters of a gallon per square yard, application being made by a pressure distributor, the surface then to be covered with a layer of three-quarter inch broken stone and thoroughly rolled. During the rolling, additional three-quarter inch stone shall be applied and broomed about until the voids in the top course are well filled; all loose stone shall then be swept from the surface and a sealing coat of one-half gallon of binder per square yard shall be applied and immediately covered with a layer of one-half inch stone and again rolled; surface will then be ready for traffic. This treatment is probably better adapted to waterbound macadam than to the penetration bituminous type; however, if found necessary to break up and reshape penetration macadam, it is suggested that the latter loosening by the weeder be omitted and a spread, one stone thick, of two-inch broken stone be applied and the first application of binder be increased to one gallon or one and a quarter gallons. This method is not applicable to an extended mileage as it is generally better to resurface when a road reaches this stage. Concrete Roads with Thin Bituminous Surfaces. The second-class concrete with thin bituminous wearing surface is a difficult type to maintain; the bituminous surface under traffic patches off, and as the concrete is usually not strong enough to resist abrasion, holes develop in the concrete, patching results in a rough riding surface and probably the best way to secure a smooth riding road is to resurface, using a two-inch bituminous mixing type top. Asphalt, Topeka Mix, Amiesite, etc. The holes which develop in the bituminous mixing method type wearing surfaces should be repaired as follows : — Excavate the old material at the defective spot the entire depth of course, so that the edges will present clean, vertical surfaces, these surfaces and the exposed foundation to be swabbed or painted with hot asphaltic cement or paving pitch, the hole then to be filled, with a mixture similar to that used in original construction, whenever practicable, using sufficient quantity so that after consolidation by rolling (or tamping in case the extent of repairs is limited) the surface of the new patch will be flush with the adjacent pavement. In case there is no local mixing plant available, or the limited extent of repairs do not justify expense of treatment as above, holes may be repaired with the mixture of crushed stone and cold patch asphaltic emulsion, as outlined for macadam surfaces. Concrete Pavements. The cracks which develop in concrete pavements may be the result of either frost action, settlement of foundation or contraction, and are properly treated by being poured with hot paving pitch or asphalt binder. If spots disintegrate, the defective material should be removed and replaced by new concrete. 102 TOP COURSES Block Pavements. Block pavements of brick, stone, asphalt, etc. properly constructed should not require repairing for a considerable term of years; cracks which develop should be grouted with hot paving pitch or asphalt binder; areas which settle, thereby breaking the bond of the grouted joints resulting in crushing or cobbling the blocks, should be taken up, the sand cushion reformed, all sound blocks cleaned and re- laid, being turned over where necessary, any broken blocks to be replaced by new whole ones, joints then to be grouted with Portland cement grout preferably, if the original pavement was so constructed, otherwise the joints may be poured with hot paving pitch. It should be noted that repairs with fresh cement grout require to be protected by barricades for a period of about a week, so that such repairs should be confined to one side of the pavement in long stretches, leaving the other side available for traffic; where the repairs are limited in extent and barricades are especially undesirable, the patch may be covered with two inches of earth and further protected by planking during the time required for the grout to set. Where joints are poured with paving pitch, traffic need be diverted only during the time of actually making the repair; this is a decided advantage. Observation demonstrates that horse traffic on steep grades leave the pavement and seek the earth shoulder, so that so far as practi- cable these shoulders should be improved by widening,' and by gravelling or covering with broken stone to avoid excessive rutting, also that on sharp curves the tendency of motor vehicles is to cut close to the inner edge, making it well for this reason to stone or gravel the shoulders at these points. Along the edges of the rigid types of pavement, block and con- crete especially, traffic usually develops a deep rut which if neglected becomes dangerous, to rapidly moving traffic; this rut should be kept filled with gravel or broken stone. Excess material when removed from the shoulders should be so disposed of as to widen embankments and flatten slopes. General Organization Methods. There are three general plans for performing the work of general maintenance: the patrol system, the repair gang and by contract. The nature of the work renders it difficult to estimate in terms of labor and material with precision, so that except in the case of surface treatments, repair by contract is not advised. By the patrol system the roads patrolled are under constant supervision and the responsi- bility for neglect is fixed. The repair gang may be used to supple- ment the patrol system when it is desired to expedite extensive small repairs, and also to perform all necessary repairs upon any roads not patrolled. A patrolman living in the vicinity of his work, equipped with a single horse, one yard wagon and small tools, cost- ing three dollars a day, can make all minor repairs on a section of between five and seven miles of macadam. The repair gang should be equipped with a small motor truck, say of one and a half tons capacity, to be used in transporting the men and tools within MAINTENANCE OF IMPROVED HIGHWAYS 103 a radius of about twenty-five miles from their headquarters base; this truck can also assist by hauling som^ materials required in the work. It is concluded that a combination of the patrol and repair gang systems is an improvement over the adoption of either plan of organ- ization exclusively, also that the success of either plan depends entirely upon the experience, good judgment and ability of the man in direct charge and control of this work. As nearly all of the hauling in connection with maintenance work is over hard-surfaced roads, motor equipment for delivering stone, oil, etc. would naturally be considered. The writers' opinion is that for short hauls teams are economical, also that the motor tractor and trailers system of equip- ment are more efficient than the complete single unit system. Summarized Costs for the Season of 1915 New York State, In order that the following figures may be more easily understood, it is well to outline the development of the use of the different types of pavement. From 1898 when State Highway Improvement began until 1909 to which time 1787 miles had been constructed, practically the entire mileage consisted of Waterbound Macadam. Up to this time there had been no systematic maintenance, which resulted in a large mile- age of road requiring more than ordinary expenditure to bring it up to standard. Beginning in 1909, Penetration Bituminous Macadam was gen- erally used on the main roads with Brick near cities and villages. About 191 2 the department tried out Concrete roads with thin Bituminous Oil Tops. This type proved unsatisfactory in that the bituminous surface peeled in spots and the concrete used was not sufficiently strong to stand the traffic directly. The high cost of maintenance can be seen from the following table. The type has not been used since 19 14. The department is now designing first- class Concrete roads where roads of that class are economical. In the following tabulation of maintenance and renewal costs, therefore, the average per mile represents approximately a fair sample of both yearly maintenance and renewal for Waterbound Macadam, Gravel and Concrete Bituminous and represents only ordinary yearly maintenance for Bituminous Macadam, Concrete Pavements, Brick and other high-class rigid pavements; none of these latter classes have been down long enough to yet require renewal, which makes their cost as shown much less than will ultimately be required. io4 TOP COURSES to H On H CO O +■>•£_ c£ K5T3 O bfl & £ m m O O O O O •OlOHOO O fON VO M O H O O O 1O1OH00 tJ- O lO H O M M O 't ^-no nO « VO00 00 r? IO00 0\ Q\ r-^. fo .-d o fa > o d d — -P s 0) co aj 4-> C/) Q. tu be M u > d MAINTENANCE OF IMPROVED HIGHWAYS 105 Of the mileage shown in the preceding table, the following table shows the amount of resurfacing. 1 Table Showing Resurfacing Costs 19 15 Type of Road Resurfaced No. Miles Total Cost Cost per Mile Gravel 12.88 176.29 43-72 24.85 0.36 $ 77,686.27 997,776.66 243,760.22 160,321.37 4,003.40 $6,000 6,000 6,000 6,400 12,000 Waterbound Macadam Penetration Bituminous Concrete Bituminous Top Block Pavements Totals 258.10 $1,483,547.92 1 The type of resurfacing is not necessarily the same type as the original road as shown in column No. 1. Supplementary Explanation of Mr. Bristow's General Costs and Discussion The authors wish to call attention to two points in the general cost tabulation. The average cost of maintenance and renewal for 191 5 is given as $750 per mile for the total system. This system includes approximately 1,000 miles of road recently built on which there is practically no charge except minor repair aggregating not over $200 per mile per year. For a completed system of this character all of which is under normal maintenance and renewal, the average cost per mile would be approximately $900 per mile, as is evident by excluding the thousand miles from the tabulation of total cost. In the resurfacing table it is evident from the cost per mile that better grades of top courses were generally placed on the Waterbound and Gravel roads than originally constructed; this means that in some cases the original design was not proper for the class of traffic the road served. The most evident faults of the usual maintenance are in delaying the work till late in the season and in careless mending of ruts and depressions before the application of surface treatments. It is well to emphasize the necessity of using a coarse grade of stone preferably 1^ to 2 \" size in mending noticeable depressions. The hole should be dug out, the edges squared up, the depression filled, bound with heavy binder and screened and rolled. Carelessness in this regard has resulted in a large amount of justifiable complaint. Typical Maintenance Costs of Different Types. From a detailed study of 600 miles of road in Western New York with which we are personally familiar, the following typical costs are derived. 106 TOP COURSES It is assumed that the type used is suitable to the class of traffic served as indicated on page 96. The maintenance system is a combination of patrol, gang work and contract. A one-man patrol with horse and wagon is used to keep the shoulders in shape, the ditches clean and small holes repaired. Gang work with proper machinery under state control to paint guard rail and make more extensive surface repairs and contract work for oiling and surfacing. Detail oiling costs are given under cost data, (page 333). This system is not highly efficient as the executive heads are changed at short intervals for partisan reasons; the depart- ment is a convenient means of dispensing minor patronage and the maintenance money is rarely available early enough in the spring to be used to the best advantage, but it represents about as good a method as can be expected in doing public work on a large scale and as such is of more practical value as a guide of costs than figures based on maximum efficiency. 1 6' Waterbound Macadam Class II and III Traffic. Life of Top Course 4 to 10 years. Say 6 years for Class II Traffic and 8 years Class III Traffic under the following maintenance. Class II Traffic. Yearly patrol including materials for Minor Repairs and Painting Guard Rail @ $200 per mile per year $1,200 Calcium Chloride 1st and 7th year @ $125 per mile per year 250 Cold Oiling 2nd year 250 " 3rd " 300 " 4 th " 300 Hot Oiling 5th " . . - 1,000 Cold Oiling 6th " 250 Resurfacing with Waterbound Mac. 7th year 4,000 7 year total $7,55° Cost per mile per year Maintenance and Renewal. . . .$1,000 " " " " " for Maintenance 500 Class III Traffic. Yearly Patrol $1,600 Calcium Chloride 1st and 9th years @ $125 per year 250 Cold Oil 2nd year 250 " 3rd " 250 " 4 th " 300 " 5th " 300 Hot Oil 6th " 1,000 Cold Oil 8th " 250 Resurfacing 9th " 4,000 9 year total $8,200 Cost per mile per year Maintenance and Renewal $900 " " " " " for Maintenance 500 MAINTENANCE OF IMPROVED HIGHWAYS 107 16' Penetration Bituminous Macadam, Class II and IV Traffic. Life of Top course 5 to 10 years. Say 8 year average. Yearly patrol @ $150 per mile per year $1,200 Cold Oil 3rd year 250 4-tn 250 :> tn 3°° Hot Oil 6th " i ? ooo Cold " 8th " 250 Resurfacing 9th year Bit. Mac 6,000 9 year total $ 9,250 Cost maintenance and renewal per mile per year St, 000 per mile per year 400 a a 16' Brick Pavement, Class I Traffic. Probable life based on Medium Traffic Rochester City Streets 10 to 25 years. Say 18 year average. Yearly patrol and minor repairs ©$150 per mile per year .... $2 , 700 Renewal of surface and edging 15 ,000 18 year total $17,700 Cost of maintenance and renewal per mile per year. . .$1,000 " per mile per year approx 200 .. a 16' 1st Class Concrete Pavement, Class II A Traffic. None of these pavements have been down a sufficient length of time to give us any reliable data as to probable length of life before resurfacing. We will assume 10 to 15 years or an average of 12 years when they will be resurfaced with Asphalt, Brick, or Clay Cubes at a cost of $9,000 to $15,000 per mile. Yearly patrol $150 per mile per year $1,800 Resurfacing and minor renewals 10,000 13 year total $11,800 Cost of maintenance and renewal per mile per year .... $900 " " " per mile per year approx $200 Maintenance Conclusion. The indications are that the yearly cost of maintenance and renewal of a well-designed road system will run about $900 per mile per year. The effect of bad design is evident from resurfacing costs, for if Waterbound Macadam is built on a Class I Traffic Road the life is easily halved, increasing the maintenance and renewal cost to $1,500 per mile per year and causing continual inconvenience to the travel- ing public by repairs and reconstruction. Probably the most feasible method of reducing maintenance cost will be by utilizing Prison labor to manufacture and in a limited way apply the maintenance materials. io8 TOP COURSES IL-s o o o o o ugS o O <^ ^CO — >> lO l^ Tf H 3 v n °.52 O o o o o o o o o CO r^-O io -3- 3 *£££ S^ W N fn CJ t-i c Q o o o o o o o o o o Ph ^^ N M 't lO 't o ^■1 €£f s 1— 1 2 tf w ~ tf r°N o o o o o p^ £ ^ U O O m ^00 H O MD io ^ m o £ u 5x h3 tf < o o o o o W J« c« c .Is o ^ O o o o >< O^ 0^ O^ O^ ^o Pl,U u l-O io ro M o ^ Pi (N M M M H €*? < a t— i X o ft 3 l-l 1-1 o w CO s* vb • • tf M < . ^ * ^ * g U MD -£-< 8'0"- "»■*■ p-"-I "kJ Rails and Posts Painted }-\ lJ _* 3 Coats White Lead and Oil. l_J i — i i i L.J Iron Pin, %'Diam., 6"lonq'' Fig. 29 The posts are cedar, white oak, or chestnut, and the rails are hem- lock, yellow pine, or white pine. Such guard-rail costs from twenty- five to forty cents per foot, about five cents per foot per year for maintenance, and needs renewal every eight to ten years: the capi- talized cost at 4% is approximately $1.25 as figured by the New York State Highway Commission, and on this basis they have decided that it is cheaper to use a fill slope of 1 on 4 up to a seven-foot depth, eliminating the guard-rail, than it is to use a 1 on 1^ fill slope with guard-rail. The wooden guard-rail as built acts as a warning only. If a machine or rig becomes unmanageable and hits the rail, it generally breaks or the posts tear out, allowing the vehicle to turn turtle on the fill IOQ no MINOR POINTS slope. So many accidents of this kind occur that there is a demand for a rail that actually gives protection as well as a warning. Concrete Guard-Rail. Because of this demand and the high cost of maintenance and renewal of the common wooden trail, concrete guard-rail is being adopted. The simplest and best design of this kind that the author has seen was tried out by the New York State Department of Highways on the Ridge Road, near Rochester, N.Y., in 1910. A sketch is given below. This construction has been specially com- mended by the automobile associations. H ---- J V ~-8 -=* ,J <7i , I . ' » 'Diaphragm to Prevent Slipping Endwise^ <7- fc? Mesh Rein- forcement to I I I L-J I I 1 J Fig. 30 The rail was invented by J. Y. McClintock, County Engineer of Monroe County, N.Y. It is neat in appearance, durable and strong, and is specially adapted for a combination bridge and approach - jr - 2 0%* -------- j Section C-D. End Elevation. Fig. 31. — Showing Raised Parapet on Skew Bridge extended over Straight Parapet Retaining Wall RETAINING WALLS in Round tol{Rad. Found to li'ffad. Weep-Holes of3"V(e\—\- to be Spaced 6' 1^ CI ass Concrete '- 0.39 sa./n. Rods Spaced 18" C.toC. -0.-55H-H-2* x: Type W. "t"(ininches)=2H-7 Type R. Fig. 32. — New York State Standard Retaining Walls Reinforcing Steel Bars of Deformed Section H n 12 13 U 15 16 I? iS iQ 20 Stem 1 Net Spacing Area C-C O.601 6}" O.601 32 O.601 .s" O.601 4i" O.601 4" O.994 6" 0.994 52 0.994 5' ; O.994 42 0.994 4s Length 12-2 " i3 ; -3r i4 r -5 " i6'-8 " i7 r -9 ' i8'-ioi" 2o , -o " 2I , -li" 2 2 '-< " Heel Toe Net Area Spacing C-C Length Net Area Spacing C-C Length O.442 1\" 4 _II 2 0.442 Q§" 3'-3*' O.442 W 5 52 O.442 8J" 3-8" O.442 .si" 6'-o" O.442 7i" 4 _0? > O.601 6i" 6'-6" 0.601 8|* 4 # -4*' O.601 -3'/ 5s 7'-o" 0.601 ~3V /8 4'-0" O.601 .3 if 44 7-6i' / O.601 6i" s'-4' O.785 52 8'-o|* O.785 ~5" /J 5'-6" O.785 A 7 " 4s 8'- 7 " O.785 6J" 5'-io|" 0.785 4f" 9'-i" 0.785 6i" 6'-2|" O.785 4 " 9 , -7 ,/ 0.785 -3'/ 54 6'-7" 1 In each set of 3 bars in stem, first bar which is of length given, extends to top of wall, second bar to height f H, third bar to height \ H. When Type W is used as a bank wall (that is, above the roadway), max. H= 20'; min. X = 2' for H of 5 to 10'; and 0.2 H for H greater than 10'. When Type W is used as a sustaining wall (that is, below the roadway), max. H = 13'; and min. X = 3', except where foundation is rock or entirely below frost. When Type R is used as a bank wall, max. H = 20'; min. X = 0.15 H for H greater than 10'. When Type R is used as a sustaining wall, max. H = 13'; min. X = 0.25 H for H greater than io'. 112 MINOR POINTS rail. The old design of an iron bridge rail connected with a wooden road rail has been an eyesore. § . The actual cost of manufacture and setting was from fifty to sixty cents per foot The contract price for such rail would, probably, run from eighty cents to one dollar, depending upon the length of the haul, freight rate, and difficulty of digging post holes but even at the high figure it is cheaper than the wooden rail and is a safe construction. The anchor and rod shown on the sketch is used on curves or even on straight stretches where new fill is encountered, to prevent the posts being torn out by impact from runaway machines The rail proper has a web and bar reinforcement; it is designed to stand a six-ton horizontal load at the center of the panel The rails and posts are molded separately and allowed to set for, at least, a month; they are then put together in much the same manner as the wooden rail. The rounded top of the post makes it possible to e TconsXa g bTe d mileage of this rail has been erected in New York and New Jersey and has prevented many serious accidents. It has been hit by autos, tar kettles, rollers traction-engines and 1 rigs i and in no case has the vehicle gone over the bank -which is the general cause of fatal accidents. The rails and posts will break when hit by a heavy machine, but the reinforcement merely bends (does not snap) and continues to exert enough resistance to hold the machine on the roadway. Guard Rail has two distinct purposes : first, as merely a warning, at culverts, curves, low embankments, etc. where the danger is not great, and second, as an actual protection in dangerous places. Concrete guard rail is not advocated where warning alone is sufficient. Bridge Rail and Raised Parapets. Bridge rail for small span bridges is of two types, iron pipe rail (see figure 29) or solid raised parapets (see figure 31). The solid parapet is to be preferred. Exj mansion Joint Retaining Walls. In unusual cases retaining walls are needed in road construction. Plain or reinforced concrete walls are generally used, the selec- tion depending upon the relative cost. The plain concrete wall is considered the best type for heights up to twelve feet; the re- inforced cantilever form from twelve feet to eighteen feet, and above eighteen feet the buttressed design. We give page 1 1 1 examples and rules for the plain and reinforced cantilever types only, as the necessity for walls higher than eighteen feet is very rare, .bor tne Plan. Key Expansion Joint. Fig. 32 a CURBS 113 design of buttressed walls the reader is referred to the standard works on reinforced concrete. Retaining walls are usually built in monolithic sections of 20' to 25' in length; expansion joints are provided between these sec- tions. The expansion joints may consist of simply a plane of weak- ness between the sections produced by allowing one section to set before building the adjacent wall, or it may be a key joint as shown in figure 3 2 a, and the plane of separation may be made more pro- nounced by coating the concrete with a thin layer of asphaltum or pitch. Brick Gutter., 1)8* Tile Drain Fig. $3. — Showing Concrete Toe Wall Toe Walls. Toe walls are nothing more than low retaining walls or very sub- stantial curbs. They are used in cuts on the outside of the gutters to prevent unstable side slopes from filling the gutters or heaving them out of shape by sliding pressure. Figure 33 gives a section of Eden Valley Hill near Buffalo, N. Y., where a clay quicksand cut was successfully protected in this manner. k b *,■ Round to I "Rad. Cinders, 6 ravel or Crushed Stone Round to I "Rad. ~ ("Aspha/tJoinK vrfe Brick f\<-l2- Sand- Cinders, Oravel or Crushed Stvne Cinders, Grave/ Z" Porous OWne - or Crushed Stone- Tile Stone Curb. Concr.Curb and Gutter. 2 Porous Tile- Concr.Curb and Gutter. Round to I "Rad. - 4 ~i "Joints Filled with Paving Pitch X7.it: - -P'fjySand Simple Stone Curb For Light Traffic Village Streets. k/0>i s 5' 'Concrete Base Simplest Form of Concrete Curb. (5howing also Form of Expansion Joint where Brick are Laid Longitudinally in Gutter.) Fig. 34 Curbs. Curbs are constructed of stone and of concrete. H4 MINOR POINTS Stone Curbs. The cuts given show the methods of setting; the size of curb- stones for first-class work range from 16" to 22" in depth, 5" to 6" in thickness and 3' to 5' in length. For small villages, curbstone of 4" width, set in the simplest manner shown, is satisfactory. The stones most used are granites, Milestones of New York State, and the tougher sandstones such as Medina, Berea, Kettle River, etc. The prices range widely, depending on the locality of the work. Mr. William Pierson Judson, in his " Roads and Pavements," gives the following range of costs : C.I.Filleh- Bolts Straplmng'thick Bolts. 7 " 8 Pine Board <6~> ::::! 30' Blade, Letters on I Red Field CM 'Screws <22 m-HUDSONT- '.ALBANY IOnDj I^ <43*m- NASSAU ! -Both Sides to be Lettered if Ordered. X.H.No.lZl lTL Letters to be of cm t*> Aluminum on a Black Field. 1. 9" Gal v. Iron Pipe, (Outside Diam.) Cylinder of 2nd Class „ WMf'i' Concrete-. Diam. 6.. ".?. ■:■ ! : i-'^.' ' a I---J3 Og Iron Pin through Pipe $$ K- 6" >i Cast Iron, Filler. <6$ 1_ J* ■I "Rivet la **•+- ^"Carriage Bolt *■ Upset Thread after Nut is in Place. Fig. 35 Straight curbs set, cost about as follows, with 30 per cent to 50 per cent added for curves: GUTTERS 115 Granite $0.50 to $0.90, unusual case $1.25 per foot. Ulster and Oxford bluestone, $0.40 to $0.80, unusual case $1.00 per foot. Medina and Berea sandstone, $0.35 to $0.90. Shoo/Gf er ^o^f'" Concrete usually costs from $.040 to $0.50 with $0.35 added 4?Sand f° r a combined gutter, though Cushion combined gutter and curb have \ < - been built for $0.50. r5w^/^^^^ The s i m pi e concrete curb - . . . „ , ' (figure No. 34) has been built Cobble Gutter. - , \ • • A -cc 4. *. t\"& &? business sec- Ditch Lininq tions of small villages. Jbor the s residential portions or where \$§^ the cost of stone curbing is high, ~~~^^^r 5$!^ a concrete curb of the simplest ^^^^^Fr^^^E^T^ design is the best practice, as l^AjDrr TOC-* c ^y conditions and require- i'^^s^i ments are neither necessary nor 't'(ulJ°Wd4 expected. Concrete orSand Foundation-, Grouted or Sand Joints. Guide Signs and Danger Signs. Brick Gutter. . £2 72m ■ II Leaching Basin. Fig. 40 Upset \ Ends of Bars ^ and flat- ten to-g" Standard (jratin 3 Diam. * ; Circular Opening-'' Catch Basin. REFACING OLD WALLS 117 Where cobblestones are not available, ordinary building brick may be used or No. 4 crushed stone, as shown page 115. "Riprap and Dykes. Well-constructed riprap protects stream banks and bridge ap- proaches from stream wash except in unusual cases where a solid masonry or concrete protection is required. The sizes of stone suitable for riprap are usually specified at a minimum of f cubic foot and 50 per cent or more of the material to be over 2 cubic feet. Where the road is located in bottom land and is covered with back- water in the Spring, it can be protected by riprap paving on both sides or a dyke and riprap paving on one side as shown in figures No. 38 and No. 39. Grates. Cost of cast-iron grates about $0.05 per pound. Cost of wrought-iron grates about $0.08 per pound. Repointing Masonry and Refacing Old Walls. Old masonry structures can often be used complete or in part by repointing the joints; they should be cleaned out thoroughly with a chisel and filled flush with a 1 to 1 Portland Cement mortar. The author does not believe in facing up old masonry abutments 2 Reinforcing Bars^J\ Spaced IZ'X.toC. Fig. 41. Facing for Old Masonry if it can be avoided; however, if it seems advisable, because of shortage of funds, the old joints should be well cleaned out and hook dowels used as shown in cut No. 41. One dowel every 6 square feet is good practice. The concrete facing should be at least 12 inches thick and reinforced to prevent settlement and temperature cracks. CHAPTER VII MATERIALS The selection of materials is an important part of the design. Most municipal and State Departments have well equipped labora- tories for testing stone, gravels, brick, bitumens, cements, etc. The object of these tests is to determine the physical and chemical prop- erties that have a particular bearing on the action of the materials under construction conditions. While these conditions are not attained they are approximated and by a comparison of the laboratory results with the actual performance of the different materials in practice a relation can be established that is useful as a basis for judgment : We are greatly indebted in this 2nd edition to Mr. H. S. Matti- more and Mr. J. E. Myers who have rearranged and brought up to date much of the material on tests and their significance. This chapter gives a brief statement of the desirable qualities and the tests for: 1. Top course, macadam stone. 2. Screenings. 3. Bottom course, macadam stone. 4. Bottom course and sub-base fillers. 5. Brick. 6. Bituminous binders. 7. Concrete materials. 1. Stone for the Surfacing of Macadam Roads Stone for use in the surfacing of a macadam road should be hard and tough to withstand the abrasive action of team traffic and the vibratory action of high-speed motor vehicles and should not contain any minerals that are likely to disintegrate rapidly under influence of weather conditions. To determine the relative hardness, toughness and power to resist abrasive and impact action of traffic, stones are subjected to the following tests: 1. Abrasion. 2. Hardness. 3. Toughness. 4. Specific Gravity. 5. Absorption. 6. Fracture. 7. Geological classification. 1 Abrasion Test. The machine shall consist of one or more hollow iron cylinders; closed at one end and furnished with a tightly fitting iron cover at 1 American Society of Testing Materials. 118 DEVAL RATTLER 119 the other; the cylinders to be 20 cm. in diameter and 34 cm. in depth, inside. These cylinders are to be mounted on a shaft at an angle of 30 deg. with the axis of rotation of the shaft. At least 30 lb. of coarsely broken stone shall be available for a test. The rock to be tested shall be broken in pieces as nearly uniform in size as possible, and as nearly 50 pieces as possible shall constitute a test sample. The total weight of rock in a test shall be within 10 g. of 5 kg. All test pieces shall be washed and thoroughly dried before weigh- ing. Ten thousand revolutions, at the rate of between 30 and 2>2> per minute, shall constitute a test. Only the percentage of materials worn off which will pass through a 0.16 cm. ( T V in.) mesh sieve shall be considered in determining the amount of wear. This may be expressed either as the percentage of the 5 kg. used in the test, or the French coefficient, which is in more general use, may be given; that is, coefficient of wear = 20 X — = — , where w is the weight in w w grams of the detritus under 0.16 cm. (jg in.) in size per kilogram of rock used. 7P7T Fig. 42.Deval Rattler Hardness. Hardness is determined by a Dorry machine. A stone cylinder 25 cm. in diameter, obtained by a diamond core drill from the material to be tested, is weighed and placed in the machine so that one end rests on a horizontal cast-iron grinding disk with a pressure of 25 grams per sq. cm. The disk is revolved 1,000 times during which standard crushed quartz sand about \\ mm. in diameter is auto- matically fed to it. The cylinder is then removed and weighed and the coefficient of hardness obtained by the formula 20 — \ the loss in weight, expressed in grams. In order to get reliable results two cylinders are generally used, each one being reversed end for end during the test. 1 Test for Toughness. 1. Test pieces may be either cylinders or cubes, 25 mm. in diam- eter and 25 mm. in height, cut perpendicular to the cleavage of the rock. Cylinders are recommended as they are cheaper and more easily made. 2. The testing machine shall consist of an anvil of 50 kg. weight, and placed on a concrete foundation. The hammer shall be of 2 kg. 1 American Society of Testing Materials. 120 MATERIALS weight, and dropped upon an intervening plunger of i kg. weight, which rests on the test piece. The lower or bear-surface of this plunger shall be of spherical shape having a radius of i cm. This plunger shall be made of hardened steel, and pressed firmly upon the test piece by suitable springs. The test piece shall be adjusted, so that the center of its upper surface is tangent to the spherical end of the plunger. 3. The test shall consist of a 1 cm. fall of the hammer for the first blow, and an increased fall of 1 cm. for each succeeding blow until failure of the test piece occurs. The number of blows necessary to destroy the test piece is used to represent the toughness, or the centi- meter-grams of energy applied may be used. 1 Determination of the Apparent Specific Gravity of Rock. The apparent specific gravity of rock shall be determined by the following method: First, a sample weighing between 29 and 31 g. and approximately cubical in shape shall be dried in a closed oven for 1 hour at a temperature of. no degrees C. (230 degrees F.) and then cooled in a desiccator for 1 hour; second, the sample shall be rapidly weighed in air; third, trial weighings in air and in water of another sample of approximately the same size shall be made in order to determine the approximate loss in weight on immersion; fourth, after the balances shall have been set at the calculated weight, the first sample shall be weighed as quickly as practicable in distilled water having a temperature of 25 degrees C. (77 degrees F.); fifth, the apparent specific gravity of the sample shall be calculated by the following formula: W Apparent specific gravity = — — Tt^ - " 1 wmcn W = tne weight W — MI in grams of the sample in air and Wi = the weight in grams of the sample in water just after immersion. Finally, the apparent specific gravity of the rock shall be the average of three determinations, made on three different samples according to the method above described. 2 Determination of the Absorption of Water per Cubic Foot of Rock. The absorption of water per cubic foot of rock shall be determined by the following method: First, a sample weighing between 29 and 31 g. and approximately cubical in shape shall be dried in a closed oven for 1 hour at a temperature of no degrees C. (230 F.) and then cooled in a desiccator for 1 hour; second, the sample shall be rapidly weighed in air; third, trial weighings in air and in water of another sample of approximately the same size shall be made in order to determine the approximate loss in weight on immersion; fourth, after the balances shall have been set at the calculated weight, the first sample shall be weighed as quickly as possible in distilled water having a temperature of 25 degrees C. (77 degrees F.); fifth, allow 1 American Society of Testing Materials. 2 American Society of Testing Materials. COST OF TESTS 1 21 the sample to remain 48 hours in distilled water maintained as nearly as practicable at 25 degrees C. (77 degrees F.) at the termination of which time bring the water to exactly this temperature and weigh the sample while immersed in it; sixth, the number of pounds of water absorbed per cubic foot of the sample shall be calculated by the fol- lowing formula: W2 — Wi Pounds of water absorbed per cubic foot = '-= ^—x 62.24 in which W =the weight in grams of sample in air, Wi =the weight in grams of sample in water just after immersion, W2 =the weight in grams of sample in water after 48 hours' immersion, and 62.24 — the weight in pounds of a cubic foot of distilled water having a tem- perature of 25 degrees C. (77 degrees F.). Finally, the absorption of water per cubic foot of the rock, in pounds, shall be the average of three determinations made on three different samples according to the method above described. Fracture. Stone suitable for road work should crush in cubical shapes rather than in thin, flat pieces and preferably with rough, jagged fracture that it may interlock firmly under action of the roller. Geological Classification. The geological classification is determined from an examination with a microscope or powerful hand glass, and a consideration of its origin. Great refinements are avoided as the general classification is all that is necessary to the highway engineer after the physical qualities are ascertained by test. Cost of Tests. The cost of collecting and testing stone as given in the 1909 Report of the New York State Department of Highways is $8.55 per sample. The following tables show tests on the more common rock: 122 MATERIALS Table 21. Taken from Bulletin No. 31, United States Office of Public Roads Rock varieties Granite Biotite-granite Hornblende-granite . . Augite-syenite Diorite Augite-diorite Gabbro Peridotite Rhyolite Andesite Fresh basalt Altered basalt Fresh diabase Altered diabase Limestone Dolomite Sandstone Feldspathic sandstone Calcareous sandstone Chert Granite-gneiss Hornblende-gneiss . . . Biotite-gneiss Mica-schist Biotite-schist Chlorite-schist Hornblende-schist . . . Amphibolite Slate Quartzite Feldspathic quartzite Pyroxene quartzite . . Eclogite Epodosite Per cent Tough- Hard- Cementing wear ness ness value 3-5 15 18.I 20 4 4 IO 16.8 17 2 6 21 18.3 3° 2 6 IO 18.4 24 2 9 21 18.I 4i 2 8 19 17.7 55 2 8 16 17.9 29 4 12 15-2 28 3 7 20 17.8 48 4 7 II 137 189 3 3 23 17. 1 in 5 3 17 15.6 239 2 30 18.2 49 2 5 24 17.5 156 5 6 IO 12.7 60 5 7 IO 14.8 42 6 9 26 17.4 90 3 3 17 15-3 119 7 4 15 ^•3 60 10 8 15 19.4 27 3 8 12 17.7 26 3 7 IO 17.1 30 3 2 19 17.5 4i 4 4 IO 17.8 30 4 — 16 4 2 — 24 3 7 21 16.5 53 2 9 IO 19.0 29 4 7 12 "•5 102 2 9 19 18.4 17 3 2 17 18.3 21 2 3 27 18.6 17 2 4 31 17-4 21 3 6 16 16.0 47 Specific gravity 2.65 2.64 2.76 2.80 2.90 2.98 3.00 3-40 2.60 2.50 2.90 2-75 3.00 2.95 2.70 2.70 2-55 2.70 2.66 2.50 2.68 3.02 2.76 2.80 2.70 2.90 3.00 3.00 2.80 2.70 2.70 3.00 3-3° 3-03 PROPERTIES OF ROCKS 123 Tables 22 and 23 Combined From Annual Report N. Y. State Highway Comm. 1914 County Number of com- plete tests Erie Saratoga. . . . Steuben .... Clinton Dutchess Essex Franklin . . . Fulton Herkimer . . . Monroe Montgomery Niagara Saratoga. . . . St. Lawrence Washington . Dutchess . . . Herkimer . . . Montgomery Niagara .... St. Lawrence Washington . Wayne Essex W T arren Clinton Dutchess Essex Franklin. . . . Fulton Hamilton . . . Jefferson. . . . Lewis Orange Putnam Saratoga. . . . St. Lawrence Warren Washington . Westchester . Essex Franklin .... Hamilton . . . Jefferson .... Lewis Oneida St. Lawrence Warren. .... 46 4 Number of partial tests (no core piece) Weight lbs. per cu. ft. Water ab- sorbed, lbs. per cu. ft. French coeffi- cient of abrasion Hard- ness Tough- ness Weighted value Calcareous Sandstone 5 6 4 1 167 169 162 0.65 9.5 0.31 10. 1 1.44 9.4 12.9 15-9 i5-i 13-4 13-8 131 Dolomite 6 175 0.41 11.9 15.8 12.7 4 1 174 0.43 12.4 17-3 11.9 4 173 0.42 13-5 16.9 15.8 4 174 0.51 9-5 14.9 12. 1 4 176 0.15 11.8 16. 1 14.4 17 173 0.67 8.4 I3-I 6-7 13 2 171 1.07 10.3 14.8 8.2 8 174 o.39 10.6 14.7 II-3 11 168 1.50 6-5 14.0 7.0 8 174 o.33 8.6 15-5 9.2 31 174 0.65 10.5 15-7 9.9 6 175 0.29 10.7 i5-i 10.5 Dolomitic Limestone 8 1 176 0.46 9.0 14.9 10.9 4 1 170 0.47 H-3 16.7 8.2 8 1 175 0.41 13.0 15-8 12.4 7 166 2.19 9-5 131 7-8 7 168 0.38 9.2 16.8 6.8 4 175 0.36 13-7 16. 1 10.8 4 173 o.59 10.2 15.5 8.7 Gabbro 176 I 0.29 183 I 0.37 Gneiss 7-6 10. 1 17-3 17.7 6.9 9.8 5 185 0.27 10.5 17.2 H-3 8 1 172 058 7.0 17. 1 9.1 29 2 176 0.31 8-4 17. 1 8.1 8 178 0.50 6.2 16. 1 7-8 12 1 169 0.25 11. 1 17.8 11. 5 11 173 o.37 8.2 17.0 5-8 26 1 171 0.23 11. 1 17-3 12.0 6 167 0.27 9.6 17.9 10.6 7 179 0.38 7-i 17. 1 6.4 10 1 172 0.32 8-5 16.6 7-5 7 180 0.20 10.0 17.0 8.5 52 172 0.27 9-7 17.5 10.2 30 2 173 0.30 7-5 17-3 6-5 4 170 0.29 8-5 17. 1 10.9 37 2 171 o.39 8-3 16.9 7-8 Granite 1 5 171 0.38 7-5 18.0 5-i 6 165 0.31 8.7 17.9 9-4 5 165 0.36 9.9 18. 1 9.0 23 1 166 0.23 12. 1 18.4 10. 1 8 166 0.36 10.9 18.4 9.2 6 166 0.13 10.2 18.9 8.2 30 165 0.25 9.9 18.3 8.1 5 165 0.45 7-9 17.9 7-7 68 76 72 80 84 90 70 82 58 69 73 55 66 73 73 68 76 83 63 68 84 7i 64 75 78 64 68 59 80 64 80 75 62 66 72 74 64 7i 67 64 7i 75 83 79 77 74 67 124 MATERIALS From Annual Report N. Y. State Highway Comm. 1914. — Cont. County Number of com- plete tests Number of partial tests (no core piece) Weight, lbs. per cu. ft. Water | French Hard- ness Tough- ness Albany Cayuga Clinton Columbia . . . Erie Fulton Genesee .... Greene Herkimer . . . Jefferson .... Lewis Madison .... Monroe Montgomery Niagara Oneida Onondaga . . . Ontario Otsego Rensselaer. . Saratoga .... Schoharie . . . Seneca Ulster Warren Washington . Dutchess.. Columbia . . . | Dutchess Rensselaer . . Washington . | Allegany. . Broome . . . Cayuga . . . Chenango . Clinton. . . Delaware . Erie Franklin. . Greene . . . Herkimer . Jefferson. . Livingston Madison . . Niagara. . . Orleans. . . Otsego Saratoga . . 16 8 10 12 8 n 4 IS 14 53 8 5 6 4 8 4 5 7 8 21 5 Limestone 13 7 168 0.60 7 9 14-3 6.4 34 6 170 0.49 8 8 14.9 7-8 14 2 170 0.28 8 2 14.1 5-3 12 170 0.28 9 1 15-3 9.2 9 3 167 o.57 8 1 16.6 8-3 6 1 168 0.21 7 7 15-5 6-5 6 3 169 0.26 8 15.0 8.2 11 169 0.36 11 1 16.4 8.9 17 9 169 0.26 8 7 14.8 8.2 105 44 169 0.28 7 6 I5-I 6.4 26 20 169 0.32 6 9 14.1 6.2 16 1 169 0.23 8 4 14.7 7-7 4 168 0.27 8 1 14.1 7-4 12 2 169 0.24 8 5 15-3 8.0 11 1 168 0.84 7 1 12.8 6-5 31 19 169 0.29 7 8 13.8 6.6 25 1 170 0.38 8 9 15-7 8.4 11 169 o.39 10 2 15-9 10.2 7 2 169 0.32 8 1 14. 1 6-3 4 1 171 0.21 7 5 15.0 5-3 5 170 0.24 8 7 13-7 7.0 29 2 169 o.34 8 1 14.9 6.7 7 3 169 0.21 9 4 15-3 7-9 12 3 170 0.25 8 1 15.6 7-4 5 170 0.24 8 9 15-7 7-4 5 3 169 o.34 7 9 15-5 6.9 Marble 178 I 0.30 I QUARTZITE 7-3 I 14.2 Sandstone 6.0 168 0.28 16.5 18.3 17. 1 2 166 0.36 13.5 18.8 11. 8 1 ... 166 0.49 12. 1 18.7 14.8 i ... 167 0.40 I4.6 18.9 16.3 156 2.10 8.4 13.4 9.1 165 1.29 7-8 12.9 10.5 167 1.16 7-8 12. 1 io.s 164 1.58 8.7 11. 2 10.4 163 0.71 11. 7 18.5 11.0 167 1-45 7.0 12.7 8.5 159 2.10 6.3 s.i 7-8 i57 1.06 9-7 17.9 7-1 169 0.62 8.6 14-5 8.1 160 2.50 10.9 16.4 10.7 156 1.46 8.3 16.2 6.3 160 3.02 8.8 9.6 8.8 163 2.15 9-9 13-9 8.6 158 1.78 9.0 16.4 8.2 155 2.18 n. 8 14.4 8.1 162 1-75 8.4 11. 9 9.6 163 0.36 10.7 18.0 8.7 PROPERTIES OF ROCKS 125' From Annual Report N. Y. State Highway Comm. 1914. — Coat. County Number of com- plete tests I Number of partial tests (no core piece) Weight, lbs. per cu. ft. Water ab- sorbed, lbs. per cu. ft. French coeffi- cient of abrasion Hard- ness Tough- ness Weighted value Sandstone. — Continued Schoharie . . . Schuyler .... Seneca Steuben St. Lawrence Sullivan .... Ulster Wyoming . . . Albany Columbia . . . Dutchess. . . . Greene Montgomery Rensselaer . . Saratoga .... Schenectady Ulster Essex Franklin .... Herkimer. . . Jefferson. . . . Rockland ... I 12 6 3 165 1. 21 9-4 15.2 11. 7 4 162 2.14 8.1 ' 11. 6 10.6 5 165 0.86 IT.O 13.9 IS.8 22 3 157 2-79 8-3 9-3 10. 16 159 0.79 IO. O 17.8 7.2 30 4 164 1.26 6.5 14.9 8.2 8 166 0.64 8.0 14-3 8.1 7 ... 159 2-54 6.0 5-1 7-9 5 ; ... 167 0.75 7-5 13-2 7-2 1 - 12 168 0.32 10.7 15.9 11. 7 10 2 168 0.57 8.1 16.2 11. 5 13 169 0.48 7-1 15.6 9-5 4 166 i-39 10. 1 n*3 11. 8 10 169 0.44 9.1 159 9.4 5 168 0.99 11. 8 15-2 11.9 4 165 1. 10 9.2 14.6 9 5 7 I ... 169 o.59 7-5 13.8 10.2 Syenite 7 ' 184 0.52 7-7 17. 1 6.7 4 171 o.45 10. 1 18.3 8.0 13 174 0.16 12.5 18.0 11. 6 7 176 0.34 12.4 18.1 14.5 I 183 Trap I Q-39 13.2 | 17.6 I 1.64 I 70 58 77 54 73 58 61 36 56 76 68 62 65 69 78 66 60 64 75 85 88 _9L Table 24. L Geological Classification Class Type Family I Igneous 1 Intrusive (plutonic) a Granite b Syenite c Diorite d Gabbro k e Peridotite 2 Extrusive (volcanic) I a Rhyolite b Trachyte c Andesite d Basalt and diabase 1 Calcareous j a Limestone b Dolomite II Sedimentary 2 Siliceous I a Shale b Sandstone c Chert (flint) ( 1 1 Foliated a Gneiss b Schist < c Amphibolite III Metamorphic ! a Slate 2 Nonfol lilted 1 b Quartzite c Eclogite d Marble 1 Bulletin No. 31, United States Department of Public Roads. I2 6 MATERIALS The following quotation from this same bulletin describes the characteristics of the three groups: Igneous Rocks. "All rocks of the igneous class are presumed to have solidified from a molten state, either upon reaching the earth's surface or at varying depths beneath it. The physical conditions, such as heat and pres- sure under which the molten rock magma consolidated, as well as its chemical composition and the presence of included vapors are the chief features influencing the structure. Thus, we find the deep- seated plutonic rocks coarsely crystalline with mineral constituents well defined, as in case of granite rocks, indicating a single, prolonged period of development, whereas the members of the extrusive or volcanic types, solidifying more rapidly at the surface, are either fine-grained or frequently glassy and vesicular, or show a prophyntic structure This structure is produced by the development of large crystals in a more or less dense and fine-grained ground mass, and is caused generally by a recurrence of mineral growth during the effusive period of magmatic consolidation. .1-1 "In the arrangement of the rock families from a mineralogical standpoint it will be noted that the plutonic rock types, granite, syenite, and diorite, are represented by their equivalent extrusive varieties rhyolite and andesite, and that diabase has been included somewhat arbitrarily, with basalt, as a volcanic representative of gabbro. These latter rocks are of special interest, owing to their wide distribution and general use in road construction. They occur in the forms of dykes, intruded sheets, or volcanic flows and vary in structure from glassy-porphyritic (typical basalt) to wholly crys- talline and even granular (diabase). Their desirable qualities for road-building are caused to a large extent by a peculiar interlocking of the mineral components (ophitic structure), yielding a very tough and resistant material well qualified to sustain the wear of traffic. "Igneous rocks vary in color from the light gray, pink, and brown of the acid granites, syenites, and their volcanic equivalents (rhyolite, andesite, etc.) to the dark steel-gray or black of the basic gabbro, peridotite, diabase, and basalt. The darker varieties are commonly called trap. This term is in very general use and is derived from trappa, Swedish for stair, because rocks of this kind on cooling fre- quently break into large tabular mas es, as may be seen in the ex- posures of diabase on the west shore ol the Hudson River from Jersey City to Haverstraw. Sedimentary Rocks. "The sedimentary rocks as a class represent the consolidated products of former rock disintegration, as in the case of sandstone, conglomerate, shale, etc., or they have been formed from an. accumu- lation of organic remains chiefly of a calcareous nature, as is true of limestone and dolomite. These tegmental or clastic materials have been transported by water and 'deposited m^^^nnS °* the sea or lake bottoms, producing a very characteristic bedded or stratified structure in many of the resulting rocks. METAMORPHIC ROCKS 127 "In the case of certain oolitic and travertine limestones, hydrated iron oxides, siliceous deposits, such as geyserite, opal, flint, chert, etc., the materials have been formed chiefly by chemical precipitation and show generally a concentric or colloidal structure. 1 Oolitic and pisolitic limestones consist of rounded pealike grains of calcic car- bonate held together by a calcareous cement. Travertine is the so-called 'onyx marble' of Mexico and Arizona. It is a compact rock, concentric in structure and formed by the precipitation of car- bonate of lime from the waters of springs and streams. "Loose or unconsolidated rock debris of a prevailing siliceous nature comprise the sands, gravels, finer silts, and clays (laterite, adobe, loess, etc.). Shell sands and marls, on the other hand, are mainly calcareous, and are formed by an accumulation of the marine shells and of lime-secreting animals. Closely associated with the latter deposits in point of origin are the beds of diatomaceous or infusorial earth composed almost entirely of the siliceous casts of diatoms, a low order of seaweed or algae. "This unconsolidated material may pass by imperceptible grada- tions into representative rock types through simple processes of in- duration. Thus clay becomes shale, and that in turn slate, without necessarily changing the chemical or mineralogical composition of the original substance. "Such terms as flagstone, freestone, brownstone, bluestone, gray- stone, etc., are generally given to sandstones of various colors and composition, while puddings tone, conglomerate, breccia, etc., apply to consolidated gravels and coarse feldspathic sands. "The calcareous rocks are of many colors, according to the amount and character of the impurities present. Metamorphic Rocks. "Rocks of this class are such as have been produced by prolonged action of physical and chemical forces (heat, pressure, moisture, etc.) on both sedimentary and igneous rocks alike. The foliated types (gneiss, schist, etc.) represent an advanced stage of metamorphism on a large scale (regional metamorphism), and the peculiar schistose or foliated structure is due to the more or less parallel arrangement of their mineral components. The non-foliated types (quartzite, marble, slate, etc.) have resulted from the alteration of sedimentary rocks without materially affecting the structure and chemical com- position of the original material. "Rocks formed by contact metamorphism and hydration, such as hornfels, pyroxene marble, serpentine, serpen tineous limestone, etc., are of great interest from a petrographical standpoint, but are rarely of importance as road materials. "The color of metamorphic rocks varies between gray and white of the purer marbles and quartzites to dark gray and green of the gneisses, schists, and amphibolites. The green varieties are com- monly known as greenstones, or greenstone schists." 1 G. P. Merrill's " Rocks, Rock Weathering, and Soils," 1897, pp. 104-114. 128 MATERIALS Interpretation of Tests. It has been found impractical to specify definite qualities of stone for use in macadam highways. Economy and practical engineering demand that all available sources be considered. Tests are made to determine the relative qualities of stone from these different sources and the results used as a guide for selection. In the work of the New York State Highway Commission all tests are tabulated geographically, using a county as a unit. Table Nos. 22 and 23 are compiled from the records of this department. It will be noted that comparisons are made in different classifications only, as it is considered that conclusions should not be drawn from a comparison of tests procured from materials having different origins and composed of different minerals. For the purpose of ready comparison, there has been introduced a figure known as the "weighted value." (See last column Tables 22 and 23.) This is computed by giving relative weights of three to the French coefficient, two to the hardness, one to the toughness values and adding the three together. These relative weights were determined from a consideration of the amount of material used in the different tests and the personal equation in running them. By consulting these tables the available rocks of different classi- fications in various sections throughout New York State can be determined readily, and as new tests are completed they are compared with good average material from that section. Conclusions. Trap (diabase), granite, gneiss, quartzite, sandstone and limestone are the most common rocks and when found in a good state of preser- vation make good surfacing materials. As generally found, trap is uniform in hardness and toughness, making an excellent material for use in top course. Granite and gneiss, where they occur with hornblende replacing a large percentage of the quartz, make an excellent surfacing stone. Quartzites when found in good state of preservation are hard and tough. They should not be confused with crystalline quartz which is hard but brittle. Sandstones are extremely variable and only the better varieties should be used. Limestones range from the fine grained dense products which are hard and tough to the coarse grained soft products which are not suitable for surfacing. Screenings. Screenings act as a filler and binder for waterbound macadam and as a partial filler for bituminous macadam. For use in waterbound construction the main mineral constituent is the most essential feature to be considered as this must be a material that will form a binder and "puddle" readily when subjected to the action of a road roller and water. VITRIFIED BRICK 129 Limestone screenings have proved the most efficient as a binder in waterbound construction, although trap and some other igneous rocks can be bound with their own dust by repeated puddling. Screenings consisting mainly of quartz have never been used success- fully in waterbound construction except by the addition of some lime- stone screenings. The use of a percentage of clay or loam as a binder is not advisable except where the cost of limestone screenings would be prohibitive. Laboratory methods for testing the cementing power of rock powders are available but the results obtained are erratic and unde- pendable. In plain waterbound roads it is often necessary to mix some lime- stone screenings, fine sandy loam, or even a small percentage of clay loam with trap, granite, sandstone, quartzite, or gneiss screenings to get a good bond and prevent raveling in dry weather. 3. Bottom Course Macadam Stone As the bottom stone simply spreads the wheel loads transmitted through the top course and is not directly subjected to the traffic action, almost any stone that breaks into cubical irregular shapes that will not air or water slake and that is hard enough to stand the action of the roller during construction will be satisfactory. Any of the materials listed above in Table No. 24 except shale and slate can be used, provided that they are not rotten from long ex- posure in the air. The different available varieties are usually tested in the same manner as for top stone in order to pick the best. 4. Fillers Fillers are used in the bottom course to fill the voids between the crushed stone and to prevent rocking or sidewise movement of the larger pieces. They should be easy to manipulate in placing, should not soften when wet, or draw water up from the subgrade by capillary action. The materials most used are Coarse sandy loam Coarse sand Gravel with large excess of fine material Stone screenings The fitness of the material can be determined by inspection and by wetting a handful; if it gets sticky or works into a soft mud it should not be used. 5. Vitrified Brick Bricks must withstand the same destructive agencies as described for top stone. They must be uniform in size, tough, hard, dense, evenly burned, and, on account of their peculiar shape, must have a high resistance against rupture. These properties are tested by the standard methods adopted by the American Brick Manufacturers' Association, as described in the New York State specifications on page 390. 130 MATERIALS It should be understood that bricks suitable for paving are manu- factured in a different way and of different materials than ordinary building bricks. "The materials for molding any paving brick must be of a peculiar character which will not melt and flow when exposed to an intense heat for a number of days but will gradually fuse and form vitreous combinations throughout while still retaining its form. "The resulting brick must be a uniform block of dense texture in which the original stratification and granulation of the clay has been wholly lost by fusion which has stopped just short of melting the clay and forming glass. "The clay while fusing must shrink equally throughout, thus causing the brick to be without laminations or of any exterior vitrified ' crust differing from the interior." L The great majority of paving brick are made in Ohio, Illinois, Indiana, Pennsylvania, West Virginia, and New York. They are classed as shale or fire-clay brick. 6. Bituminous Binders The subject of bitumens is an intricate one and the reader is re- ferred to the works of Clifford Richardson, Prevost Hubbard, and others, for detailed information, as a book of this character can give only an outline. There are a number of dust preventives and road binders on the market which depend for their effectiveness on a bituminous binding base. The term bitumen is applied to a great many substances. Hubbard arbitrarily defines bitumens as "consisting of a mixture of native or pyrogenetic hydrocarbons and their derivatives, which may be gaseous, liquid, a viscous liquid, or solid, but if solid melting more or less readily upon the application of heat, and soluble in chloroform, carbon bisulphide, and similar solvents." 2 The bitumens may be classified as native and artificial. The native bituminous materials, that are used in road work, are the asphaltic and semi-asphaltic oils (dust layers), Malthas (the binding base of Rock Asphalts), Trinidad, Bermudez California, and Cuba asphalts, Gilsonite, and Grahamite (which, however, are too brittle in their natural state and require fluxing with a suitable residual oil before they can be used as binders). The natural asphalts are refined to remove water and any objectionable amount of impurities by heating until the gases are driven off, skimming the vegetable matter which rises to the surface, and removing the mineral constit- uents which fall to the bottom. The artificial bituminous materials are derived by the destructive distillation of coal, or by fractional distillation of crude coal tars, or the native petroleum oils. They comprise the crude coal and water gas tars, the refined tars, the residual oils and semi-solid binders derived from the petroleum oils. They vary greatly in consistency and binding power. 1 Judson's " Roads and Pavements," page 87. 2 "Dust Preventives and Road Binders." John Wiley & Sons. BITUMINOUS BINDERS 131 The following material is briefed from Bulletin No. 34, United States Office of Public Roads: The light oils and tars have a relative small percentage of bituminous base and are effective only so long as it retains its binding power; the more permanent binders contain a larger percentage of bitumen; these are the heavy oils and semi-solids. Artificial Bitumens. Crude Tars. — Coke ovens and gas plants produce most of the coal tars in use. These tars contain various complex combinations of carbon, hydrogen, and oxygen and small amounts of nitrogen and sulphur. They vary in composition according to the material from which they are made and the temperature at which they are distilled. The percentage of free carbon ranges from 5 per cent to 35 per cent, and the bitumen from 60 per cent to 95 per cent, depending on the temperature of manufacture. Tars produced at high temperatures contain free carbon in excess which weakens their binding power; they, also, contain a large amount of anthracine and naphthalene, two useless materials from the standpoint of road w r ork. Tars produced at low temperatures are to be preferred. Coke tar is low temperature tar; gas tar is high temperature tar. Refined Tars. — Much of the road tar is refined tar — that is, it has been subjected to fractional distillation to remove the valuable volatile compounds. The residuum from this process is a thick viscous material known as coal-tar pitch, and if the crude tar from which it is obtained was produced at a low temperature it is nearly pure bitumen; the dead oils obtained from the distillation are of little value and are often run back into the pitch, which makes it liquid when cold. The following table gives the approximate com- position of water-gas tar, crude coal-tar, and refined tar. Table 25. Specific Gravity and Composition oe Tap Products Table from Bulletin No. 34 United States Office of Public Roads Kind of Tar Specific Gravity Ammo- niacal Water Total Light Oils to 170 C. Total Dead Oils i7o°27o°C. Residue . (by Difference) Water-gas tar . . . Crude coal tar . . Refined coal tar . 1. 04 1 1. 2IO 1. 177 % 2.0 O.O % 021.6 dij.2 612.8 % 652.O £26. £47-6 % C24.O /S4-8 /39-6 a Distillate mostly liquid. b Distillate all liquid. c Pitch very brittle. d Distillate mostly solid. e Distillate one-half solid. / Pitch hard and brittle. g Distillate one-third solid. Table 25 a gives a more up-to-date analysis of the coal tars on the market. The tests and detailed requirements for light, medium, and heavy bitumens are given in specifications, page 377. If the tar is used as a temporary dust-layer only, it should be a low- temperature, dehydrated tar, liquid when cold. If used as a 13 MATERIALS Table 25 a. Circular No. 97, U. S. Office of Public Roads Analysis of crude coke-oven tars produced in the United States and Canada. Serial No. 5126 5123 5124 5i37 5121 5125 5128 5200 5189 5160 5074 5081 5005 5083 5159 5107 5086 5078 5087 5109 5122 5188 5404 5I08 5127 5089 General Information Company and location Solvay Process Co., Syracuse, N.Y Semet-Solvay Co., Pennsylvania Steel Co., Steelton, Pa Semet-Solvay Co. National Tube Co., Ben wood, W.Va Semet-Solvay Co., Milwaukee Coke & Gas Co., Milwaukee, Wis Semet-Solvay Co. Pennsylvania Steel Co., Lebanon, Pa By-Products Coke Corporation, South Chicago, 111 Semet-Solvay Co., Detroit, Mich Semet-Solvay Co., Empire Coke Co., Geneva, N.Y. . Semet-Solvay Co., Dunbar Furnace Co., Dunbar, Pa Semet-Solvay Co., Central Iron & Coal Co., Tuscaloosa, Ala Philadelphia Suburban Gas & Electric Co., Chester, Pa Type of Oven Semet-Solvay Semet-Solvay Co., Ensley, Ala The X. E. Gas & Coke Co.. Everett, Mass Lackawanna Steel Co., Lackawanna Iron & Steel Co., Lebanon, Pa Dominion Tar & Chemical Co., Sydney. Xova Scotia Hamilton Otto Coke Co., Hamilton, Ohio. Carnegie Steel Co., South Sharon, Pa.. . Maryland Steel Co., Sparrows Point, Md. Citizens' Gas Co., Indianapolis, Ind Pittsburg Gas & Coke Co., The United Coke & Gas Co., Glassport, Pa Zenith Furnace Co., Duluth, Minn. Otto Hoffman United Otto Maximum temperature of firing retorts 1650-1450 C. 1050-1450 C. 1050-1450 C. 1050-1450° C. 1050-1450 C. io5o-i45o v 1050-1450' Illinois Steel Co., Joliet, 111.. Illinois Steel Co., Indiana Gary, Ind Steel Co. Camden Coke Co., Camden, N.J. Cambria Steel Co., Johnstown, Pa. Lackawanna Steel Co., Buffalo, N.Y. Koppers Otto Hoff- man United Otto Otto Hoff- man United Otto United Otto Roth berg 1050- 1450° C. 1050-1450° C. I2 5 0°C. 1050° C. i25o°C. ^icx^C. ( 1000° C. hi8oo°F.) ( 2 ) (iiii°C. ( (2000° F.) ( 1666° C. H30oo°F.) J 1333° C. ( (2400 F.) ( 1222° C. \ (2200 F.) { ( 2 ) I I222-I277°C. \ 2 2OO-230O°F. I 1444° C. i( 2 6oo°F.) noo°C. I 1000 C. t(i8oo°F.) I 1222° C. ( (2200° F.) (IIII°C. ( (2000° F.) (iiii°C. ( (2000° F.) ' 1000° C. (1800'-' F.) ooo° C. 800° F.) CIRCULAR ON PUBLIC ROADS Table 25 a — Continued *33 Answers to Questions Examination Per Maximum temperature Specific gravity Per cent of free carbon in tar Specific gravity Per cent Per cent cent soluble in CS2, includ- ing to which coal of crude tar of tar, of free of is brought 25 C. carbon ash H2O 050-1150 C. 1. 12-1. 21 3-12 1. 195 7.76 0.12 92.12 g5o-ii50° C. 1. 12-1. 21 3-12 1.206 S-77 •07 91.16 050-1150 C. 1. 12-1. 21 3-12 1. 176 7-14 .04 92.82 050-1150 C. 1. 12-1. 21 3-12 1. 168 6.10 •05 93-85 050-1150 C. 1. 12-1. 21 3-12 1. 1/3 4-71 .06 95-23 950-1150° C. 1. 12-1. 21 3-12 1. 191 7-49 •03 92.48 950-1150° C. 1. 12-1. 21 3-12 1. 169 6.56 .11 93-33 950-1150 C. 1. 12-1. 21 3-12 I-I59 6.07 .08 93-85 950-1150 C. 1. 12-1. 21 3-12 1. 181 8.85 .02 9I-I3 ii5o°C. 1. 17 5.72 I-I59 5-05 .02 94-93 1000 C. ( 1. 16 I (20°C) — 1. 141 3-96 •05 95-99 1150 c. { I- 17 I (i5° C.) 8 1. 175 6.90 .06 93-04 1 I200°C. 1. 17 8-10 1. 160 43-94 .00 86.06 1000 C. ) (i8oo°F.)J 1. 10 16-24 1. 214 14-05 • 13 85.82 m 1. 170 10-15 1 -H3 10.81 •05 89.14 ^ iin° c. (2000 F.) 1. 14 6 16.0 1. 160 8.37 .06 91-57 1444° C. (2600° F.) , 2 7. 09-10.64 1. 191 7.89 •03 92.08 1222° C. , (2200 F.) 3 1. 19 3 8-IO 1. 179 8.49 •03 91.48 1222° C. (2200 F.) J I. I4-I. 15 f (50° F.) 4-5 1 1. 133 5-21 .07 94.72 ( 2 ) < 1.207 I io° C. r 16.59 1. 176 10.53 .04 89.43 ( 2 ) ( 2 ) (*) 1195 12.18 •05 87.77 1388 C (2500 F.) | 1. 16-1. 20 12-15 1. 171 3.89 .06 96.05 ( 880-950 X c. 4 1. 174 1. 169 4-35 1. 169 2.73 .04 97.23 833° c. 1 (i5oo°F.) 1055° c. 1 1. 20-1. 30 5 (1. 221) 7-9 5 (7-3) J 1. 182 11.30 .06 88.64 (i900°F.) J l ini°C (2000 F.) 1 l iin°C. } 1. 12 1 15 1. 211 12.40 .16 87-44 (2000 F.) iooo C. 1 (1800 F.) icoo° c. j> 1. 16 16-24 1. 210 16.80 .00 83.20 (1800 F.) i 1 134 MATERIALS Table 25 a — Continued Serial No. 5126 5123 5124 5137 5121 512S 5128 5200 5189 5160 5074 5081 5095 5083 5159 5107 5086 5078 5087 5109 5122 5188 5404 5I08 5127 5089 Company and Location Examination, Public Roads Solvay Process Co., Syracuse, N.Y. .... Semet-Solvay Co., Pennsylvania Steel Co., Steelton, Pa . . • • • • • Semet-Solvay Co., National Tube Co., Benwood, W.Va • Semet-Solvay Co., Milwaukee Coke & Gas Co., Milwaukee, Wis Semet-Solvay Co., Pennsylvania Steel Co., Lebanon, Pa • • • • By- Products Coke Corporation, South Chicago, 111 ••••:• ■■:■:•' Semet-Solvay Co., Detroit, Mich. Semet-Solvay Co., Empire Coke Co., Geneva, N.Y • • • ■ Semet-Solvay Co., Dunbar Furnace Co., Dunbar, Pa • • - ■ ;• • • •• Semet-Solvay Co., Central Iron & Coal Co., Tuscaloosa, Ala ... Philadelphia Suburban Gas & Electric Co., Chester, Pa Semet-Solvay Co., Ensley , Ala . . . The New England Gas & Coke Co., Everett, Mass • Lackawanna Steel Co., Lackawanna Iron & Steel Co., Lebanon, Pa Dominion Tar & Chemical Co., Sydney, Nova Scotia • • ••••••••• Hamilton Otto Coke Co., Hamilton, O.. . Carnegie Steel Co., South Sharon, Pa . Maryland Steel Co., Sparrows Point, Md. Citizens' Gas Co., Indianapolis, Ind.. . . . Pittsburg Gas & Coke Co., The United Coke & Gas Co., Glassport, Pa Zenith Furnace Co., Duluth, Minn Illinois Steel Co., Joliet, 111. ............ Illinois Steel Co., Indiana Steel Co., Gary,. Ind A'A'^'i Camden Coke Co., Camden, N.J. Cambria Steel Co,, Johnstown, Pa . . . Lackawanna Steel Co., Buffalo, N.Y Distillation results Water ^ o 1. a 1.0 1.1 1.8 .6 (*) 6.9 4.0 2.0 3-2 2.3 3-3 2.2 5-4 3-2 3-4 1.0 ^1 0.8 .8 1.0 1. 5 •5 ( 7 ) 5-9 3-4 1.7 2.8 2.0 2.8 2.0 4.4 2.8 1.0 1.6 13 1.2 I.I I.I I.O 3-6 3.0 1.9 1.6 3-5 3.0 2.2 1.9 IO.I 8.3 2.7 2.2 References to Table 25 a Light oils up to no° C. ^ p > *o. 3 • 4 1.9 1.4 1.6 • 4 9 2.8 2.6 i-7 2.4 2.3 8 1.4 2-9 9 1.4 1.9 3-i 9 1.6 1.3 I.I I.I 1.7 9 1.7 9 1.3 1.8 9 3-i 10 c 1 Approximately. 2 No information. . , 3 Varies with coal. Coal with 28 per cent of volatile matter used. 4 With H2O. 5 At present. 6 Variable. i Trace. 0.3 .3 1.5 1.2 1-3 •3 2-3 2.1 1.4 1.9 1.3 1.0 2.3 1.4 1-5 2-5 1.2 •9 9 1.3 1.2 1.0 1.4 2-3 •3 s Trace of solids. 9 Distillate, solid. 10 Distillate, one-fourth solid. 11 Distillate, nine-tenths solid. 12 Distillate, three-fourths solid. 13 Distillate, eight-ninths solid. M Distillate, one-half solid. CIRCULAR ON PUBLIC ROADS 135 Table 25 a — Continued Examination, Office of Public Roads Distillation results Middle oils, iio°-i7o C. Heavy i7o°-2 oils, 70 C. Heavy oils, 2 7 o -3i5° C. Pitch 1 Serial No. i* &3 • -° ^ fcjD ■4-1 xi # y> vO°5 -° Xi ,&> 0.8 0.7 »I3-I II-5 1 9 8.2 7-3 25 76.6 79-1 5126 9 2.0 1.7 9 14.0 12.3 2 " 7.9 6-9 26 74.7 77-6 5123 • 7 .6 14.9 13-2 21 11. 9 10.6 27 69.5 73-1 5124 .8 .6 13 2 1. 1 18.9 20 5-5 4.9 25 69.4 72.5 5137 .8 .6 14 17.5 15-5 19 9-4 8.4 25 70.I 73.7 5121 12 1.1 * -4 •9 •3 15 23.6 11 I4.6 20.7 13.0 9 9.8 8 6.9 8.9 5, 27 65.I 26 68.4 68.9 72.O 5125 5128 .6 •5 io I7 .6 15-5 22 11.4 10.4 27 63.8 67.7 5200 .2 .2 i° 20.0 17.8 21 6.5 5-7 25 69.6 73-1 5189 •3 •3 18.6 16.3 i0 7-5 6.8 27 68.O 7i. 5 5 160 1.2 .2 .8 .2 22.8 17 16.5 19-5 14.1 15 13.6 14 9-3 12.5 8.2 57-8 27 69.3 62.0 73-2 5074 5081 .6 •5 23.5 20.4 17 15.6 14.4 27 55-2 59-7 5095 9 .1 .1 11 13-0 10.9 21 9.4 8.1 25 7O.7 74-6 5083 .6 • 7 9 .6 .6 •4 27.2 27.9 16 12. 1 24.2 24.4 10.2 19 7-3 19 3-8 19 11.0 6.7 3-5 9-7 27 59-8 27 6l. I 25 73-7 63-5 64.9 77-5 5159 5io7 5086 - .6 •4 12 17.2 151 21 9.6 8.5 2869.7 73-2 5078 1.4 5 •4 9 .2 9 -4 .6 p -3 9 2.2 i-3 •4 •3 .2 •3 •5 .2 1.7 23-9 18 26.9 11 18.1 9 20.0 9 20.6 14 20.5 9 7-1 9 11. 7 21.4 23.6 15-9 18.0 18.5 18.2 6.1 9.9 10 n.6 I* 6.9 19 12.5 11 13.4 9 7-1 23 8.5 12 7.4 24 1 1.8 10.4 6-3 11. 1 12.0 6-5 7-5 6.9 10.2 27 60.8 27 63.5 27 63.7 26 62.8 26 67.1 2566.4 26 72.O 27 71. 1 64.7 67.6 67.8 66.3 70.2 70.1 74-8 75-o 5087 5109 5122 5188 5404 5108 5127 5089 References to Table 25 a 15 Distillate, two-thirds solid. 16 Distillate, four-fifths solid. _ 1 7 Distillate, seven-eighths solid. 18 Distillate, one-ninth solid. 1 9 Distillate, one-third solid. 20 Distillate, one-sixth solid. »i Distillate, one-fifth solid. 22 Distillate, two-fifths solid. 23 Distillate, one-seventh solid. 24 Distillate, three-fifths solid. 25 Pitch, soft and sticky. 26 Pitch, very soft and sticky. 27 Pitch, hard and brittle. 28 Pitch, plastic. 136 MATERIALS more permanent binder and applied hot, it should have a larger percentage of pitch, should contain no water, and be free from an excessive amount of free carbon. If used as a mastic in bituminous macadam, it should contain a high percentage of pitch and be free from the defects mentioned. Natural Bitumens and Artificial Residual Oils and Semi-Solids. Mineral oils can be classed as paraffin petroleums, mixed paraffin and asphaltic petroleums, and asphaltic petroleums. The relative value of oils as a source of supply for road materials depends on their percentage of asphaltic residue. The eastern oils found in New York, Pennsylvania, West Virginia, etc., are paraffin petro- leums; the western oils vary from light to heavy asphaltk petro- leums, and the southern oils have a mixed paraffin and asphaltic base. The crude petroleum is refined by fractional distillation to obtain its valuable products, such as kerosene, etc. The character of the residue depends, as for the tars, on the crude material and the method of manufacture; the operation known as "cracking," which is used to increase the yield of the inflammable oils, produces an excess of free carbon. Tlie paraffin petroleum residuums are soft and greasy and are not suitable for road work; they contain a large amount of the paraffin hydrocarbons and paraffin scale (crude paraffin) . The California petroleum residuums resemble asphalt, and if care- fully distilled without cracking should contain little or no free carbon. They are suited to road work. The Texas, or semi-asphaltic petroleums contain some paraffin hydrocarbons and about 1 per cent of paraffin scale. Residuums from these oils, if containing a relatively small amount of paraffin, can be successfully used. The tests and required properties of residuum bituminous binders used on the New York State roads in 19 14 are given in specifications, page 377. The following tables give a general idea of the relative character- istics of the crude petroleums and petroleum residuums. Table No. 26. Results of Tests oe Crude Petroleum Tables from Bulletin No. 34 United States Office of Public Roads 4j +j 9i oS a «3 +J .a -So • 5° &u3 £o| « 3 Kinds of Oil 'G > ^° 3 •rj offi ^ *55 ^ > > > Pennsylvania, paraffin. .. . Texas, semi-asphaltic .... 0.801 .904 (a) 43 % 47.3 % 58.0 % 68.0 % 632.0 California, asphaltic ...... •939 26 20.0 27.0 49.0 J42.7 C51.0 «57-3 a Ordinary temperature b Soft c Quick flow d Volatility at 200' e Soft maltha; sticky 7 hours. ADHESIVENESS RESULTS OF PETROLEUM RESIDUUM 137 Kinds of Oil "G > a '0 Volatility at 200° C. 7 Hours u 3 a C/2 as Pennsylvania, paraffin. . . . Texas, semi-asphaltic .... California, asphaltic 0.920 •974 1.006 186 214 191 % 14.2 6.2 17-3 y< i Cor. Post in Fen c tf '<&t6L Barn !2"MapleTree *£ 4 & ^o 15 'Oak Fig. 43. — Alignment Notes The deflection angles at the tangent intersections are usually read to the nearest minute, taking a double angle to avoid mis- takes; the magnetic bearing of each course is recorded. For all deflection angles over 4 it is good practice to figure and run in on the ground the desired curve. Curves with central angles of less than 4 can be run in with the eye during construction. The center line is marked at intervals of either fifty or one hundred feet (see cross-section, page 121) in any convenient manner; the alignment of these points should be correct to within 0.2 and the distance along the line to within 0.1 per 100 feet of the length; any attempt to get more accurate sta- tioning is a waste of time. The chaining may be done on the surface of the ground up to a grade of 5 per cent with no objec- tionable error; beyond that slope, however, the tape should be leveled and plumbed. Steel tapes should be used for chaining the center line and referencing the hubs. A convenient method of marking the actual center-line sta- tions is to use a nail and piece of flannel; red flannel for the ioo' stations and white flannel for the intermediate 50' stations, if needed. Where the soil is sandy, or muddy, and these nails would be kicked out or covered, a line of stakes can be set out- side of the traveled way on a specific offset from the center line. However, if an offset line is used the chaining of all curves should be done on the center line to insure a correct center-line distance and the stakes placed radially on the desired offset. Railroad 146 THE SURVEY spikes make good permanent transit points and are easily placed. At the same time that the line is run it is just as well to paint the 100' station numbers on any convenient place where they can be readily seen, as stations marked in this manner make it much easier to sketch in the topography than if marked in chalk on stakes. Also, if the stations are permanently marked it is easier for the construction engineer to pick up the transit points at some future time. A party of five men will run from two to four miles of center line a day, the speed depending upon the number of curves and length of tangents, if the hubs have been previously placed and referenced. If the hubs are placed at the same time the line is run, the work is greatly delayed. Two men can place and reference the transit points at the tangent intersections at the rate of from four to ten miles per day. Sta. D.M.# I B.M.*£ B.5. + F. 5. 7TZT 4.ZCI B.IH -6TKT 3-*H ice: H.I i n s- q -y 4\?>.7U Elev. ~v Tyszn*- wt . sP "Hg.KZ Spike in 15" Elm, Righ t o f S tq .frrfrft- Top qf S t one Hi t ihing Pos t , le ft . S t a. 15+ 60: y^ Fig. 44. — Bench Level Notes Levels and Cross-Sections. Bench levels are run in the usual manner; the levels will be sufficiently accurate if the rod is read to the nearest o.oi'; for such work any good level and a self-reading rod graduated to hundredths are satisfactory. Benches are established at intervals of 1,000-1,500 feet; they must be substantial, well marked, and so situated as not to be disturbed during construction. A small railroad spike in the root of a tree, a large boulder, or the water table of a building make good benches. The bench levels may be referred to some local datum in general use or to the U. S. levels, or the datum can be assumed. LEVELS AND CROSS-SECTIONS 147 In running bench levels it is better to use each bench as a turn- ing point, as side-shot benches may be wrong even if the line of levels is correct. Cross-sections are taken at either 100' or 50' intervals, at all culverts, possible new culvert sites, and any intermediate breaks not shown by the normal interval. Enough sections are taken to show the constantly changing shape of the road. The distance of the shots from the center line of the road is read to the nearest 1.0' where the ground has no abrupt change of slope and to the nearest 0.5' where there is a well- defined abrupt change. The elevations are read to the nearest o.i'. The sections should extend from fence line to fence line, or in villages from sidewalk to sidewalk, and the position of the pole lines, tree lines, curbs, etc., noted. Engineers differ as to whether the sections should be taken at a normal interval of 50' or ioo'. Table 27 gives the difference in the computed quantity of earthwork using 50' and ioo' sections with intermediate sec- tions at well-defined breaks in the grade. Table 27 Name of Road Scottsville Mumford . . Scottsville Mumford . . Leroy Caledonia . . . . K Leroy Caledonia Clarence Center Clarence Center Lockport Tonawanda . . "East Henrietta Rochester Length Figured mile Charac- Excava- ter of , tion 50' Road Section flat hilly rolling flat rolling flat flat rolling Cu. Ft. 61.444 in. 109 57,840 77,841 73,727 38,037 59,096 37,275 Excava- tion ioo' ; Section J Appro - ! ximate Cu. Ft. 6i,995 111,700 60,560 78,659 73,048 39-415. 59,47o 36,075 Differ- Cu. Ft. 55o 600 2700 800 700 1400 400 1200 Per cent of Differ- ence + t 9 o% + % + 1 % -1 % + 3iu% + i 7 o% ~2,\ % 148 THE SURVEY The following tabulation shows the variation for shorter sections of the starred roads. Name Station of and to road Station Quantities by 50' Sec- tions Quantities by ioo' Sections Approx- imate Difference Per cent of Difference Leroy Caledonia, 80- 90 . . . Cu. Ft. 19,151 Cu. Ft. 19,525 Cu. Ft. 400 + 2 % 90-100 . . . " 100-110 . . . 21,915 2i,555 23,415 20,689 1500 900 + 7 % -4 % " II0-I20 . . . Total and averages . East Henrietta 15,220 77,84i 15,030 78,659 200 800 ~~ I T0% + 1 % Rochester, 0-19 . . . 14,625 14,300 300 -2 % 32-49 • • • n,95o n,575 350 -3 % 49-66 . . . Total and averages . 10,700 37,275 10,200 36,075 500 1200 -s % -si % The question of quantities is not the only factor in determining the interval. Where it is important to fit the local conditions, as in a village, or to utilize an old hard foundation, the designer is helped by 50' sections. Sra. B.S. F.S. H.I. Elev. Left ^ ^ ^ N U> \£> £ JO If) v& VC> £1 CM VSJ *JS U> VSl in CM CM CM JV* cvj CD CD CD CT> CD / 5.4 5.5 53 5.3 6.5 B.M.*3 10 tOO 5.41 931.73 ezezz /OtSO 40 14 12 5 C CM K CM ^ K N £> ir> JO Jo ui irj M CM £j £f CVj CM CD CD Ci ^ 0^ 55 6.0 6.5 6.3 60 6. 5 3/1/9 24 M> N N ^ K ^ ^ $"' ^ *5 CM cm CM cm CM CD CD S\ CD eft 6.3 7.0 7.0 7.6 8.0 T.Rt65 Rock on lit 00 1.32 2.10 330.35 929.63 26 20 14 II 8 (y CQ *0 CJ> (VJ > Lf) f\i CM < Cvj t\i C\j CM C\J CVJ CVj C\J t\j O O) cd CD CSv CD 8.2 6.7 9.1 8.8 8.6 8. ► 8 II 12 20 23 ^ K - v> cvj ^- CM ^ fVj CM CM CM $) CD CD CD 5 3.0 9.3 89 6.0 v J 30 20 13 9 5 C IO 14 13 30 Fig. 45. — Cross-Section Notes DRAINAGE 149 In taking cross-sections the work becomes mechanical, and unless the engineer in charge is unusually alert to all the inter- mediate changes better results will be obtained by the use of the shorter interval. For these reasons the author believes that a 50' interval is advisable except on long uniform stretches of road. A party of three men will run from 4,000 to 7,000 feet of 50' cross sections per day; a party of four men from 5,000 to 9,000 feet, depending on the country. DRAINAGE The drainage notes show the position and size of all the exist- ing culverts; the area of the watersheds draining to them and a Drainage Old Structures V^ Sta.lfT-f-25 Present 12" V.TP Bad Condition £l S+a. 24+00 Present Concrete To w n i n 1 211 Culvert Built by Water Satisfactorily 2'x 2 '* 3Q j C arries £L Sta . 45+5Q - 49 + 00 Fl ood Backwater Covers Present Rngd 1. 5 ' i n Sp ri ng of. Y e ar y n o C urren t . Raise R oad 2.5 ' a nd m ak e Fil l of Boulder Stone or 6 ravel Sta. 55+10 Present 24" V.T. P does not C arry W ater i n F reshets -O- Notes New Structures ~ 5t a J5 +25. Drainage Area 40 Acres Slope 20' to 1000 Use l8 " C \t.P. ?rox. 5ta. 24+0 N o N ew \Culvert Needed. £l 5ta. 55-t 10 Draina ge Area 300 A . -O Rolling Farm Land, Slope a bout 30' per 10 00 Use Xx7> Cnnr.rhte Box. ft >K Fig. 46 recommendation of the size culvert to be built; the location, drainage area, and size of desirable new culverts; the necessity for outlet ditches and their length, if required; the elevation of flood water near streams, and the condition of the abutments and superstructure of long-span bridges. The cross-section levels are supplemented to show these points fully. Where the U. S. geological maps are available the areas of watersheds can be easily determined; where no such maps have been made the drainage areas can be easily mapped with a small 15" plane table ISO THE SURVEY oriented with a magnetic needle; the distances can be paced and the divides determined with a hand level. One inch to 2,000 feet is a convenient scale. The drainage scheme should be carefully worked out by the Chief of Party, as the possibilities of friction with local people are greater on this part of the design than any other. In the chapter on Drainage this fact was mentioned and designers were cautioned not to use new culverts unless necessary. TOPOGRAPHY The topography notes show the features of the adjacent terri- tory that might affect the design. These include the location of buildings, drives, intersecting roads, streams, railroads, poles, trees, sidewalks, crosswalks, and property lines. The names of property owners are recorded. +50 -o- Sta.lZ +50 Sta.ll +50 \SfaJO Fig. 47 A simple method of locating these points is to refer them di- rectly to the previously run center line by right-angle offsets; such notes are easily taken and quickly plotted. In taking the topography the plus stationing along the center line and the offset distances to all points inside of the road fences should be measured by tape to the nearest foot; the distances to and the dimensions of buildings, etc. outside of these limits, can be paced or estimated; the bearings of the property lines can be read near enough with a pocket compass, except for Right of Way surveys which are described on page 154. The instruments needed for work of this kind are a pocket compass reading to 2 , steel picket, and metallic tape. FOUNDATION SOILS 151 Two experienced men will take from two miles to four miles of topography a. day except in villages, where from one half to a mile is average speed. DIRECTION AND AMOUNT OF TRAFFIC is determined by inspection and inquiry of the residents along the road. To illustrate the information required, an extract from the sur- vey report of the Fairport Nine Mile Point Road is given below: FAIRPORT NINE MILE POINT ROAD TRAFFIC REPORT Heavy Hauling. The direction of heavy hauling on this road is approximately as follows: 1. Station No. 195 to station o towards Fairport. 2 " " 195 " " 400 " Webster. 3. " " 580 " "- 400 This divides the road into three sections for the determination of the ruling grades. The ruling grade for section 1 will be determined by the hills at station 10 and station 48 and probably will be limited to 5 per cent. The ruling grade for section 2 will be determined by the knolls at stations 267, 285, and 300. The ruling grade for section 3 will be determined by the hills at stations 445 and 494. The team traffic is medium heavy station 90 to station o; light, station 270 to 90; medium, station 270 to 375; heavy, station 375 t° 386; very heavy, equivalent to city street, station 386 to 408; medium heavy, station 408 to 450, and light, station 450 to 580. Macadam construction will not be suitable stations 386 to 408. The automobile pleasure traffic will be largely through traffic and probably fairly heavy. FOUNDATION SOILS The notes on soils show the character, width, and depth of the existing surfacing material and the kind of underlying mate- rial. This feature of the survey is important, as it governs the thickness of the bottom course, and, to a certain extent, the posi- tion of the grade line where an existing solid foundation can be utilized and the thickness of the improved road reduced to a minimum. Even with a careful soil examination it is impossible to make the design of the foundation definite, as mentioned on page 73, but the quantity of the material that will be needed can be esti- mated very closely. The subsoil can be readily examined by driving a i|" or 1" steel bar to the required depth, which is usually not over 4.0' to 5.0' even in cuts, removing the bar and replacing with a §" gas pipe, which is driven a few inches and withdrawn. The core will give a fair idea of the material to be encountered. 152 THE SURVEY Soil Notes Foundation Recommendations Sta.toSta. Surface Mat SubSurface 30 Sand &Gravel Sand 8c Gravel Total Thickness Metalling 6" » »? ?t 9" 30 3/ Clay & 6 ravel Clay I'd own 31 36 Clay Clay » » v 12" 3d 40 Gravel 8 "deep Wef Clay Underdrain on J?t Stone I5"deep 40 41 »i 4" u Clay Loam Fill at this Point n 6 " v V s ^ J Fig. 48 Where rock is encountered the elevation of the outcrop is shown, and if the rock underlies the road for any distance within two or three feet of the surface this depth is determined by driving bars. Sample notes below: Station Left Center Line Right 62 3^ 20 00 20 63 1*1 1.2' 1.0' 25 00 22 The note 3.5' means that 20' to the left of the proposed center line 20 of the improvement, the rock is 3.5' below the present surface; from these notes the rock can be readily plotted on the cross- sections. Its character can be determined from adjacent out- crops, or from test pits, if required. LOCATION AND CHARACTER OF MATERIALS The selection of materials and the estimate of the construction cost depend on a knowledge of the available materials and their location relative to the road. LOCATION AND CHARACTER OF MATERIALS 153 Unloading Points for Freight. Provided U. S. geological maps are obtainable, the position of sidings may be marked on the sheets. The notes for each siding show its car capacity; whether or not an elevator unloading plant can be erected, and if hand unloading is necessary whether teams can approach from one side or two. They should also show any coal trestles that can be utilized in unloading, and the location and probable cost of any new sidings that will materially reduce the length of the haul. Canal or river unloading points are shown in the same manner. Stone Fs+. /. Oeo.Barber lOOOc.u.yds. Fence _ Boulders 20% Granite 40%Sanasfc 10% Limestone 3Q% Soft Rock 50% of the Granite must be Blasted or Sledged 2. Patrick Don/In 2SOO cuyds. same as above 3 Mike O' Donne. 1 1 5QO cu yd: Large Granite Boulders 75% must be Blasted ■4-. Old Limestone Quarry 2Q 1 face ~ Samples taken^ looks good for Top Stone ™ Fig. 49 Sand, Gravel, and Filler Material. The position of sand and gravel pits and filler material are noted with their cost at the pit; if no local material is available the cost f.o.b. at the nearest siding is given. Stone Supply. Provided imported stone is to be used the work is simplified to determining the rate f.o.b. to the various sidings for the product of the nearest commercial stone-crushing plant that produces a proper grade of stone. In case local stone is available the location of the quarries or outcrops is shown; the amount of stripping, if any, and the cost of quarry rights. If the estimate will depend upon rock owned by a single person an option is obtained to prevent an exorbitant raise in price. In the case of field or fence stone a careful estimate is made of the number of yards of boulder stone available, the owners' 154 THE SURVEY names, what they will charge for it, the position of the fences or piles relative to the road, or side roads, and if the fences are not abutting on a road or lane the length of haul through fields to the nearest road or lane. As fer^es are usually a mixture of different kinds of rock, the engineer estimates the percentage of granite, limestone, sandstone, etc., and the percentage that will have to be blasted or sledged in order to be crushed by an or- dinary portable crusher. The amount of field stone required per cubic yard of macadam is given in estimates, page 274. If there is a large excess of stone a careful estimate need not be made, only enough data being collected to determine the probable position of the crusher set-ups and the average haul to each set- up. If a sufficient supply is doubtful a close estimate is made as outlined above and options obtained from the various owners. Samples of the different rocks are tested. (See materials.) Preliminary surveys of the above description should be made at a speed of from two to four miles per week at a cost of from $35 to $70 per mile, allowing $6 per day for the engineer; $3.50 for the instrument man; $2 per man for three laborers; $1 per day board per man and $4 per day for livery. Right of Way and diversion line surveys are often needed but are usually not made at this time; if the designer believes that additional land must be acquired or that a diversion line is necessary, he indicates the information desired and the surveys are made. RIGHT OF WAY SURVEYS These surveys are used not only to show the amount of land to be acquired but, also, the damage to property from altering the shape of a field, cutting a farm in two, changing the position of a house or barn relative to the road, etc. The acreage to be taken is shown by an ordinary land survey in which the road lines, property lines, corners, etc., are located in relation to the proposed center line of the improvement, and their lengths and bearings carefully determined. It is often difficult to locate the road boundaries, as town records are care- lessly kept and there is a general tendency to encroach on the road. As the amount paid for new Right of Way is rarely settled on an acreage basis, it is customary to take the existing fence lines as the road line unless it is very evident that the fence has been moved. This produces better feeling on the part of the property owner and does not affect the price paid. The lines between adjoining properties are usually well defined. In cases where an orchard is damaged the position and size of the trees are noted; where a field or farm is cut the whole field is shown, with the shape and acreage of the pieces remaining after the land actually appropriated has been taken out. As is usually done in all land surveys, the parcel to be bought is traversed and the survey figured for closure error to insure the description against mistakes. The standard form of map and description of the N. Y. State Department is shown in the following illustration: WAY SURVEYS 155 "5" 2 a >>rt ra -la G ~ H <" , fi^ -!-> ^ -^3 (J *T3 o „, G ~ « G+ 2 ^ SiG en. o k" . o o o a> ^ ^O Oh Q, C-G 0) T3 X O^ ^ O^^O g-G=3 O g ° * ti ~~* en -13 co « O g >:5 3 o - ?n oi Is Mm -.,. ^G^-a^ 1 ^^ en "tj qj o G fe CO ^ r? OT ~ e o m § « - cl G^ ^ \*c\ uO hoi r«S CVI o <3- ivo Z (S f5 !5 i ki ru N £ si '-V * i*> Oh G Ln «*i UJ CO CM ^ $ a? *j| CVJ N CV4N. Q ■5 CD ^ ^ UJ Uj h ^ ? Ci ? < (8 f Si 2: CM CD ^ u o en O - co a sG^jo,G W4)0 ^ a o-o G " m eo^J il" 1 XI O to t3 G "± t3 « O .-m « Jo 2 G^S « ?1 f) +J C O en c rt ^ B,°' oj5 730.o 4,297.5 3,820.0 3 5 438-o 2,865.0 2,455-7 2,292.0 2,148.8 1,910.0 00.15 00.2 00.25 00.3 00.4 00.45 00.5 00.6 00.7 00.75 00.8 00.9 — — O O O O O O O O O O O O 07-5 10.0 12.5 15.0 20.0 22.5 25.0 30.0 35-o 37-5 40.0 45-o 1 68 THE SURVEY Table 29. — Continued, Degree of Radius of Deflection per foot of Arc Deflection per Deflection per 50' Curve Curve 25' of Arc of Arc 1 Feet Minutes j Deg. Minutes ' Deg. Minutes 3° 20' . . 1,719.0 01.0 , 1 50.0 3° 30' • • 1,637-1 01.05 — 52.5 3° 40'.. 1,562.7 01. 1 — O 55-o 4° 00' . . 1,432.5 OI.2 — — 1 00.0 4° 20' . . 1,322.3 01.3 — I 05.0 4° 30' • • 1,273-3 01.35 I o7-5 4° 40;..: 1,227.9 01.4 — — I 10. 5° 00' . . 1,146.0 01.5 — I 15.0 s° 30' . . 1,041.8 01.65 — — I 22.5 6° 00' . . 955-o OI.8 — I 30.0 6° 30' . 881.5 01.95 — I 37-5 7 00' . 818.6 02.1 — — I 45-0 7° 3o' . . 764.0 02.25 — I 52.5 8° 00' . . 716.3 02.4 — — 2 00.0 8° 3°' • • 674.1 02.55 — — 2 07-5 9 00' . . 636.6 02.7 — — 2 15.0 9° 3o' . . 603.2 02.85 — — 2 22.5 io° 00' . . 573-o 03.0 — 2 30.0 io° 30' . . 545-7 03-15 — ■ — 2 37-5 11° 00' . . 520.9 03-3 — — 2 45.o 11° 30'.. 498.3 03-45 — — 2 52.5 12° OO' . 477-5 03.6 — — 3 00.0 12° 30' . 458.4 03-75 — — 3 07.5 13° 00' . . 440.8 . 03-9 — — 3 15.0 13° 3o' . ■ 424.4 04.05 — — 3 22.5 14 oo' . . 409-3 04.2 — — 3 30.0 14° 30' . . 395-2 04.35 — — 3 37-5 iS° 00' . . 382.0 04.5 — — 3 45-o 15° 3o' . . 369.6 04.65 — — 3 52.5 16° 00' . . 358.1 04.8 2 OO.O 4 00.0 16° 30' . . 347-3 04.95 2 03.8 4 07.5 17° 00' . . 337-o 05.1 2 07-5 4 15.0 17° 30' . . 3274 05-25 2 II. 2 4 22.5 18° 00' . . 318.3 05.4 2 15.O 4 30.0 18° 30' . . 309-7 05.55 2 18.7 4 37-5 RADII AND DEFLECTIONS 169 Table 29. — Continued Degree of Radius of Deflection per ft. of Arc Deflection per 25' of Arc Deflection per Curve Curve 50' of Arc Minutes Degree Minutes 19 00' 301.6 05-7 2 22.5 19° 3o' 293.8 05.85 2 26.2 20° Oo' 286.5 06.0 2 30.O 20° 30' 279-5 06.15 2 33-7 21° OO' 272.9 06.30 2 37-5 21° 30' 266.5 06.45 2 41.2 2 2° Oo' 260.5 06.6 2 45-o 22° 30' 2547 06.75 2 48.7 23° 00' 249.1 06.9 2 52.5 23° 3°' 243.8 07.05 2 56.2 24° oo' 238.8 07.2 3 00.0 24° 30' 233-9 07-35 3 03-7 25 00' 229.2 07-5 3 07.5 26 00' 220.4 07.8 3 15.0 2 7 oo' 212.2 08.I 3 22.5 28 oo' 204.6 08.4 3 30.0 29 oo' 197.6 08.7 3 37-5 30 00' 191. 09.O 3 45-o Deflection per 10' of Arc 31° 00' 184.8 09-3 3 52.5 3 2° 00' I79-I 09.6 4 00.0 1° 36' 33° 00' 173-6 09.9 — — 1° 39' 34 00' 168.5 I0.2 — — 1° 42' 35° 00' 163.7 IO.5 — — 1° 45' 36 00' 159.2 IO.8 — t — 1° 48' 37° 00' 154-9 II. I — — 1° 5i' 38 00' 150.8 II.4 — — 1° 54' 39 00' 146.9 11. 7 — — 1° 57' 40 00' 143.2 12.0 — — 2° 00' 42 00' 136.4 12.6 — — 2° 06' 44 oo' 130.2 13.2 — 2° 12' 46 00' 124.6 13.8 — — 2° 18' 48 00' 119.4 14.4 — — 2° 24' 5o° 00' 114. 6 15.0 — • — 2° 3o' 52° 00' no. 2 15.6 — — 2° 36' 54° 00' 106. 1 16.2 — — 2° 42' 56° 00' 102.3 16.8 — — 2° 48' 170 THE SURVEY For all practical purposes the value of 5,730 can be used. In the same manner a two-degree curve is one having such a radius that 100 feet of arc will subtend two degrees of central angle, and its radius is 27T R = ^— X 100 2 _ 18900 27T or J of the radius of a one-degree curve. The radius of a three-degree curve will be J of 5,730. The radius of a four-degree curve will be \ of 5,730. The formula for the radius of any degree of curve is therefore tt (2) R The degree of curvature for any specified radius is therefore D = 5.73Q R. (3) In general the degree of curvature is expressed by the central angle subtended by 100 feet of arc, and the radius for that degree of curve is found by dividing 5,730 feet, the radius of a one- degree curve, by the degree of curve desired expressed in degrees and decimals of a degree. That is, if the radius of a 3 30' curve is wanted, divide 5730 by 3.5, which equals 1637. i'. The radii given in Table No. 29 are computed in this manner. Length of curve. For a 5 curve a central angle of 5 subtends 100' of arc; a central angle of io°, 200' of arc; a central angle of 12 30', 250' of arc. That is, for a specified central angle the length of any specified curve equals that central angle expressed in degrees and decimals of a degree divided by the degree of curve expressed in degrees and decimals multiplied by 100; i.e., the length of a io° 15' curve for a central angle of 20 45' = 20. 7 ^ X 100' = 202.4/and is expressed by the formula 10.2 Table 30. Functions of a One-Degree Curve Figured on a Basis of R = 5730' and Tabulated to Tenths of Feet Use 100' chords up to 8° Curves Use 50' chords up to 16 Curves Use 25' chords up to 3 2 Curves Use 10' chords above 32° Curves 5 c rr O" 1 3 g B It- Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. O 0.0 0.0 0.2 50.0 0.9 100.0 2.0 150.1 O IO 0.0 8.3 0.3 S8.3 1.0 108.4 2.2 158.4 IO 20 0.0 16.7 0.4 66.7 1.2 116. 7 2.4 166.8 20 30 O.I 25.0 0.5 75.o 1.4 125.0 2.7 I75-I 30 40 O.I 33-3 0.6 83.3 1.6 133-4 2.9 183.4 40 50 0.2 41.7 0.7 91.7 1.8 I4I-7 3-2 191. 7 So 6O 0.2 50.0 0.9 100.0 2.0 150.1 3-5 200.1 60 .. .__ _,... 1 FUNCTIONS OF THE ONE-DEGREE CURVE 171 Use ioo' chords up to 8° Curves Use so' chords up to i6° Curves Use 25' chords up to 32 ° Curves Use 10' chords above 32 Curves c O S . t-i O tifl 4 5 6 7 C Ext. Tan. Ext. Tan. Ext. Tan. 300.3 Ext. Tan. .0000 3-5 200.1 5-5 250.2 7-9 10.7 350.4 I .0167 3-5 200.9 5-5 251.0 7-9 301. 1 10.8 351.3 I 2 •0333 3-6 201.8 5-5 251.8 8.0 302.0 10.8 352.1 2 3 .0500 3-6 202.6 5-6 252.7 8.0 302.8 10.9 352.9 3 4 .0667 3-6 203.4 5-6 253-5 8.0 303.6 10.9 353-8 4 5 •0833 3-6 204.3 5-6 254-3 8.1 304-5 11. 354-6 5 6 .1000 3-7 205.1 5-7 255-2 8.1 305-3 11.0 355-5 6 7 .1167 3-7 205.9 5-7 256.0 8.2 306.1 11. 1 356.3 7 8 .1333 3-7 206.8 5-8 256.8 8.2 307.0 11. 1 357.1 8 9 .1500 3-8 207.6 5-8 257.7 8-3 307.8 11. 2 358.o 9 10 .1667 3-8 208.4 5-8 258.5 8.3 308.6 11. 2 358.8 10 11 .1833 3-8 209.3 5-9 259.3 8.4 309.5 11.3 359-6 11 12 .2000 3-9 210. 1 5-9 260.2 8.4 310.3 11.3 360.5 12 13 .2167 3-9 210.9 5-9 261.0 8.4 311.1 11.4 361.3 13 14 .2333 3-9 211.8 6.0 261.9 8.5 312.0 11.4 362.2 14 IS .2500 3-9 212.6 1 6.0 262.7 8-5 312.8 H-5 363-0 15 16 .2667 4.0 213.4 6.1 263.5 8.6 313.7 11. 5 363.8 16 17 .2833 4.0 214.3 6.1 264.4 8.6 314.5 11.6 364.7 I7 a 18 .3000 4.0 215. 1 6.1 265.2 8.7 315.3 11. 7 365.5 I l8 19 .3167 4.1 215.9 6.2 266.0 8-7 316.2 11. 7 366.3 19 20 •3333 4.1 216.8 6.2 266.9 8.8 317.0 11.8 367.2 , 20 21 •3500 4.1 217.6 6.2 267.7 8.8 317.8 11.8 368.0 21 22 .3667 4.2 218.4 6.3 268.5 8.9 318.7 11.9 368.8 .22 23 .3833 4.2 219.3 6.3 269.4 8.9 319-5 11.9 369-7 23 24 .4000 4.2 220.1 6.4 270.2 9.0 320.3 12.0 37o.5 24 25 .4167 4-3 220.9 6.4 271.0 9.0 321.2 12.0 371.4 ! 25 26 •4333 4-3 221.8 6.4 271.9 9.0 322.0 12. 1 372.2 26 27 .4500 4-3 222.6 6-5 272.7 9.1 322.8 12. 1 373-0 27 28 .4667 4.4 223.5 6-5 273-5 9.1 323-7 12.2 373-9 28 29 •4833 4.4 224.3 6-5 274.4 9.2 324-5 12.2 374-7 29 30 .5000 4.4 225.1 6.6 275-2 9.2 325.4 12.3 375-5 30 3i •5167 4-5 226.0 6.6 276.1 9-3 326.2 12.4 376.4 31 32 •5333 4-5 226.8 6.7 276.9 9-3 327.0 12.4 377-2 32 33 •5500 4-5 227.6 6.7 277-7 9.4 327-9 12.5 378.1 33 34 .5667 4-6 228.5 6.8 278.6 9-4 328.7 12.5 378.9 34 35 .5833 4.6 229.3 6.8 279-4 9-5 329-5 12.6 379-7 35 36 .6000 4-6 230.1 6.8 280.2 9-5 330.4 12.6 380.6 36 37 .6167 4-7 331.0 6.9 281. 1 9.6 331.2 12.7 381.4 37 38 •0333 4-7 231-8 6.9 281.9 9.6 332.0 12.7 382.2 38 39 .6500 4-7 232.6 7.0 282.7 9-7 332-9 12.8 383.1 39 40 .6667 4.8 233-5 7.0 283.6 9-7 333-7 12.9 383.9 |40 41 .6833 4.8 234-3 7-i 284.4 9-8 334-6 12.9 384.7 41 42 .7000 4.8 235-1 7-i 285.2 9.8 335-4 13.0 385.6 42 43 .7167 4.9 236.0 7-i 286.1 99 336.2 13-0 386.4 43 44 •7333 4.9 236.8 7-2 286.9 9.9 337-1 I3-I 387.3 44 45 .7500 4.9 237.6 7.2 287.7 10. 337-9 I3-I 388.1 45 46 .7667 5.o 238.5 7.3 288.6 10.0 338.7 13-2 388.9 46 47 •7833 5-0 239-3 7-3 289.4 IO.I 339-6 13.2 389.8 47 48 .8000 5-0 240.1 7-3 290.3 10. 1 340.4 13.3 390.6 48 49 .8167 5-i 241.0 7-4 291. 1 10.2 341.2 13-4 391.4 49 SO .8333 5-1 241.8 7-4 291.9 10.2 342-1 13-4 392.3 50 51 .8500 5-i 242.6 7-5 292.8 10.3 342-9 13.5 393-1 51 52 .8667 5-2 243-5 7-5 293-6 10.3 343-7 13.5 394-o 52 53 .8833 5-2 244-3 7-5 294.4 10.4 344-6 13-6 394-8 53 54 .9000 5-2 245-2 j 7-6 295-3 10.4 345-4 13-7 395-6 |54 55 .9167 5-3 246.0 7-6 296.1 10.5 346.3 13-7 396.5 |55 56 •9333 5-3 246.8 7-7 296.9 10.5 347-1 13-8 397-3 56 57 .9500 5-3 247.7 i 7-7 297.8 10.6 347-9 13-8 398.1 57 58 .9667 5-4 248.5 \ 7-8 298.6 10.6 348.8 13-9 399-0 58 59 .9833 5-4 249-3 ! 7-8 299.4 10.7 349-6 139 399-8 l» 172 THE SURVEY Use ioc ' Chords up to 8 3 Curves Use 25' Chords up to 32 Curves Use 50' Chords up to 16 3 Curves Use io' Chords above 3 2 Curves m -4-> 3 G § O Z a 3 O 0) 20° 21° 22° 23° 3 -fi 3 C S O Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. .0000 88.4 1010.4 97.6 1062.0 IO7.2 1113.8 117.4 1165.8 1 .0167 88.5 1011.2 97-7 1062.8 IO7.4 1114.6 117. 6 1166.6 I 2 .0333 88.7 1012.1 97-9 1063.7 IO7.6 1115.5 117. 7 1167-5 2 3 .0500 88.8 1012.9 98.1 1064.5 IO7.7 1116.4 117.9 1 168.3 3 4 .0667 89.0 1013.8 98.2 1065.4 IO7.9 1117.3 118. 1 1169.2 4 5 •0833 89.1 1014.6 98.4 1066.3 I08.0 1118.1 118.3 1170.1 5 6 .1000 89.3 1015-5 98.5 1067.2 I08.2 1119.0 118.4 117 1. 6 7 .1167 89.4 1016.3 98.7 1068.0 I08.4 1119.8 118.6 1171.8 7 8 •1333 89.6 1017.2 98.8 1068.9 IO8.6 1120.7 118.8 1172.7 8 9 .1500 89.7 1018.1 99.0 1069.7 I08.7 1121.5 118.9 H73-5 9 10 .1667 89.9 1019.0 99.2 1070.6 I08.9 1122.4 119. 1 H74-4 10 11 .1833 90.0 1019.8 99-3 1071.5 IO9.O 1123.3 H9-3 H75-3 11 12 .2000 90.2 •1020.7 99-5 1072.4 IO9.2 1124.2 II95 1176.2 12 13 .2167 90.3 1021.5 99.6 1073.2 IO9.4 1125.0 II9-7 1177.0 13 14 •2333 90.5 1022.4 99-8 1074.1 IO9.6 1125.9 119.8 1177.9 14 15 .2500 90.6 1023.2 99-9 1074.9 IO9.7 1126.7 120.0 1178.8 15 16 .2667 90.8 1024. 1 100. 1 1075.8 IO9.9 1127.6 120.2 1179.7 16 17 •2833 90.9 1024.9 100.2 1076.6 IIO.O 1128.5 120.4 1180.5 17 18 .3000 91. 1 1025.8 j 100.4 1077.5 HO.2 1129.4 120.5 1181.4 18 19 .3167 91.2 1026.7 100.5 1078.4 IIO.4 1130.2 120.7 1182.2 19 20 ■3333 91.4 1027.6 100.7 1079-3 I IO.6 1131.1 120.9 1183.I 20 21 •3500 91.6 1028.4 100.9 1080. 1 HO.7 1131.9 1 21.0 1184.0 21 22 .3667 91.7 1029.3 IOI.I 1081.0 no. 9 1132.8 121. 2 1184.9 22 23 •3833 91.9 1 030. 1 | IOI.2 1081.8 III.O II33.7 121. 4 1185.7 23 24 .4000 92.0 103 1. IOI.4 1082.7 in. 2 1134.6 121.6 1186.6 24 25 .4167 92.2 1031.8 IOI.5 1083.5 111.4 "35.4 121. 7 1187.5 25 26 •4333 92.3 1032.7 IOI.7 1084.4 111.6 1136.3 121.9 1188.4 26 27 •4500 92.5 IQ33-5 101.8 1085.3 in. 7 1137.1 122. 1 1189.2 27 28 .4667 92.6 1034.4 102.0 1086.2 111.9 1138.0 122.3 1190.1 28 29 •4833 92.8 1035.2 102. 1 1087.0 112. 1 1138.8 122.4 1190.9 29 30 .5000 92.9 1036.1 102.3 1087.9 112. 3 H39-7 1 2 2.6 1191.8 30 31 •5167 93-1 1037.0 102.5 1088.7 112.4 1140.6 122.8 1192.7 3i 32 •5333 93-2 1037.9 102.7 1089.6 112. 6 1141.5 123.0 1 193.6 32 33 •55oo 93-4 1038.7 102.8 1090.4 112.7 1142.3 123.2 1194.4 33 34 •5667 93-5 1039.6 103.0 1 09 1. 3 112. 9 H43-2 123.3 "95-3 34 35 36 .5833 93-7 1040.4 103. 1 1092.2 113. 1 1 144.0 123.5 1196.2 35 .6000 93-9 1041-3 103.3 1093. 1 II3-3 1144.9 123.7 1197.1 36 37 .6167 94.0 1042. 1 103.4 1093.9 H3-4 H45-8 123.9 H97-9 37 38 •6333 94.2 1043.0 103.6 1094.8 113-6 1146.7 124. 1 1198.8 38 39 .6500 94-3 1043.9 103.8 1095.6 H3-7 II47.5 124.3 1199.6 39 40 .6667 94-5 1044.8 104.0 1096.5 II3-9 1 148.4 124.4 1200.5 40 41 .6833 ! 94.6 1045.6 104. 1 1097.4 114.1 1149.2 124.6 1201.4 4i 42 .7000 j 94-8 1046.5 104.3 1098.3 H4-3 1150.1 124.8 1202.3 42 43 .7167 94.9 1047.3 104.4 1099. 1 114.4 1151-0 124.9 1203. 1 43 44 •7333 95-1 1048.2 104.6 .1100.0 II4.6 1151.9 125. 1 1204.0 44 45 •750O 95-2 1049.0 104.7 1 100.8 II4.8 1152.7 125.3 1204.9 45 46 .7667 95-4 1049-9 104.9 1101.7 115.0 H53-6 125-5 1205.8 46 47 .7833 95-6 1050.8 105. 1 1102.5 II5- 2 II54.5 125-7 1206.7 47 48 .8000 95-7 1051.7 105.3 1103.4 II5.3 II55.4 125.8 1207. 5 48 49 .8167 95-9 1052.5 105.4 1 104.3 H5-5 1156.2 126.0 1208.3 49 50 •8333 .8500 96.0 1053-4 105.6 1105.2 H5-7 1157.1 126.2 1209.2 50 51 96.2 1054.2 105.7 1106.0 115.8 H57-9 126.4 1210.1 51 52 .8667 96.3 1055. 1 105.9 1 106.9 116. 1158.8 126.6 1211.0 52 53 •8833 96.5 IQ55-9 106. 1 1107.8 116. 1 H59-7 126.7 1211.8 53 54 .9000 96.7 1056.8 106.3 1108.6 116.3 1160.6 126.9 1212.7 54 55 .9167 96.8 IQ57-7 106.4 1 109.4 116.5 1161.4 127. 1 1213.6 55 56 •9333 97-o 1058.6 106.6 1110.3 116.7 1162.3 127.3 1214-5 56 57 .9500 97.1 1059.4 106.7 IIII. 2 116.8 1163.1 127.5 1215.3 57 58 .9667 97-3 1060.3 106.9 1112.1 117. 1 164.0 127.6 1216.2 58 59 •9833 97-4 1061.1 107.0 1112.9 117. 2 1164..9 127.8 1217.1 59 176 THE SURVEY Use 50 ' Chords up to it Curves Use 10' Chords above 32 Curves (A +-> O QQ 24° 25° 26 27° w c 9 Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. 1375-6 .0000 128.0 1218.0 139.1 1270.3 150.7 1322.9 162.8 I .0167 128.2 1218.8 139.3 1271.1 150.9 1323.7 163.0 1376.5 I 2 .0333 128.4 1219.7 139.5 1272.0 151. 1 1324.6 163.2 1377-4 2 3 .0500 128.5 1220.5 139-7 1272.9 151.3 1325.5 163.5 1378.3 3 4 .0667 128.7 1221.4 139-9 1273-8 I5I-5 1326.4 163.7 1379.2 4 5 ■0833 128.9 1222.3 1 40. 1 1274.6 151.7 1327.3 163.9 1380.0 5 6 .1000 129. 1 1223.2 140.3 1275-5 151.9 1328. 1 164. 1 1380.9 6 7 .1167 129.3 1224.0 140.4 1276.4 152. 1 1329 164.3 1381.8 7 8 •1333 1295 1224.9 140.6 1277.3 152.3 1329.9 164.5 1382.7 8 9 .1500 129.7 1225.8 140.8 1278.2 152.5 1330.7 164.7 1383.6 9 IO .1667 129.8 1226.7 141.0 1279. 1 152.7 1331.6 164.9 1384.5 10 ii .1833 130.0 1227.5 141. 2 1279.9 152.9 1332.5 165. 1 1385.3 11 12 .2000 130.2 1228.4 141.4 1280.8 I53-I 1333.4 165.3 1386.2 12 13 .2167 130.4 1229.3 141. 6 1281.6 153-3 1334-3 165.5 J387.1 13 14 •2333 130.6 1230.2 141.8 1282.5 153.5 1335.2 165.7 1388.0 14 15 .2500 130.7 1231.0 142.0 1283.4 153-7 1336.0 165.9 1388.9 15 16 .2667 130.9 1231.9 142.2 1284.3 153-9 1336.9 166. 1 1389.8 16 17 .2833 131-1 1232.7 142.3 1285.2 I54-I 1337-8 166.3 1390.6 17 18 .3000 131-3 1233.6 142.5 1286. 1 154-3 1338.7 166.5 I39I.5 18 19 .3167 I3I-5 1234-5 142.7 1286.9 154-5 1339.5 166.7 1392.4 19 20 •3333 131.7 1235-4 142.9 1287.8 154-7 1340.4 167.0 1393.3 CO 21 •35oo i3i-9 1236.2 143. 1 1288.7 154-9 I34I-3 167.2 1394.1 21 22 .3667 132.0 1237. 1 143-3 1289.6 155. 1 1342.2 167.4 I395-0 22 23 .3833 132.2 1238.0 143-5 1290.4 155-3 I343-Q 167.6 1395-9 23 24 .4000 132.4 1238.9 143-7 1291.3 155.5 1343.9 167.8 1396.8 24 25 .4167 132.6 1239.7 143-9 1292.2 155-7 1344.8 168.0 1397-7 25 26 •4333 132.8 1240.6 144.1 1293-1 155-9 1345.7 168.2 1398.6 26 27 .4500 i33-o 1241-5 144-3 1293.9 156.1 1346.5 168.4 I3Q9-4 27 28 .4667 133. 1 1242.4 144-5 1294.8 156.3 1347.4 168.6 1400.3 28 29 .4833 133-3 1243.2 144.7 1295-7 156.5 1348.3 168.9 1401.2 29 30 .5000 133-5 1244-1 144.9 1296.6 156.7 1349.2 169. 1 1402. 1 30 31 .5167 133-7 1244.9 I45-I 1297.4 156.9 I350.I 169.3 1403.0 31 32 •5333 133-9 1245.8 145-3 1298.3 157. 1 I35I.O 169.5 1403.9 32 33 .55oo 134.0 1246.7 145-5 1299.2 157-3 I35I-8 169.7 1404.7 33 34 .5667 134.2 1247.6 145.6 1300.1 157-5 1352.7 169.9 1405.6 34 35 •5833 134-4 1248.4 145-8 1300.9 157.7 1353.6 170.1 1406.5 35 36 .6000 134-6 1249.3 146.0 1301.8 157.9 1354-5 170.3 1407.4 36 37 .6167 134-9 1250.2 146.2 1302.7 158.1 1355-3 170.5 1408.3 3 l 38 •6333 I35-0 1251.I 146.4 1303.6 158.3 1356.2 170.8 1409.2 38 39 .6500 135.2 1251.9 146.6 1304.4 158.5 1357. 1 171.0 1410.0 39 40 .6667 135-4 1252.8 146.8 1305.3 158.7 1358.0 171. 2 1410.9 40 41 •6833 135-6 1253-7 147.0 1306.2 158.9 1358.9 I7I-4 1411.8 41 42 .7000 135-7 1254.6 147.2 1307. 1 I59.I 1359-8 171.6 1412.7 42 43 .7167 135-9 1255-4 147.4 1307.9 159-3 1360.6 171.8 1413-6 43 44 •7333 136.1 1256.3 147.6 1308.8 159-5 1361.5 172.0 I4I4.5 44 45 •75oo 136.3 1257.2 147-8 1309.7 159-7 1362.4 172.2 I4I5.4 45 46 .7667 136. 5 1258.1 148.0 1310.6 160.0 1363-3 172.5 1416.3 46 47 •7833 I 136.7 1258.9 148.2 1311-5 160.2 1364.2 172.7 1417.1 47 48 .8000 136.9 1259.8 148.4 1312.4 160.4 1365-1 172.9 1418.0 48 49 .8167 I37-I 1260.7 148.6 1313.2 160.6 1365-9 I73-I 1418.9 49 50 •8333 137.2 1261.5 148.8 1314.1 160.8 1366.8 173-3 1419-8 50 51 .8500 ! 137-4 1262.4 149:0 1315-0 161. 1367.7 j 173-5 1420.7 51 52 .8667 137-6 1263.3 149.2 I3I5-9 161. 2 I368.6 ; 173-7 1421.6 52 53 .8833 137-8 1264. 1 149.4 1316.7 161. 4 1369-5 ! 173-9 1422.4 53 54 .9000 138.0 1265.0 149-5 1317-6 161.6 1370.4 174.1 1423.3 54 55 .9167 138.2 1265.9 149.7 1318.5 161.8 I37I-2 174-4 1424.2 55 56 •9333 138.4 1266.8 149.9 1319-4 162.0 1372. 1 174-6 1425. 1 56 57 .9500 138.6 1267.6 150.1 1320.3 162.2 1373-0 j 174.8 1426.0 57 58 .9667 138.7 1268.5 150.3 1321.1 j 162.4 1373-9 i75.o 1426.9 58 [59 .9833 138.9 1269.4 150.5 1322.0 162.6 1374-7 175-2 1427.7 59 FUNCTIONS OF THE ONE-DEGREE CURVE 177 Use 100' Chords up to 8° Use 50' Chords up to 16 Curves Use 25' Chords up to 32 Curves Curves Use 10 ' Chords above 32 Curves en a 3 O . 1-1 CJ bC 32° 33 ° 34° 35° i Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. .0000 230.9 1643. 1 | 246.1 1697.3 261.8 I75I.8 278.1 1806.7 I .0167 231-1 1644.0 | 246.3 1698.2 262.0 1752.7 278.4 1807.6 1 2 •0333 231.4 1644.9 246.6 1699. 1 262.3 1753.7 278.6 1808.5 2 3 .0500 231.6 1645.8 246.8 1700.0 262.6 1754.6 278.9 1809.4 3 4 .0667 231.9 1646.7 247.1 1700.9 262.9 1755.5 279.2 1810.3 4 5 .0833 232.1 1647.6 247.4 1701.8 263.1 1756.4 279-4 1811.2 5 6 .1000 232.4 1648.5 247.7 1702.7 263.4 1757-3 279.7 1812.2 6 7 .1167 232.6 1649.4 247.9 1703.6 263.7 1758.2 280.0 1813.1 7 8 • 1333 232.9 1650.3 248.2 1704-5 264.0 1759- 1 280.3 1814 8 9 .1500 233-1 1651.2 248.4 1705.4 264.2 1760.0 280.6 1814.9 9 10 .1667 233-4 1652. 1 248.7 1706.4 264.5 1761.0 280.8 1815.8 10 11 •1833 233-6 1653-0; 248.9 1707.3 264.7 1761.9 281. 1 1816.7 11 12 .2000 233-9 1653-9 ! 249.2 1708.2 265.0 1762.8 281.4 1817.7 12 13 .2167 234.1 1654.8 ; 249.4 1709. 1 265.3 1763.7 281.6 1818.6 13 14 .2333 234-4 1655.7 249.7 1710.0 265.6 1764.6 281.9 1819.5 14 15 .2500 234-6 1656.6 249.9 1710.9 265.9 1765.5 282.2 1820.4 15 16 .2667 234-9 1657.5 250.2 1711.8 266.1 1766.4 282.5 1821.3 16 17 .2833 235-1 1658.4 250.5 1712.7 266.4 1767.3 282.7 1822.2 17 18 .3000 235-4 1659-3 250.8 1713-6 266.7 1768.3 283.0 1823.? 18 19 .3167 235-6 1660.2 251.0 I7I4-5 266.9 1769.2 283.3 1824.1 19 20 •3333 235-9 1661.1 251-3 I7I5-5 267.2 1770.1 283.6 1825.0 20 21 •3500 236.1 1662.0 251.5 1716.4 267.4 1771.0 283.9 1825.9 21 22 .3667 236.4 1662.9 251.8 1717.3 267.7 1771.9 284.2 1826.8 22 23 .3833 236.6 1663.8 252.0 1718.2 268.0 1772.8 284.4 1827.7 23 24 .4000 236.9 1664.7 252.3 1719.1 268.3 1773-7 284.7 1828.7 24 25 .4167 237.1 1665.6 252.6 1720.0 268.6 1774-6 285.0 1829.6 25 26 •4333 237-4 1666.5 252.9 1720.9 268.8 1775.6 285.3 1830.5 26 27 .4500 237-6 1667.4 253-1 1721.8 269.1 1776.5 285.6 1831.4 27 28 .4667 237-9 1668.3 253-4 1722.7 269.3 1777-4 285.9 1832.3 28 29 .4833 238.1 1669.2 253-6 1723.6 269.6 1778.3 286.1 1833.2 29 30 .5000 238.4 1670. 1 253-9 1724.6 269.9 1779.2 286.4 1834.2 30 31 .5167 238.7 1671.0 254.1 1725.5 270.1 1780.1 286.7 1835. 1 3i 32 •5333 239.0 1671.9 l 254-4 1726.4 270.4 1781.0 287.0 1836.0 32 33 .5500 239.2 1672.8 254-7 1727.3 270.7 1781.9 287.2 1836.9 33 34 .5667 239.5 1673.7 255.o 1728.2 271.0 1782.9 287.5 1837.8 34 35 .5833 239-7 1674.6 1 255-2 1729. 1 271.2 1783.8 287.8 1838.7 35 36 .6000 240.0 1675-5, 255-5 1730.0 271-5 1784.7 288.1 1839.7 36 37 .6167 240.2 1676.4 ) 255-7 1730.9 271.7 1785.6 288.4 1840.6 37 38 •6333 240.5 1677.4 256.0 1731.8 272.0 1786.5 288.7 1841.5 38 39 .6500 240.7 1678.3 256.2 1732.7 272.3 1787.4 289.0 1842.4 39 40 .6667 241.0 1679.2 256.5 1733-6 272.6 1788.4 289.2 1843.4 40 41 .6833 241.2 1680.1 256.8 1734-5 272.9 1789.3 289.5 1844-3 41 42 .7000 241.5 1681.0 257-1 1735-5 273-1 1790.2 289.8 1845.2 42 43 .7167 241.7 1681.9 257.3 1736.4 273-4 1791-1 290.1 1846. 1 43 44 •7333 242.0 1682.8 257-6 1737-3 273-7 1792.0 290.4 1847. 1 44 45 .7500 242.2 1683.7 257-8 1738.2 274-0 1792.9 290.6 1848.0 45 *6 .7667 242.5 1684.6 258.1 I739-I 274-2 1793-9 290.9 1848.9 46 47 .7833 242.7 1685.5 258.3 1740.0 274-5 1794-8 291.2 1849.8 47 48 .8000 243 -o 1686.4 258.6 1740.9 274.8 1795-7 291.5 1850.7 48 49 .8167 243-2 1687.3 258.9 1741.8 275.o 1796.6 291.8 1851.6 49 50 .8333 243-5 1688.2 259-2 1742.7 275-3 1797-5 292.0 1852.6 50 51 .8500 243-8 1689. 1 259-4 1743-6 275-6 1798.4 292.3 1853.5 51 52 .8667 244.1 1690.0 259-7 1744.6 * 275-9 1799.3 292.6 1854.4 |52 53 .8833 244-3 1690.9 259-9 1745-5 276.1 1800.2 292.9 1855-3 53 54 .9000 244.6 1691.8 260.2 1746.4 276.4 1801.2 293-2 1856.3 54 55 .9167 244-8 1692.7 260.5 1747-3 276.7 1802. 1 293-4 1857.2 55 56 •9333 245-1 1693.7 260.8 1748.2 277.o 1803.0 293-7 1858.1 56 57 .9500 245-3 1694.6 261.0 I749-I 277-3 1803.9 294.0 1859-0 ,57 58 .9667 245-6 1695-5 261.3 1750.0 277-5 1804.8 294-3 18599 58 59 •9833 245-8 1696.4 261.5 1750.9 277-8 1805.7 294.6 1860.8 59 FUNCTIONS OF THE ONE-DEGREE CURVE 179 Use 100' Chords up to 8° Curves Use 50' Chords up to 16 Curves Use 25' Chords up to 32° Curves Use 10' Chords above 32° Curves en a 9 "SB u to CD CD QP 36° 37° 38° 39° I en a Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. .0000 294.9 1861.8 312.3 I9I7-3 330.2 1973.0 348.7 2029.1 1 .0167 295.2 1862.7 312.5 1918.2 330.5 1973-9 349-Q 2030.0 1 2 •0333 295.4 1863.6 312.8 1919.1 330.8 1974.9 349-3 2031.0 2 3 .0500 295-7 1864.5 3I3-I 1920.0 33i. 1 1975-8 349-6 2031.9 3 4 .0667 296.0 1865.5 313-4 1921.0 331.4 1976.7 349-9 2032.9 4 5 •0833 296.3 1866.4 313-7 1921.9 331-7 1977.6 350.3 2033.8 5 6 .1000 296.6 1867.3 314.0 1922.8 332-0 1978.6 350.6 2034.7 6 7 .1167 296.9 1868.2 314.3 1923-7 332.3 1979-5 350.9 2035.6 7 8 •1333 297.2 1869.2 1 314-6 1924.7 332.6 1980.5 351-2 2036.6 8 9 .1500 297.5 1870.1 314-9 1925.6 332.9 1981.4 351.5 2037-5 9 10 .1667 297-7 1871.0 315-2 1926.5 333-2 1982.3 351-8 2038.5 10 11 .1833 298.0 1871.9 315.5 1927.4 333-5 1983-2 352-1 2039.4 11 12 .2000 298.3 1872.9 315.8 1928.4 333-8 1984.2 352.4 2040.4 12 13 .2167 298.6 1873.8 316. 1 1929.3 334-2 1985. 1 352.8 2041.3 13 14 -2335 298.9 1874-7 316.4 1930.2 334-5 1986.1 353-1 2042.3 14 15 .2500 299.2 1875.6 316.7 1931-1 334-8 1987.0 353.4 2043.2 15 16 .2667 299-5 1876.5 ! 317.0 1932. 1 | 335-1 1987.9 353-7 2044.1 16 17 .2833 299.7 1877.4! 317-2 19330 335-4 1988.8 354-o 2045.0 17 18 .3000 300.0 1878.4 317.5 1933-9 | 335-7 1989.8 354-3 2046.0 18 19 .3167 300.3 1879-3 317.8 1934-8 336.0 1990.7 354-6 2046.9 19 20 •3333 300.6 1880.2 ! 318.1 1935-8 336.3 1991.7 354-9 2047.9 20 21 •35oo 300.9 1881.1 318.4 1936.7 ! 336.6 1992.6 355-3 2048.8 21 22 .3667 301-2 1882.1 318.7 1937.6 ! 336.9 1993-6 355-6 2049.8 22 23 .3833 301.5 1883.0 319.0 I938.S 337-2 1994-5 355-9 2050.7 23 24 .4000 301.8 1883.9 319-3 1939-5 337-5 1995-4 356.2 2051.7 24 25 .4167 302.0 1884.8 319.6 1940.4 337-8 1996.3 356.6 2052.6 25 26 •4333 302.3 1885.8 319-9 I94I-3 338.1 1997.3 356.9 2053-5 26 27 .4500 302.6 1886.7 320.2 1942.2 338.4 1998.2 357-2 2054.4 |27 28 .4667 302.9 1887.6 320.5 1943-2 338.7 1999.2 357-5 2055-4 28 29 .4833 303.2 1888.5 320.8 I944-I 339-1 2000.1 357-8 2056.3 29 30 .5000 303.5 1889.5 321. 1 1945.0 339-4 2001.0 358.1 2057.3 30 3i •5167 303.8 1890.4 321.4 1945.9 339-7 2001.9 358.4 2058.2 I 31 32 •5333 304.1 1891.3 321.7 1946.9 340.0 2002.9 358.8 2059.2 132 33 •5500 304-3 1892.2 322.0 1947.8 340.3 2003.8 359-1 2060.1 33 34 .5667 304.6 1893.2 322.3 1948.8 340.6 2004.8 359-4 2061. 1 34 35 .5833 304-9 1894-1 322.6 1949.7 340.9 2005.7 359-8 2062.0 35 36 .6000 305-2 1895.0 322.9 1950.6 341.2 2006.6 360.1 2063.0 36 37 .6167 305.5 1895.9 323.2 I95I-5 341-5 2007.5 360.4 2063.9 37 38 .6333 305.8 1896.9 323.5 1952.5 , 1953.4 341.8 2008.5 360.7 2064.8 38 39 .6500 306.1 1897.8 323.8 342-1 2009.4 361.0 2065.7 39 40 .6667 306.4 1898.7 324.2 1954.4 342.4 2010.4 361.3 2066.7 [40 41 •6833 306.7 1899.6 324-5 1955.3 342.8 2011.3 361.6 2067.6 41 42 .7000 307.0 1900.6 324-8 1956.2 J 343-1 2012.3 362.0 2068.6 42 43 .7167 307.2 1901.5 325.1 I957-I 1 343-4 2013.2 362.3 2069.5 43 44 •7333 307-5 1902.4 325-4 1958.1 ; 343-7 2014. 1 362.6 2070.5 44 45 .7500 307.8 1903.3 325-7 1959.0 344-0 2015.0 363.0 2071.4 45 46 .7667 308.1 1904.3 326.0 1960.0 344-3 2016.0 363-3 2072.4 46 47 .7833 308.4 1905.2 326.3 1960.9 344-6 2016.9 363-6 2073.3 47 48 .8000 308.7 1906. 1 326.6 1961.8 344-9 2017.9 363.9 2074.2 48 49 .8167 : 309.0 1907.0 326.9 1962.7 345-3 2018.8 364-2 2075.1 49 50 •8333 309.3 1908.0 327.2 1963-7 345-6 2019.7 364-5 2076.1 50 51 .8500 309-6 1908.9 327.5 1964.6 345-9 2020.6 364-9 2077.0 51 52 .8667 309.9 1909.8 ] 327.8 1965-5 346.2 2021.6 365-2 2078.0 52 53 .8833 3^0.2 1910.7 328.1 1966.4 346.5 2022.5 365-5 2078.9 53 54 .gooo 310.5 1911.7 328.4 1967.4 346.8 2023.5 365.8 2079-9 54 55 .9167 310.8 1912.6 328.7 1968.3 347-1 2024.4 366.2 2080.8 55 56 •9333 311. 1 1913-5 329-0 1969.3 347-4 2025.4 366.5 208I.8 '; 56 57 .9500 3H-4 1914-4 329-3 1970.2 347-8 2026.3 366.8 2082.7 57 58 •9667 3II-7 1915-4 329-6 1971-1 348.1 2027.2 367.1 2083.7 I 58 59 .9833 312.0 1916.3 329-9 1972.0 348.4 2028.1 367.4 2084.6 1 ! 59j i8o Use ioo' Chords up to 8° Curves Use 50' Chords up to 16 Curves THE SURVEY Use 25' Chords up to 3 2° Curves Use 10' Chords above 32 Curves en P § Dec. of Degree 68° 69° 70° 7i° 1 "> ! 3 C § Ext. Tan. Ext. Tan. 1 Ext. Tan. Ext. Tan. .0000 1181.6 3864.9 1222.9 3938.1 1265.0 401 2.1 1308.4 4087.1 I .0167 1182.3 3866.1 1223.6 3939-4 1265.7 4013.4 1309.2 4088.4 I 2 •0333 1183.0 3867.3 1224.3 3940.6 1266.4 4014.6 1309.9 4089.7 2 3 .0500 "83.7 3868.5 1225.0 3941-8 1267.2 4015.9 1310.6 4091.0 3 4 .0667 1184.4 3869.7 1225.7 3943-Q 1267.9 4017. 1 1311-3 4092.2 4 5 .0833 1185.1 3870.9 1226.4 3944-2 ! 1268.6 4018.4 1312.1 4093-5 5 6 .1000 1185.7 3872.2 1227. 1 3945-5 1269.3 4019.6 1312.8 4094.7 6 7 .1167 1186.4 3873-4 1227.8 3946.7 1 1270.1 4020.8 1313-5 4096.0 7 8 •1333 1187.1 3874-6 1228.5 3947-9 1270.8 4022.1 1314-2 4097.2 8 9 .1500 1187.8 3875-8 1229.2 3949-2 1 1271-5 4023.4 1315-0 4098.5 9 10 .1667 1188.5 3877-0 1229.9 3950.4 1272.2 4024.6 1315.7 4099.8 10 11 •1833 1189.2 3878.2 1230.6 3951-6 ' 1272.9 4025.8 1316.5 4101.1 11 12 .2000 1189.8 3879-5 1231.3 3952.9 j 1273-6 4027.1 1317-2 4102.3 12 13 .2167 1190.5 3880.7 1232.0 3954-1 1274.4 4028.4 I3I7-9 4103.6 13 14 •2333 ; 1191.2 3881.9 1232.7 3955-3 1275.1 4029.6 1318.6 4104.8 14 IS .2500 1191.9 3883.1 1233-4 3956.6 1275-8 4030.8 1319-4 4106. 1 15 16 .2667 1192.6 3884.3 1234-1 3957-8 : 1276.5 4032.1 1320.1 4107-3 16 17 .2833 "933 3885.6 1234.8 3959-Q j 1277-3 4033-4 1320.8 4108.6 17 18 .3000 H93-9 3886.8 1235-5 3960.2 i 1278.0 4034.6 1321.5 4109.8 18 19 •3l67 1194.6 3888.0 1236.2 3961.5 ; 1278.7 4035-9 1322.3 4111.1 19 20 •3333 "95-3 3889.2 1236.9 3962.7 1279.4 4037.1 1323.0 4112.4 20 21 •3500 1196.0 3890.4 1237-6 3964.0 1280.1 4038.4 1323.7 4113-7' 21 22 .3667 1196.7 3891-6 1238.3 3965-2 1280.8 4039.6 1324.4 4II4-9 22 23 •3833 1197.4 3892.9 1239.0 3966.4 ! 1281.6 4040.9 1325-2 4116.2 23 24 .4000 1 198.0 3894.1 1239.7 3967-6 j 1282.3 4042.1 1325-9 4II7-4 24 25 .4167 1198.7 3895-3 1240.4 3968.9 1283.0 4043.4. 1326.7 4118.7 25 26 •4333 1199.4 3896.5 1241.1 397o.i 1283.7 4044.6 1327-4 4119.9 26 27 .4500 1200. 1 3897.7 1241.8 397L3 1284.5 4045-9 1328.2 4121.2 27 28 .4667 1200.8 3898.9 1242.5 3972.5 1285.2 4047.1 1228.9 4122.4 ! 2 8 29 •4833 1 201.5 3900.2 1243.2 3973-8 1285.9 4048.4 1329.7 4123.7 I 29 30 .5000 1202. 1 3901.4 1243.9 3975-0 1286.6 4049.6 1330.4 4125.0 30 3i •5167 1202.8 3902.6 1244.6 3976.3 ! 1287.3 4050.9 i33i-i 4126.3 j 31 32 •5333 1203.5 3903-8 1245-3 3977-5 1288.0 4052.1 I33I-8 4127-5 32 33 •55oo 1204.2 3905-0 1246.0 3978.8 J 1288.8 4053-4 1332.6 4128.7 33 34 .5667 1204.9 3906.3 1246.7 3980.O 1289.5 4054.6 1333.3 4130.0 34 35 •5833 1205.6 3907-5 1247.4 3981.2 1290.2 4055.9 I334-I 4131.5 35 36 .6000 1206.2 3908.7 1248.1 3982.4 1290.9 4057-1 1334.8 4132.6 36 37 .6167 1206.9 3909.9 1248.8 3983.7 1 291.7 4058.4 1335-6 4I33.9 37 38 •6333 1207.6 3911.2 1249.5 3984.9 1292.4 4059-6 1336.3 4I35-I 38 39 .6500 1208.3 3912.4 1250.2 3986.1 1293-1 4060.9 1337. 1 4136.4 39 40 .6667 1209.0 39I3.6 1250.9 3987.4 1293.8 4062.1 1337-8 4137.7 40 41 .6833 1209.7 3914-9 1251.6 3988.7 : 1294.6 4063.4 1338.5 4139-0 41 42 .7000 1210.3 3916. 1 1252.3 3989-9 1295.3 4064.6 1339-2 4140.2 42 43 .7167 1211.0 3917.3 1253-0 399I-I 1296.0 4065.9 1340.0 4i4i-5 43 44 •7333 1211.7 3918.5 1253-7 3992.3 1296.7 4067.1 1340.7 4142.7 44 45 •7500 1212.4 3919-8 1254-4 3993-6 1297.5 4068.4 I34I-5 4144.0 45 46 .7667 1213.1 3921.0 I255-I 3994-8 1298.2 4069.6 1342.2 4145-3 46 4 l .7833 1213.8 3922.2 1255-8 3996.0 ; 1298.9 4070.9 I343-Q 4146.6 47 48 .8000 1214.5 3923.4 1256.5 3997-3 1299.6 4072.1 1343-7 4147.8 48 49 .8167 1215.2 3924-7 1257.2 3998.6 1300.4 4073-4 1344-5 4149.1 49 50 •8333 1215.9 3925-9 1257-9 3999.8 1301.1 4074-6 1345-2 4150.4 50 51 .8=500 1216.6 2927.1 1258.6 4001.0 1301.9 4075-9 1346.0 4151.7 51 52 .8667 1217.3 3928.3 1259-3 4002.2 i 1302.6 4077.1 1 1346.7 4152.9 52 53 •8833 1218.0 3929-6 1260.0 4003.4 1303-3 4078.4 1347.5 4154-2 53 54 .9000 1218.7 3930.8 1260.7 4004.7 1304.0 4079.6 1348.2 4155-4 54 55 .9167 1219.4 3932.o 1261.4 4006.0 1304.8 4080.9 13490 4156.7 56 •9333 1220. 1 3933-2 1262. 1 4007.2 \ 1305-5 4082.1 1349-7 4158.0 56 57 9500 1220.8 3934-4 1262.8 4008.5 1306.2 4083.4 1350.5 4159-3 57 58 .9667 1221.5 3935-7 1263.5 4009.7 1306.9 4084.6 I35I-2 4160.5 58 59 •9833 1222.2 3936.9 1264.3 4010.9 [307.7 4085.0 1352.O 4161.S 59 1 88 THE SURVEY Use ioo' Chords up to 8° Curves Use 25' Chords up to 32 Curves Use 50' Chords up to 16 Curves Use 10' Chords above 32 Curves -.0 _„o _.o o a 3 C Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. .0000 2235.6 5533-3 2303.6 5630.8 2373-4 573o.o 2445.1 5830.9 O I .0167 2236.7 5535-o| 2304.7 5632.5 2374.6 5731.7 i 2446.3 5832.6 I 2 •0333 2237.8 5536.6 1 ! 2305.6 5634-1 2375-8 5733-3 ! 2447.5 5834.3 2 3 .0500 2238.9 5538.2 2307.2 5635.8 2377.0 5735-o 2448.8 5836.0 3 4 .0667 2240.1 5539-8 2308.1 5637.4 2378.2 5736.7 2450.0 5837.7 4 5 •0833 2241.2 5541-5 2309-4 5639-1 2379-4 5738.4 2451.2 5839.4 5 6 .1000 2242.3 5543-1 2310.5 5640.7 2380.5 5740.0 2452.4 5841. 1 6 7 .1167 2243-5 5544-7 2311.6 5642.4 2381.7 5741.7 | 2453.6 5842.8 7 8 •1333 2244.6 5546.3 2312.8 5644.0 2382.9 5743-4 2454-8 5844.5 8 9 .1500 2245.7 5547-9 2314-0 5645.7 2384-1 5745-1 2456.0 5846.2 9 10 .1667 2246.8 5549-5 2315. 1 5647.3 1 2385.3 5746.7 2457-2 5847-9 10 n •1833 2248.0 5551-2 2316.3 5649.0 ! 2386.4 5748.4 2458.5 5849-6 11 12 .2000 2249.1 5552-8 2317-4 5650.6 2387.6 5750.0 2459-7 5851.3 12 13 .2167 2250.2 5554-4 2318.6 5652.3 2388.8 5751-7 2460.9 5853-o 13 14 .2333 2251.3 5556.o 23I9-7 5653.9 2390.0 5753-4 2462.1 5854.7 14 15 .2500 2252.5 5557-6 2320.9 5655-5 2391.2 5755-1 2463-3 5856.4 15 16 .2667 2253.6 5559-2 2322.0 5657-1 2392.4 5756.7 2464.5 5858.1 16 17 •2833 2254-7 5560.9 2323.2 5658.8 2393-5 5758.4 2465.8 5859.8 17 18 .3000 2255.8 5562.5 2324-3 5660.4 2394-7 576o.i 2467.0 5861.5 18 19 •3167 2257.0 5564.1 2325.6 5662.1 ; 2395-9 5761.8 2468.2 5863.2 19 20 .3333 2258.1 5565.7 2326.7 5663.7 ! 2397-1 5763.4 2469.4 5864.9 20 21 •3500 2259-3 5567.3 2327.9 5665.4 ! 2398.3 5765-1 2470.6 5866.6 21 22 •3667 2260.4 5568.9 2329.0 5667.0 2399-5 5766.8 2471.9 5868.3 22 23 .3833 2261.5 5570.6 ; 2330.1 5668.7 2400.7 5768.5 2473-1 5870.1 23 24 .4000 2262.7 5572.2; 2331-3 5670.3 2401.9 577o.i 2474-3 5871.8 J24 25 .4167 i 2263.8 5573.8 i 2332.5 5672.0 2403.1 5771-8 2475.5 5873-5 *5 26 •4333 2264.9 5575.4 ' 2333-7 5673-6 2404.3 5773-5 2476.7 5875.2 26 27 .4500 2266.0 5577.0; 2334.8 5675-3 2405-5 5775-2 2478.0 5876.9 27 28 .4667 i 2267.2 5578.6 2336.0 5676.9 2406.6 5776.9 2479.2 5878.6 128 29 .4833 2268.4 5580.3 2337-1 5678.6 2407.8 5778.6 2480.4 5880.3 29 30 .5000 2269.5 5581.9 2338.3 5680.2 2409.0 5780.2 2481.6 5882.0 30 31 .5167 2270.6 5583.5 2339-5 5681.9 2410.2 5781.9 2482.9 5883.7 31 32 •5333 i 2271.7 5585.1 2340.7 5683.5 2411.4 5783.6 2484.1 5885.4 32 33 •55oo 2272.8 5586.8 2341-9 5685.2 2412.6 5785-3 2485.3 5887.2 33 34 •5667 , 2273.9 5588.4 2343-Q 5686.8 2413.8 5787.0 2486.5 5888.9 34 35 .5833 2275.1 5590.1 2344-1 5688.5 2415.0 5788.7 2487.8 5890.6 35 36 .6000 2276.2 5591.7 2345-3 5690.2 2416.2 5790.3 2489.0 5892.3 36 37 .6167 2277.3 5593-3 2346.5 5691.9- 2417.4 5792.0 2490.3 5894.0 37 38 •6333 2278.5 5594-9 2347.7 5693-5 2418.6 5793-7 2491.5 5895-7 138 39 .6500 2279.7 5596.6 2348.9 5695-2 2419.8 5795-4 2492.7 5897.5 '39 40 .6667 2280.8 5598.2 2350.0 5696.8 2421.0 5797-1 2493.9 5899.2 40 41 •6833 2281.9 5599-8 2351-2 5698.5 2422.2 5798.8 2495.2 5900.9 4i 42 .7000 2283.0 5601.4 2352.3 5700.1 : 2423-4 5800.4 2496.4 5902.6 42 43 .7167 2284.1 5603.1 2353-5 5701.8 ' 2424.6 5802.1 2497.7 5904-3 43 44 •7333 2285.3 5604.7 2354-7 5703.4; 2425.8 5803.8 2498.9 5906.0 44 45 .750O 2286.5 5606.4 2355-8 5705.1 ' 2427.0 5805.5 2500.1 5907-7 45 46 .7667 2287.6 5608.0 2357-o 5706.8 2428.2 5807.2 2501.3 5909.4 46 47 •7833 2288.7 5609.6 2358.1 5708.5 ; 2429.4 5808.9 2502.6 59H.2 47 48 .8000 2289.9 5611.2 2359-3 5710.1 2430.6 5810.6 2503.8 5912.9 48 49 .8167 2291. 1 5612.9 2360.5 57ii-8i 2431-8 5812.3 2505.1 59146 49 50 .8333 2292.2 5614.5 2361.7 5713.4 2433-0 5814.0 2506.3 5916.3 SO 51 .8500 2293.3 5616.2 2362.9 57I5-I 2434.2 5815.7 2507-5 59i8. 1 5 1 52 .8667 2294.4 5617.8 ; 2364.0 5716.7 2435-4 5817.3 2508.7 5919.8 52 53 .8833 2295.6 56l9.4i 2365-1 57i8.4 2436.6 5819.0 2510.0 5921-5 53 54 .9000 2296.7 562I.O 2366.3 5720.0 2437-9 5820.7 2511-2 5923-2 54 55 •9167 2297.9 5622.7 2367.5 5721.7 2439.1 5822.4 2512.5 5925.o 55 56 •9333 ; 2299.0 5624.3 i 2368.7 5723-4 2440.3 5824.1 2513-7 5926.7 56 57 .9500 ; 2300.2 5625.9 2369.9 5725.1 2441.5 5825.8 2515-0 5928.4 57 58 .9667 1 2301.3 5627.5 2371.0 5726.7 2442.7 5827.5 2516.2 5930.I 5» 59 •9833 J 2302.4 5629.2 2372.2 5728.4 2443.9 5829.2 2517-5 5931-9 ., 59 FUNCTIONS OF THE ONE-DEGREE CURVE 193 Use ioc >' Chords ud to 8 3 Curves Use 25' Chords up to 3 2 Curves Use 50' Chords up* to 16 Curves Use 10' Chords above 32 Curves m I a O !U . tl be d 9 Ext. Tan. Ext. Tan. Ext. Tan. ! Ext. Tan. .0000 i 2518.7 5933-6 2594.2 6038.2 2671.8 6144.7 ; 2751.5 6253.2 1 .0167 i 2520.0 5935-3 i 2595-5 6040.0 2673.1 6146.5 2752.9 6255.1 I 2 •0333 2521.2 5937-0 2596.8 6041.7 2674-4 6148.3 2754-2 6256.9 2 3 .0500 2522.4 5938.8 ; 2598.1 6043.5 i 2675-7 6150. 1 2755-6 6258.7 3 4 .0667 2523.6 S940-5 i 2599-3 6045.2 | 2677.0 6151.9 2756.9 6260.5 4 5 .0833 2524.9 5942.3 2600.6 6047.O ■ 2678.4 6153-7 2758.3 6262.4 5 6 .1000 2526.1 5944-0 2601.9 6048.7 i 2679.7 6155-4 2759.6 6264.2 6 7 .1167 2527.4 5945-7 2603.2 6050.5 26S1.0 6157.2 2761.0 6266.0 7 8 •1333 2528.6 5947-4 2604.4 6052.2 j 2682.3 6159.0 2762.3 6267.8 8 9 .1500 2529.9 5949-2 2605.7 6054.O ! 2683.6 6160.8 2763.7 6269.7 i 9 10 .1667 2531-1 5950.9 2607.0 6055.8 2684.9 6162.6 2765.0 6271.5 10 11 .1833 2532.4 5952.7 2608.3 6057.5 2686.3 6164.4 2766.4 6273.4 II 12 .2000 2533-6 5954-4 2609.6 60S9.3 2687.6 6166.2 2767.7 6275.2 12 13 .2167 2534-9 5956.1 2610.9 6o6l. I 26S8.9 6168.0 2769.1 6277.0 13 14 • 2333 2536.1 5957-8 2612. 1 6062.8 269O.2 6169.8 2770.4 6278.8 14 IS .2500 2537.4 5959-6 2613.4 6064.6 ! 2691.5 6171.6 2771-8 6280.7 l g 16 .2667 2538.6 5961.3 2614.7 6066.4 ; 2692.8 6173-4 2773-1 6282.5 16 17 .2833 2539-9 5963-1 2616.0 6068.2 ; 2694.2 6175-2 2774-5 6284.4 17 18 .3000 2541. 1 5964-8 2617.3 6069.9 2695.6 6177.0 2775-8 6286.2 18 19 •3167 ; 2542.4 5966.5 2618.6 6071.7 2696.9 6178.8 2777.2 6288.0 19 20 ■3333 2543-6 5968.2 2619.8 6073.4 ' 2698.1 6180.6 2778.5 6289.8 20 21 •35oo 2544.9 5970.0 2621. 1 6075.2 2699.5 6182.4 2779.9 6291.7 21 22 .3667 2546.1 5971-7 2622.4 6077.0 2700.8 6184.2 2781.2 6293-5 22 23 .3833 2547.4 5973-5 2623.7 6078.8 2702.1 6186.0 2782.6 6295.4 23 24 .4000 2548.6 5975-2 2625.0 6080.5 2703-4 6187.8 2784.0 6297.2 24 25 .4167 2549-9 5977-0 ' 2626.3 6082.3 2704.8 6189.7 2785.4 6299.1 ' 25 26 •4333 2551-2 5978.7 1 2627.6 60S4.I 2706.1 6191.5 2786.7 6300.9 26 27 .4500 2552.5 5980.5 , 2628.9 6085.9 ij 2707.4 6193-3 2788.1 6302.7 27 28 .4667 2553-7 5982.2 II 2630.2 6087.6 1 2708.7 6195-1 2789.4 6304.6 28 29 .4833 j 2555-Q 5983-9 2631.5 6089.4 2710. 1 6196.9 2790.8 6306.4 29 30 .5000 2556.2 5985.6 2632.7 6091.2 2711-4 6198.7 2792.1 6308.2 30 3i .5167 2557.5 5987-4 2634.0 6093.0 2712.7 6200.5 2793-5 6310.1 31 32 •5333 2558.7 5989.1 ; 2635.3 6094.7 2714.0 6202.3 2794-9 6311.9 32 33 .5500 2560.0 5990.9 2636.6 6096.5 2715-4 6204.1 2796.3 6313.8 33 34 •5667 , 2561.2 2562.5 5992.6 i 2637.9 6098.3 2716.7 6205.9 2797-6 6315.6 34 35 .5833 5994-4 2639.2 6100. 1 2718.0 6207.7 2799.0 6317.5 35 36 .6000 2563.8 5996.1 2640.5 6101.8 2719.3 6209.5 2800.3 6319-3 36 37 .6167 2565.1 5997-9 2641.8 6103.6 2720.7 6211.4 2801.7 6321.2 37 38 •6333 2566.3 5999.6 2643.1 6105.4 2722.0 6213.2 2803.1 6323.0 Li 38 39 .6500 | 2567.6 6001.4 2644.4 6107.2 2723-4 6215.0 2804.5 6324.9 J 39 40 .6667 ! 2568.8 6003.1 2645.7 6109.0 ' 2724.7 6216.8 2805.8 6326.7 40 41 .6833 2570.1 6004.9 2647.0 6110.8 j 2726.0 6218.6 2807.2 6328.6 41 42 .7000 2571.3 6006.6 1 2648.3 6112.5 1 2727.3 6220.4 2808.6 6330.4 42 43 .7167 i 2572.6 6008.4 2649.6 6114-3 2728.7 6222.3 2810.0 6332.3 43 44 •7333 J 2573.9 6010.1 2650.9 6116.1 1 2730.0 6224.1 2811.3 6334-1 |44 45 .7500 ! 2575.2 6011.9 1 2652.2 6117.9 1 2731.4 6225.9 2812.7 6336.0 45 46 .7667 1 2576.4 ! 6013.6 ' 2653.5 6119.7 2732.7 6227.7 2814. 1 6337.8 46 47 .7833 ; 2577-7 6015.4 ; 2654.8 6121.5 2734-1 6229.5 2815.5 6339.7 ^7 48 .8000 2578.9 6017. 1 ; 2656.1 6123.2 2J35-4 6231.3 2816.8 6341.5 48 49 .8167 2580.2 6018.9 2657.4 6125.O 2736.7 6233.2 2818.2 6343-4 149 50 .8333 2581.5 6020.6 2658.7 6126.8 ' 2738.O 6235.0 2819.6 6345-2 SO 5i .8500 2582.8 6022.4 2660.0 6l28.6 ; 2739.4 6236.8 2821.0 6347.I 51 52 .8667 2584.0 6024.1 ! 2661.3 6130.4 2740.7 6238.6 2822.3 6349-0 52 53 .8833 2585.3 6025.9 2662.6 6132.2 2742.1 6240.5 i 2823.7 6350.9 53 54 .9000 2586.6 6027.6 2663.9 6133.9 2 743-4 6242.3 i 2825.1 6352.7 54 55 .9167 2587.9 6029.4 2665.3 6135.7 2744.8 6244.2 ' 2826.5 6354-6 55 56 •9333 2589.1 6031. 1 2666.6 6137.5 2746.1 6246.0 2827.8 6356.4 56 57 .9500 2590.4 6032.9 2667.9 6139-3 2747-5 6247.8 2829.2 6358.3 57 58 .9667 2591-7 6034.6 2669.2 6141.1 2748.8 6249.6 2830.6 6560.1 58 59 > .9833 : 2593-0 | 6036.4 , 2670.5 6142.9 2750.2 6251.4 2832.0 6362.0 59 194 THE SURVEY Use ioo' Chords up to 8° Curves Use 25' Chords up to 32 Curves Use 50' Chords up to 16 Curves Use 10' Chords above 32 Curves d 3 O C 3_ O Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. .0000 2833-4 6363.8 2917-5 6476.6 3004.0 6591.6 3092.9 6709.0 1 .0167 2834-8 6365.7 2918.9 6478.5 3005.5 6593-6 3094.4 6711.0 I 2 •0333 2836.1 6367.5 2920.3 6480.4 3006.9 6595.5 3095-9 6712.9 2 3 .0500 2837-5 6369.4 2921.8 6482.3 3008.4 6597.5 3097.4 6714.9 3 4 .0667 2838.9 6371.3 2923.2 6484.2 3009.8 6599.4 3098.9 6716.9 4 5 •0833 2840.3 6373.2 2924.6 6486.1 3011.3 6601.3 3100.4 6718.9 5 6 .1000 2841.7 6375-0 2926.0 6488.0 3012.8 6603.2 3101.9 6720.8 6 7 .1167 2843.1 6376.9 2927.5 6489.9 3014-3 6605.2 3103.4 6722.8 7 8 • 1333 2844.5 6378.7 2928.9 6491.8 3015.7 6607.1 3104.9 6724.8 8 9 .1500 2845.9 6380.6 2930.3 6493.7 3017.2 6609.1 3106.4 6726.8 9 10 .1667 2847.2 6382.5 2931-7 6495.6 3018.6 6611.0 3107.9 6728.8 10 11 .1833 2848.6 6384.4 2933.2 6497.5 3020.1 6613.0 3109.5 6730.8 11 12 .2000 2850.0 6386.2 2934.6 6499.4 3021.6 6614.9 3111.0 6732.7 12 13 .2167 2851.4 6388.1 2936.1 6501.3 3023.1 6616.9 3112.5 6734-7 13 14 •2333 2852.8 6389.9 2937.5 6503.2 3024.5 6618.8 3114-0 6736.7 14 15 .2500 2854.2 63918 2938.9 6505.2 3026.0 6620.8 3II5-5 6738.7 x 5 16 .2667 2855-6 6393.7 2940.3 6507.1 3027.5 6622.7 3ii7.o 6740.7 16 17 .2833 2857.0 6395-6 2941.8 6509.0 3029.0 6624.7 3118.5 6742.7 17 18 .3000 2858.4 6397-4 2943.2 6510.9 3030.4 6626.6 3120.0 6744.6 18 19 .3167 2859.8 6399-3 2944.7 6512.8 3031.9 6628.6 3121.5 6746.6 19 20 •3333 2861.2 6401.2 2946.1 6514-7 3033.3 6630.5 3123-1 6748.6 20 21 •3500 2862.6 6403.1 2947.5 6516.6 3034-8 6632.5 3124.6 6750.6 21 22 .3667 2864.0 6404.9 2948.9 6518.5 3036.3 6634.4 3126. 1 6752.6 22 23 .3833 2865.4 6406.8 2950.4 6520.4 3037-8 6636.4 3127.6 6754-6 23 24 .4000 2866.7 6408.7 2951.8 6522.3 3039-3 6638.3 3129.1 6756.6 24 25 .4167 2868.1 6410.6 2953-3 6524-3 3040.8 6640.3 3130.7 6758.6 25 26 •4333 2869.5 6412.4 2954-7 6526.2 3042.2 6642.2 3132.2 6760.6 26 27 .4500 2870.9 6414.3 2956.2 6528.1 3C43.7 6644.2 3133-7 6762.6 27 28 .4667 2872.3 6416.2 2957.6 6530.0 3045-2 6646.1 3135-2 6764.6 28 29 .4833 2873.7 6418. 1 2959.0 6531.9 3046.7 6648.1 3136.7 6766.6 29 30 .5000 2875.1 6419.9 2960.4 6533.8 3048.1 6650.0 3138.3 6768.6 30 3i .5167 2876.5 6421.8 2961.9 6535-8 3049.6 6652.0 3139-8 6770.6 3i 32 • 5333 2877.9 6423.7 2963.3 6537-7 3051-1 6653.9 3I4I-3 6772.6 32 33 .55oo 2879.4 6425.6 2964.8 6539-6 3052.6 6655.9 3142.9 6774-6 33 34 .5667 2880.8 6427.5 2966.2 6541.5 3054-1 6657.8 3144-4 6776.6 34 35 .5833 2882.2 6429.4 2967.7 6543-4 3055-6 6659.8 3145-9 6778.6 35 36 .6000 2883.6 6431.2 2969.1 6545-3 3057-0 6661.7 3147.4 6780.6 36 37 .6167 2885.0 6433-1 2970.6 6547-3 3058.5 6663.7 3149.0 6782.6 37 38 •6333 2886.4 6435.0 2972.0 6549-2 3060.0 6665.7 3150.5 6784.6 38 39 .6500 2887.8 6436.9 2973.5 6551-1 3061.5 6667.7 3152.0 6786.6 39 40 .6667 2889.2 6438.8 2974.9 6553.0 3063.0 6669.6 3153-5 6788.6 40 4i .6833 2890.6 6440.7 2976.4 6555.o 3064.5 6671.6 I3I55.I 6790.6 41 42 .7000 2892.0 6442.5 2977.8 6556.9 3066.0 6673.5 3156.6 6792.6 42 43 .7167 2893.4 6444.4 2979.3 6558.8 3067. 5 6675-5 3158.2 6794-6 43 44 •7333 2894.8 6446.3 2980.7 6560.7 3068.9 6677.4 3159.7 6796.6 44 45 .7500 2896.3 6448.2 2982.2 6562.7 3070.4 6679.4 3161.2 6798.6 45 46 .7667 2897.7 6450.1 2983.6 6564.6 3071.9 6681.4 3162.7 6800.6 46 47 .7833 2899.1 6452.0 2985.1 6566.5 3073-4 6683.4 3164.3 6802.6 47 48 .8000 2900.5 6453-9 2986.5 6568.4 3074-9 6685.3 3165-8 6804.6 J48 49 .8167 2901.9 6455.8 2988.0 6570.4 3076.4 6687.3 3167.4 6806.6 49 50 .8333 2903.3 6457.6 2989.4 6572.3 3077.9 6689.2 3168.9 6808.6 SO 51 .8500 2904.7 6459-5 2990.9 6574-3 3079-4 6691.2 3170.5 6810.6 51 52 .8667 2906.1 6461.4 2992.3 6576.2 3080.9 6693.2 3172.0 6812.6 52 53 .8833 2907.6 6463.3 2993.8 6578.1 3082.4 6695.2 3173-6 6814.7 53 54 .9000 2909.0 6465.2 2995.2 6580.0 3083.9 6697.1 3I75-I 6816.7 54 55 .9167 2910.4 6467.1 2996.7 6582.0 3085.4 6699.1 3176.6 6818.7 55 56 •9333 2911.8 6469.0 2998.1 6583.9 3086.9 6701. 1 3178.1 6820.7 56 57 .9500 2913-3 6470.9 2999.6 6585.8 3088.4 6703.2 3179-7 6822.7 57 58 .9667 2914.7 6472.8 3001. 1 6587.7 3089.9 6705.2 3181.2 6824.7 58 .9833 2916. 1 6474.7 3002.6 6589.7 3091.4 6707.1 3182.8 6826.8 59 FUNCTIONS OF THE ONE-DEGREE CURVE 195 Use ioo' Chords up to 8 Use 50' Chords up to i6' Curves Curves Use 25' Chords up to 3 2° Curves Use 10' Chords above 32 Curves en ^S IOO° ; co 1 O Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. N 4386.4 8337-2 4516.9 8495.1 4651.6 8657.1 4790.7 8823.4 4934-4 8994o O IO 4407.9 8363.2 4539-1 8521.8 4674-5 8684.5 4814.4 8851.61 4958.9 9023.2 IO 20 4429.5,8389.4 4561.3 8548.6 4697.5 8712.0 4838.1 8879.9 4983.4 9052.3 20 30 4451.2)8415.6 4583.7 8575-6 4720.6I8739.7 4862.0 8908.3! 5008.1 9081.5 30 40 4473.0 8442.0 4494.9J8468.5 4606.2 8602.6 4743.9I8767.5 4885.0 8936.8,5032.9 9110.8 40 50 4628.9 8629.8 4767.2 8795.4 4910.2 8965.51 5057-9 9*40.3 50 60 4516.9 8495.1 4651.6 8657.1 4790.7 8823.4 4934-4 8994-3 5083.0 9169.9 60 116 H7° 118 ii9° 120 Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. Ext. Tan. 5083.0 9169.9 5236.6 9350.5 5395-4 9536.3 5559-7 9727.6 5730.0 9,924-6 IO 5108.2 9199-7 5262.6 9381. 1 5422.4 9567.8 5587.7 9760.0 5758.9 9,958.i 10 20 5133-6 9229.6 5288.9 94H-9 5449-5 9599-5 5615-8 9792.6 5788.0 9,990.6 20 30 5I59-I 9259.6 5315-3 9442.8JJ5476.8 9631-3 5644.1 9825.4 5817.3 10,025.6 30 40 5184.8 9289.8 534i-8|9473-8 5504.3 9663.2 5672.6 9858.3 5846.8 10,059.7 40 50 5210.6 9320.1 5368.5 9505.0 !5532.o 9695-3 5701.2 9891.4 5876.4 10,093.7: 50 60 5236.6 9350.5 5395-4 9536.3'5559.7 1 I' 9727.6 5730.0 9924.6 5906.1 i 10,127.7 60 METHODS OF RUNNING CURVES 197 r w A central angle L = 100 X-=:= ~ 1 ^— X 100. D Degree of curvature For the convenience of the field engineer column 1, Table 30, gives the central angle (A) in degrees and minutes (as read by the transit) ; column 2 gives the same angle expressed in degrees and decimals for figuring curve lengths. Externa^ PC. \ PI -— \*r Center of Curve Fig. 52 Tangent length and externals. Sketch No. 52 shows a general curve problem. The deflection angle between the tangents at the point of intersection (P.I.) = the central angle of the curve that will fit these tangents; it is referred to as A. The tangent distances equal the distance from the P. C. (be- ginning of curve) to the P. I. or P. I. to P. T. (end of curve) and is expressed by the formula T = Radius X tangent of — (4) Therefore, for a given central angle A, the tangent length is directly proportional to the radius. If the tangent lengths of a i° curve for different A's are tabulated, the tangent length for any desired degree of curve equals tangent length for i° curve for the specified A divided by the degree of the desired curve expressed in degrees and decimals of a degree. Expressed as a formula this reads: „, . . Tangent i° curve for specified A (5) Tangent for desired curve = — and reversing the formula we can determine the desired degree of curve for a specified tangent length by the formula Tangent i° curve for specified A (6) Specified tangent length desired. The external is the distance from the P. I. to the curve arc on the line between the P. I. and the center of the curve. It is determined by the formula: \ (7) Ext = — — : t — Radius = Radius! — — ; s — 1 land is directly Cosine A \ Cosine A / 198 THE SURVEY proportional to the radius in the same manner as the tangent length; therefore, the external of any desired curve for a specified A equals the external of a i° curve for that A divided by the degree of curvature. - Point of Tangency Tangent^ A ng/e -i Central Angle A Xenter of Curve Fig. 53 4 Curve. The Distances along the Arc between (HZ) (2-3) (3 - 4 J etc. are each 100'. Fig. 54 Expressed as a formula this reads: . '. , Ext. i° curve for specified A (8) External for desired curve = ■= METHODS OF RUNNING CURVES 199 and reversing, as for tangents, the desired degree of curvature is obtained that gives a specified external distance, by the formula, Ext. i° curve for specified A (9) Specified Ext. distance desired. Methods of running curves. Curves are run in the field by tangent offsets, middle ordinates or deflection angles. Deflec- tion angles is the simplest method and is almost universally used. It is based on the principle that the angle 5 between the tangent and arc chord, one end of which is at the point of tangency, is equal to \ the central angle subtended by that chord. Sup- pose the angle A is 4 and the arc length ST = 100 feet. This curve would then be a 4 curve. From the previous definitions locate the point T (Fig. 53) by turning the deflection angle S = 2 from the tangent and measuring 100 feet of arc in such a position that the end of the arc would be on the line of the chord ST. It is impossible to conveniently measure the arc distance and for all practical purposes a chord length of ioo' will answer for a 4 curve (see discussion, page 173). Suppose we wish to locate the points 2, 3, 4, 5, and 6 on the 4 curve from point 1 or the P. C. of a curve (Fig. 54). A Set the transit at the P. C; if we turn a deflection— = 2 from 2 the tangent xy the line of sight will pass through the point 2; if we turn — = 4 the line of sight will pass through point 3; 6°, point 4, etc.; it only remains to measure to these points to locate them definitely. This can be done in two ways, by measuring the distances 1-2, 1-3, 1-4, 1-5, etc., or by measuring 1-2, 2-3, 3-4, 4-5, etc. In the first case the difference between the length of arc and the chord length becomes so great that, unless a correction is made, the points are not exactly located; that is, the length of arc between points 1, 2, 3, 4, 5, 6, = 500' while the chord length 1-6 = 497.5'; also, it takes longer to measure the distances 1-2, 1-3, 1-4, 1-5, 1-6, etc., than it would 1-2, 2-3, 3-4, 4-5, etc. In the second method we can use chords of ioo' from 1-2, 2-3, etc., with no appreciable error, as the distance measured by chords 1, 2, 3, 4, 5, 6, = 499-94'. Therefore, the method usually adopted is to turn the deflection ^4 angle— and measure the chord 1-2, which locates the point 2; 2 D then turn the deflection angle — and measure the chord distance 2-3, locating point 3, etc. The fact has been mentioned that the use of the chord distance as equal to the arc introduces an error but that this error is of no importance for a 4 curve: As the degree of curvature increases, the difference between an arc length of ioo' and the chord length becomes greater, and it is necessary to determine the limit of curvature that will allow the use of ioo' chords in locating curve 2oo THE SURVEY points. On page 145 the statement is made that center line chain- ing should be correct to within 0.1' per 100' of length, which allows a difference in arc and chord of 0.1' This occurs when the degree of curvature reaches g° per 100'. The difference can then be reduced by the simple expedient of using 50' chords, which reduces the error for this degree of curvature from 0.10' per 100' of length using 100' chords to 0.02' using 50' chords; 50' chords can be used up to 18 curves and beyond that point 25' chords. It is better not to use the full limit of allowable error, and a good working rule is ioo' chords up to 8° curves, 50' chords up to 1 6° curves, 25' chords to 32 and beyond that io' chords. For any given curve the deflection angle and central angle are directly proportional to the length of the arc, and if the de- flection angle for 100' arc of io° curve equals 5 the deflection angle for one foot of arc of io° curve equals-^— = = 100 100 3 minutes. An example of a typical simple curve problem can now be given: k Tan.Ler?gth-—^ PL=5t ^ 22f4Z6 B Fig. 55 To determine the degree of curvature desired from a fixed external distance At station 23 + 42.6 we have a deflection angle of 25 10' be- tween tangents AB and B'C; suppose upon examining the ground it is decided that to fit the old roadbed and give good alignment the curve should be located somewhere between 13. 5' and 14. 5' to the right of the transit point at station 23 + 42.6. Proceed as follows: from table 30 pick out the external for a i° curve for A = 25 10', this equals 141.0'. The problem is to determine the degree of curvature that will give an external of between 13.3' and I4-5'- Use formula (9). = Ext. x° curve for 25° 10' = 141^ = . curve> 13-5 , 13.5 Ext. i° curve for 25° 10 = 14^0 = . curye 14.5' 14.5 To fit the conditions some curve must be selected between a SIMPLE CURVE PROBLEM 20I 10.44 and a 9.72 . A io° curve would be naturally selected as being the simplest to figure. To determine the required degree of curvature for a fixed tangent length Take the same problem as above except there must be a tangent length of between 127' and 12c/. Use formula (6). Tangent i° curve for 25 10' 1279.1' D = — — — , = 10. 07 curve. 12 D = Tangent i° curve for 25° 10 127 1279. 1 9. 91 curve. 129 129 Table No. 30 gives tangent for 25 10' — 1 279.1'. These limiting values would result in the selection of a io° curve. The degree of the desired curve is usually selected in one of these two ways; ordinarily it is determined by the external distance. A=25°W D= 10° R= 573.0 T~ 127.9 L- 251.7 RC=5ta.22H4.7 P.T.^Sta. 24+66.4 Fig. 56 Simple Curve Problem. Case 1. To compute the notes for a io° curve for a deflection angle of 25 10' between tangents at station 23 + 42.6. Central angle = 25 10'. Table No. 30 gives the tangent i° curve for 25 10' = 1 279.1. 1 2 70. 1 Tangent io° curve = — = 127.91 10 The station of the P.C. then equals station 23 + 42.6 P.I. minus 127.9' = station 22 + 14.7. rpi , .* r A 25.16667° , , The length of curve = — = -^ — 5 — X 100 = 251.7 feet. D 10 The station of the P. T. (Tangent point, or end of the curve) as measured around the arc is then station (22 + 14-7 P.C.) + 251.7' = station 24 + 66.4. The rule for running curves requires the use of 50' chords for a io° curve. We must, therefore, figure the deflections for the even stations and the 50' stations as follows: Station 22 + 50, 23 + 00, 23 + 50, 24 + 00, 24 + 50, and to check the curve station 24 -f- 66.4. For a io° curve, Table No. 29. 202 THE SURVEY The deflection for ioo' of arc = 5 1' " " = ° 03' The distance from the P.C. station 22 + 14.7 to station 22 + 50 is 35.3'; the deflection per foot = o° 03', for 35-3' = 35-3 X o° 03' = 105.9 minutes = i° 46'. The distance P.C. to station 23 +00 equals 85.3', or 50' farther than for station 22 + 50; the deflection per 50' of arc equals 2 30'; therefore, the deflection for station 23 + 00 equals the deflections for station 22 + 50 (i° 46') plus 2 30', the de- flection for 50' of arc or 4 16'; in a like manner the deflection for station 23 + 50 is 6° 46'; for 24 + 00, 9 16'; for 24 + 50, ii° 46'; the distance from station 24 + 50 to the P.T. station 24 + 66.4 is 16.4'; the deflection for 16.4' equals 16.4 X o° 03' = 49. 2'; the deflection for station 24 + 66.4 is, therefore, (n° 46' + 49O = 12° 35'; if the deflection notes have been properly figured this last deflection to the P.T. should always be § the central angle of the curve; in this case \ of 25 io', which equals 12 35', checking the notes. To run the curve. Set up the transit at the P. I.; sight along the tangent (B.A.), measure off the distance 127.9 (tangent length) along this line and set the P. C. exactly on the line. In a like manner set the P.T. on the forward tangent (B'.C.) 127.9' from the P.I. Then set up the transit on the P.C. and with the vernier at o° 00' sight on the P. I., using the lower plate motion. Loosen the upper motion and deflect i° 46'; measure along this line 35.3', which locates station 22 + 50 on the curve arc; then loosen the upper motion and set the vernier to read 4 16'; measure 50' from the just located station 22 + 50, so that the forward end of the tape is in line with the transit deflection of 4 16'; this locates station 23 + 00 on the curve arc. In a like manner deflect 6° 46' and measure forward 50' from station 23 + 00 to station 23 + 50, etc., until the P.T. is reached. If the curve has been correctly run the last deflection of 12 35' will strike the previously located P. T. and the distance from station 24 + 50 to this P.T. will be 16.4'; if the distance checks within 0.2' it is sufficiently close. The above problem and method of laying out a curve is the simplest form encountered; in it we assume that the P. I., P.T. and all intermediate points on the curve are visible from the P.C. and that the P.I. is accessible. In nine cases out of ten this method is applicable to road curves, but where the P.I. occurs outside of the road fences it sometimes is located in a stream, pond, building, etc., and cannot be occupied. This is known as the problem of the inaccessible P.I. More often it is impossible to see the P.T., or some inter- mediate point on the curve from the P.C, which necessitates intermediate transit points on the curve. The problem of inac- cessible P.C.s or P.T.s is so rare it will not be illustrated. I PROBLEM OF THE INACCESSIBLE P. I. 203 Problem of the Inaccessible P. I. Case 2. The point H {P-L) cannot be occupied. Locate any two convenient points, <> and t on the tangents A.B. and B'.C. and measure the distance st equals, say, 110.5'. Set the transit at s and measure the angle between the line A.s. produced and st, say, 5 10'; in a similar manner measure the angle at t between st produced and the forward tangent /C, say 20 00'. The total deflection then between the tangent AsB and B'tC or the central angle of the curve to be run is the sum of these two deflections, angles (5 10') + (20 oo') = 25 10'. Assuming a io° curve is desired we must locate the P.C. from the point s and the P.T. from the point t. k _ 127.9- >j W—39.0'-** 88.9'— -A , , LI o [PC \S=Sta. 22+53.7. H jt ?.T. inaccessible D Fig. 57 In the preceding simple curve problem the tangent length of a io° curve with a central angle of 25 10' was figured to be 127.0/; it, therefore, remains to compute the distance sH which subtracted from 127.9' will give the distance from s along the tangent sA to the P.C, of the curve. In a similar manner compute tH, which subtracted from 127.0/ gives the distance along the forward tangent tC to the P. T. of the curve. Knowing the station of the point s as measured along the tangent A.B. the station of the P.C. is determined;, then figure the deflections in the usual manner and run the curve. For the values given the computations are as follows: To determine sH and Ht. Use the law of sines (see Trigono- metric formulae, page 477). sH:st: sin 20 oo': sin 25 10' _ st sin 20 oo r = 110.5 X 0.34202 = gg , s ~ sin 25 io 7 " ' 0.42525 = st sin 5 io' = 110.5 X 0.09005 = / sin 25 io' 0.42525 Therefore, the distance from ^ to the P.C. is 127.9' — 88.9' = 39. o'. The distance from t to the P.T. is 127.9 — 23.4 = 104.5. Having these distances the P.C. and P.T. are located. As- sume that station of s was measured along the tangent AB and found to be station 22 -f- 53.7. 204 THE SURVEY The station of the P.C. then equals 22 -\- 14.7 " " P.I. " " 23 + 42.6 " " P.T. " " 24 + 66.4, using the length of curve figured in Case 1. The deflections are figured and the curve run as in Case 1, assuming that all the curve points are visible from the P.C. Case 3. Where the P.T. or intermediate points on the curve are not visible from the P.C. (a) Where an intermediate set-up is required. Use the same curve as in Case 1. The deflections for the different curve points were figured as follows: Deflections. Instrument at P.C. , foresight on P.I. P.C. Station 22 + 14-7 Deflection o° OO' 22 + 5o i° 46 23 + 00 4° 16' 23 + so 6° 46' 24 + 00 9° 16' 24 + 5o n° 46' 24 + 66.4 12° 35' P.C. 22+ 14.7 RI. Z2+50 P.T. 24+66.4- FlG. 58 Set up the instrument at the P.C. and locate the points 22 + 50, 23 + 00 and 23 + 50; suppose 24 + °° is not visible, set up at station 23 + 50, set the vernier at o° 00' and back sight on the P.C; transit the telescope and finish the curve, using the same deflections as figured for the instrument set up at the P.C; that is, turn the deflection of 9 16' for station 24 + 00, ii° 46' for 24 + 50, and 12 35' for the P.T. In general it can be said that whenever the P.C is used as a back- sight from the intermediate set-up, set the vernier at o° oo' when sighting on the P.C; transit the telescope and use original notes for the balance of the curve. (b) Where two or more intermediate set-ups are required. For the first set-up, say, at 23 + 50, proceed as above and set station 24 + 00; suppose 24 + 50 is not visible from station 23 + 50; set up at station 24 + 00 and with the vernier reading 6° 46' back sight on station 23 + 50; transit the telescope, set the vernier to read n° 46' for station 24 + 5°> and proceed, using the same deflections as originally figured. In general, where the P.C is not visible from, the intermediate set-up, set the PROBLEM OF THE INACCESSIBLE P. I. 205 vernier to read the deflection figured for the point used as a backsight; transit the telescope and proceed with the curve, using the notes originally figured. That is, if the instrument is set up at station 24 -f- 00 and 22 + 50 used as a backsight, the vernier is set at i° 46', and using the lower motion the wire is set on station 22 + 50; then transiting the telescope the curve is run by setting the vernier at n° 46' for station 24 + 50, etc. If station 23 -f- 00 is used as a backsight, set the vernier at 4° 16' when sighting the machine; then transit and proceed as above. These three cases cover any ordinary road curve problems. CHAPTER IX OFFICE PRACTICE Under office practice we include i. Mapping the preliminary survey. 2. Designing the improvement and estimating the quantities. 3. Producing a finished set of plans from which the road can be constructed. 1. Mapping the preliminary survey. The mapping of the preliminary survey serves as a base from which the design of the new work, and the quantities necessary thereto, can be built up. It consists of three views of the road: the plan, showing the topographic features; the profile, showing the longitudinal differences of elevation, and the cross-sections, showing the constantly changing transverse shape. The scales in general use are as follows: Plan Profile Cross-sections 1" = 100' 1" = 1" = 100' horizontal io' vertical 1" = 10' 1"= so' 1* = 1" = 50' horizontal io' vertical 1" = 5' or 1" = 4' i" = 20' 1" = 1" = 20' horizontal 5' vertical 1*= 5' or 1" = 4' I" = I0' 1" = 1* = 10' horizontal 10' vertical 1" = 2' The 100' scale is too small for convenience in design, and earth- work quantities figured from cross-sections plotted 1" to 10' are not reliable. For work on ordinary country roads, the 50' scale is generally adopted, using cross-sections plotted 1" to 5' or 1" to 4 r ; this scale is satisfactory for laying the grade line and computing the earthwork. The larger scales of 1" = 20' or 1" = io' are useful in village work where a large amount of detail must be shown. Plotting the center line. The survey center line can be plotted by deflection angles at the transit points, using a table of natural tangents, a vernier 206 PLOTTING THE CENTER LINE 207 protractor or an ordinary paper protractor graduated to 15 minutes. Where the center line has been well located in the field and there seems to be no necessity for a paper re-location, no great care need be taken in plotting the deflection angles, as in such a case the map serves more as a picture of the topographic features than as a basis for alignment. Where a random line has been run in the field and some shifting of the center line is necessary, both angles and distances must be accurately plotted. If any extensive change of alignment is made, the new deflections and distances should be checked by figuring the difference of latitude and longitude for both the survey line and the office line between the points of equality. Where the consideration of sight distance (see page 17) governs, Table No. 31 will be of service. For convenience in plotting the topography, the 100' survey stations are plainly marked. The most common mistakes in plotting the map are made by reversing the deflection, as right instead of left and vice versa, or in adding or omitting 100' in scaling long-tangent distances. The work should be checked for mistakes of this nature. All curve data is marked plainly on the map near the P.I. and shows The deflection angle A The degree of curve D The radius of curve R The tangent length T The length of curve L The station of the P.I. The station of the P.C. The station of the P. T. If the curves have been figured in the office and have not been run in the field it is good practice to scale the offsets from the tangent to the curve and mark them on the map. These offsets from the center line as run are then transferred to the cross-sections and the profile plotted from center line elevations on the cross-sections. Table No. 31 gives the approximate distance that an auto- mobile driver can see an approaching car, assuming that he is driving in the center of the macadam and that the approaching car is also in the center. Two distances are given for each curve, the first assuming that the line of sight is six feet from the ground, which is about right if the curve is on a straight grade, and makes the line of sight tangent to the cut slope of 1 on ij 19 feet off center for the narrow section shown in Fig. No. 7, page 30, and, second, assuming that the line of sight is close to the ground, as occurs on rounding the top of a hill, in which case the line of sight will be tangent to the side slope at, approximately, n' off center. 208 OFFICE PRACTICE Table 31 Degree of Curvature Radius of Curve Feet Sight Distance Case One. Feet Sight Distance Case Two. Feet 5 1146.0 400 310 6 7 955-o 818.6 375 35o 290 270 8 9 716.3 636.6 330 3io 250 235 10 573-o 295 220 12 477-5 270 200 14 16 18 20 409.3 358.1 318.3 286.5 245 230 220 210 18s 17s 165 160 30 191. 170 130 40 143.2 145 no So 114. 6 130 IOO Plotting the topography. If the topography has been recorded on a system of right-angle offsets, as suggested and illustrated on page 123, it can be easily and quickly plotted by using the transparent scale shown here. This scale gives the plus distance along the survey base line, or center line, and the offset distance from the line in one operation. As a general rule the plotting of the topog- raphy need not be checked. Level Computations. The survey computations of the Bench Levels are checked and a list of bench eleva- tions prepared; these elevations are used in cross-section level notes and from them the notes are computed between benches. As each bench is reached these notes are cor- rected to agree with the elevation adopted for that bench and then carried forward on the corrected basis. The allowable error for cross-section levels, as mentioned in the chap- ter on surveys, is less than 0.1 feet. The cor- rection of the levels at each bench prevents any cumulative error and makes the eleva- Fig. 59. — Conven- tions of the cross-section shots agree with ient Transparent the adopted bench elevations with an error Scale for Plotting of less than 0.1'. This is as close as the Topography PLOTTING THE PROFILE 209 readings can be plotted and as close as they can be read in the field. The computation of the bench levels and the adjustment of the cross-section notes should be checked by a competent man. The most common mistake in figuring the cross-section read- ings is to use the wrong height of instrument for a section. Such a mistake cannot be detected in plotting the sections, but is generally discovered when the profile is plotted. In checking the notes particular care should be taken on this one point. Pldtting the cross-sections. The cross-sections must be very carefully plotted, as the re- liability of the earthwork computation depends largely on their accuracy. The cross-section paper used should be exact in the divisions and should be printed or engraved from plates. Ruled paper is inaccurate. The plotting is checked by reliable men. Reading the shots back from the plotted cross-section is preferable to reading them from the book. The elevations of the center line and of the ditch line are written over the section. The station number or plus of each section is written on the right margin. The fact that the section has been graveled within the traveled way, that stone has been spread to a certain thickness, or any other fact that would influence the designer when laying a grade line, is noted on the section. See Fig. 60. Fig. 60 It is common practice to allow the inexperienced men to plot and check the cross-sections. We believe this is a mistake. This part of mapping is the most important of the preliminary plans; and the work should be plotted and checked so that the points are correct to the nearest 0.1 feet in elevation. These points are then connected with a fine ink line. Plotting the profile. The profile is plotted from the center-line elevations given in the cross-section notes unless the proposed center line does not coincide with the survey center line, in which case the elevations 210 OFFICE PRACTICE of the proposed line are projected from the previously plotted cros-s sections. It is not necessary to spend so much time for accuracy in plotting as on the sections, as the profile only serves as a guide in laying the grade line and no quantities depend upon its correctness. An error of 0.2 feet is allowable. The elevation of each plotted center-line point is recorded with its stationing. See Fig. 63. The Design. The completion of the profile finishes the preliminary mapping. The first operations of the office design are as follows : A . The selection of section. B. The depth of metalling. C. The laying of the grade line. These three points are so dependent on each other that they can- not be separated. The most experienced man available should do this part of the work. He should be thoroughly familiar with the road from field inspection, and in designing he follows the general principles dis- cussed in the chapters on Grades, Sections, and Foundations. His selection depends on a report of this character. PRELIMINARY DESIGN REPORT, NEW CONSTRUCTION Dec. 10, 1914. Division Engineer Dept. of Highways Dear Sir: In accordance with your request of Nov. 25th find enclosed report on a reasonable cost for the Town Line-Manitou State-County Highway. General Report and Estimate, Town Line — Manitou State-County Highway With a proper use of local materials a satisfactory road can be built at a cost of $94,000 or approx. $11,000 per mile including Engineering and Contingencies. An expenditure of $12,000 per mile would not however be excessive. The Braddocks Bay crossing is the expensive feature of this road; it raises the cost of the entire road about $1,000 per mile. Design No. 1 is recommended (see page 217). A detail report follows. Signed Designing Engineer. DESIGN REPORT 21 1 Detail Report and Estimate, Town Line — Manitou State-County Highway Length. 8.51 miles from the Ridge Road to Manitou Beach. Foundation Soil. Heavy soil not particularly good foundation Sta. o to 133; sandy soil balance of distance except across Braddocks Bay. A 9" thickness of some form of macadam is advisable Sta. o to 133; 7" or 8" the balance of the distance should be satisfactory except across Braddocks Bay where it is safe to figure on 12" to 15* of Stone. Grade. The present surface can be followed closely. The excavation should not exceed 2800 cu. yds. per mile except across Braddocks Bay; a rough estimate of borrow excavation for this fill is 15,000 cu. yds. Alignment. Good; no right of way required except possibly at Sta. 350 near the schoolhouse at the turn to Manitou. Traffic and Section. There is a heavy volume of automobile pleasure traffic and a light volume of heavy hauling traffic on this road. The large amount of pleasure travel requires from 16' to 18' of stone surface; the heavy hauling does not require over 12' to 14' full depth metaling. We recommend a graded section 26' to 28' wide between ditches in cut with an' width of full depth metal with 6' of extra width of local crusher run on the shoulders Sta. o to 133; a 14/ width with 4/ of stone on shoulders Sta. 133 to 260; a width of i2 r of full depth metal with 6' of stone on shoulders the balance of the distance except across Braddocks Bay where the entire width of metaling 16' should have the full depth. This road carries so much high speed traffic that it requires some form of bituminous macadam or if Waterbound is selected, it should be treated with calcium chloride immediately and have a surface coat of bitumen applied within three months. Railroad Crossings. Sta. 223 R. W. & O. Ry. crossing; no gates or flagman. In the summer time the crossing should have a flagman as the orchards cut off the view. The crossing is not par- ticularly dangerous, but during this season of the year the traffic on this road is entitled to better protection at this point. The approach grade from the south should be made easier. Drainage. No special features; approx. cost $3,500 exclusive of bridges above 5' span to be built by the towns. Dangerous Places. The Braddock Bay crossing is a dangerous one as the fill is high and the swamp is full of semi-fluid muck from 6' to 12' deep; a first-class concrete guard rail protection should be provided. MATERIALS Filler Sand. In abundance along road and from roadbed excava- tion. Gravel. The only good gravel is Lake Gravel; this can be ob- tained up to approx. 6000 cu. yds. 1 \ miles north of Sta. 350 and 3000 212 OFFICE PRACTICE yds. i mile west of Sta. 450. Probably this gravel can be used to advantage (screened or selected beach run) as bottom course Sta. 350 to 450 or as filler for sub-base bottom and on the shoulders. Stone. 15,000 cu. yds. of fence stone are available within a mile and a half of the road Sta. o to 133. There is practically no local stone Sta. 133 to 350. 4,000 cu. yds. of- fence stone are available within ij miles of Sta. 35o. This material runs about 20 % granite fit for top and the balance soft sandstone fit for bottom either as a sub-base bottom or crushed stone bottom. There is sufficient stone at the south end of the road to build a sub-base bottom with crushed stone filler; a local granite top with crushed stone on the shoulders from Sta. o to 133 and a local crushed stone bottom 5" thick Sta. 133 to about Sta. 200. There is sufficient stone at the north end to build about if miles of crushed stone bottom with stone on shoulders or ij miles of sub- base bottom with crushed stone filler and crushed stone on shoulders. I do not think there is enough granite to make it worth while to try and use a local top on any part of the north end. It is probably better to use an imported top from Sta. 133 to 450 and imported bottom Sta. 200 to 280. (See detail Stone Statement and Computations.) Crusher set up at Sta. 100. 15,000 cu. yds. field stone available within 3 miles maximum haul. Average haul 1 J miles. Assume for safety that only 11,000 cu. yds. are available with an average haul to crusher of 1 mile. Of this 11,000 cu. yds. field stone. 3,000 c. y. used for sub-base bottom average haul \ mile u u i 1 . n, / haul to crusher \ mile 1,000 c. y. crushed stone filler < tl frQm „ i u 700 c.y. " " " " shoulders { h ^ ul fr t ^ rUS V r I m ^ e u a rry f haul to crusher 1 mile 2,500 c. y. Top course < « frQm « 3 « 7,200 c. y. field stone used for local macadam, from Sta. o. to 133, leaving 3,800 c. y. available for crushed bottom and shoulder stone for road north of Sta. 133. 3,800 cu. yds. will produce approx. 3,000 cu. yds. of crushed bottom loose measure or about 2,300 cu. yds. of rolled measure. This will build 10,600 lin. ft. of 5" bottom 14' wide. We can therefore safely specify local bottom to Sta. 200 which will leave enough shoulder stone to use as far north as Sta. 300 if necessary. Crusher set up at Sta. 350. 4,000 cu. yds. available within i| miles, say average haul 1 mile. Assume for safety that 3,000 cu. yds. only are available average haul 1 mile. This will produce about 2400 cu. yds. crushed bottom stone loose measure or approx. 1,800 cu. yds. rolled measure. 1800 cu. yds. will build approx. 90 sta. of 12' bottom 5* deep which makes it safe to specify a local bottom using crushed stone and lake gravel DESIGN REPORT 213 as far south as Sta. 280 with either gravel or crusher run the entire length of road on the shoulders. Imported bottom should be used Sta. 200 to 280. Imported Stone. $1.25 per ton f. o. b. switch. Switch can be built at Sta. 233 for $300 to $400. Water. Can be obtained at all seasons at intervals from 1 mile to 1 J miles all along the road. Cost of Different Types Grubbing and clearing $ 300.00 23,000 cu. yds. Roadbed excavation @ $.50 11,500.00 15,000 " " Brow Exc. across Braddock Bay @ $.45 6,750.00 800 " " Sub. -base @ $1.25 1,000.00 4,000 lin. ft. Concrete G. R. across Braddock's Bay @ $1.00. . 4,000.00 Drainage of system 3,500.00 Minor points @> 400 per mile 3,400.00 Engineering and Contingencies 8,000.00 Total cost of items other than metalling $38,450.00 Schedule of Unit Prices Imported Waterbound Top Sta. 133 to 450 $5.00 per cu. yd. rolled Bit. Mac. " " " " " 7-30 * Local Granite Bit. Mac. Top Sta. o to 133 6.00 * Imported Limestone Water Mac. Sta. o to 133 5.50 Sub-base Bottom Crushed stone filler o to 133 1.50 Local crushed Bottom Sta. 133 to 200 2.50 Imported Mac. Bottom Sta. 200 to 280 3.20 Local Crushed Bottom Sta. 280 to 350 2.30 Lake Gravel Bottom Sta. 350 to 450 1.90 Crushed stone or gravel on shoulders 1.50 ' loose Tarvia B 0.08 per gal. in place Table of Comparative Cost Type Approx. Cost including Eng. and Contingencies Cost per mile Total Cost Design No. 1 (For details see Cost. Est. Sheet) Design No. 2 ( " " " " " " ) Design No. 3 ( " " " " " " ) Design No. 4 ( " " " " " " ) $11,000 11,300 12,000 12,500 $ 93,5oo 96,200 102,200 106,000 Overhead approx. 30^ per cu. yd overhead estimated on other items. Computation of Unit Prices of Bottom and Top Stone. No * Note. — There is no difference in cost Sta. 0-133 between a local granite Bit. Mac. Top and an imported limestone Waterbound Top when treated with Tarvia B. 214 OFFICE PRACTICE Sub-base Bottom Course Crushed Stone Filler Sta. 0-133 Cost of Stone in fences $0.10 Loading 0.15 Hauling j mile. . 0.12 Placing and Sledging 0.20 Rolling 0.05 Crushed Stone Filler (See Filler) 0.35 cu. yd 0.40 $1.02 20% profit 20 Overhead 0.30 Estimate $1.52 Say $1.50 Crushed Stone Filler (Crusher Run) per cu y( j Cost of stone in fences $0.10 Loading 0.15 Haul to crusher 1 mile 0.35 Crushing 0.10 Cost in bins $0.70 Loading to wagons 0.01 Haul to road f mile. 0.22 Spreading and brooming 0.20 $1.13 0.35 cu. yd. per yd. of Sub-base = $0.40 Local Crushed Stone Bottom Sta. 133 to 200 Cost in bins $0.70 Loading to wagons 0.01 Hauling to road 1 1 miles 0.40 Spreading 0.06 Rolling 0.05 $1.22 Consolidation 0.3 0.37 $i.59 Filler 0.20 $i.79 20% profit 0.36 Overhead 0.30 Say $2.50 $ 2 -45 Stone on shoulders $1.50 per cu. yd. loose. Local Granite Bit. Mac. Top Sta. o to 133 Stone in fences $0.10 Loading 0.15 Blasting and sledging 0.15 Hauling to crusher 0.35 Crushing 0.15 $0.90 in bins Loading to wagons 0.01 Hauling to road f 0.22 Spreading 0.06 Rolling 0.08 $1.27 Consolidation 0.38 $1.65 Screenings No. 2 and Bit 310 Profit 0.90 Overhead 0.30 Estimate $5-95 Say $6.00 DESIGN REPORT 215 No. 2 Screenings and Bitumen. Note. There should be enough local screenings for about f of the top course. Use imported for the balance. Cost 0.45 cu. yd. screenings and No. 2 at bin $0.40 Hauling f mile 0.10 Spreading 0.12 Manipulation 21 gals, bitumen @ i\% " 3" Imported Waterbound Top @ $5.00 . . 4350.00 270 " " Stone on Shoulder @ $1.50 405.00 5400 gals. Tarvia B @ $0.08 430.00 Sec. 3. Sta. 200 to 260 1300 cu. yds. 5" Imported Bottom @ $3.20 $4150.00 780 " " 3" Water Mac. Top @ $5.00 . . 3900.00 240 " Stone on Shoulders @ $1.50 360.00 4800 gals. Tarv'a B @ $0.08 385.00 Sec. 4. Sta. 260 to 280 370 cu. yds. 5" Imported Bottom @ $3.20 $ 1185.00 230 " " 3" " Water Mac. Top @ $5.00 . . 1150.00 no " Stone on Shoulders @ $1.50 165.00 1600 gals. Tarvia B @ $0.08 130.00 Sec. 5. Sta. 280 to 310 560 cu. yds. 5" Local Bottom @ $2.30 $ 1290.00 340 " " 3" Water Mac. Top @ $5.00 1700.00 170 " " Stone on Shoulders @ $1.50 255.00 2400 gals. Tarvia B @ $0.08 190.00 Sec. 6. Sta. 310 to 335 1130 cu. yds. 9" Sub-base Bottom @ $1.75 $ 1980.00 380 " " f Water Mac. Top @ $5.00 1900.00 1800 gals. Tarvia B @ $0.08 i45-oo Sec. 7. Sta. 335 to 350 280 cu. yds. 5" Local Bottom @ $2.30. ". $ 645.00 170 " " f Water Mac. Top @> $5.00 850.00 80 " " Stone on Shoulders @ $1.50 120.00 1200 gals Tarvia B @ $0.08 95-oo Sec. 8. Sta. 350 to 450 1900 cu. yds. 5" Lake Gravel Bottom @ $1.90 $ 3600.00 1150 " " f Water Mac. Top @, ^5.00 5750-oo 550 " " Gravel on Shoulders @ $1.50 825.00 8000 gals. Tarvia B @ $0.08 640.00 Totals $55100.00 Items other than Metal 38450.00 Total Estimate $93550.00 Design No. 2. Same widths and foundation construction as Design No. 1 except that a 2\" Imported Limestone Bituminous Macadam is substituted for the 3" Waterbound Top treated with Tarvia B. Cost of 3" Water Mac. Top Design No. 1 $19,600 " " Tarvia B on Mac. Top Design No. 1 1,500 Total $21,100 Cost of 2\" Bit. Mac. Top 23,800 Increased Cost Design No. 2 over No. 1 % 2,700 Design No. 3. 16' road entire distance local bottom Sta. o to 200 and 280 to 450 and imported bottom Sta. 200 to 280 with 3" Imported Waterbound Macadam treated with 0.4 gal. Tarvia B or 0.25 gal. No. 4 Road Oil. SHRINKAGE OF EARTHWORK 219 9200 cu. yds. local bottom 5" thick @ $2.25 $ 20,700 1970 " " imported bottom 5" thick @ $3.20 6,300 6700 " " Top 3" thick @ $5.10 34,200 32000 gals. Tarvia B @ $0.08 2,560 $ 63,760 Items other than metalling 38,450 $102,210 Design No. 4. Substitute a 2\" Bit. Mac. Top for the 3" Waterbound Top of Design No. 3. This increases the cost approx. $4000. Signed Designing Engineer. Maximum gradients for the various types of pavement are as follows : Wooden block 2 % Asphalt block 4 % Brick 5% Concrete 5 % Bituminous macadam with flush or squeegee coat 4 % (In sandy country, six per cent when coarse sand is sprinkled on surface.) Bituminous macadam without squeegee 8% Waterbound macadam 8 % ''Hillside" brick 12% Stone block with open joints 12% Shrinkage of Earthwork. We have made no mention heretofore of the shrinkage of earth cut when placed in fill. This is an important factor of an eco- nomical grading design. Trautwine states that for railroad work it takes 1.08 cu. yds. gravel or sand excavation to make 1 cu. yd. em- bankment. 1. 10 cu. yds. clay excavation to make 1 cu. yd. embankment. 1. 1 2 cu. yds. loam excavation to make 1 cu. yd*, embankment. 1. 1 5 cu. yds. vegetable surface soil excavation to make 1 cu. yd. embankment. The quantities 1.08 cu. yds. gravel, etc., refer to the volume occupied by the material before removal. Trautwine also states that in loosening earth and loading into wagons or cars 1 cu. yd. of earth swells about one-fifth and measures loose practically 1.2 cu. yds. These values, however, cannot be used in roadwork, as a certain percentage of the excavation is sod or vegetable matter that is not suitable for embankment and must be wasted. This waste material raises the percentage of cut necessary to make the fill. The correct ratio for roadwork has been a source of contention among engineers, and we believe that the use of too high a value has resulted in a needless waste of thousands of dollars during the last five years in New York State alone. Under this head it may be stated that on several roads under the supervision of W. G. Harger, a careful study of this point 2 20 OFFICE PRACTICE was made, taking unusual care with the original and final cross-sections, the plotting and planimeter work, and it was found that for the cases investigated, the ratio of cut to fill varied from 1. 1 5 in heavy cuts to 1.27 in light skimming work. It is the general opinion among engineers of Division 5, N. Y. S. Dept. of Highways, that the percentage formerly used (namely 1.35) is too high. In nearly all cases where the work was at all heavy, a large excess of dirt had to be wasted. There have been some roads designed on a basis of 1.35 where more dirt was needed, but in the authors' opinion this was due to dis- crepancies in the field or office work or by allowing the con- tractor to use the roadbed excavation for filler or concrete material. If the soil encountered is suitable for such purposes, it is plainly up to the contractor to furnish other material for the places excavated. ~^\ k ~^^-^_ *! 0.3 0.6 0.9 1.2 1-5 5 8.3 8.6 8.g 9.2 95 9.8 IO.I 0.3 0.7 1.0 1.3 i.e I.Qj 16.O 8.5 8. 9 9.2 9-5 9.8 IO.I 0.4 0./ I.C 1-3 1.6 i.g 2.2 5 8.8 9.1 9.4 9.8 IO.I 10.4 10.7 II. c H-3 11. 7 «•? ► 12.3 12.6 17.0 g.i 9.4 9-7 IO.I 0.4 0.7 I.C i-3 1-7 2.C 2-3 2.6 3-c 5 9-3 9-7 10.0 0.3 0.7 1.0 i-3 1-7 2.0 2-3 2.7 3.0 33 18.0 9.6 9.9 0.3 0.6 1.0 1-3 i-7 2.C 2.3 2.7 3-c 3-3 3-7 5 9.8 10.2 0.5 0.9 i-3 1.6 1.9 2-3 2.7 3-c 3-3 37 4.1 19.0 IO.I 10.5 10.8 11. 2 11.6 11.9 12.3 12.6 13.0 13-3 13.7 14.1 14.4 5 0.4 0.7 1.1 1-5 1.9 2.2 2.6 2.9 3-3 3-7 4.1 4-5 4.8 20.0 0.9 1-3 1-7 2.1 2.5 2.8 3-2 3-6 4.0 4.4 4-8 5-2 5-5 1 1.4 1.8 2.2 2.6 3-1 3-4 3.9 4.2 4-7 5-1 5.5 5-9 6.3 2 1.9 2.4 2.8 3-2 3-6 4.1 4-5 4.9 5-3 5.7 6.2 6.6 7-1 3 12.5 12.9 13-3 13.8 14.2 14.7 15. 1 15.5 16.0 16.4 16.9 17.3 17.8 4 2.9 3-4 3-9 4-4 4.8 5-3 5-7 6.2 6.7 7-i 7.6 8.1 8.5 25.0 3-5 3-9 4.4 4.9 5-4 5-9 6.4 6.8 7.3 7-9 S.3 8.8 93 6 4.0 4-5 5.o 5-5 6.0 6-5 7.0 7-5 8.0 8.5 9.0 9-5 20.0 7 4-5 5-1 5-5 6.1 6.6 7-1 7-6 8.1 8.7 9.2 9.7 20.2 0.7 8 15. 1 15.5 16.1 16.6 17.2 17.7 18.3 18.8 19-3 19.9 20.4 20.9 21.5 9 5-5 6.1 6-7 7.2 7-8 8.3 8.9 9.4 9.9 20.6 1.2 1-7 2.2 30.0 6.1 6.6 7.2 7-8 8.4 8.9 9-5 20.1 20.7 1.2 1.8 2.4 2.9 1 6.6 7.2 7-7 8-3 8.9 9.6 20.2 0.6 1-3 1.9 2-5 3-2 3-7 2 7-i 7-7 8-3 8.9 9-5 20.2 0.7 1.4 1.9 2.6 3-2 3-8 4-4 3 17.6 18.3 18.9 19-5 20.2 20.8 21.4 22.0 22.7 23-3 23.8 24.6 25.2 4 8.1 8.7 9-5 20.1 0.7 i-3 2.1 2.7 3-3 3-9 4-7 5-3 5-9 35.o 8.7 9-3 20.0 0.7 1.3 1.9 2.7 3-3 3-9 4.6 5-3 6.0 6.7 6 9.2 99 0.6 1.2 1.9 2.6 3-2 39 4-6 5-3 6.0 6-7 7-4 7 9-7 20.4 1. 1 1.8 2.5 3-2 3-9 4.6 5-3 6.0 6.7 7-4 8.1 8 20.2 20.9 21.7 22.3 23.2 23.8 24.6 25.3 259 26.7 27.4 28.2 28.9 9 0.7 1.4 2.2 2.9 3-7 4.4 5-2 5-9 6.6 7-4 8.1 8.8 9.6 40.0 1-3 1.9 2.8 3-6 4-3 5-i 5-7 6-5 7-3 8.1 8.8 9.6 30-4 1 1.8 2.6 3-3 4.2 4-8 5-6 6.4 7.2 7-9 8.7 9.6 30.3 1.0 2 2.3 3-1 3-8 4-7 5-4 6.2 7-1 7-8 8.6 9.4 30-2 1.1 1.8 3 22.8 23.6 24.4 25.2 26.1 26.8 27.7 28.4 29-3 30.2 30.9 31.7 32.6 4 3-3 4.2 5-o 5-8 6.6 7-5 8.4 9.2 9.9 0.8 1-7 2.5 3-4 45.0 3.8 4-7 5.6 6. 4 7.2 8.1 8.9 9.8 30.6 i-5 2.4 3-2 4.1 6 4-3 5-2 6.1 6.9 7-8 8.7 9.6 30.4 1-3 2.2 3.o 39 4.8 7 4.8 5-7 6.7 7-6 8.4 9-3 30.2 1.1 2.0 2.8 3-7 4-7 5-5 8 25-4 26.3 27.2 28.2 29.0 29.9 30.8 31-7 32.6 33-6 34-5 35-4 36.3 9 5-9 6.8 7-8 8.7 9.6 30.5 1.4 2.4 3-4 4.2 5-2 6.2 7-i 50.0 6.9 7-8 8.8 9.8 30.8 1-7 2.7 3-7 4.6 5-6 6.6 7.5 8.5 2 7-9 8.9 9.9 30.9 2.0 3-o 4.0 5-o 5-9 6.9 8.0 9.0: 40.0J 4 9.0 30-1 31. 1 2.1 3-2 4.2 5-2 6.2 7-3 8-3 9.4 40.4 1-5 6 30.1 311 32.2 33-2 34-4 35-4 36.5 37-5 38.6 39-6 40.8 41.8 42.9 8 1.1 2.2 3-3 4.4 5-6 6.6 7-7 8.9 40.0 41. 1 2.2 3-4 4.4 60.0 2.1 3-2 4.4 5-6 6.7 7-8 9.0 40.1 1.3 2.4 3-6 4.8 5-9 2 3-2 4.4 5-5 6.6 7.8 9.1 40.2 1.4 2.6 3-8 5.0 6.2 7-4 4 4.2 5-4 6.6 7.8 9.1 40.2 1.5 2.7 3-9 5-2 6.4 7.6 8.8 6 35-2 36.4 37-7 39-o 40.2 41-5 42.8 44.0 45-2 46.5 47-8 49.0 5©-4 8 6.3 7-6 8.8 40.1 1.4 2.8 4.0 5-4 6.6 8.0 9.2 50.5 1.8 70.0 7-3 8.7 9.9 1-3 2.6 4.0 5-3 6.6 8.0 9.2 50.6 2-0, 3.3 2 8.4 9-7 41. 1 2-5 3-8 5-2 6.6 7-9 9.4 50.7 2.1 3.4; 4-8 j 4 EARTHWORK COMPUTATION TABLES 245 Distance Horizontal Table 37. — continued Sum of Areas Vertical Quantities in Cubic Yards 41 42 43 44 45 46 47 48 49 50 75 100 D'uble Areas II.O n-3 H.5 11.8 12. 1 12.3 12.6 12.9 I3-I 13.4 20.2 26.8 14-5 1.4 1-7 1.9 2.2 2-5 2.8 3-0 3-3 3-6 3-9 0.8 7-8 — 15-0 1.8 2.1 2-3 2.6 2.9 3-2 3-5 3-8 4.1 4.4 i-5 8-7 — 5 2.1 2.4 2-7 3-o 3-3 3-6 3-9 4.2 4-5 4.8 2.2 9.6 — 16.0 2.5 2.8 3-1 3-4 3-7 4.1 4-3 4-7 4.9 5-3 2.8 30.5 — 5 12.9 132 13-5 13-8 14.2 14-5 14.8 I5-I 15-4 15-7 23.6 31.4 — 17.0 3-3 3-6 3-9 4-3 4.6 4.9 5-2 5-5 5-9 6.2 4-3 2.4 — 5 3-7 4.0 4-3 4-7 5-o 5-3 5-7 6.0 6.3 6-7 4.9 3-3 — 18.0 4.1 4.4 4-7 5-i 5-4 5-7 6.1 6.4 6.7 7-1 5-7 4.2 — 5 4.4 4-7 5-1 5-5 5-8 6.1 6.5 6. 9 7.2 7.6 6.4 5-2 — 19.0 14.8 I5-I 15-5 15-9 16.3 16.6 16.9 17-3 17.7 18.1 27.1 36.1 — 5 5-2 5-5 5-9 6-3 6.7 7-1 7-4 7-7 8.1 8.5 7.8 7.0 — 20.0 5-9 6-3 6.7 7-1 7-5 7-9 8-3 8.7 9.0 9-5 9.2 8.8 — 1 6-7 7-i 7-5 7-9 8.3 8.7 9.1 9-5 9.9 20.4 30.6 40.7 — 2 7.5 7-9 8.3 8.7 9.1 9.6 20.0 20.4 20.8 1-3 1.9 2.6 — 3 18.2 18.6 19.1 19-5 20.0 20.4 20.8 21.3 21.8 22.2 33-2 44-4 — 4 9.0 9-5 9.9 20.3 0.8 1-3 1-7 2.2 2.7 3-2 4-7 6.2 — 25.0 9.7 20.2 20.7 1.2 i-7 2.2 2.6 3-1 3-6 4-1 6.1 8.2 — 6 20.5 1.0 1-5 2.0 2-5 3-0 3-5 4.0 4-5 5.0 7-5 50.0 — 7 1-3 1.8 2.3 2.8 3-3 3-8 4-4 4.8 5-4 5-9 8.8 1.8 — 8 22.0 22.6 23.1 23-7 24.2 24.7 25.2 25-8 26.3 26.8 40.3 53-7 — 9 2.8 3-3 3-8 4-4 5-o 5-5 6.1 6.7 7.2 7-8 1.6 5-5 — 30.0 3-5 4.1 4-7 5-2 5-8 6.4 6.9 7.6 8.1 8.7 3-2 7-3 — 1 4-3 4.8 5-4 6.1 6.6 7.2 l* 8.4 9.1 9.6 4.4 9.2 — 2 5-0 5-7 6.3 6.8 7-5 8.1 8.7 9-3 9.9 30.6 5.8 61.0 — 3 25.7 26.4 27.0 27.7 28.3 28.9 29.6 30.2 30.8 31.4 47.2 62.9 — 4 6.6 7-2 7-8 8.5 9.2 9.8 30.4 1.1 1-7 2.4 8.6 4-8 — 35-0 7-3 8.0 8.6 9.3 30.0 30.6 i-3 2.0 2.6 3-3 9.9 6.7 — 6 8.1 8.8 9.4 30.1 0.8 1-5 2.2 2.8 3-6 4-3 5i.4 8.5 — ! 7 8.8 9.6 30.2 1.0 1.6 2.4 3.1 3-7 4-5 5-2 2.8 70.4 8 29.6 30.4 31.0 31.7 32.5 33-2 33-8 34-6 35-4 36.1 54-i 72.1 9 30.4 1.2 1.8 2.6 3-3 4.0 4.8 5.6 6.3 7.0 5-5 4.0 — i 40.0 1.2 1.8 2.6 3-4 4.2 4.8 5-6 6.4 7.2 8.0 6.9 5.8 — 1 1.8 2.7 3-4 4.2 5-o 5-8 6.6 7-3 8.2 8.9 8.3 7-8 — 2 2.6 3-4 4.2 5.o 5-8 6.6 7-4 8.2 9.0 9.8 9.6 9.6 — 3 33-4 34-2 1 35.o 35-8 36.7 37-4 38.3 39-2 399 40.8 61. 1 81.5 — 4 4.2 4.9 5-8 6.7 7-4 8.3 9.2 9.9 40.7 1.6 2.4 3-4 — 45-0 4.8 5-7 6.6 7-4 8.3 9-1 40.0 40.9 1-7 2.6 3-8 5-i — 6 5-7 6.6 l 4 8.3 9.2 40.0 0.9 1.8 2-7 3-5 5-3 7.0 — 7 6.4 7-3 8.2 9.2 40.0 0.8 17 2.6 3-5 4.4 6.7 8.9 — 8 37-2 38.1 ' 39.o 399 40.8 41.7 42.6 43-5 44-4 45-3 68.1 90.6 — 9 79 8.8 9.8 40.7 1.6 2.6 3-5 4.4 5-3 6.4 9-3 2.6 — 50.0 9-5 40.4 41-3 2-3 3-3 4.2 5-2 6 o' 2 7.2 8.2 72.2 6-3 — 2 41.0 2.0 3-o 4.0 5.o 6.0 7.0 8.0 9.0 50.0 5-0 100. 4 2-5 3-5 4.6 5-6 6.6 7-7 8-7 9-8 50.8 1.8 7-8 03.6 6 44.1 45-2 46.2 47-3 48.4 49-4 50.5 51-5 52.6 53-7 80.5 107.3 — 8 5.6 6.7 7.8 8.9 50.0 5I 'o 2.2 3-3 4.4 5-5 3-4 II.O — 60.0 7-1 8.2 9.4 50.5 1.6 2.8 4.0 5-1 6.2 7-4 6.1 14.8 — 2 8.6 97 50.9 2.1 3-3 4-5 5-7 6.9 8.1 9-3 8.9 18.6 — 4 50.2 5i.3 2.5 3-7 5.o 6.1 7-4 8.6 9.9 61. 1 91.7 22.1 — 6 5i.6 52.8 54-i 55-4 56.7 58.0 59-2 60.4 61.7 63.0 94-5 125.9 8 3-i 4.4 5-7 7.0 8.3 9.6 60.9 2.2 3-5 4.8 7.2 29-5 — 70.0 4-7 6.0 7-3 8.6 60.0 1 61.3 2.6 4.0 5-4 6.6 100. 33-2 — 2 6.2 7.6 8.9 60.3 1.7 3-1 4.4 5.8 7.2 8.61 02.9 37-1, - 4 246 OFFICE PRACTICE Distance Horizontal Table $1- — continued Sum of Areas Vertical Quantities in Cubic Yards 2 3 4 5 6 7 8 9 10 11 12 13 14 D'uble Areas 2.8 4.2 H 7.0 8.5 9.9 n-3 12.7 14. 1 15-5 16.9 18.3 19.7 76.0 2.9 4-3 5-8 7.2 8.7 IO.I 1.6 3-o 4-5 59 7-4 8.8 20.2 8 3.0 4.4 5-9 7-4 8.9 0.4 i-9 3-3 4.9 6.3 7-8 9-3 0.8 80.0 3-o 4.6 6.1 7.6 9.1 0.7 2.2 3-7 5-2 6.7 8.2 9.8 1-3 2 3-i 4-7 6.2 7-8 9-3 0.9 2.5 4.0 5-6 7-1 8.7 20.2 i-7 4 3-2 4.8 6.4 8.0 9.6 11. 2 12.7 14-3 15-9 17.5 19.1 20.7 22.3 6 3-3 4.9 6.5 8.1 9.8 1.4 3-1 4-7 6.3 7-9 9-5 1.2 2.8 8 3-3 5-o 6.7 8.3 10. 1-7 3-3 5-o 6.7 8.3 20.0 1-7 3.4 90.0 3-4 5-1 6.8 8-5 0.2 1.9 3-6 5-4 7-i 8.7 0.4 2.1 3.8 2 3-5 5.2 7.0 8.7 0.5 2.2 3-9 5-7 7-4 9.2 0.8 2.6 4.4 4 3-5 5-3 7-1 8.9 10.7 12. s 14.2 16.0 17.8 19.5 21.3 23.1 24.9 6 3-6 5-4 7-3 9.1 0.9 2.7 4-5 6.4 8.2 9-9 1.8 3.6 5.4 8 3-7 5-6 5-8 7 i 9-3 1.1 3.o 4.8 6.7 8.5 20.4 2.2 4.1 6.0 100. 3-9 7.8 9-7 1-7 3-6 5-5 7-5 9-5 1.4 3-3 5-3 7.2 05 4.1 6.1 8.1 10.2 2.2 4-3 6.3 8.4 20.4 2.4 4.4 6.5 8.6 10 4-3 6.4 8.5 10.7 12.8 14.9 17.0 19.1 21.3 23.4 25-5 27.7 29.8 15 4.4 6.7 8.9 1.1 3-3 5-5 7.8 20.0 2-3 4-4 6.6 8.8 31.2 20 4.6 6.9 9.2 1.6 3-9 6.2 8-5 0.8 3-2 5-4 7-7 30.2 2.4 125.0 4.8 7.2 9.6 2.1 4-5 6.9 9-3 1-7 4.1 6.5 8.8 1.4 3-7; 30 5.o 7-5 10. 2-5 5-o 7-5 20.0 2.5 5-o 7-5 9.9 2.5 5.0 35 5-2 7.8 10.4 12.9 15.5 18.2 20.8 23-4 25-9 28.5 31. 1 33-7 36.4' 40 5-4 8.0 0.7 3-4 6.1 8.8 i-5 4.2 6.8 9-5 2.2 4.8 7-6 45 5-6 8.3 1.1 3-9 6.7 9-5 2-3 5-o 7.8 30.6 3-4 6.2 8.8 150.0 5-7 8.6 1-5 4-3 7.2 20.1 2.9 5-8 8.7 1.6 4-5 7-3 40.2' 55 5-9 8.9 1.9 4.8 7.8 0.7 3-7 6.7 9.6 2.6 5-6 8.5 '1 60 6.1 9.2 12.3 15-3 18.3 21.4 24.4 27.5 30.6 33-6 36.6 39-7 42.8 65 6.3 9-5 2.6 5-8 8.9 2.1 5-2 8-3 i-5 4-6 7-7 40.9 4.1J 70 6.5 9-7 3-o 6.2 9-4 2.7 5-9 9.2 2.4 5-6 8.9 2.1 5-5, i75-o 6.7 10. 3-3 6.7 20.0 3-3 6.7 30.0 • 3-3 6.7 40.0 3-3 6.6, 80 6.9 0.3 3-7 7-1 0.6 4.0 7.4 0.9 4.2 7-7 1.2 4.5 8.0 85 7.0 10.5 14.1 17.6 21.2 24.6 28.2 31.7 35-2 38.7 42.2 45-7 49-3 90 7.2 0.8 4-5 8.1 1.6 5-2 8.8 2.5 6.2 9-7 3-3 7.0 50.5 95 7-4 1.1 4.9 8-5 2.2 5-9 9.6 3-4 7-1 40.8 4.4 8.2 1.9 200.0 7-8 i.7 5-6 9-5 3-4 7.2 3i-i 5-0 8.9 2.8 6.7 50.5 4-4 10 8.1 2.2 6.3 20.4 4.4 8-5 2.6 6.7 40.8 4.8 8.8 3-o 7-1 20 8-5 12.7 17. 1 21.3 25.6 29.8 34-1 38.4 42.6 46.8 5i. 1 554 59-6 30 8.9 3-3 7-7 2.2 6.6 3I.I 5-6 40 4-4 a 8 3-3 7.8 62.1 40 9.2 3-9 8.5 3-i 7.8 2.4 7.0 1-7 6.4 50.9 5.6 60.2 4.8 250.0 9.6 4-5 9-3 4.1 8.9 3-7 8.5 3-4 8.2 2.9 7-8 2.7 7-41 60 10. 5.o 20.0 5-o 30.0 S.o 9.9 5.o 50.0 5.0 60.0 4.9 70.0 70 10.4 15.6 20.7 25.9 31. 1 36.3 41.4 46.7 51-9 57-o 62.2 67.4 72.5 80 0.8 6.1 1-4 6.8 2.2 7.6 3-0 8.3 3-8 9.1 4.4 9.8 5-2 90 1.1 6.7 2.2 7.8 3-3 8.8 4-4 50.0 5-6 61. 1 6.8 72.2 7-9 300.0 i-5 7.2 2.9 8.6 4.4 40.2 5-9 1.8 7-4 3-i 8.8 4-5 80.4 10 1.9 7-8 3.6 9.6 5-5 1.4 7-4 3-3 9-3 5-2 71. 1 7.0 3-0 20 12.2 18.3 24.4 30.6 36.6 42.8 48.8 55-0 61.2 67.2 73-3 79-4 8 5 .6 30 2.6 8-9 5-2 1.4 7-7 4.1 50.5 6.7 2.9 9.2 5-5 81.8 8.2 40 3-o 9-5 5-9 2.4 8.8 5-3 1.8 8.3 4-8 71.3 7.8, 4.2 90.8 350.0 3-3 20.0 6.6 3-3 9.9 6.6 3-4 60.0 6.8 3-3 8o.o| 6.8 3-3 60 3-7 0.6 7-4 4-3 41. 1 7-9 4-8 1.6 8.5 5-3 2.1 9.0 6.0 70 14.1 21.2 28.2 35-2 42.2 49-2 56.3 63.3 70.3 77-4 84.3 9i-5 98.5 80 4-4 1.6 8.8 6.1 3-3 50.6 7-8 5-o 2.3 9.4 6.6, 3-9 IOI.I 90 4.8 2.2 9.6 7-1 4.4 1.8 9.2 6.7 4.1 81.5 8.9 6.3 3.6 400.0 EARTHWORK COMPUTATION TABLES 247 Table 37. — continued Distance Horizontal Sum of Areas Vertical Quantities in Cubic Yards 15 16 17 18 25.3 19 20 21 22 23 24 25 26 27 D'uble Areas 21.2 22.5 23.9 26.7 28.2 29.6 31.0 32.4 33-8 35-2 36.6 38.0 76.0 1-7 3-i 4-6 6.0 7-4 8.9 30.3 1.8 3-2 4-7 6.0 7-5 9.0 8 2.2 3-7 5-2 6.7 8.2 9.6 1.1 2.6 4.1 5-6 7.o 8.5 40.0 80.0 2.8 4-3 5-8 7-3 8.9 30.4 1.8 3-4 4.9 6.4 7-9 9-4 1.0 2 3-3 4.8 6.4 8.0 9.6 1.2 2.7 4.2 5.8 7-3 8.8 40.4 2.0 I 4 23.8 25.5 27.1 28.6 30.2 31.8 33.4 35.o 36.6 38.2 39-8 41.4 43 -o 6 4.4 6.1 7-7 9-3 0.9 2.6 4.2 5-8 7-5 9.1 40.7 2.3 4.0 8 5.o 6.7 8.3 30.0 1-7 3-4 5.o 6.6 8.4 40.0 1.6 3-3 5.o 90.0 5-6 7.2 8.9 0.6 2-3 4.1 5-8 7-5 9.2 0.9 2.6 4-3 6.0 2 6.1 7-8 9.6 1-3 3-1 4-8 6-5 8.2 40.0 1.8 3-5 5-2 7.C, 4 26.7 28.4 30.2 32.0 33-7 35-5 37.3 39.o 40.8 42.7 44.4 46.2 48.0 6 7.2 9.0 0.8 2.6 4.4 6-3 8.0 9.8 1-7 3-6 5-4 7.i 9.o | 8 7-8 9-6 1-5 3-3 5-2 7.o 8.9 40.8 2.6 4-4 6.3 8.1 50.0 100. 9.2 31.2 3.0 5.o 7.0 8.9 40.8 2.7 4-7 6.6 8.5 50.5 2.5 05 30.6 2.6 4-6 6.6 8.7 40.7 2.7 4-7 6.8 8.8 50.9 2.9 S, °l 10 32.0 34-1 36.2 38.4 40.5 42.6 44.7 46.8 48.9 51.0 53-2 55.4 57.5! 15 3.3 5-5 7-7 9.9 2.2 4-4 6.6 8.8 51.0 3-3 5-5 7.8 60.0 20 4-6 7.0 9-3 41.6 3-9 6-3 8.6 50.9 3-2 5.5 7-8 60.1 2.5' 125.0 6.1 8-5 40.9 3-3 5-7 8.2 50.5 2.9 5-3 7.8 60.2 2.6 5.o 30 7-5 40.0 2-5 5.o 7-5 50.0 2.5 5-o 7.5 60.0 2.5 5.o 'I 35 38.8 41.4 44.0 46.6 49.2 51.8 54-4 57.o 59-6 62.2 64.8 67.4 70.0 40 40.2 2.9 5-6 8.3 51.0 3-7 6.3 9.0 61.7 4.4 7.0 9.8 2.5 45 1.6 4.4 7.2 9-9 2.8 5-7 8.2 61.1 3-9 6.6 9-3 72.2 5-o, 150.0 3.0 5.9 8.7 51.6 4-5 7-6 60.3 3-1 6.0 8.8 71.8 4-6 7-5 55 4.4 7-3 50.4 3-3 6.2 9-3 2.1 5-1 8.1 71. 1 4.0 7.0 80.0 60 45.8 48.8 52. 55.o 58.0 61.0 64.2 67.1 70.3 73-4 76.2 79-3 82.5 65 7.2 50.5 3-5 6.6 9.8 3.o 6.0 9-2 2.3 5-5 8.7 81.8 5-cj 70 8.6 1.8 5.o 8.3 61.5 4.8 8.0 71.2 4.6 7-7 81.0 4.3 7-5, i75.o 50.0 3-3 6.6 60.0 3-3 6.8 9.9 3-2 6.7 80.0 3-2 6.7 90.0 80 1.4 4.8 8.1 1.6 5.o 8.5 71.9 5-3 8.8 2.2 5-6 9.0 2.5 85 52.8 56.2 59-8 63.3 66.8 70.4 73.8 77-3 80.9 84.3 87.9 91.7 95 -o. 90 4.0 7-8 61.3 5-o 8.7 2.1 5-7 9-4 3-o 6.7 90.2 3-9 7-5 95 5-5 9.2 2.9 6.7 70.3 4.0 7-7 81.4 5-1 8.9 2.8 6.4 100. o } 200.0 8.3 62.1 6.0 70.0 3-8 7-8 81.7 5-5 9.4 93-4 7.2 101.2 05.0 10 61.0 5-1 9-3 3-2 7-4 81.5 5-5 9-5 93-7 7-8 101.9 06.0 10.0 20 63-9 68.0 72.4 76.6 80.9 85.2 89.4 93-7 98.0 102.2 106.5 110.8 115.0: 20.0 30 6-5 71.0 5-5 9-9 4-3 8.9 93-2 7-8 102.2 106.8 11. 1 15.5 40 9.4 4.0 8.6 83.2 7-9 92.5 7-1 101.9 06.5 11. 1 15.9 20.5 25.0; 250.0 72.1 7.0 81.8 6.5 91.4 6.5 101.2 06.0 10.7 15.5 20.4 25.3 30.C 60 5.o 80.0 5-o 90.0 4-9 100.0 05.0 10.0 i.S-i 20.0 25.0 30.0 35.C 70 77-8 82.9 88.1 93-2 98.4 103.8 108.9 114.1 119. 2 124-5 129.5 134.8 140.0 80 80.6 5-9 91.2 6-5 102. 1 07.4 12.9 18.1 23.5 28.9 34-3 39-8 45.0 90 3-2 8.9 4-4 100. 05.7 II. I 16.8 22.2 29.7 33-3 38.9 44.5 50.C 300.0 6.0 91.8 7-5 03 -3 09.2 14.8 20.6 26.3 32.1 37-9 43-5 49-3 55.0 10 9.0 4-7 100.9 06.8 12.7 18.6 24.3 30.3 36.2 42.2 48.1 54-1 60.0 20 91.5 97-7 103.9 IIO.O 116. 1 122.2 128.3 134.5 140.6 146.8 152.8 158.9 165.0 30 4-5 100.7 07.1 13.2 19.6 26.O 32.2 38.5 44-9 5ii 57-2 63.8 70.0 40 7.2 03.8 10.2 16.7 23.2 29.7 36.2 42.5 49-1 55-5 62.0 68.5 75-q 80.0 350.0 100.0 06.6 13.4 20.0 26.8 33-3 40.0 46.6 53-2 60.0 66.8 73.4 60 02.8 09.6 16.4 23.2 30.2 37-0 43-8 50.7 57-5 64-5 71.2 78.1 85.0 70 105.6 112. 5 119. 6 126.8 133.8 140.8 147.8 154.8 161.8 168.9 175.9 183.0 190.0 80 08.2 15-5 22.9 30.0 37-2 44-3 51.7 58.9 66.1 73-4 80.5 87.8 95-o 90 II. I 18.5 25-9 33-3 40.7 48.1 55.6 63.0 70.4 77.8 85.2 92.6 200.0 400.0 248 OFFICE PRACTICE Table 37. — Continued Distance Horizontal Sum of Areas Vertical Quantities in Cubic Yards 28 29 30 31 32 33 34 35 49-3 36 50.7 37 38 39 40 D'uble Areas 39-4 40.8 42.3 43-6 45-1 46.5 47-9 52.1 53-5 54-9 56.3 76.0 40.4 1.8 3-3 4.8 6.2 7.7 9.1 50.5 2.0 3-5 4.9 6.4 7-8 8 1-5 3-o 4.4 5-9 7-4 8.8 50.3 1.8 3-3 4.8 6.3 7-8 9.2 80.0 2-5 4.1 5-6 7-1 8.6 50.1 i-7 3-2 4-7 6.2 7-7 9.2 60.8 2 3-6 5-2 6.7 8.2 9.8 1.3 2.9 4-5 6.0 7-5 9.1 60.7 2.3 4 44.6 46.2 47.8 49-3 51.0 52.5 54-2 55-7 57-3 58.9 60.5 62.1 63.7 6 5.6 7.2 8.8 5o.5 2.1 3-7 5-4 7.o 8-7 60.3 1.9 3-5 5-2 8 6.7 8.3 50.0 1-7 3-3 5.o 6-7 8.3 60.0 1.6 3-3 5.o 6.6 90.0 7-7 9.4 1.1 2.8 4-5 6.2 7-9 9.6 1-3 3.o 4-7 6.4 8.2 2 8.8 50.5 2.2 4.0 5-7 7-5 9.2 61.0 2.7 4.4 6.2 7-9 9-7 4 49.8 5i.5 53.3 55-1 56.9 58.7 60.4 62.2 64.0 65.8 67.6 69.4 71. 1 6 50.8 2.6 4.4 6.3 8.1 9.9 1.7 3-5 5-3 7.2 8.9 70.8 2.6 8 1.8 3-7 5-5 7-4 9-3 61. 1 3-o 4.8 6.6 8.5 70.3 2.2 4.0 1 00.0 4.4 6.4 8.3 60.3 62.2 4.2 6.1 8.1 70.0 72.0 3-9 5.8 7-8 05 7.0 9.1 61.1 3-2 5-2 7.2 9-3 71.3 3-4 5-4 7.4 9-5 81.5 10 59-7 61.8 64.0 66.0 68.1 70.4 72.5 74-5 76.7 78.8 81.0 83.1 85.2 15 62.3 4-5 6.7 8.9 71. 1 3-4 5-5 7-8 80.0 82.2 4-5 6.7 9.0 20 4.8 7.2 9-5 71.8 4.1 6.5 8.8 81.0 3-4 5-6 8.0 90.3 92.7 125.0 7-3 9.8 72.1 4-5 7.0 9.4 81.8 4.2 6.6 9.0 91.4 3-9 6.1 30 70.0 72.5 5.o 7-5 80.0 82.5 5-o 7-5 90.0 92.5 5.o 7-5 100.0 35 72.6 75-2 77.8 80.5 83.0 85.6 88.1 90.7 93-4 96.0 98.5 IOI.I 103.7 40 5-2 7-9 80.5 3-2 5-9 8.6 9i.3 4.0 6.7 9.4 102.0 04.8 07.5 45 7-8 80.6 3-4 6.1 8.9 91.7 4-5 7.2 100. 102.8 05.6 08.3 II. I 150.0 80.4 3-2 6.2 9.0 91.9 4-7 7-6 100.5 03.4 06.2 09.1 12.0 14.8 55 2.9 5-9 8.8 91.8 4-7 7-7 100.8 03.8 06.7 09.7 12.6 15.6 18.5 60 85.5 88.5 91.6 94.7 97-7 100.9 103.9 107.0 110.0 113. 1 116. 1 119. 2 122.2 65 8.1 9i.3 4.4 7-6 100.8 03-9 07.1 10.2 13.3 16.6 19.7 22.8 26.O 70 90.8 4.0 7.3 100.5 03.8 70.0 10.2 13.4 16.7 20.0 23.2 26.4 29.6 i75.o 3-4 6.6 1 00.0 03.3 06.8 in. 1 13.4 16.8 20.0 23-3 26.7 30.1 33.4 80 6.0 9.4 02.8 06.2 09.6 13.2 16.5 19.9 23-4 26.8 30.2 33-7 37-1 85 98.6 102. 1 105.6 109.2 112.8 116. 1 H9-7 123.2 126.7 130.2 133.8 1373 140.8 90 101.2 04.8 08.4 12.0 15-6 19.2 22.9 26.4 30.0 33-8 37-2 40.9 44-5 95 03-7 07.4 11. 1 14.9 18.6 22.2 26.0 29.6 33-3 37-0 40.6 44.4 48.I 200.0 08.8 12.8 16.6 20.6 24.4 28.4 32.2 36.2 40.0 44.0 47.8 51.6 55-6 10 14.0 18.2 22.2 26.4 30.4 34-4 38.6 42.6 46.8 50.8 54-8 59-o 63.O 20 119.4 123.6 127.8 132.0 136.2 140.6 145.0 149.0 153.4 157.6 162.0 166.2 I70.4 30 24.6 29.0 33-4 37-8 42.2 46.8 51-0 55-6 60.0 64.4 69.0 73-4 78.O 40 29.6 34-4 39-o 43-6 48.2 53-o 57-6 62.0 66.8 71.4 76.0 80.6 85.4 250.0 34-6 39-4 44.2 49.0 54.o 58.8 63.6 68.4 73-2 78.0 82.8 87.8 92.5 60 40.0 45.o 50.0 55-0 60.0 65.0 70.0 75.o 80.0 85.0 90.0 95-0 200.0 70 145-2 150.4 155-6 161. 166.0 171. 2 176.2 181.4 186.8 192.0 197.0 202.2 207.4 80 50.4 55-8 61.0 66.4 71.8 77.2 82.6 88.0 93-3 98.8 204.0 09.6 I4.6 90 55-6 61.2 66.8 72.2 77-8 83.4 89.0 94.4 200.0 205.6 11. 2 16.6 22.2 300.0 60.8 66.4 72.4 78.0 83.8 89.4 95-2 201.0 06.8 12.4 18.2 24.0 29.6 10 65.8 71.8 77-6 83.6 89.4 95-4 201.6 07.6 13.6 19.4 25.2 31.2 37-0 20 171. i77.o 183.2 189.4 195-4 201.8 207.8 214.0 220.0 226.2 232.2 238.4 244.4 30 76.2 82.6 88.8 95.2 201.6 07.8 14.2 20.4 26.7 33-2 39-4 45-6 52.0 40 81.6 88.0 94.6 201.0 07.6 14.0 20.4 26.8 33-4 40.0 46.4 52.8 59-2 350.0 86.8 93-2 200.0 06.6 136 22.2 26.8 33-6 40.0 46.6 53-4 60.2 66.8 60 92.0 98.8 05.6 12.4 19.2 26.4 33-0 39-8 46.8 53-6 60.4 67.4 74.2 70 197.2 204.2 211. 2 218.4 225.6 232.2 239-4 246.4 253-4 260.4 267.6 274-6 281.6 80 202.4 09.6 16.8 24.0 312 38.4 45-8 52.8 60.2 67.6 74-4 81.8 89.0 90 07.4 14.8 22.2 29.6 37.o 44.4 51.8 59-2 66.6 *74-o 81.4 88.8 96.2 400.0 EARTHWORK COMPUTATION TABLES 249 Table 37. — Continued Distance Horizontal Sum of Areas Vertical Quantities in Cubic Yards 41 42 43 44 |45 46 47 48 49 5o 75 1 100 D'ubl Areas 57-7 59-2 60.6 61.9 63-4 64.8 66.2 67.6 69.0 70.4 105.7 140.7 76.0 9.2 60.7 2.1 3-6 5-0 6. 4 7.8 9-3 70.7 2.3 08.4 44-4 — 8.0 60.8 2.2 3-7 5-2 6.7 8.2 9-7 71.2 2.6 4.1 11. 2 48.1 — 80.0 2.2 3-8 5-3 6.8 8-3 9.8 71.4 2.9 4-4 5-9 14.0 51.8 — 2 3-8 5-4 6.9 8.4 70.0 71.6 3-2 4.7 6.2 7.8 16.8 55-5 — 4 65.3 66.8 68.4 70.1 71.7 73-3 74-8 76.4 78.0 79-6 H9-5 159.2 6 6.8 8.4 70.1 17 3-4 5.o 6.6 8.3 9.8 81.5 22.3 63.0 — 8 8.4 70.0 1-7 3-3 5-o 6.7 8.4 80.0 81.7 3-4 25.1 66.7 — 90.0 9.8 1.6 3-3 4.9 6.7 8.4 80.1 1.8 3-5 5-2 27.9 70.6 — 2 71.4 3-2 4.8 6.6 8.4 80.2 1.9 3-6 5-3 7-i 30.7 74-1 — 4 72.8 74-7 76.4 78.2 80.0 81.8 83.6 85.3 97-1 88.9 133-4 177.7 — 6 4.4 6.3 8.1 9.9 1-7 3.5 5-3 7-2 8.9 90.8 36.2 81.6 — 8 5.9 7-7 9-7 81.4 3-3 5-2 7.o 8.8 90.7 2.6 38.9 85.1 — 100.0 9.8 81.7 83.6 5-6 7-5 9.4 91.4 93-4 95-3 7-2 45-9 94-4 — 05 83-5 5-6 7-6 9.6 91.7 93-7 5-7 7-8 9.8 101.9 52.8 203.7 — 10 87.3 89.4 91.6 93-7 95-8 97-9 100. 1 102.2 104.4 106.5 159.8 212.9 — 15 91.2 93-4 5-6 7-8 100.0 102.2 04.5 06.7 08.9 11.8 66.7 22.2 — 20 5-o 7-3 9-7 101.9 04.2 06.5 08.9 11. 2 13-5 15.6 73-7 31-4 — 125.0 8.8 101.2 103.6 05-9 08.4 10.7 13. 1 15.5 17.9 20.2 80.5 40.7 — 30 102.5 05.0 07.5 10.0 12.5 15-0 17.5 20.0 22.5 25.0 87.5 50.0 — 35 106.3 108.9 111.5 114. 2 116. 7 H9-3 121. 9 124.4 127. 1 129.8 194.5 259.2 — 40 11. 2 12.9 15.5 18.2 20.8 23-6 26.2 28.9 31.7 34-3 201.5 68.5 — 45 14.0 16.7 19.5 22.3 25.0 27.8 30.6 33-4 36.2 38.9 08.5 77-7 — 150.0 17.8 20.6 23.5 26.3 29.2 32.1 34-9 37-8 40.8 43-6 15-4 87.0 — 55 21.5 24-5 27.4 30.4 33-4 36.4 39-4 42.3 45-2 48.2 22.4 96.3 — 60 25-3 128.4 I3I.5 134-4 137.6 140.6 143.8 146.8 149.8 152.8 229.3 305.5 — 65 129. 1 32.3 35-4 38.5 41.7 44-9 48.0 51.2 54-3 57-4 36.2 14.8 — 7o 32.9 36.1 39-4 42.5 45-8 49.0 52.2 55-5 58.8 62.0 43.o 24.0 — I75-0 36.8 40.0 43-4 46.8 50.0 53-4 56.8 60.0 63-4 66.8 50.0 33-3 — 80 40. s 439 47-4 50.8 54-2 57.7 61.0 64.5 67.9 71.4 56.9 42.6 — 85 144.4 147.8 ISI.4 154.9 158.4 161.9 165.4 168.9 172.4 176.0 263.9 351.8 — 90 48.1 51.8 55-3 58.9 62.5 66.2 69.8 73-4 77.0 80.5 70.8 61. 1 — 95 51.8 55.6 59-3 63.0 66.8 70.4 74.1 77-8 81.4 85.2 77-7 70.3 — 200.0 59.5 63.4 67.3 71.1 75.o 78.9 82.8 86.7 90.5 94-5 91.6 88.9 — 10 67.0 71.2 75-2 79-3 83.4 87.5 91.5 95-6 997 203.7 305.5 407.4 — 20 174-7 178.9 183.3 187.5 191.8 196.0 200.1 204.4 208.8 213-0 319.4 425.9 — 30 82.2 86.8 91.1 95-5 200.0 204.5 08.8 13.3 17.8 22.2 33-3 44.4 — 40 89.8 94-5 99.0 203.7 08.3 13.0 17.5 22.2 26.8 31-5 47.1 62.9 — 250.0 97.4 202.3 207.0 li.g 16.6 21.6 26.3 3I-I 36.0 40.8 61.0 81.5 — 60 205.0 10.0 15.0 20.0 25.0 230.0 35-0 40.0 45 -o 50.0 75.o 500.0 — 7o 212.7 217.8 223.0 228.2 233.3 238.6 243-7 248.9 254-1 259.2 388.7 5i8.5 — 80 20.2 25.6 30.9 36.3 41.7 47.1 52.4 57-8 63.2 68.5 402.7 37.o — 90 27.8 33-4 38.9 44-5 50.0 55-6 61. 1 66.7 72.3 77-8 16.5 55-5 — 300.0 35-4 41.2 46.9 52.6 58.3 64.2 69.8 75-6 81.3 87.0 30.4 74.0 — 10 43 -o 49.0 54-8 60.8 66.7 72.7 78.5 84.4 90.4 96.3 44-3 92.6 — 20 250.6 256.7 262.8 268.9 275.0 281.2 287.2 293-3 299-5 305.5 458.2 611. — 30 58.2 64-5 7o.7 77-1 83.3 89.7 95-9 302.2 308.5 14.8 72.1 29.6 — 40 65.7 72.3 78.7 85.2 91.7 98.2 304.6 11. 1 17.6 24.1 85.9 48.1 — 35o.o 73-3 80.1 86.7 93-4 300.0 306.8 13-3 20.0 26.7 33-o 99.8 66.0 — 60 80.9 87.9 94.6 301.5 08.3 15-3 22.0 28.9 35-8 42.6 513.7 85.2 — 70 288.5 295.6 302.6 309.7 316.7 323.8 330.7 337-8 344-8 351-8 527.6 703.6 — 80 96.1 303.4 10.5 17.8 20.0 32.3 339-4 46.7 53-9 61. 1 41.5 22.2 — 90 303.7 111 18.5 25-9 33-3 40.7 48.1 55-5 62.9 70.3 55-5 40.6 ~~ 400.0 250 OFFICE PRACTICE o H H W w Hi Hi PQ O CO > O u OO CO H Hi m On OnOO 00 t^vO VO iO V) (/I fONH IT) ON CO t^ M VO &) ONOO 00 t^ vo vO vo vo ^f o • 1— I oo r^\o vo tJ- CO - M M H H o~ ON i o o o o o o o o o O O O O O O O o o" 4-> o o o o M CD o o o o^ o HH o~ d d d o~ o~ o~ o~ o~ H (N fO't vovO X^OO ON ON ONOO 00 t^vO vO vo vo t/3 CONH lO On co r^- m io • o dodo 6 M M H M ON 1 o o o H +-> o o o o o o o o o o o o o o o o o 0) 88888 HH H N CO^j" vonO t^OO ON rtOO -0 vo vo ■^ _ OO t^O vo T}- CO o H O O O O Q o o o o o o o Pn h n foi* VONO t^OO ON ON ONOO OO t^NO NO VO VO C/3 00 t^-O vo Tf co -nO IO i" (N N H t/3 m roio N On m co vo t^. <3J o o .o Q O O O O M M H M r^; O O O O O § o o o o O O O O O ON 6 6 6 6 6 6 6 6 6 o -*-> H \» 3l" Si" 6\» 2250 2300 2350 2400 2450 2500 2550 2600 20,800 21,300 2 1 ,800 22,200 2 2 , 70O 23,200 23,600 24,IOO 26,000 26,600 27,IOO 27,700 28,200 28,800 29,400 30,000 32,300 33,000 33 , 7oo 344oo 35,2oo 35,900 36,600 37,300 43,700 44,700 45,7oo 46,700 47,700 48,700 49,600 50,600 54,200 55,300 56,600 57,800 59,000 60,200 61,400 62,600 1 Note. — The quantities in this table are figured by slide rule but are sufficiently close for the purpose to which the table is put. EARTHWORK CHART 253 OK 254 OFFICE PRACTICE Table 39. — Continued 14-FOOT ROAD Weight of Depth of Loose Spread 1 cu. yd. Stone, Loose Measure 2\» 3i" 3l" S\» 6f" 2250 24,300 30,400 37,7oo 51,000 63,200 2300 24,800 31,000 38,500 52,200 64,600 2350 25,400 31,700 39,3oo 53,300 66,100 2400 25,900 32,400 40,200 54,400 67,500 2450 26,400 33,000 41,000 55,6oo 68,900 2500 2 7,000 33,700 41,800 56,700 70,300 2550 2 7,600 34,4oo 42,700 57,800 71,600 2600 28,IOO 35,ioo 43,5oo 59,000 73,ooo 15-FOC )T ROAD 2250 26,000 32,600 40,400 54,700 67,700 2300 26,600 33,2oo 41,300 55,900 69,200 2350 27,200 34,000 42,200 57,200 70,800 2400 2 7,800 34,7oo 43,IOO 58,400 72,200 2450 28,400 35,4oo 44,000 59,600 73,800 2500 29,000 36,100 44,800 60,800 75,200 2550 29,500 36,900 45,800 62,000 76,700 2600 30,100 37,600 46,700 63,200 78,200 16-FOC >T ROAD 2250 27,800 34,7oo 43,!oo 58,400 72,300 2300 28,400 35,5oo 44,000 59,600 73,900 2350 29,000 36,300 45,ooo 60,900 75,5oo 2400 29,600 37,ooo 45,900 62,200 77,200 2450 30,200 37,800 46,900 63,600 78,700 2500 30,900 38,600 47,800 64,900 80,300 2550 31,500 39,400 48,800 66,200 82,000 2600 32,IOO 40,100 49,800 67,400 83,600 The computation of earthwork is the longest operation of the quantity estimate. When this is finished the quantity estimate is considered as practically complete. YARDS OF MACADAM PER 100 FEET 255 Table 40 Giving the Number of Cubic Yards of Macadam PER IOC > Feet of Road for Different Widths and Depths Depth Width of Macadam 2" 2\» 3" zh" 4" 5" 6" r 10' .... 6.17 7.71 9.26 10.80 12.34 15.43 18.52 21.61 12' .... 7.41 9.26 II. II 12.96 14.82 18.52 22.22 25.93 14' ... • 8.64 I0.80 12.96 15.12 17.28 2I.6l 25.92 30.25 15' .... 9.26 11.58 13.89 16.20 18.52 23.16 27.78 32.41 16' .... 9.88 12.35 I4.81 17.28 19.76 24.70 29.63 34-57 18' .... 11. 11 I3.90 16.67 19.44 22.22 27.79 33-34 38-89 20' .... 12.35 15.44 18.52 21.60 24.70 30.87 37.04 43.21 22' .... 13.58 16.98 20.37 23.76 27.16 33.96 40.74 47-53 The other quantities figured are: length of road in miles. Table 38 converts lineal feet to miles. Quantities of macadam, sub-base, concrete paving foundations, square yards of resurfacing, which are simple computations involv- ing length, width, and depth: Tables 39, 40, and 41 can be conven- iently used. Quantities of oil or other surface or penetration treatments, which are usually specified as gallons, per square yard: Table 42 is developed with this in view. Concrete for culverts or retaining walls. Where a large amount of work is done it generally pays to compile a table of quantities for standard culverts of different sizes and lengths. The quan- tities can then be picked from these tables sufficiently close for a preliminary estimate. There would be no object in including in a book of this character any table suitable for certain culverts, as each department has a different standard. Expanded metal and reinforcing bars, Tables 15 and 16 cover these features. Weights of cast-iron pipe: Table 14 can be used. Incidentals requiring ordinary arithmetical computations only. The quantity estimated being completed, the estimate of cost is made. This is considered in chapter X. Final Design Report On the completion of the design a report of this nature is filed for reference. 256 OFFICE PRACTICE Rochester, January 31, 19 16 REPORT ON DESIGN AND ESTIMATE of the VARYSBURG-WARSAW PART II COUNTY HIGHWAY No. 1349 WYOMING COUNTY (FORMERLY known as the Orangeville-Warsaw Road) Length, 4.74 Miles Type of Construction. Top: From Orange ville Center to Corporation line Village of Warsaw (Sta. 0+00 — 249 + 57) 3" waterbound macadam 14' wide. Imported limestone top and screenings, surface application of calcium chloride using 2^# per sq. yd. in two applications. Bottom: Orangeville Center Sta. 0+00 to corporation line Village Warsaw Sta. 249 = 57, 5" macadam 14' wide. Screened gravel local. Section 26' wide with 18" ditch. Estimated Cost $49,858.00 Engineering and Contingencies 5,042.00 $54,900.00 Cost per Mile $11,600.00 Survey. By H. TenHagen, rodman, Spring, 1915. Weather con- ditions favorable. Design. By F. W. Mills, Leveler. Field Inspection. By George A. Wellman, County Assistant; F. W. Mills, Leveler. Estimate. By George A. Wellman, County Assistant; George G. Miller, Chief Draughtsman. Status and Connections. This road is a section of the old Cherry Valley Trail which is claimed to be the shortest highway between Buffalo and New York. It is the beginning of a system of improve- ments to connect the western part of Wyoming County with Warsaw. At Halls Corners (Sta. 145 + 00) it will connect by a proposed County System road with the east end of C. H. 1267. With the proposed Varysburg- Warsaw Part I Road and proposed Route 19 it will connect with Route 16 already built. It will also connect with the proposed County System road south through Johnsonburg, North Java, to Arcade. Foundation Soil. Is for the most part clay loam and is very unstable when wet. This calls for an unusual estimate for sub-base. Traffic. The heaviest traffic is eastward and down hill. It con- sists principally of wagon hauled farm traffic. There is considerable automobile traffic both ways. This will greatly increase as the high- ways West towards Buffalo are improved. Grades. The ruling grade for eastbound traffic is 8% between Sta. 23+00 and 29+00. The ruling grade in the opposite direction is a short 8.54% at Sta. 149. FINAL DESIGN REPORT 257 Alignment. The alignment is generally straight and curves easy. The minimum radius is 573' at Sta. 162+00. Right of Way. No land taking is necessary. Type of Metaling. The bottom course throughout the entire length of the road is to be of screened gravel as per B. R. sample G-2095 5" deep, to be obtained on the property of Henry Roth, one mile south of Sta. 145+00. Top course is to be 3" imported limestone from Buffalo Crushed Stone Co. with screenings from same source. Drainage. All existing culverts have been replaced by either concrete or cast-iron pipe structures. The former have been given preference wherever head room permitted. The bridge at Sta. 15+40 is to be rebuilt by the Town of Orange- ville and that at Sta. 175 + 19 by the Town of Warsaw. Materials The materials suitable to be used in the work have all been tested and accepted by the Department. The filler for the bottom course to be the fine material screened out at the pit. Sand for use in concrete has been accepted from the Roth pit, one mile south of Sta. 145+00. Unloading Points. Erie R. R. at Warsaw Station siding, 1200' from Sta. 249+57- Respectfully submitted, Perry Filkin, Division Engineer Construction Plans. The construction plans should give suf- ficient information to show the contractor what he is expected to do and to enable the constructing engineer to stake out and to build the road. A finished set of plans consist of a map, profile, and cross-sections showing the alignment in relation to the preliminary survey line, the proposed grade elevations, the shape of the finished road, the widths Fig. 65 and depths of road metaling, the crowns to be used, the existing struc- tures and the proposed structures, and all the minor points of design. Each Department has its own method of giving this information, and it makes little difference how it is shown so long as it is complete and clear. In general it may be said that the scales used are the same 258 OFFICE PRACTICE as in mapping the preliminary survey and that the size of sheets or rolls must be convenient to handle in the field; sheets larger than 24 ,/ X3o // are clumsy. Miscellaneous Points. A point often overlooked in lying a grade line is the proper approach to a railroad grade crossing where the track is on a curve and has a superelevated rail. Where the road grade is level, or nearly level, the solution is comparatively simple, as shown in Fig. 65; but where the grade of the road is in an opposite direction to the elevation of the rail it is more difficult and sometimes impossible to make an easy riding crossing. ^ Edge of Macadam Fig. 66 Also, where a road, on a steep grade, crosses the railroad track on a large skew angle, care must be taken to flatten the grade nea,r the track to avoid distorting the road section due to the difference in the rate of grade of the track and road. See Fig. 66. Grade Crossing Eliminations. In grade crossing elimination designs the following minimum clearances have been adopted, Div. 5, N. Y. S. Dept. of Highways. Where a highway is to be built under a railroad the crown eleva- tion is made 13.5' below the bottom of the bridge girder, and the minimum right-angle distance between abutments is taken as 26 feet. For solid floor railroad through girder bridges a clearance of 1 3.5' below the bottom of the girder means a distance of from 16.5' to 1 7.0' below the top of the rail. The tables (pp. 259, 260) are taken from Spofford's " Theory of Structures," and a pamphlet issued by Heath & Milligan, of Chicago. They show the approximate weight of through girder railway bridges with the depth of floor system. They are useful for preliminary estimates of grade-crossing elimination. The weights given are for the steel only; the weight of the floor system must be added. For purposes of a rough prelim inary esti- mate of cost the superstructure can be assumed to cost $60.00 per ton in place including all erection costs. Where the highway crosses over the railroad a minimum clearance of 21.0' is used from the top of rail to the bottom of the highway GRADE CROSSING ELIMINATIONS 259 bridge; the span varies with the number of tracks. In determining the length required it is best to get in touch with the railroad engineers. Right of Way Computations. The form of traverse computa- tion and closure was shown on page 155. The areas of rights of way are generally figured by dividing the parcel into rectangles, trapezoids, triangles, sectors, or segments, and figuring these shapes from the formulae given in Table 57. These areas are checked by planimeter. They are usually figured to the nearest 0.01 acre. 260 OFFICE PRACTICE The method of double-meridian distances can, however, be used if desired. The following formula and example are given to illustrate this method. It is not often necessary and is a tedious computation : The rule is: Twice the area of the figure is equal to the algebraic sum of the products of the double-meridian distances of each course multiplied by its latitude. In which the double-meridian distance equals the sum of the meridian distances of the two ends of each course referred to the THROUGH PLATE GIRDER SPANS, STRINGER FLOOR. WEIGHTS, DIMENSIONS, ETC. 1^ Base of Rail I I I I Low Iron 771(77ZW7777777777777777777777777777} • Top of Masonry LOADING-COOPER'S E. 60 Total end Shear is given in thousands of pounds per rail and includes both live and dead loads. 230,000 220,000 210,000 200,000 190,000 270,000 260,000 250,000 210,000 130,000 120,000 110,000 90,000 80,000 Erector's Note. W= Total weight of one single track span with two light girders. 45 14L7 50 156.0 55 167.5 3'4' 3'11" 60 176 3 65 '• 191.1 3'4^" 70 204.7 75 220.4 5'0' 80 232.8 85 248.6 c u o O « s * & O ID O 90 262.3 3'5' 30' to 50' .24 W .42 W .52 W 95 277 4 100 291.2 3'6' 55' to 80' 25 W .46 W 50 W 110 319 6 85' to 110' :s w .51 W .44 W RIGHT OF WAY AREAS 261 3 be < 10 10 a cT too .S o U s- to a o u to < u d to U2 O P o TO " W bx) to a 3 • H COOO CN t^-00 coO O Tf to Ml pq "0 *0 *o *^o M ^t" CO ^ 0000 COOO CN CN cn r^oo 00 O 3 H f) fCt cr . m cn CD (N U CJ M 1 1 c3 ~ X>- O O **- s II ^ O 4->.a II '— +J 7 ° O to *-i CN 10 M CO M TO t- oT X 1 8 II "£ Id Hi +■> u j-i TO CO CT O a; CN nj mH < >< 262 OFFICE PRACTICE meridian drawn through the most westerly point of the parcel, and the latitude of each course is reckoned as plus if the course runs north and minus if it runs south. Take as an example the right of way parcel shown in Fig. 50, page 155, for which the traverse has been figured and refer the meridian distances to the meridian drawn through the corner 3.1' distant from station 194+ 71.7. Parabolic Crowns for . Pavements. It is often convenient to have the following data on parabolic crown ordinates in making templets for pavement work. Divide the distance from the center of the road to the curb or edging into ten equal parts and call the total crown 1.0; the dis- tance down to the surface of the pavement from the crown elevation at each of these ten points expressed in terms of the total crown will be Center of Road, point No. o o . 00 1 0.01 2 o . 04 3 0.09 4 0.16 5 025 6 0.36 7 0.49 8 0.64 9 0.81 Curb point 10 1 .00 fointio Curb SUMMARY OF POINTS 263 Summary of Points to be Considered in Making an Economical Design Justifiable economy in grading is limited to the intermediate grades and to variations in the cross-sections. A well designed road in these particulars may easily save $700 to $900 per mile over a careless design. Economy in widths of hard paving is attained by the selection of a width suitable to each particular road or part of road. A uniform width of 16 ft. for all roads is unnecessary and a waste of money both in first cost and in maintenance and renewal. For class I and II. traffic nothing less than 15' to 16' should be considered; for class III and IV the widths may vary from io' up. The cost per ft. width per mile for different types is approx. as follows: Type of Road . f ir ^ ost P er .. J ^ foot width per mile. Brick $1200 Asphalt 1 100 Concrete 700 Bituminous Macadam 600 Waterbound " 500 Economy in foundations is limited to utilizing local materials to their best advantage with short hauls. Economy in top courses lies in the selection of the cheapest type suitable to the traffic conditions and the use of the minimum thick- ness. Economy in maintenance is attained by preventing rather than by repairing damage. Table 41 Square Yards per 100 Feet and per Mile eor Different Width of Surface Width Number of Square Yards Width Number of Square Yards Feet Per 100 Feet Per Mile Feet Per 100 Feet Per Mile 8 SS.S8g 4,693 26 288.889 15,253 10 in. in 5,867 28 311.III 16,427 12 *33-333 7,040 30 333-333 17,600 14 I55-556 8,213 32 355-556 ^,773 15 166.667 8,800 34 377-778 19,947 16 177.778 9,387 36 400.000 21,120 18 200.000 10,560 3^ 422.222 22,293 20 222.222 n,734 40 444.444 23,466 22 244.444 12,907 42 466.667 24,640 24 266.667 14,080 44 488,889 25,813 264 OFFICE PRACTICE Q < w < p o» W W H O H m Z O a < w M a p on d 80.00 lOO.OO I20.00 I40.00 I50.00 160.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00 320.00 340.00 88888 06066 VO 00 O m IO N00 O M fN CO IO N00 10 to m n 06 fOOO fN N H to rfvo N H M H H M O M tN Tj-VO vq rt- cn 00 tN n tN r-~ m ON O tN CO tO M tN tN tN tN N O O M fN VO "t tO M O VO M VO M tO VO00 ON M fN CN tN fN CO tO NO d tONO ^O covO O fOO covO O fOO tovo 00 ao NO i to 00 O M W tO tJ-vO NOO On Tf tovO noo 4" tovd r^oO rj- tovO N00 200.00 211. II 222.22 233-33 244.44 <* d 10 rf- to tN \0 "5 4" to CN vO to •<+ tO\© O W O ON00 N m 00 nvo h o'co nvo NCO 00 On O w VO Th to tN M tO ^t fO tN M to 4- to W M M tN tO T}- tO O OvOO NVO q 00 nvo «o d 06 nvo to vO vO NOO ON CO CO d to rj- 10 tovO vq q ^00 to VO N N00 ON tN vq q "too ONVO 4" M OO vovO KO0O0 ON O M M tN CN t>- M tO O VO tO M 00 tO ON O M M tN CO co rf tovO co N m to O co d 06 »o « fO^"^ tovO PO d NfOO NO VO to -00 00 O 0« HNN O tO M VO fN O to M VO tN O O M M tN tN d 00 0 H (O vO rf ON to ^ O -too to OO M NHO N fN VO M to Q -* ON to N 5 -tOO CO N N tN VO M CO M CN CN CO <0 to O TfOO to tO "*t Tf "t to N tN vO W »0 tOVO \Q NN O -^-00 tO N 00 OOOO ON ON d O* m tovO n 00 H to iflO 00 m to lOVO M M M M OOONrhN N O * w < 0* C/3 W M H O H O 1-) 3 < O O Pi w PQ P 00 160.00 200.00 240.00 280.00 300.00 O Q O Q O OOOOO dodo* 6 MOO "tOO tO co t "* -t 520.00 560.00 600.00 640.00 680.00 720.0c 760.00 800.00 840.00 880.00 N H OlN 't f) MOOvq 'ttO H 00 O 4 CO IO00 CI O 00 M M o co O ^fNM IO CO co CO t t M 0> N -t d W O0 O "t CI m 06 O -t ci 0> o »o too N O COOO M O 1^ r-~ t^oo co d n ->t O O -t r* m O N N t>O0 O M N00 fOO^N CI N COOO O ci N COOO O i-t^H TfO M H N M Ci t OO M N *t O to M O 46 ioh>o 00 CS to O* 0 M O CI O to m ci ■t r- m -too O O n i>. r-» coo co q coo d co d cOO O CO to IH M CI CI CI NO «)NO q coo O O coO O O O COO O O co co co "t CO N O co N cOO O coO COO O coO COO O cOO •t t to to to O co N O CO O coO O co d coo d co O coO O co O O O N N tO noO co ■0 t^-o to »o COO t>N "O cs cs ci co co ON CI -tOO M M 00 't O N 10 t t tYl 00 M -t N O CO N O CI to co ONO CI 00 PO CI ci CI M POO On ci to to to too O CO to rj- CO ci O M "t N M M M M CI CI M O ©N00 N M O 00 t^O H O06 l^-O cOO 00 m *t CI CI IN CO CO O -t" CO CI M »0 't CO CI M tO 't CO CI M N O coO ON CO -t 't 't -t ON00 NO O 00 NO to O 00 NO »o « tNO fO to to too O to ci M O N ■<*• CO M 00 O "t CO mooo -too M COO O O M M M 1H IN CN O0 O CO M O t^ to CO pi 6 Nioto CS CS CI CO CO M M N TJ" CI M 00 O "t CI m 00 O -t- ci 000 M 't N CO CO -t "t ^J" O O0 O CO M O N tO CO M ONlfltOH O CI IO00 M 10 to to too CI M NOO NO co OO q 0" co do d O coo 00 |_| M W M CI (OOMOO co 00 co q co d co d M ^-vO O* CI CI CI CI CI CO N co O N co vo fo q co 0' co d O co ■t- N O ci to CO CO ■* -t "t" OmoOn OO co OO O O CO do O0 O COO 00 Tj- to to to to 2 00 NNM («5 N CI O M CO n ci O m co ON ci Tt N00 _ _ - - O O "t On co to q ■too co to O 'tOO CO O^ ci -to 0> M CI CI CI CI OO CI N M O t^ CI O M lO r*. ci t-i to M tO ON M co co co co 't ota cooo q -too co n d -tod co n ■f O 00 M CO Tj" -t "t »o »o q On m coO N 00 m co "pO 00 M CO »0O 00 *■* fO too O0 O CI "t N n O ci "tO i-"- d ci 4 m coO 00 00 H fOlON OO M CO lO N 00 M CO to t^ CI CO co CO CO O « "t N ON q ci -to 00 d ci -to 06 O ci -tooo •t -t t -t -t 4-1 ai 00 « ^>o OOO O N t mhn n« O00 O N t CI CI CO co CO vOOOONt cC CO t t t CHAPTER X COST DATA AND ESTIMATES 1 New methods of construction have so changed the cost of road improvements that engineers just going into this work, or those not familiar with present methods, are often handicapped in making estimates. The cost data given in this chapter has been gathered chiefly since 1907 and covers most of the items necessary for estimating the cost of any ordinary road improvement. Such data must be used intelligently or it will be misleading. Local conditions should always govern in making estimates, and in presenting costs it is best to describe the conditions under which the work was performed, leaving their special application to the one using the data. An engineer's estimate should represent the probable average bid price. In the following examples of actual cost those have been selected that are considered to be average cases. Contractors who have an unusually good plant and a well-or- ganized force can often do the work cheaper than is shown; on the other hand, those new to the work will spend more. Where machinery is used it is more satisfactory to include the items of depreciation, repairs, and interest in a lump-sum item for the whole contract than to try to reduce it to a yardage basis. These charges will be considered under the heading of " Plant and Pay Roll." BITUMINOUS AND WATERBOUND MADACAM CON- STRUCTION Cost of Earth Excavation. Table 43 shows the cost of earthwork on four roads in New York State, which represent easy, average, and difficult work. The cost per cubic yard includes excavation and placing in fill, shaping the subgrade for the stone, and trimming the shoulders and ditches. For heavy fills with short hauls wheeled scrapers were used, but the largest part of the work was done by wagons. Cost of Rock Excavation. The writer has no reliable personal data on ledge rock excava- tion. Rockwork on roads is usually a small item; the cuts are small and consequently expensive. Perhaps there is no item more variable in cost than small rock cutting. It is therefore safer to take as a basis of estimate the bids of experienced road 1 Much of the data in this chapter was contributed by the author to the Engineering News and published July 13, 191 1. 266 EARTH EXCAVATION 267 to o M H < > < O X w w H < W 1-1 n S3 a >H J-l !r1 ^ _, >_ *_, 0) t; , ^^ H «J 5 3 ^g « G §? s Vh ,5 rt g £ x K G 13 Ctf (1) *— 1 ~T> >, £ G gravel ay, har nd, cla; cavatic :avatio "0 T3 rt X tt G ° ^ -^ NNir)N | 2 M H M M 6 6 6 6 t& ,1 Ex- ition Yds. 0000 0000 0^ 0^ 0^ 0^ 3>^ CXf 00~«f 0^ rf /\ CN M M *G wi upioq q S* *T3 +-> 4-> 4-> ?5 -i-> ^. rf w « tfl in « tn: $ a a g a a g g ~ cj ._ ._ 53 .G WM h3 C/3 mh m3 C/3 O rt H|N H|C4 <♦ 00 OO NN Q^hoO CO M s *g| OO <0 CO CO cs t^\0 MD C/3 d *0 1 T3 £ r* u> (J 5*3 (NOOOOcnOO NNNNIOC^HVO M M ■*-> o^a iH|N rO TtGO OO t^vO I a 3 wS| CO <0 co co co 1 ■9 ."S z ^ p S ^tOO On -00 vO O CO CO M JS s *£g M MM oi 3 £ 00 O ^tOO 00 O CN rt MMMMCSMCNCN u +J 1=5-3 3 r-l|N r-i\C* 0£ 0, On t^CO OO O Q\ OO ^| MMMMMMMM XI a en 3 T3 1 T3 fc 3 -° vO ~ * 10 r^-vO O CO On TO 1 OnOO On On r^OO NO u gg 6 M M M M M M M ' 6 fc V a a *55 cu. yds. Amount broken per hour 7.7 " " Divided as follows: Weight per cu. yd. loose Tailings, 1,004 cu. yds 31-8% 2,457 lbs. 2\" stone, 1,618 cu. yds $*•$% 2,383 " 1" stone, 323 cu. yds 10.2% 2,277 " Screenings, 210 cu. yds 6.7% 2,585 " Cost per cu. yd. in bins at crusher $0,898 Cost per ton in bins at crusher 0.745 Divided as follows: Cost Per Cent Labor, steam, drilling $0,092 10.3 Coal, oil, waste, powder, etc 0.084 9-4 Sharpening drills and tools 0.069 7-7 280 COST DATA AND ESTIMATES Breaking stone for crusher 0.279 31.0 Loading stone for crusher 0.098 n.o Hauling stone for crusher 0.072 8.0 Feeding crusher ' °-°53 5-9 Engineer of crusher 0.031 3.4 Coal, oil, waste, and repairs of crusher 0.079 8.8 Other repairs 0.0 41 4.5 Total $oT8o8 100 W. E. McClintock, Engineer, Chelsea, Mass., season 1887: Labor $0. 20 per hour Teams 0.45 " " Material. Trap rock. Amount broken 1,718 tons Stone delivered at crusher bf subcontractor for $0.75 per ton. Cost. Tools $0,013 Oil, waste, etc 0.016 Fuel 0.050 Stone at crusher 0.750 Crushing (labor) 0.194 Total per ton $1,023 Dustless Screenings. The construction of bituminous mac- adams requires a dustless screening product referred o in the beginning of the chapter as No. iA; it is obtained by re- screening the ordinary screenings (f" product) to remove the dust; the percentage of dust in the ordinary screenings will vary according to the stone crushed and the setting of the crusher jaws. The author has no reliable data for small crushing plants, but through the courtesy of the Buffalo Cement Company the following data is given for their output of limestone screenings at Buffalo, N.Y. Size of screen opening for ordinary screenings f" Size of dust screen openings \" Cu. yds. of dust for 1 cu. yd. ordinary screenings 0.35 " " dustless screening 1 cu. yd. ordinary screening. 0.65 The same data from the Leroy plant of the General Crushed Stone Company gives: Size of screen openings for ordinary screenings f " to |£" " dust screen openings \" u T 5 g" Cu. yd. of dust per cu. yd. ordinary screenings 33% " " " Dustless screenings per cu. yd. ordinary screenings 67% Percentage of screenings to total output for Leroy limestone approximates 15%. The above furnished to the writer through the courtesy of the General Crushed Stone Company, of Easton, Pa. COST OF APPLYING BINDER 281 COST OF STONE FILL BOTTOM COURSE The following data is taken from Road 5,021, season of 1910; labor cost $0,175 P er hour, teams $0.40 per hour. The amount placed was 10,000 cu. yds. rolled measure. The average rolled depth was 1.1 ft. The surface was carefully brought to line and grade, allowing a variation of 1 in. either above or below, which inequality was taken out with the top stone. A 3 in. bituminous top course was placed directly on this fill. The top layer of bottom stone was sledged to reduce all stones to 8 in. or under. Flint stone was used to "fill the top 6 in. and to surface the rough fill. The bottom course was of fence stone, hauled, on an average, about one-half mile. I estimate that one cubic yard rolled measure requires 1.25 cu. yds. loose. The cost of the bottom course per cubic yards rolled measure was $1.03, divided as follows: Loading 1.25 cu. yds $0.19 Hauling 1.25 " " \ mile 0.20 Placing 1.25 " " and rolling 0.24 Sledging 0.15 Flint * 0.10 Cost of fence stone 0.15 Total, per cu. yd $1.03 Cost of Sub-base Bottom Course. Road 495, Parma Corners- Spencerport. E. E. Kidder, Engineer. 1,082 cu. yds. placed, average depth 6". Not much sledging required. Cost of stone, 1 cu. yd $0.10 Loading, per 1 " " 0.184 Hauling 1 mile • 0.30 Laying, sledging and spreading filler . . . 0.136 Rolling 0.02 Superintendence 0.02 Cost of filler in pit nothing (gravel used). 0.00 Loading J cu. yd 0.04 Hauling J cu. yd. 1 mile 0.10 Total 0.90 COST OF APPLYING BITUMINOUS BINDER The following data is taken from Road 5,021, season of 1910. Bituminous macadam, penetration method: Labor. Kettleman $0.20 per hour Spreaders 0.20 " " Plain labor 0.175" Teams 0.45 " " Apparatus. 4 bbl. kettle (coal burner) Bitumen heated 2 bbl. " (wood burner) to 400 F. 282 COST DATA AND ESTIMATES 1 2 ton Kelly roller Spreading pots having a vertical slot \" wide. Organization. Rollerman acting as foreman i Spreader i Kettleman 3 Laborers Average speed 350 ft. of 16 ft. road, per day. Quantities. 16,850 gals, laid in one coat covered 13,330 sq. yds., or 1.26 gals, per sq. yd. Cost per gal. Unloading and hauling | mile $0.0015 Heating 0.0032 Spreading 0.0032 Rolling and supervision 0.0051 Total $0.0130 Bituminous material f.o.b. Caledonia 0.0950 Total per gal $0.1080 Second quantity. Forty-two thousand gallons covered 24,000 sq. yds. in one coat, an average of 1.75 gals, per sq. yd. Cost per gal. Unloading and hauling if miles $0.0032 Heating 0.0040 Spreading 0.0039 Rolling and supervision 0.0042 Total $0.0153 Bituminous material f.o.b. Caledonia , m 0.0950 Total per gal $0.1103 Cost of Applying Bituminous Binder. Road 5,046, Penetra- tion Method. 18,890 gals, spread on 12,378 sq. yds. in one coat, of 1.52 gals. per sq. yd. Apparatus. 5 2 bbl. kettles (wood burners) Fuel. Used bbl. staves and some extra wood. 1 10- ton Buffalo Pitts Roller. Spreading hods. Organization. Per Hour 1 Foreman $0.30 2 Pourers, each 0.25 5 Kettlemen, each 0.20 2 Spreaders of No. 2, each ; 0.20 4 Helpers, each , 0.175 Labor of Placing. Cost per gallon. Fuel $0,001 Kettlemen 0.005 Pouring 0.003 PUDDLING WATERBOUND ROADS 283 Helpers 0.007 Supervision 0.002 Total $0,018 Material f.o.b. Scottsville 0.093 Total per gal $0,111 Kentucky Rock Asphalt. I have the following data from the Clarence Center Road, Mr. John D. Rust, Engineer, col- lected during the season of 1910. In this work an 8-ton tandem roller was found to do better than a 6-ton tandem. The cost of handling, spreading, and rolling this material, from data of five days selected, varied from $0,033 to $0,036 per sq. yd. ; the average being $0,034. The following may be taken as a typical analysis of this cost: Abbreviations. L. Laborers. F. Foreman. T. Teams. E. Roller engineer. Asphalt $10.25 per ton f.o.b. unloading point. Run of July 20, 1909. 69.22 tons hauled and placed. 1,730 sq. yds. covered. 80 lbs. asphalt per sq. yd. 5 L. at cars, 10 hours, at $1.50 each $ 7.50 \ F. at cars at $2.25 per day B . . 1.12 5 T. haul 2 miles at $4.00 per team 20.00 5 L. on wheelbarrows, n hours, each $0.15 per hour .... 8.25 1 T. at shredding machine 4.40 3 L. on rakes, 11 hours at $0.15 per hour ' 4.95 3 L. shoveling, n hours, at $0.15 per hour 4.95 1 F. at shredder, 11 hours at $0,225 P er hour 2.48 1 E. on roller, n hours at $0.30 per hour 3.30 Total , $56.95 Cost per square yard, $0,033. PUDDLING WATERBOUND ROADS There are two methods of puddling : First, by Pipe Line and Hose. Second, by Sprinkling Carts. In the first method a i^-in. or 2-in. pipe line is laid along the road with taps every 200 to 300 feet. The road is wet down by a hose fastened to these taps and sprayed on by a nozzle, or the hose is fastened to a sprinkling attachment on the roller, which throws the water directly onto the wheels; this method is cheaper and more satisfactory than using sprinkling carts, but to work well a pressure of 125 lbs. should be maintained at the pump, which requires a better pumping apparatus than contractors usually have. A very satisfactory plant, used near Rochester, N.Y., 284 COST DATA AND ESTIMATES consisted of a Gould Triplex Pump, operated by a 6-H.P. gasoline engine; the relief valve at the pump was set at 120 lbs. The cost of such puddling on Road 492 for 3,000 cu. yds. of top course was $0.05 per cu. yd.; on Road 294 for 4,000 cu. yds. of top course it was $0.06. This cost includes pumping, helper tending hose, and rollerman. Brooms on the roller were used which materially reduced the cost of brooming the screenings. No charge for water, no allowance made for laying the pipe line; this last charge is included in the lump-sum item of installing plant for a waterbound road, page 311. Gillette, in his handbook, gives sprinkling by carts approxi- mately $0.10 per cu. yd. of top course, which includes sprinkling the subgrade as well as puddling the top course. As the sub- grade is rarely sprinkled, his data reduced to the conditions cited on roads 492 and 294 would give approximately $0.06 per cu. yd. of top course. To this is added the cost of rolling, or about $0.04, which makes the cost of puddling by this method about $0.10 to $0.12, or about twice the amount of the first method. Mr. E. A. Bonney, on the Hamburg-Buffalo road, from a metered supply of water, states the amount required to first puddle a 3-in. top course varies from 50 gals, to 55 gals, per cu. yd. of top course, and the amount needed for the second puddle will be considerably less. Mr. H. P. Gillette states, in a monograph on the Economics of Road Construction, that 30 gals of water per cu. yd. will puddle a road. Mr. E. E. Kidder states that approximately 80 gals, are required per cu. yd. of top course for two puddles. The author's experience agrees with the larger quantities. McClintock Cube Pavement. The general costs of this ex- perimental pavement were given in chapter V. We here give the detailed cost of the vitrified clay cubes and clay-ash cubes only, as the concrete cubes have not worn satisfactorily. Vitrified Shale Cubes. During 1909, 74,000 2|-in. vitrified shale cubes manufactured at Reynoldsville, Pa., were laid at a cost as follows: Teams at $0.50 per hour. 74,000 cubes f.o.b. Reynoldsville $231.25 Freight 68.41 Carting 67.00 Laying 20.00 Total . $386.66 Note. 331 sq. yds. were covered at a cost of $1.17 per sq. yd. Clay and Ash Cubes. In 19 10, cubes made of a local clay mixed with ashes and burned were tried in the effort to get a cheap, tough clay product. As far as known, this is the first time bricks made in this way have been used on roadwork. The ash-clay process has been worked out and patented by Karl Langenbeck, of Boston, Mass. Many local clays used for ordinary brick or farm tile will not stand up under vitrification PUDDLING WATERBOUND ROADS 285 without the addition of expensive, imported refractory clays; but the substitution of coal ashes for the more expensive clays has a similar effect and the cost is materially reduced. Some of the local clay was sent to Mr. Langenbeck, who turned out a few cubes that compare favorably in toughness with the best paving bricks on the market. The Standard Sewer Pipe Company, of Rochester, N.Y., undertook to furnish 400,000 2-in cubes of this description for Mr. McClintock. It was necessary for them to experiment to determine a practical method of molding, the correct tempera- ture to use, and the best proportion of ashes, which naturally raised the price above ordinary practice. In molding they used a modification of the ordinary pipe-molding machine, which produced a hollow square of cubes, at the rate of 30,000 cubes per hour. The scoring knives were so set that the cubes were nearly cut apart, leaving just enough uncut clay to hold them together during the burning, after which a light blow sepa- rated them cleanly. The toughness of the resulting cubes can probably be increased by further experiment; but the product was good, although not up to the standard of the sample cubes made by Mr. Langenbeck. The cost of the ash-clay cubes was as follows: 400,000 cubes f.o.b. Rochester, N.Y. $1,200.00. . .$0,711 per sq. yd. Carting, six miles 247.75. • -0.147 " Filler 27.00. . .0.016 " Labor of laying 191.77. . .0.113 " Roller 12.9 4. . .0.008 " Total $1,679.46 $0,995 " Note. 1,688 sq. yds. covered Labor, $0.22 an hour I f j . and carti Teams, $0.50 an hour ) J to to Mr. McClintock has stated, in discussing the cost, that in large quantities he believes the cubes can be delivered f.o.b. at the plant for $1.50 per 1,000, which would reduce the cost as shown above to about $0.60 per sq. yd., and that the high cost of laying was due to the irregular shape of the first batch, due to not scoring the cubes deeply enough. Amiesite Cost Data. Road 1319, Honeoye Village, Season of 1915. H. W. Baker, Eng. in charge. 4700 sq. yds. laid 16' wide i\" deep. Laid in two courses. Bottom course 2\" thick coarse ma- terial; surface \" thick fine material. Material 588 tons @ $4.00 per ton f.o.b. plant $2352 .00 588 tons freight $0.54 per ton 341 .04 Total cost materials $2693.04 286 COST DATA AND ESTIMATES Labor Force at cars unloading 6 Laborers @ $2.00 per day $12 .00 1 Foreman @ $2.50 per day 2 . 50 1 Fireman @ $3 . 00 per day 3 . 00 1 Night Fireman @ $2 . 50 per day 2 . 50 $20.00 Equipment at Cars 1 boiler and pipe line per day 8 . 00 \ ton coal and oil 2.50 Total daily cost of unloading $30 . 50 Hauling \ mile 4 teams @ $5.00 per day $20 . 00 Spreading and Compacting 1 Asphalt raker 3 . 00 4 Laborers @ $2 . 00 per day 8 . 00 1 Rollerman 3 . 00 per day $14 . 00 Equipment 1 roller (10 ton tandem) 10 . 00 Coal, oil, etc o . 80 Daily cost spreading and compacting $24.80 Summary Daily Force Account Unloading 30 . 50 Hauling 20 . 00 Spreading and compacting * 24.80 $75- 3o Number of days worked, 15 Total labor 1 5 days $1129.50 " materials cost. 2693 . 04 Total cost $3822 . 54 Cost per sq. yd 0.81 Bid price sq. yd o . 85 Conditions This work was done under bad weather conditions the night tem- perature being below freezing and only two days with an air temper- ature above 55 F. It was necessary to keep a night fireman at the cars to keep up steam and to move the steam pipes to different parts of the cars to insure the amiesite being in a condition to loosen and shovel in the morning. The material was loosened by bars and sledges to the bottom of the cars steamed 10 to 30 minutes and then shoveled into dump wagons, covered with canvas and hauled to the street. Under favorable weather conditions the cost of unloading from the cars would probably be reduced 40%. Table of Amounts of Amiesite required for different thicknesses and materials. DEPTHS AND WEIGHTS 287 «" s •8 A W U \0 NO to O O COO lOO x^. Cfl "-< m co ^ -0 O* 06 r^-o' O* *o CO J>»0 O LO a 8 J « A S3 *-* c s a Si 5; St 5; St St St St St St St St Si St St B £ ih|-*hmmI-* i-tl-^Hwcol-* i-iNMnmN< N N N N ro CN CN CN CN CO CN CN CN CN CO «4-l <4H Pcj J3 *& — ; r. • W-* "~ ' «H< ^1-* co|-* H-* «M m|->*Hoo ih|oo»h|th hMw!* w|oo«|-^ ^1- 1— 1 O O M t^ CS CO m O 00 O N't n ao iH|N CN rj- *>■ Qs CN ^NO N 10 coo an Tt 4-J +-> CJ w O « W fO CN ss! *e a O (U Si St & Si St St S; St St St St St St St St i-i|-*»o!oo coloowl^ r-'-u-x eolaowJT* ■H|Tj(m|oo «|aoco|-* -rB, a-> CO co - \C> ~^o ^0 ^0 "1" "1" 'I' "t "1" r^. j>. j>. r^. i>- 2-6 * fe cr S CN CN CN CN . r^ r^ r-» IO W0 1OIOIO co co co CO co O O O O O t^ J>- J>. t^. J^ N N J>. !>. N ^•^ §3^ St St St St St St St St St St St St St St St JS PS to: M M M M M M M M M M M M 1— 1 M M fe^ oj u d cn .-^ w miesite [j. Yds. erTon coco J>-00 CO H OnOO *>» M M 00 cn aaH M O CO J>- *^» M M CN -3- M M CO CN O On CO* *>• M M (U O > jT O.C* O >% g N|» tc c COOO ^t" On lo CO O 10 co m tJ- m OnO co 10 r^ O , W W w TO '^3 ^JD 8 u H c/3 5 < in < -2 «T73^ fl ifl C 3 H in 0^0 « H.^h^ ^J J> 4-> c 288 COST DATA AND ESTIMATES Hassam Concrete Pavement. Cost of Grouting. Road No. 5529. Lyell Ave., Spencerport. Season 1915. E. E. Kidder, Eng. in charge. Road 1 6' wide 5 " deep compacted measure and approx. 6 miles long. 9880 cu. yds. of Has- sam were grouted in 71 working days with the organization shown in Fig. 66 A. Inspectors Inquisitives Natives Vistors Etc. o °O0qO° o ocfto o o fc 2g« Gas Engine g a Man on Top / _: Tongue .5* o? n I 1 Roller *f~at Rest 1 '(as usual) Progress of Work Daily Force Account 5 Tamp & Broom-men $12.50 1 Spout-man 2.00 n 8 Operator 1 2.75 Passers Etc. 12.00 l Roller-man 3.50 l Foreman 3.50 Total 1*36.75 Fig. 66 a 71 days @ $36.25 per day $2573-75 Amount grouted 9880 cu. yds Cost of labor per cu. yd $0.26 Conditions Administration and superintendence good. Temper of crew rather bad as they were not receiving the wages that they expected to get for the first half of the job. Could have made a better record. Speed of Work Averaged 450 lin. ft. per day. Materials Cement. 8500 bbls. used or an average of 0.86 bbls. per cu. yd. of Hassam. This varied from 0.75 in the beginning to 0.95 during the latter part of the work when a liberal spread of stone was used to compensate for rough grading and a desire to end the work before winter. Sand. Royalty on sand was $0.30 per load or $0,052 per cu. yd. of Hassam. Cost of haul corresponds to average costs given in previous cost data. Stone. 16050 tons of limestone ij" to 3 J" in size were used. This amounts to 3250 lbs. per cu. yds. compacted measure which is high for this grade of stone. This was due to a liberal use of stone over poorly shaped sub-grade and to excess depth where wet material was removed. Water. Metered supply. 70 gals per cu. yd of Hassam; this includes water for engines, leakage in a long line and considerable waste at the grout mixer. Concrete Roads. Cost Data. Rd. 5423, Hartland Medina Pt. 2. Season 1914. F. W. Bristow, Eng. in charge. 9550 cu. yds. 1: 1^: 3 concrete pavement laid 16' wide 6" deep. CONCRETE PAVEMENT 289 Materials and Equipment Cement. Knickerbocker @ $1.18 net bags returned f.o.b. siding. 4 mile average haul. Sand. Excellent local sand, if mile haul. Stone. Local crushed stone (Medina Sandstone and granite \" to 2\" in size) \ mile haul to crusher, 1 mile haul crusher to road. Concrete Mixer. Koehring with boom and bucket delivery | cu. yd. batch. Speed of Work. 500 to 550 lin. ft. of road or 148 to 165 cu. yds. mixed and placed per 10 hour day. Actual Amount of Materials Used Cement 1.85 bbls. per c. y. concrete Sand 0.4 c. y. per c. y. concrete Stone 0.80 c. y. per c. y. concrete Joints Wooden joints used for J the work. Steel and felt joints used for J the work. Labor Cost of Mixing and placing Concrete Labor $0,175 per 10-hour day. The force at the mixer comprised: 1 Foreman. 2 Laborers setting forms, shoveling stone. " sand, on stone wheelbarrows. " sand passing cement, emptying " 1 Mixer runner. 1 " fireman. 4 Laborers placing concrete. 2 " on screed. 2 " floating. 1 " preparing joints. 1 " sprinkling, brooming, etc. The cost of setting forms, mixing, placing and finishing the concrete including coal ranged from 0.48 to 0.51 per cu. yd. This does not include overhead or plant charge. The water cost per cu. yd. concrete was approx. $0.04 and includes laying pipe line and pumping from creeks. The overhead charge per cu. yd. of concrete were approx. as follows: Bond $0,036 per cu. yd. Employers Compensation Insurance $2.92 per . $100 payroll 0.096 " " " Public Liability Insurance 0.016 " " " Machinery and tools, freight hauling, erection, interest, depreciation and repairs 0.600 " " " $.748 Say $0. 75 per cu. yd. 10 3 6 2 1 1 290 COST DATA AND ESTIMATES MM u JO O -ft* £ 5 «! < hi « H w kr < « H Hi H m W H W H* « < <: 111 teH w rt < 'I lunouiy J3A03 pu-Bg ^unoray 08 o o o o o o 10 000 00 t^ 000 00 o o o o Amj MOO CO limoray 000 O O to IPO 00 OO ■<* 10 ^unoray SJ3H113 000 . O O to O tovO O f- o 006 4 %& o o o o lunoray S19A.VJ 88 O O o o 88 8 O O M o 000 o o O O to t vO VO vOMD vO vO *=t 10 s 3 ^unoray O VO OOO OON 6\ M * ^-oo NCOM ;99j {-egui^ GO C/I CO Otoji ^cob w 0» fO CM IO O >H OOOOO d d d 11 II II II II 10O moo NO ^100 rf CS Chute 1 2.00 s Placing cone 3 2.00 F Screeding 2 2.00 G Floating 1 2.50 H J Brooming and general help Cement (to hopper) 1 2 1.85 2.00 K Cement (getting ready ) 1 1.85 L Stone (barrows) 4 or 5 1.85 M Stone (shoveling) 8 .1.85 N Sand (barrows) 2 or 3 1.85 O Sand (shoveling) 3 1.85 P Forms 2 2.00 R Pump 1 2.25 T Trimming grade 3 1.85 W Watchmen, covering and wetting down 2 1.85 Also charge X A Supt. and Va timekeeper Trimmers U T" place expansion joints Fig. 66 b Manipulation Exclusive of water, forms and trimming sub-grade. Supt. (J) . _ 3 . Time keeper {\) Foremen (2) (see Fig. 66 B) Engineer (1) ( " " " ) Fireman (1) ( " " " ) Mason (1) ( " " " ) Laborers (8 at 2 . 00) (see Fig. 66 B) 16 . 00 Laborers (21 at 1.85) ( " " " ) 38.85 Total ' $73-85 00 1 .00 7.00 3.00 2.50 2.50 294 COST DATA AND ESTIMATES Based on an average day's work of 182 cu. yds. (10-hour day), the manipulation of a cubic yard of concrete would cost with the above organization, $.406. Note. The above organization has laid over 780 lin. ft. of 16^ ft. pavement (outside dimensions), in a 10-hour day. (The 182 cu. yds. is based on a length of 600' of pavement with a cross-section of 8.2 sq. ft.) Water Plant should be capable of supplying 30 gallons per minute. Pipe: 10,000 (at least) lin. ft. 2" pipe, galvanized at 16 cents $1600 . 00 Black, at 14J cents,. .' 1450.00 35 "Ts" for same (one each 300') at 50^ 17 .50 Pumps: 3 to 25 H.P., dependent on conditions. For lower power, gas engine O. K.; for higher, steam the best (latter, best for surety of supply). Outfits $150.00 to $1000.00 Where wells were necessary, 2" supply pipe. Driving same $1.40 per ft. for depths not greater than 40'; $1.50 to 70' depth. This includes pipe and point. Shaughnessy paid a lump sum for water from Bay shore to I slip ($300.00, I believe) from hydrants. On 5232A, water was bought from private parties for part of the work at $3.00 per day. Cost of running steam pump located at well or surface water supply, including operator, varies from $5.00 to $8.00 per day de- pendent on weather conditions. 200' of rubber hose at $0 . 34 per ft. necessary for connection with mixer and sprinkling road. Of this a 15-foot section should be con- nected on intake pipe of mixer, with which to sprinkle sub-grade. Forms . 1. 6" Channel forms (steel) 32^ per ft. including pins, 8' sections. 2. Patent steel forms with bevel, 24^ per ft. Pins for same $1.00 each, one necessary for each section. Sections 12' long. 3. Wooden forms with bevel about 12 cents per lin. ft. Cost of placing same, 2 men at $2.00 per day, $4.00 per day (see "P," Fig. 66 B). At least 1200 ft. of forms necessary, so that 600 lin. ft. of road can be built without forms being moved. Based on (2) forms would cost 1200' of forms, $288.00 Say no pins 110.00 Total $398.00 Trimming Sub-grade Three men generally necessary at $1.85 per day (see T, Fig. 66 B). CONCRETE COST DATA 295 General A steam roller ($2200.00) might justly be, partially at least, charged to concrete. Cost of operating same, including rollerman, not greater than $12.00 per day if owned by contractor. CONCRETE COST DATA Name of Road, Main Street, Sec. Ill, County Highway No. 130. (Erie County, New York State) Length, 3.68 miles. Thickness, average 7" parabolic crown. Width, 16'. Proportions of mix, 1-1^-3. Total No. Cu. Yds., 7038. 1. Labor, exclusive of water supply, including supervision. .6818 2. Plant forms and tools 3091 f Steel plates .1782 ) 3. Expansion joint material \ Tarred paper .0295 \ 2077 I .2077 J 4. Water supply, including labor 0625 5. Cement placed on roadside ready for mixer 2. 3379 6. Sand " " " " " " 8359 7. Stone " " " " " " 1. 0518 8. Reinforcement, if any 0000 Total cost per cu. yd 5 .4867 Labor, rate per hour, 16 J^; Teams, rate per hour, 50^; Hours in day worked, 10. Remarks: Work done by State day Labor. Materials unloaded by hand. Plant Charges included proportionate to life of plant. Seven-tenths mile average haul. Name of Road, Huntington- Ami tyville, Pt. 2, No. 12 19 (Suffolk County) . Length, 4.69 miles. Thickness, 4!" and 6f" = av. 5! ". Width, 1 6'. Proportions of mix, 1-1J-3. Total No. Cu. Yds., 7409. 1. Labor, exclusive of water supply, including supervision ... $.52 2. Plant forms and tools 58 3. Expansion joint material 05 10 36 23 05 4. Water supply, including labor 5. Cement placed on roadside ready for mixer 2 6. Sand " " " " " " 7. Stone Gravel " " " " " 1 Reinforcement, if any, 0000 Total per cu. yd $4 . 89 Contractor's Bid Price 5-3° Labor, rate per hour, 20^; Teams, rate per hour, 55^; Hours in day worked, 10. 296 COST DATA AMD ESTIMATES Remarks: Auto truck for most of haul. Gravel furnished by large screening and washing plant accounts for high plant cost. Only proportionate part charged for this plant as it is to be used to produce commercial output. Road No. 1 20 1 — Nassau County (1) Sand: In bins Heling Bros, per cu. yd $.15 Haul by auto (contractor owner) 2\ mi. at .15 .38 Sand on road per cu. yd .53 Gravel: In bins Heling Bros, per cu. yd , .85 Haul by auto (contractor owner) 2\ mi. at .15. . . .7,8 Gravel on road per cu. yd $1 . 23 Cement: Cement stored at $1.00 per day for 150 days $150.00 Approximately 11 70 bbls. stored. Storage per bbl. . 13 F.o.b. Farmingdale (est.) 1 . 30 Handling and hauling (.10 est.) double handling in most cases .10 Cement per bbl. on road $1 • 53 This price also approx. cost of cement bought from Parker, Hassam Co. Plant: (2) 1 auto truck (Sauer) $6,500.00 1 Concrete mixer 1,200 . 00 2 doz. shovels 2 1 . co 2 teams at $700., 2 bottom dumpers at $400 2,200.00 Forms (wooden) 800 lin. ft 40 . 00 8 barrows 24 . 00 2 doz. picks 42 . 00 1 bucket conveyor, loader 600.00 1 Screed 15.00 Incidentals 20 . 00 $10,672.00 (3) Pressure water from fire plugs Pavement per cu. yd. cane. .10 $100.00 1500 ft. (lin.) 2" pipe at .08 120.00 100 ft. 2" rubber hose 50.00 300 ft. 1" rubber hose 45.00 $315 00 (4) Manipulation per cu. yd. in place. This does not include covering, uncovering, sprinkling .92 (5) Forms: Setting and re-setting forms per lin. ft. of road .... .03 3X 1.75 X 20 days = .03 3093 CONCRETE COST DATA 297 (6) Trimming: Per cu. yd. of concrete .28 6 men @ $1 .75 20 days = $210 roller @ 12. 6 " = 72 990 cu. yd. of cone. Expansion joints at 40^ apiece every 30 ft. Covering and uncovering and wetting concrete during curing season 2 men to cover ©$1.75 $3 . 50 2 men to uncover at $1 . 75 3-5° 1 man to sprinkle 1 . 75 8.75x20 R , ^^ .18 per cu. yd. cone. 990 - Road No. 1203 — Nassau County (1) Sand: In Pit of Mr. Bennett per cu. yd $.15 Screening and loading (estimated) .22 Haul 2 mi. (auto truck) contractor owner @ .12. . .24 Cost per cu. yd. on road $.61 Trap Rock (Imported) F.o.b. Baldwin $1.59 per cu. yd $1.59 Unloading 15^ .15 Haul 1.6 mi. at 25^ teams and auto truck .40 Stone per cu. yd. on road $2 . 14 Cement: F.o.b. Baldwin per bbl $1 . 38 Handling and hauling per bbl. .05 .05 Cement per bbl. on road $1 . 43 (2) Plant: 1 Screed $ 20 . 00 1 Concrete Mixer 1800.00 1 Steam Roller 3000.00 1 doz. Wheelbarrows 36 . 00 2 doz. Shovels 21 . 00 3 Teams at $700, 3 Bottom Dumpers at $400 3300.00 1 Auto Truck 5000 . 00 2 doz. Picks 42 . 00 Forms (wooden $20.00; steel $126.00,) 146.00 Incidentals 50 . 00 Water Wagon 400 . 00 $13,815.00 298 COST DATA AND ESTIMATES Water: Pressure line fire plugs, total. 4000 ft. 2" pipe .06 100 " " rubber hose 3OO I (4) Manipulation: Includes all work, sprinkling, covering, uncovering, in place complete per cu. yd (5) Forms: Setting and resetting forms per lin. ft. of road (6) Trimming (Sub-grade) Per cu. yd. concrete in place Expansion Joints at 40^ apiece every 30 ft. $100.00 240 . 00 50.00 45.00 $435 00 $.67 •05 .24 Road No. 12 1 9 — Suffolk County (1) Sand: In bins Heling Bros, pit cu. yd Haul by auto (contract) 1.5 mi. est. .30. Cost per cu. yd. on road Gravel: In bins Heling Bros, pit cu. yd Haul by auto (contract) estimated .30. . Cost per cu. yd. on road Cement: Bbl. f.o.b. Farmingdale Handling and hauling Cement on road per bbl. (2) Plant: Same as on No. 1218 (3) Water: Cost of water 1 mile 2" pipe at .06 per. ft.. 100 ft. 2" rubber hose 400 ft. 1* 1 pump and gas engine (est.), (4) Manipulation: Includes all work; sprinkling, covering, uncovering, in place complete per cu. yd (5) Forms, setting and reset ing per lin. ft. of road (6) Trimming: Per lin. ft. of road Expansion joints at 40^ apiece every 30 ft. $ . 10 • 30 $ .40 $ •75 •30 $1 •05 $1 .27 08 $1 35 $700 . 00 320.00 50.00 60.00 1,000.00 $2,130.00 $.50 .02 .06 CONCRETE COST DATA 299 Road No. 1218 — Suffolk County (1) Sand: In bins Heling Bros, pit cu. yd $.10 Haul by auto (by contract) 1.7 mi. approx .40 Sand per cu. yd. on road $ . 50 Gravel: • In bins Heling Bros, pit cu. yd $ . 75 Haul by auto (by contract) approx .40 Cost per cu. yd. on road $1.15 Cement: F.o.b. Farmingdale $1.27 Handling and hauling .08 Cement on road per bbl $1 . 35 (2) Plant: Gravel and sand screening complete, including various set ups $15,000.00 1 Concrete Mixer 1,800.00 1 Roller 2,800 . 00 1 Screed 20 . 00 18 Wheelbarrows 54 . 00 2 doz. Shovels 21 . 00 3 teams @ $700, 3 bottom dumpers @ $400 3,300.00 2 doz. Picks 42 . 00 1 rd. Planer and Scarifier 600.00 Forms (steel) 630 . 00 Tarpaulins 100 . 00 Incidentals 75 . 00 $24,442.00 (3) Water: Total paid for water approx $130.00 1 mile 2" pipe at .06 320.00 100 ft. 2" rubber hose 50.00 400 ft. 1" " " 60.00 $560.00 (4) Manipulation: Includes all work, sprinkling, covering, uncovering, in place complete per cu. yd $ . 54 (5) Forms: Setting and resettimg forms per lin. ft. of road. . . .02 (6) Trimming: Per lin. ft. of road .06 Expansion joints 40^ apiece every 30 feet. 300 COST DATA AND ESTIMATES COST DATA Name of Road, Huntington Town Line-Farmingdale, Part i (Suffolk County) Length, 1.27 miles. Thickness, 5" & 7"= Av. 6". Width, 16'. Proportions of mix, 1-1J-3. Total No. Cu. Yds. 2051. 1. Labor, exclusive of water supply, including super- vision $ . 56 2. Plant forms and tools .61 3. Expansion joint material .06 4. Water supply, including labor .12 5. Cement, Placed on roadside ready for mixer 2 . 13 6. Sand, " " " " " " .22 7. Stone, " " " " " " 1. 01 8. Reinforcement, if any 0000 Total per cu. yd $4.71 Contractor's Bid Price 5 . 30 Labor, rate per hour, 20^; Teams, rate per hour, 55^; Hours in day worked, 10. Remarks: Auto truck for most of haul. Gravel furnished by large screening and washing plant accounts for high plant cost. Only proportionate part charged for this plant as it is to be used to pro- duce commercial output. Road No. 1202 — Nassau County (1) Sand: Estimated at .10 per cu. yd. in bins $ . 10 Haul (by contract) estimated at .40 per cu. yd. . . .40 $.50 Gravel: Stiff leg derrick set up (in bins) $2 . 00 Haul (by contract) estimated at .40 per cu. yd. . . .40 1st set up per cu. yd. on road $2 . 40 Drag line set up per cu. yd $1 . 20 Haul (by contract) estimated .40 per cu. yd .40 2nd set up per cu. yd. on road $1 . 60 Imported gravel per cu. yd. scow L. I. City .85 Unloading from scow L. I. City .15 Freight L. I. City to Central Park .78 Unloading at Central Park .10 Haul (by contract) at .20 per cu. yd .20 Cost per cu. yd. on road $2 .08 Average cost per cu. yd. on road $1 .92. CONCRETE COST DATA 301 Cement: Bbl. f .o.b. Central Park ( P^ed ) $1 • 20 \ by market Haul by contract per. Bbl .06 Handling .02 Per Bbl. on road $1.28 (2) Plant estimated $12,000.00 (3) Water: Cost of water $1,000 . 00 4 mile 2" pipe at $0.06 per ft 1,270.00 100 ft. 2" rubber hose at $0 . 50 50 . 00 400 ft. 1" " " at $0.15 60.00 1 pump, boiler, etc 1,000 . 00 $3,380.00 (4) Manipulation: Includes all work, sprinkling, covering and uncover- ing; in place complete per cu. yd $.64 (5) Forms: Setting and taking up per lin. ft. of road .04 (6) Trimming: Per lin. ft. of road . 075 Expansion joints at 40^ apiece every 30 ft. COST DATA Name of Road, Little Valley-Cattaraugus, Part 1 (Cattaraugus County) T ,v ., / 3900 lin ft. 16 ft. \ Length, 5-35 miles. j ^ „ „ ^ ft j -,. , /6"&8" = Av. 7" -\ Thickness, | g. & f= Ay £ |r J 2.575 miles completed. Width, 16 and 14. Proportions of mix. 1-1J-3. Total No. Cu. Yds. 8280 — This cost covers 4165 cu. yds. 1. Labor, exclusive of water supply, including super- vision . 423 2. Plant forms and tools. .662+ .158 for coal .820 3. Expansion joint material .045 4. Water supply, including labor .030 5. Cement, placed on roadside ready for mixer 1 .984 6. Sand, " " " " " " .522 7. Stone, " " " " " " 1.505 8. Reinforcement, if any 0000 Total cost per cu. yd 5 . 324 Contractor's Bid Price 6.30 302 COST DATA AND ESTIMATES Labor, rate per hour, i8|j£; Teams, rate per hour, None used; Hours in day worked, 10. Remarks: Industrial Ry. Plant delivering sand, stone and cement into hopper of concrete mixer, clam shell unloader used to take material off cars. Material delivered alongside mixer in buckets proportioned for one batch size. For plant layout see illustration Erie R.R.^ 1 Plant Layout COST OF CONCRETE WORK The following data will help in estimating the cost of small concrete jobs, such as culverts, walls, etc. This data was collected by Mr. E. E. Kidder during the season of 1908. Table 49 contains the theoretical proportions of cement, sand, and stone required for the three ordinary mixtures of concrete. These values were found by experience to agree with actual proportions very closely for \" to (Continued page 304.) CONCRETE COST DATA 3°3 .*8 o -g < oo ■fJ 73 S3 &-^ s^6 tJD bo J .a.a c -t-> ►* o ■PUn 4) > ; ~ ; y tn +-> ,3 g^ fl rT ° 3 3 d M \0 ON CO ■* ON q o q T3 O CO v> tJ- N f)N c o o o IP X3*CJ J/J co (N ^ ON ^ ON MOO gs - O o o o to co to to n O O u MO O to o m> d ro o o oo ON O CO o o" cs be c >-i W 3 >»3 • c/j c/J . a w >> w 8 8 8 o o t3 o 8 8 CO 8 00 o o ON M w u a Hf» 45 O 3 en 'c H g 3 u O 3 c .3 w i- a P tn 3 < t-i 33° Cost of paint per gal. (freight not included) $1.05 Cost of paint per lin. ft. of guard-rail 0.0084 Cost of paint and painting per lin. ft. of guard-rail 0.0165 Lin. ft. of guard-rail painted per gal 134-4 No. gals, of paint used per lin. ft. of guard-rail 0.0077 Time required to paint one foot, in decimals of an hour . 0.0269 Concrete Guard-Rail. Style of rail shown in sketch on page 87, chapter on Minor Points. Labor, $0,225 per hour. Cost of manufacturing 1,233 lin. ft. of rail of the above de- scription. Taken from the Report of the New York State Highway Commission of 1910. Lumber $ 32.46. . . .$0,026 per lin. foot. Steel 139.64 0.114 " " Cement , 57.62 0.046 " " " Gravel 10.00 0.008 " " Metal cores 77.00 0.063 " " PRICES OF VITRIFIED PIPE 307 Labor 231.83, 0.188 Miscellaneous • 5-35- • • • -^004 Total $553-9° °-449 This data applies to small quantities; if manufactured on a large scale the cost should be reduced to about $0.30 per lin. ft. The cost of setting the above rail varied from $0.09 to $0,125 per lin. ft: labor $0,225 per hour. This does not include haul- ing from the factory to the intended position on the road. Cobble Gutter. Road 5,046, W. G. Harger, Engineer. Labor, $0,175 P er nour - Foreman, $3.50 per day. Cobbles averaged 6 in. in size; no sand cushion required, as gutter was built in a sand cut. Gutter was laid by ordinary laborers using paver's tools; tamped with a paving rammer, and the top voids filled with No. 2 stone crushed on the job. 430 sq. yds. were laid at the following cost per sq. yd.: Cobbles, free .$0,000 Loading £ cu. yds. of cobbles 0.030 Hauling | " " " J mile 0.024 Laying and tamping . 0.080 Filler. Cost of 0.05 cu. yds. No. 2 stone at crusher bin, approximately 0.030 Hauling 0.05 cu. yds. 1 mile 0.015 Spreading and brooming, 0.05 per cu. yd. No. 2 stone . . . 0.010 Total $0,189 PRICES OF VITRIFIED PIPE The discounts vary, but if no quotations of current prices are available the following list will serve for an approximate estimate: 1 Eastern List Size Discount 3* to 24" 88% 24" and 30" 80% 33" and 3 6* 75% At these discounts the net prices per foot in car-load lots f .o.b. factory are: Size Price Size Price a;, $0,024 20" $0,270 4" 0.030 21" 0.325 c 0.036 2 2" O.360 6" 0.048 24" O.390 8" 0.066 27" O.900 10" 0.096 30" I. IOO 12" 0.120 33" I.560 15" 0.162 36" I-7SO 18" 0.227 — — 1 Engineering News, April 4, 191 2. 3 o8 COST DATA AND ESTIMATES g o u u < g < Hi O CO P o u u w u p< o H Hi PL) Ph O w 5 o 1 P-C CO o a PQ *> fa § 888818 88888888 OOOOO 1 ON mDOOOOO» o F^ (J M W M H H H MMMM 4) m 6ft xn 73 a | rt $3 so Hi o & o 10 6 4©= 4 10 bo c^ Hi > < U O Hi a. o PLANT AND PAY-ROLL 309 PLANT AND PAY-ROLL Table 50, page 252, shows in a convenient form the value of plants and the largest weekly force account of two months' duration on fourteen roads in New York State. From this and other information it is reasonable to assume that a contractor has tied up, outside of money on plant and materials, from $5,000 to $8,000 for the full length of time that the work is in progress, and for short periods he may have as high as $15,000 or $20,000 invested. Interest, Depreciation, Repairs, etc. To the best of my judgment the following estimates show about the amount of money required on the different styles of construction noted. These data are based on an outfit which would be capable of a speed of about 0.7 mile per month, or five miles in a season. Adopted Value of Plant Items Item 10-ton roller Brick roller Traction-engine Crusher Elevator Bin and screen 5 H.P. gasoline engine Gas engine and pump 6000 ft. of pipe Wagons Hand tools Plows Road machine Tar kettle Wheel scraper Slush scraper 10-ton roller used for hauling Hauling traction-engine .... Value $1 $2700 1800 1200 900] 200 \ 500 J 250 200 600 115 150 100 200 25-200 70 6 2700 2200 Life 20 yrs. 20 5 say 10 6 1 3 5 5 5 10 Annual Repairs $70.00 40.00 IOO.OO 400.00 50.00 50.00 10.00 IO.OO 10.00 10.00 IO.OO 200.00 200.00 1 Including new tank every three years. 310 COST DATA AND ESTIMATES 6% Interest and Depreciation on Plant Items Item Roller Traction-engine Crusher Elevator Screen Bin Gasoline engine Gasoline pump 6000 feet 1 1" pipe Wagons Hand tools Plows Tar kettle Concrete mixer Brick roller Wheel scrapers Slush scrapers Roller used for hauling Hauling engine Interest $162.00 72.00 54.OO 12.00 3.00 30.00 15.00 12.00 36.OO 6.00 9.00 6.00 I2.00 I20.00 108.00 5.00 162.OO 132.OO Depreciation $135.00 150.OO IOO.OO 30.00 50.00 40.00 30.00 40.00 60.OO 20.OO 150.00 30.OO IO.OO IOO.OO 15.OO 270.OO 300.OO Charge for bond \% total contract. PLANT FOR WATERBOUND MACADAM STONE IMPORTED Elevator unloading plant, provided more than 2,000 cu. yds. of stone is to be unloaded. Item Interest Depreciation Repairs Elevator Bin $12.00 30.00 15.OO 162.OO 36.OO 12.00 9.00 6.00 12.00 5.00 80.00 $30.00 40.00 30.00 135-00 60.00 40.00 150.00 30.00 40.00 15.00 300.00 $50.00 50.00 50.00 70.00 10.00 50.00 IO.OO IO.OO 150.00 5 H.P. gasoline engine 1 roller with broom and sprink- ling attachment 6000 ft. 1 \" pipe Gasoline engine and pump Hand tools Plows Road machine . . 2 wheel scrapers 2 slush scrapers 1 <; wagons . Totals one season's work 5 miles Total per mile $379-°o 76.00 $870.00 174.00 $450.00 90.00 PLANTS FOR MACADAM STONE 3ii Force account money out: Allow six weeks out continually for length of job at \% interest per month. Allow $6,000 out, or $40.00 interest per mile on force account. Bond charge: J of 1% contract price; approximately $25.00 per mile. Insurance charge: $2.00 per $100.00 total force account, approximately $100.00 per mile. Allow for moving plant on job, $500.00 lump sum. Plant for Waterbound Macadam Local Stone Item Interest Depreciation Repairs 1 traction engine 1 crusher and bin $ 72.00 IOO.OO IO.OO I2.00 322.OO $150.00 220.00 50.00 30.00 770.OO $100.00 400.00 80.00 20.00 300.00 1 steam drill and bits 1 small boiler for drill Roller, pipe, gasoline engine and pump, hand tools, plows, road machine, scrapers and wagons as for imported stone plant. Total of these items Total for season, 5 miles .... Total per mile $516.00 103.00 $1220.00 245.00 $900.00 180.00 Force account slightly larger on local stone roads. Approxi- mately $7,000.00 out. Interest on force account $50.00 per mile Bond charge 20.00 " Insurance 120.00 " " Moving plant on job, $500.00 lump sum. Plant for Bituminous Macadam Imported Stone Item Interest Depreciation Repairs Elevator unloading plant 2 rollers $60.00 320.OO 36.OO 112.00 $IOO.OO 270.OO 535-00 $150.00 140.OO 30.OO 170.OO 3 tar kettles Hand tools, plows, road machine, scrapers and wagons as for waterbound macadam. Total of these items Total for season, 5 miles Total per mile $528.00 106.00 $905.00 181.00 $490.00 98.OO Interest on force account $40.00 per mile Bond charge 30.00 " Insurance 100.00 " " Moving plant on job, $500.00 lump sum. 312 COST DATA AND ESTIMATES Plant For Bituminous Macadam Local Stone Item Interest Depreciation Repairs i traction engine $72.00 IOO.OO IO.OO I2.oo 468.OO $150.00 220.00 50.00 30.00 805.00 $100.00 400.00 80.00 20.00 340.00 i crusher outfit i steam drill and bits i portable boiler for drill Rollers, hand tools, plows, road machine, scrapers, wagons, and tar kettles as for imported stone. Total of these items . . Total for the season, 5 miles . . Total per mile $662.00 132.00 $1255.00 251.00 $940.00 188.00 Interest on force account $50.00 per mile Bond charge 25.00 " Insurance 120.00 " " Moving plant on job, $500.00 lump sum. FORMS FOR ESTIMATES The following forms of estimate have proved very satisfactory. The item of 6% on materials is used to cover demurrage and interest on money tied up on freight and stone. The other items of profit are what we consider a reasonable return for the risk of such contract work. Mechanical hauling is not considered, because few contractors own plants that make it possi- ble. The total item of interest, depreciation, repairs, and inter- est on force account money for the whole job is charged against top and bottom stone, as the construction quantities of the macadam will vary less from the estimated quantities than any other classes of work. Standard Estimates. ( Figured on the basis of 20% profit on labor, 6% on materials, 6% on money invested, and an allow- ance made for depreciation on different plants, as previously given. Labor at $0,175 P er hour Teams at $0,450 " FORMS FOR ESTIMATES 3*3 Earth Excavation. Class Amount per Mile Price per Cu. Yd. Easy Easy Easy Average Average Hard Hard 5,000-10,000 cu. yds. 3,000- 5,000 cu. yds. i,5oo- 3,000 " 3,000- 5,000 ' 1,500- 3,000 " " 3,000- 5,000 1,500- 3,000 ' $0.40 0.45 0.50 O.50 O.60 O.60 0.70 Rock Excavation. Large boulders (for which 10 cu. yds. a mile are allowed on all estimates) . . .$1.50 per cu. yd. ( Steam drillwork, limestone 1.25 " " " I * \ " " " granite 1.50 " " " \ Hand " " limestone 2.00 " " " 2 ' \ " " " granite 2.00 " " " 1. Large quantities 2. Small quantities Field Stone Sub-base. A sub-base course 6 in. deep made of the usual size fence stone requires 1 cu. yd. loose for 1 cu. yd. rolled; 12 in. deep requires 1.25 cu. yds. loose. Cost of cobbles per loose cu. yd $0.10 Loading cobbles per loose cu. yd 0.15 'Hauling cobbles 1 mile per loose cu. yd. 0.35 Placing cobbles per loose cu. yd 0.10 Multiply these items by 1.25 for 12-in. depth of sub-base. Rolling cobbles per loose cu. yd 0.05 Filler (see below) . — Total 20% profit $ — Estimate $ Filler. I cu. yd. per cu. yd. rolled sub-base. Cost J cu. yd. at pit or crusher $ — Loading J cu. yd 0.05 Hauling f cu. yd. 1 mile 0.10 Spreading J cu. yd 0.04 Total $ — 314 COST DATA AND ESTIMATES Sub-base Bottom Course. Same relation of loose and rolled quantities as for sub-base. Cost fence stone per loose cu. yd $0.10 Loading fence stone per loose cu. yd. ... 0.15 Hauling 1 mile per loose cu. yd 0.35 Placing and sledging 0.20 Rolling 0.05 Filler (see below) — Total $ — 20% profit 1 — Estimate — Filler. i cu. yd. per cu. yd. rolled sub-base. Cost J cu. yd. at pit or crusher $ — Loading 0.05 Hauling 0.33 cu. yd. filler per mile 0.10 Spreading and brooming 0.08 Total $ — Imported Bottom Stone Materials. 3" course, 3,05c 1 lbs. f.o.b. crusher. $ — 4" " 3,i5o " " " — 6% profit — Total $ — Freight on stone to delivery point — Total, No. 1 .$ — Labor. Unloading Under 2,000 cu. yds. (shoveling) $0.15 per cu. yd. Over 2,000 cu. yds. (elevator) 0.10 " " " Hauling (Teams) Bad conditions 0.35 " " " Average conditions 0.30 " " " Good conditions 0.25 Mechanical hauling 0.15 " Spreading 5I in. loose depth 0.06 " 4 in. loose depth 0.08 Rolling 0.05 At this point total up and add 30% of the total to change the estimate from loose to rolled measure. Filler (see below) Labor, total $ — 20% profit . — Total, No. 2 $ — 1 These weights are for limestone. See pages 268, 272. iC u u a (c tt tt (C a (( u FORMS FOR ESTIMATES 315 Filler. Cost of 0.35 cu. yd. at pit or crusher $ — Loading 0.35 " " 0.05 Hauling 0.35 " " 1 mile @ $0.35 per yd. mile . . 0.12 Spreading and brooming 0.35 cu. yd 0.07 Filler, Total $ — Summary. Total No. 1 $ — Total No. 2 $ — Interest and depreciation $ — Estimate $ — Imported Top Stone Waterbound Macadam Materials. 1 4,450 lbs. stone f.o.b $ — 6% profit $ — Total $ — Freight on stone to delivery point $ — Total No. 1 $ — Labor. Unloading (same as bottom) $ — Hauling (same as bottom) $ — Spreading 0.08 Rolling 0.04 Puddling 0.06 Total, loose measure $ — Add 30% .$ — Total rolled measure $ — Screenings. (See below) Total $ — 20% profit — Total No. 2 JT~ Screenings. 2 Unloading 0.5 cu. yd $0.07 Hauling 0.5 " " 1 mile 0.15 Spreading 0.5 " " by cross dump wagons 0.03 0.5 " " " hand 0.07 Total $ — Summary. Total No. 1 $ — Total No. 2 — Interest, depreciation, etc — Estimate $ — 1 These weights are for limestone. See pages 268, 272. 2 Screenings are usually unloaded by hand. 316 COST DATA AND ESTIMATES IMPORTED TOP STONE BITUMINOUS MACADAM. PENETRATION METHOD Materials. 2" course, 4,350 lbs. stone and screenings, f.o.b. crusher . .$ — 3" " 4,050 " " " " " " . . . — gal. bituminous binder, f.o.b. plant — 6% profit — Total . . $ — Freight on materials to delivery point — Total No. 1 .$ — Labor. No. 3 stone. Unloading 1 cu. yd. (same as given) . . . .$ — Hauling 1 cu. yd. " " " .... — Spreading 1 cu. yd. " " " .... — Rolling 1 cu. yd 0.09 Total $ — Add 20% for 2" course, 30% for 3" course — Total No. 2 $ — No. iA, No. 2, and Bitumen. Unloading 0.6 cu. yd. for 2" course (same as given) $ — (( n a z u il ct i" li u li tl Hauling at the rate of $0.30 per yd. per mile — Hauling bitumen at rate of $0,002 per gal. per mile — Spreading and brooming No. iA and No. 2 at rate of $0.30 per cu. yd — Manipulation of heating and spreading bitumen at $0,015 per gal — Total No. 3 $ — Total No. 2 — Total $ — Add 20% profit — Total No. 4 $ — Summary. Total No. 1 $ — Total No. 4 — Interest, depreciation, etc — Estimate $ — Local Stone Macadam. Field stone. 1 cu. yd. field stoAe = 1 cu. yd. crushed. 1.8 cu. yds. field stone = 1 cu. yd. No. 3 and No. 4 rolled. Cost of field stone $0.10 per cu. yd. Blasting or sledging, per cu. yd. actually blasted or sledged 0.40 " " " LOCAL STONE MACADAM 317 9 — Loading field stone 0.15 per cu< yd Hauling field stone 1 mile 0.35 Crushing Sandstone (soft) 0.10 Limestone 0.15 Granite and trap rock 0.20 Total cost in bins (loose including Nos. 1, 2, 3, and 4 stone) $ — Quarried Stone. Limestone, quarrying, small quarries $0.50 Conglomerate, " " " 0.75 Trap, " " " 0.65 Crushing (same as above) — Total cost in bins $ — The crushing does not include repairs to crusher. The crushing is taken from previously given data. The item of quarrying includes delivery to crusher. Estimate of Bottom Stone. Cost in bins $ — Loading, per cu. yd 0.01 Haul (same as bottom) — Spread (same as bottom) — Rolling (same as bottom) — Total (loose measure) $ — Add 30% — Total rolled measure $ — Filler (same as bottom) — 20% profit — Total, No. 1 $^^~ Interest and depreciation — Estimate $ — Local Top Stone. Cost in bins $ — Manipulation same as for imported stone — Total $ — 20% profit — Total No, 1 Interest and depreciation . , Estimate Material. Imported No. 2 Stone, Estimated Loose 2,400 lbs. stone $ — 6% profit — Total No. 1 $ — 318 COST DATA AND ESTIMATES Unloading (same as bottom) — Haul (same as bottom) — Spreading o.io Total $ — 20% profit — Total No. 2 $ — Total No. 1 $ — Total No. 2 — Estimate $ — Local No. 2 Stone. Cost per cu. yd. in bins $ — Haul same as above — Spreading same as above — Total $ — 20% profit — Estimate $ The following is an example of the method of using these standard forms. Estimate for Local Fence Stone Bottom Course Assume that stone will cost $0.10 per cu. yd. in the fences. " I mile average haul to crusher. " 20% of the stone has to be sledged or blasted. " f of a mile average haul from the crusher. " that filler costs $0.15 per cu. yd. in the pit. " average haul of J mile for filler. " that the interest and depreciation charge for the total job, say 4 miles, is distributed over 6,000 cu. yds. of macadam. Use Standard form for Local Bottom Stone, given on page 261. Cost 1 cu. yd. field stone $0.10 Blasting and sledging \ cu. yd. stone 0.08 Loading 1 cu. yd. field stone 0.15 Hauling 1 cu. yd. field stone \ mile " 0.18 Crushing 1 cu. yd. (Mixed granite and sandstone) 0.15 1 cu. yd. Total cost in bin $0.66 Cost 1 cu. yd. crushed stone in bins $0.66 Loading on wagons 0.01 Haul to road, average conditions, f of a mile ... 0.22 Spreading 5 \" loose 0.06 Rolling 0.05 Total $1.00 EXAMPLE OF METHOD 319 Add 30%. . , .30 Per. cu. yd. rolled measure $1.30 Filler (see below) 0.31 Labor, total $1.61 Add 20% profit 0.32 Total, No. 1 $1.93 Interest, depreciation, etc. (see below) 56 Estimate per cu. yd. rolled in place $2.49 say $2.50. Filler. As mentioned on page 240, the screenings produced in crushing bottom only, as in this case, will amount only to 50% of the required filler, therefore two estimates must be made for filler as below: Screenings for Filler. Cost of 0.35 cu. yds. screenings in bin @ $0.66 per cu. yd. . . $0,230 Loading 0.35 " " from bin 0.003 Hauling 0.35 " " " f of a mile 0.077 Spreading and brooming 0.35 cu. yds 0.070 Total ; $0,380 Sand Filler. Cost of sand in pit 0.35 cu. yds $ 0.052 Loading 0.35 cu. yds 0.050 Hauling 0.35 " " \ mile (short-haul figures) 0.060 Spreading and brooming 0.35 cu. yds 0.070 Total $0,232 Average these costs as the screenings must be utilized to use up the total output of the crusher. Average filler $0.31. Interest, Depreciation, etc. From page 255, using value adopted for, say, waterbound macadam roads, the following charge for a 4-mile road is figured: Interest on plant 4X 103.00 . . . $412.00 Depreciation on plant 4X245.00 .... 980.00 Repairs on plant 4X180.00 .... 720.00 Interest on pay-roll 4X 50.00 .... 200.00 Bond charge 4X 20.00 .... 80.00 Insurance 4X 120.00 .... 480.00 Moving plant on job 500.00 Total $3,372.00 to be spread over 6,000 cu. yds. of macadam. 3,372 ~~ = 0.56 cents 6.000 320 COST DATA AND ESTIMATES The cost of an improved highway generally depends on the item of top and bottom stone in place complete. Many of the minor items have standard prices. Such items as cast-iron pipe, the vari- ous sizes of tile, pipe railing, mesh reinforcement steel, etc., will hardly vary in price throughout the Eastern States. A table of these standard prices as used by the New York State Highway Com- mission is given below. It will be noted that all of these items have little bearing on the total cost, and that the items of Earth Excavation, Sub-base, or Sub-base Bottom Course, Macadam Bottom and Top Course, Con- crete Foundation, Brick Pavement, etc., which of necessity are not standard in price, determine whether or not the road is to be expen- sive. Unit Prices Minor Items Overhaul on excavation $ o.oi per yd sta. yd. sq a ft lin. ft ft. it a n ton lin. ft. a Third-class masonry cement joints . . 6.00 per cu Second-class concrete 9.00 " " Third-class concrete (stone ) 7.00 " " " " (gravel) 5-50 " * Pointing old masonry 0.75 " sq Riprap 1.50 " cu Paving cement joints 1.50 Cobble gutter 0.50 Expanded metal 0.08 Guard-rail 0.30 2" pipe rail 1.50 Concrete guard-rail 1.00 Cast-iron pipe in place 35-oo 6" V. T. P. in place 0.30 12" V. T. P. " " 0.60 15" V. T. P. " " 0.90 " " " 18" V. T. P. " " 1. 10 " " " 24" V. T. P. " " 2.00 " " " Relaying old pipe 0.10 " " " 4" farm tile under drain in place .... 0.10 " " " Steel in place 0.05 " lb. Oak timber in place 50.00 " M.B.M. Hemlock timber in place 40.00 " M.B.M. Danger signs 2.00 each Guide-board posts 6.00 " Highway No. signs 1.00 " Guide signs per letter 0.15 " The item of Earth Excavation as shown in Table 36 may vary between 40c and 65c. In extreme cases where material is difficult to handle, it may be estimated still higher. A particular instance of costly excavation where 70c was estimated occurs on a road near the Lackawanna Steel Plant at Buffalo. This road had been filled with slag from time to time. In the remaining variable items the length of haul is a governing factor and three actual conditions of determining the average haul are given here before proceeding farther with the estimate data. EXAMPLE OF METHOD 321 The following cases 1, 2, and 3 show also the present method of estimating where interest and depreciation are not directly considered. Case I The simplest possible conditions. Perry Village County Highway, Wyoming County, N.Y. Imported stone, delivery at middle of road — coal trestle available for unloading — no dead haul to road. Road 16 feet wide throughout. Railroad at Station 60 Station 0+00 = beginning of contract 106+ 23 = end of contract For ease of computation, say stone runs 10 yds. to mile. Station o + 00 to 60 1.1 miles average .55 miles ,55milesXn yds. = 6.05 yd. miles Station 60 + 00 to 106 + 23 ., .. .44 miles X 8. 7 yds. = 3.83 yd. miles .87 miles average .44 miles J/ , — ; —* ^ , r, — ^ — r 1 6 Total yds. 19.7 9.88 yd. miles Total 9.88 yd. miles „ . . ■ = = 0.50 miles average haul 19.7 yds. J & Completion of Perry Village Estimate. Stone from Rock Glen Quarries Stone $0.65 per ton f.o.b. Cu. yd = 2400 lbs. Freight .40 " " " Sub-base Stone .50 " " " Bottom Top Screenings Sub-base Stone 78 .78 .78 .60 Unloading 15 .15 .15 .15 Average haul § mi. at •35 ••••; 175 .175 -175 .175 Manipulation 30 .25 .20 .20 1.305 Consolidation (plus \). .468 .451 .4 + \ .22 .5220 Filler (^ cu. yd. sand Screen- at $1.00) 50 ings .522 Sand .50 Profit (20%) 474 .465 .369 Freight (40 + .08 + (40+. 08+ (40+08 .16) 64 .16+192) .832 +.096 ) .576 6% interest on freight to cover demurrage, etc 038 .05 .034 Manipulation of Bi- tuminous Material . . .60 S3.525 $4,275 te.824 Use $3.55 Use $4.30 Use $2.80 Case II The Walker-Lake Ontario Road, Monroe County, N.Y. Road extends from Station o + 00 to Station 197 + 45. . 322 COST DATA AND ESTIMATES Local stone — mostly fences. Because of location of stone as determined by engineer's inspection, it was determined to make three set-ups of crusher, at Station 40, 104 + 50 and at Station 157. The hauls from stone piles to these crushing points were figured in the regular manner. From the crusher to road, the hauls were arranged, From Station 40 — haul stone o + 00 to 77 + 00 " " 104 + 50 haul stone 77 + 00 " 130 + 00 " " 157 " " 130 + 00 " 197 + 45 Care was taken to see that enough stone was available near each crushing point to furnish macadam between stations supplied from that set-up. The widths of road were as follows: o + 00 to 40 + 00 — 12' wide 40 + 00 " 66 + 60 — 16' wide 66 + 60 " 129 + 50-14' " 129 + 50 " 197 + 45-12' " Use 10 yds. per mile for 12' road proportionally 11.7 yds. mile for 14' road " 13.3 " " " 16' " Haul on road from Station 40 + 00 12' wide o + 00 to 40 + 00 0.76 miles average .38 .38 miles X 7.6 yds. = 2.89 yd. miles 16' wide 40 + 00 to 66 + 60 .50 miles average .25 .25 miles X 6.6 yds. = 1.65 yd. miles 14' wide 66 + 60 to 77 + 00 .20 miles average .1 plus dead haul .5 .6 .6 miles X 2.3 yds. = 1.38 16.5 5.92 yd. miles Haul on road from Station 104 + 50 14' wide Station 77 + 00 to 129 + 50 (say 130) 77 + 00 to 104 + 50 .52 miles average .26 .26 miles X 6.1 yds. = 1.59 yd. miles 104 + 50 to 130 .48 miles average .24 .24 miles X 5.6 yds. = 1.34 yd. miles 1 1.7 yds. 2.93 yd. miles Haul on road from Station 157 12' wide Station 129 + 50 (say 130) to 197 + 45 130 to 157 .51 miles average .26 .26 miles X 5.1 yds. = 1.33 yd. miles 157 to 197 + 45 .76 miles average .38 .38 miles X 7.6 yds. = 2.89 yd. miles 12.7 yds. 4.22 yd, miles EXAMPLE OF METHOD 323 Average haul for entire road From Station 40 16.5 yds. 5.92 yd. miles 104 + 50 n.7 " 2.93 " " 157 1^7 " 4.22 " " 40.9 13.07 13.07 -7- 40.9 = .32 miles say .3 miles average haul Sub-base Bottom Course Stone $ .15 Sledging, blasting, and sorting 30% of stone at .35 per yd. . . .105 Loading into wagons 15 Haul to crusher at Stations 40, 104 + 50 and 157. One mile at .35 35 Haul on road. 3 mile at .35 105 Manipulation 20 Consolidation (plus i) 212 Filler (J cu. yd. sand at .80) 40 Profit (20%) -334 $2,006 Use $2.00 Local Stone Top Course — Bituminous Binder Top Course Screenings Stone 15 .15 Sledging, blasting, and sorting 60% of stone at .35 .21 .21 Loading into wagons .15 15 Haul to Crusher at Stations 40, 104 + 50 and 157. 1.1 miles at .35 385 .385 Crushing 35 .35 Haul on road .30 miles at .35 .105 .105 Manipulation .20 .20 Consolidation (plus f ) .517 1.55 Filler (.401. yd. of screenings) .620 .620 Profit (20%) 537 Manipulation Bituminous Material .60 $3-824 Use $3.85 Case III The Obi-Cuba Highway, #965, Allegany County, N. Y. 9.93 miles long. From a field inspection of this road, it was found that stone was available at both ends of road, but not in the middle. An ample supply of good gravel was found in the middle section, and it was determined to build a concrete base with bituminous top, this type of road being the only one which could be built using local material. 324 COST DATA AND ESTIMATES The hauls and freight charges on imported material would make the cost prohibitive. The road was divided into three sections as follows: Station o + oo to 330 local field stone concrete Station 330 + 00 " 460 " gravel " Station 460 + 00 " 524 + 14 " quarry stone " Haul on stone + 00 to 330. Crusher at 146, 220, and 285. These crusher set-ups were determined upon more by reason of nearness of stone supply and grade of haul than to equalize the haul- ing distance. The haul to the crusher was figured for the separate sources of supply and found to average i| miles. Haul from crusher on road, Station 146 to Station + 00 (12' wide use 10 yds. per mile) 2.76 miles, average 1.38 miles. 38 miles X 27.6 yds. = 38.09 yd. mi. 25 25 43 X 5 X 10 X 5 X 8 X 8.6 tl _ 1.25 5-o = 1.25 = 3-2 = 3-7 64.2 yds. 52.49 yd, mi. Station 146 to 170 0.5 miles, average .25 miles Station 220 to 170 1.0 miles, average 0.5 miles Station 220 to 245 0.5 miles, average .25 miles Station 285 to 245 0.8 miles, average 0.4 miles Station 285 to 330 0.86 miles, average .43 miles Total for 1st section 52.49 -r- 64.2 = 0,82 mile, average haul for 1st section. Haul from gravel pit to road. Station 330 to 460. Bank station 385 at side of road — no dead haul great enough to be figured. Station 385 to 330 1.1 miles, average haul .55 miles .55 miles X n yds . = 6.05 yd, miles Station 385 to 460 1.4 miles average haul .7 miles .7 " X 14 " = 9-8 " Total 25 yds. 15.85 yd. miles 15.85 -S- 25 = .63 miles Say .65 average haul Haul from quarry in Village of Cuba § mile from end of road. Station 460 to 524 + 14 Station 460 " 500 14' wide (use 11.7 yds. per mile) Station 500 " 524+14 16' wide (use 13.3 yds. per mile) Station 460 to 500 0.8 miles, average .4 mi. Station 524 + 14 to 500 dead haul .5 " Quarry to 524 + 14 " " _ 1 75 " 1.65 " 1.65 miles X 9.36 yds. = 15.44 yd- mi. Station 500 to 524 + 14 0.5 miles, average haul .25 mile Quarry to 524+14 dead haul .75 mi. 1 mile X 13.3 yds. = 13.3 y d. mi. i.oomi. Total 22.66 28.74 " " XS7-6 " = 165.89 X io. 4 " = 54 X i4.o " = 9 .8 EXAMPLE OF METHOD 3 2 5 28.74 -S- 22.66 = 1.27 Say 1.3 miles average haul . Haul on Sand Pits at Stations 26 and 385 Station 26 to + 00 .5 miles, average .25 mi. .25 miles X 5 yds. = 1.25 yd. mi. Station 26 to 330 5.76 miles, average 2.88 mi. 2.88 Station 385 to 330 1.04 miles, average .52 mi. .52 " Station 385 to 460 1.4 miles, average .7 mi. .7 " Station 460 to 500 0.8 miles, average .4 mi. 385 to 460 dead haul 1.4 mi. Total 78 " 1.8 " X 9-36 " = 16.8 " " Station 500 to 524 + 14 0.5 miles, average .25 mi. 460 to 500 dead haul .8 mi. 385 to 460 " " 1.4 mi. Total "245 mi. 2.45 " X 6.65 yds. = 16.3 " " 103.01 215.44 215.44 -£- 103.01 = 2.1 miles average haul. Hani on Cement Cement delivered at Cuba and Portville. Station o + 00 to 160 Say 10 bbls. to mile 3 miles, average 1.5 mi. 7.5 miles X 30 bbls. = 225 bbl. mi. dead haul, Portville 6.0 mi. to o + 00 7.5 mi. Station 160 to 460 5.68 miles, average 2.84 mi. 460 to 524 -+- 14 dead 1.3 mi. Penn. R.R. to 524 +14 .2 mi. 4.34 mi. 4.34 mi. X 56.8 bbls. = 246.5 bbl. mi. Station 460 to 500. .8 miles, average 0.4 mi. 500 to 524 + 14 dead .5 mi. Penn. R.R. to 524 + 14 .2 mi. 1.1 mi. 1.1 mi.X9.36 bbls. = 10.3 bbl. mi. Station 500 to 524 + 14 .5 average .25 mi. Penn. R.R. to 524 + 14 .2 mi. .45 mi.X6.65 bbls.= 3.0 bbl. mi. .45 102.81 484.8 bbl. mi. 484.8 -r 102.81 =4.7 miles, average haul. Having the haul figured for stone, gravel, cement, and sand, it was decided to obtain a composite price for the aggregate of the concrete instead of presenting an estimate with three prices for concrete foundation. This was done as follows: 326 COST DATA AND ESTIMATES Field Stone. Stone $.10 yd. royalty Blasting 35 " Loading 15 " Haul to crusher 1.5 @ 40c . . .60 " 40c. yd mile used as haul 1 Crushing 30 " was off steep hills and Haul to road .8 mi. @ 35c.. . .28 " hard grades $1.78 yd. Gravel. Gravel (royalty) $.50 Stripping 05 Loading (by hand) .15 Haul to Station 385, 0.1 mile @ 35c .03 Haul on road, .65 miles @ 35c .23 I96 Stone at Cuba Quarry. This stone bought from quarry owner at flat rate of 75c. in bins: Stone $.75 Haul 1.3 @ 35c 455 $1,205 Say $1.21 Sta. 0+00 to 330 = 6.25 miles @ 10 yds. = 62.5X$i.78 = $iii.25 330 " 460 = 2.46 " "10 " =24.6X .96= 23.62 460 " 500 = .8 " " n. 7 " = 9.36X1.21= 11.33 500 to 524+14= .5 " "13.3 " = 6.65 X1 21= 8.05 103. 11 ^ $154-25 $154.25 -T- 103. 1 1 = $1.49 composite price Sand Sand (screened) $1.00 yd. royalty Loading .10 " Haul to road 0.1 @ 40c 04 " 40c. used because of Haul on road 2.1 miles @ 35c. . .735 steep hard grade $1,875 Say $1.88 Cement Delivered at Cuba or Portville $1.05 per bbl. Haul .188 tons X 4.7 miles X .29 per ton mile . . . .25 " " $1.30 Concrete Inasmuch as gravel must be screened and sharp sand supplied, the proportions for stone concrete, ratio 1 — 2\ — 5, were used in place of standard gravel proportions. This is Fuller's rule for proportions of cement, stone, etc., for one cubic yard of concrete. A table of these ratios for different mixtures is found on page 248. Stone $i-49X .92 = $1.3708 Sand 1.88X .46 = .8648 Cement 1.30X1.21 = 1.573 $3.8086 1 This item is higher than noted in the previously given cost data, as this esti- mate is made according to the N. Y. S. method, which does not consider interest and depreciation as a separate item. EXAMPLE OF METHOD 3^7 Mixing $ .40 Spreading 20 Profit 20% .8817 $5-2903 Say $5.30 per cu. yd. Note : — This method of estimating does not consider deprecia- tion directly. See other method of estimating following. The method of estimating the top course for a Concrete Bitumin- ous Top road does not vary from an ordinary bituminous top course, except that under the present New York State specifications the course is figured for loose measure. Therefore the items for con- solidation and filler would be omitted. Brick Cost Data on Country Roads. The cost of brick pavements on country roads differs somewhat from similar work on city streets. There is not much data avail- able for this class of work, but through the courtesy of Mr. Wm. C. Perkins, First Assistant Engineer, New York State Department of Highways, the author is able to give some unusually reliable data obtained from fifteen miles of brick paving averaging 14 ft. wide, built near Buffalo, N.Y., in 19 10. Mr. Perkins' method of estimating, as given on page 331, assumes that 20% profit on both materials and labor will take care of the plant and pay-roll charges and give a reason- able profit. The method of estimating is different from that given on macadam roads. His results are good. Excavation. Where brick pavement is built on an ordinary unim- proved country road, the excavation is of the same class and will cost the same as given for macadam roads. Where pavements are built over macadam roads and the old sur- face must be cut into two or three inches and reshaped, the excava- tion is much more expensive. For this class of work see page 335 (scarifying and reshaping). Labor Manipulation for Different Items of Brick Pavement Laid During 1910, in the Buffalo Residency. These items figured from force accounts kept by the different engineers in charge of roads. Labor averaged $0,175 per hour. Concrete Base, 5" thick (exclusive of edging). Machine-mixing, laying same in place, including labor of tamping, etc. Road No. 2-R, Buffalo-Hamburg.. $0.0853 per sq. yd. " 128, Buffalo- Aurora 0.0991 " " " (gravel concrete) " 863, Blasdell Village. ... 0.1228 " " " " 87, Main Street, Sec. 2. 0.1129 " " " (3" base) " 862, Hamburg Village . . 0.0655 " " " (28' and 30' wide) The excessive cost on Blasdell Village due to a poor concrete mixer (gasoline) which was constantly breaking down. 328 COST DATA AND ESTIMATES On Main Street, Sec. 2, poor organization and too high priced men; also, lack of water, causing delays. On Hamburg Village low price due to width of base 28' and 30', allowing work to progress faster. On Road No. 69, Main Street, Sec. 1, edging and base were laid in one operation; gasoline mixer; plenty of water; cement, $1.12; sand $1.40; labor, $1.90 per day; stone, $1.12 per cu. yd.; base 3" thick; 8" edgings; cost in place, including edging $4,696 per cu. yd., or $0,506 per sq. yd., or $0,886 per lin. ft. of road. Assumption. If we assume, $0.09 per sq. yd. as an average cost for 1 6' road (exclusive of edging) the manipulation would be $0,648 per cu. yd. If we assume $0.0655 P er S Q- yd- for street work (Hamburg Village) the manipulation would be $0,472 per cu. yd. Concrete Edging. 8" thick. Hand-mixed; placing same, including erecting of forms, and removing same; tamping, placing steel, and all labor necessary. Road No. 2-R, Buffalo-Hamburg, $0.0730 per lin. ft. of edging 0.0821 " sq. yd. of pavement (Road 16' wide) Road No. 128, Buffalo- Aurora, 0.0555 " nn - ft. of 5" edging 0.0713 " sq. yd. pavement (Road 14' wide) Road No. 863, Blasdell Village, 0.0826 " lin. ft. edging 0.0929 " sq. yd. pavement (Road 16' wide) Road No. 87, Main Street, Sec. 2, 0.0748 " lin. ft. edging 0.0842 " sq. yd. pavement (Road 16' wide) On Road No. 862, Hamburg Village, concrete curb 6" top, 10" bottom, 15" deep; hand-mixed, exposed curbing, all labor, including erection and removal of forms, $0.1294 per lin. ft. Assumptions- If we assume $0,082 per sq. yd. of paving as' cost of edging and $0.09 per sq. yd. cost of base, the total cost per sq. yd., 16' road (including edging) would be $0,172 per sq. yd., or the manipulation would be $1,238 per cu. yd. If we assume $0,073 P er ^ n - ft- of &" edging io|" deep, the manipulation would be $3,379 per cu. yd. of the edging in place. (This high cost due to forms, etc., and the small amount of con- crete per lin. ft.) Sand Cushion. Spreading sand, rolling, and making bed ready for work. Road No. 2-R, Buffalo-Hamburg, $ 0.0102 per sq. yd. Road No. 128, Buffalo- Aurora, 0.0082" " " Road No. 863, Blasdell Village, 0.0187 " " " Road No. 87, Main St., Sec. 2, 0.0151 " " " Road No. 862, Hamburg Village, 0.0160 " " " (28' and 30' wide) On Main Street, Sec. 1, Road No. 69; sand, $1.40; labor, $1.90; cost per sq. yd. 2" thick, $0.0838, including material. BRICK PAVEMENT 329 Assumption. From the above I would assume $0,013 P er sq. yd. as cost of preparing sand cushion. Brick Pavement. Laying brick, including all labor of handling from the piles, removing all culls, and the rolling of the brick. Road No. 2-R, Buffalo-Hamburg, $0.0611 per sq. yd. Road No. 128, Buffalo-Aurora, 0.0544 " " " Road No. 863, Blasdell Village, 0.0969 " " " Road No. 87, Main St., Sec. 2, 0.0965 " " " Road No. 862, Hamburg Village, 0.0700 " " " (28' and 30' Road No. 69, Main St., Sec.i, 0.0983 " " " wide) Assumption. I consider Blasdell and Main Street, Sec. 1 and Sec. 2, too high and the engineer claims that the force was cut up and wasted time. I would assume $0,070 per sq. yd. as cost of laying brick, etc. Grouting. Necessary grouting to obtain flush joints, scoop method, in- cluding the placing of the protecting sand covering. Road No. 2-R, Buffalo-Hamburg,$o.o2i9per sq. yd. Road No. 128, Buffalo- Aurora, Road No. 863, Blasdell Village, Road No. 87, Main St., Sec. 2, Road No. 69, Main St., Sec. 1, Road No. 862, Hamburg Village, On Main St., Sec. 1, Road No. 69; sand, labor, $1.90; actual cost $0.0848 per sq. yd., including materials. Assumption. From the above I would assume $0,028 per sq. yd., as the cost of applying grout. Expansion Joints. Removing strips, cleaning joints, and pouring tar. Road No. 2-R, Buffalo-Hamburg, $0.0067 P e r tin. ft. of joint 0.0076 " sq. yd. pavement (Road 16' wide) Road No. 128, Buffalo-Aurora, $0.0057 per lin. ft. of joint 0.0073 " sq. yd. pavement (Road 14' wide) Road No. 863, Blasdell Village, $0.0115 per lin. ft. of joint 0.0129 " sq. yd. pavement (Road 1 6' wide) On Main Street, Sec. 1, Road No. 69, the expansion joints cost $0.0296 per lin. ft., or $0,033 P er sq. yd. (Road 16' wide), in- cluding material, labor, etc. Assumption. From the above I would assume $0.0075 P er sq. yd. as the cost of expansion joints. 0.0211 k a a 0.0322 cc a a 0.0321 ct tt u 0.0285 a 11 it 0.0273 it u " (28 and 30' wide) }; sand, $1.40; cement, $1.12; 33° COST DATA AND ESTIMATES Unloading. Data for unloading not reliable. Road No. 2-R Buffalo-Hamburg Road No. 863, Hamburg Village .$0,014 per sq. yd. .Contract taken for $1.50 per 1,000 brick; unloaded, haul i mile, and pile; this would be $0.06 per sq. yd. .$0,019 P er S Q- yd. Road, No. 69, Main St., Sec.i . . Assumption. I would assume $0,028 per sq. yd. as on and off. Hauling. No reliable data. If we allow 600 brick per load, $5 per day for teams, 10 loads per day, haul 1 mile costs $0,034 per sq. yd. Summary, Labor Cost of Brick Pavement. Manipulation of Concrete Pavement 16' wide; edging 8" X 10J". * Concrete base $0.09 per sq. yd. . . $0,648 per cu. yd. edge ■ 0.082 " " " .. 3.378 " " " Concrete base and edging. . .$0,172 " " " . . 1.238 " " " Brick Work Labor Preparing sand cushion $0.0130 per sq. yd. Laying brick 0.0700 Grouting 0.0280 Expansion joints 0.0075 On and off 0.0280 Haul one mile 0.0340 Cost of labor $0.1805 Useful Data for Brick Roads. 6" X 10J" edging per lin. ft. of edging 0.016203 cu - yd. 8" X 10J" " " " " " 0.021605 " " 5" X 16' concrete foundation per lin. ft. 16' road 0.24691 " "■ 2" sand cushion loose per sq. yd o»°555 1 barrel of cement will grout 36 sq. yds. of pavement. 1 barrel of paving pitch will fill 130 lin. ft. of joints 1" wide. Amount of Grout Required for Stone Block Paving. For blocks similar to Medina sandstone blocks, running about 26 to the sq. yd., Gillette states that 0.6 cu. ft. of joint filler are required per sq. yd. of pavement with joints averaging \" wide. Second quality blocks with wider joints require proportionally more. * Recent cost data indicates that $0.35 per cu. yd. is ample. a tt STANDARD ESTIMATE 331 STANDARD ESTIMATE, BRICK SURFACING, EXCLUSIVE OF FOUNDATION Materials. Per Sq. Yd. Cost of brick, f.o.b. unloading point $ " " sand for sand cushion, on job ' grout, on job cement for grout, on job " " paving pitch for expansion joints, on job Labor and Teaming. Unloading brick and piling along road $0,035 Hauling brick per mile 0.040 Preparing sand cushion 0.020 Laying brick 0.070 Grouting 0.028 Expansion joints 0.007 Total $ — Add 20% profit — Estimate $ — SAMPLE — Standard Estimate, Brick Pavement — Wm. C. Perkins Brick: $22.50 per 1,000 f.o.b. cars at Road siding, bricks lay 40 to the sq. yd. Labor, $0,175 P er hour, 10 hours. Sand, 1. 00 per cu. yd. on cars at siding. Stone, 1.25 per cu. yd. on cars at siding. Cement, 1.30 per bbl. delivered on work. Sand: f.o.b. cars $1.00 Unloading 15 Haul 1 mile @ $0.30 30 Cost cu. yd. sand $1.45 Stone: f.o.b. cars $1.25 Unloading 15 Haul 1 mile @ $0.30 30 Cost cu. yd. stone $1.70 Concrete: 1 - 2 J - 5. Use any standard mixing tables, stone 1" and under, dust screened out. Cement, 1.19 bbls. X $1.30 = $1.55 Sand, 0.46 cu. yds. X 1.45 = 0.67 Stone, 0.91 " " X 1.70 = 1.55 * Manipulation = 0.50 $4.27 20 % profit ...85 Total $5.12 * Recent cost data indicates that $0.35 is ample with labor at $0,175 per hour. 332 COST DATA AND ESTIMATES The manipulation is based on machine-mixing and is for base alone laid 5" thick. The concrete edging is estimated separately and runs from $0.13 to $0.15 per lin. ft. Material per Square Yard Brick f.o.b. cars , . . . .$0,900 Sand cushion and cover 0.080 Grout (sand and cement) 0.042 Material expansion joint 0.008 $1,030 Labor per Square Yard Unloading and piling $0,035 Haul 1 mile 0.040 Laying and rolling 0.070 Making sand cushion 0.020 Grouting 0.028 Expansion joints 0.007 Culling, replacing, etc 0.005 0.205 $I - 2 35 20 % profit .247 Total $1,482 Therefore, standard 16' road is estimated to cost, per square yard (exclusive of edging) : Concrete base $0,711 Brick 1.482 Total $2,193 P er S Q- yd. Say, $2.20 per sq. yd. In the above estimate I have allowed 20% profit on material and freight. I do this so as to cover all interest charges, inci- dentals, contingencies, etc. I consider this one of the fairest ways to take care of all general expenses. MAINTENANCE AND REPAIR COSTS Cold Oiling. The following data is furnished by Mr. Frank Bristow, Supt. of Repairs, Division No. 5, New York State Department of Highways. The work was done in 19 10. Labor averaged $0.20 per hour; teams, $0.50 per hour. Oiling. Actual Cost Data. No. 6 stock or 65% aspnaltic base oils applied cold by Studebaker Oiler upon macadam road which had been swept by horse sweeper, oil being broomed by hand where necessary and then covered by a thin coat of dustless screenings, or gravel, spread by hand. The labor costs include pumping oil from the car tank, hauling same to road, applying same, sweeping road and spreading screenings; also, demurrage on cars and moving tools and re- pairs, but not cost of the plant. MAINTENANCE AND REPAIR COSTS Table 51 333 Average cost of materials Average Quantities of Materials Used Average Cost >> -4-> 3 6 j3 7 4 12 3 > 3 O.B 55 S3 O «- . rtCO O M t- . §*■ U ft M O . M Total Labor and Material per Sq. Yd. Orleans Niagara Erie 2.48 2.24 2.00 4-43 $0.0435 0.0425 0.0437 0.0455 $1.82 i-57 1.88 1.83 0.42 0.43 0.34 0.42 0.016 0.016 0.012 0.015 $O.OI3 O.OI4 0.007 O.Oig $0,057 0.057 0.045 0.066 Erie Other information would show that cost per mile to sweep average road is $8.33; cost per gallon applying oil $0.0075; cos t ah labor sweeping, hauling, applying oil and cover about $0.25 per gal. used. Tasle 51 A Division 7 N. Y. S. Dept. Highways H. G. HOTCHKISS SUPT. MAINTENANCE Cost Data for Oiling, Surface Treatment 191 5 No. Kind Gals. No. Tons Total Cost Total Av. Cost per Miles Sq. Yds. Bit. Mat. per of Cover per Cost per Mile 16' Sq. Yd. per Mile Sq. Yd. Sq. Yd. Surface 20.54 158144 CO. 0.25 62 0.0344 1 23-63 188208 CO. 0.25 41 0.0250 17-75 19.94 146734 172775 CO. CO. 0.24 0.19 37 43 0.0237 0.0189 } 0.026 244.06 21.47 200995 C 0. 0.19 59 0.0264 16.22 199925 CO. 0.28 74 0.0287 22.51 188601 L. C 0. 0.20 3i 0.0195 \ 0.0183 171.78 41.09 382330 L. C. 0. 0.20 27 0.0177 15-44 13.42 126657 126056 H. C T. H. C T. 0.25 0.25 40 40 0.0323 0.0337 J 0.0330 309.77 17.19 143846 L. C T. 0.25 47 0.0319 0.0319 299.45 C O. = Cold Oil. See Specifications, page 386. L. C. O. = Light Cold Oil. H. C T. = High Carbon Tar. See Specifications, page 388. L. C T. = Low Carbon Tar. See Specifications, page 389. Cover Material; Slag or Stone Screenings and Pea Gravel. Cost of Applying Oil (Mechanical Spreaders) Cost of Applying Cover approx Cost of Oils on Switch C. O. $0.03 L. C. O. 0.03 H. C T. 0.07 L. C T. 0.06 $0.01 per gai. 0.25 " ton 334 COST DATA AND ESTIMATES Table 51 B Division 7 N.- Y. S. Dept. Highways Cost Data Repainting and Rebuilding Guard Rail 19 14 No. Lin. Ft. Painted One Coat Cost per Lin. Ft. No. Lin. Ft. Painted Two Coats Cost per Lin. Ft. 15325 79925 17486 42027 $.0212 •0233 .0251 .0264 26428 8433 12824 13160 $.0425 .0360 •0352 .0442 Rebuilding Wooden Guard Rail Rebuilding Concrete Guard Rail No. Lin. Ft. Cost per Ft. No.. Lin. Ft. Cost per Ft. 160 554 360 272 $.219 .189 .200 .141 100 335 $0,896 0.764 Hot Tar Flush Coats. The cost of applying hot tar flush coats by hand is practically the same as given for applying Bituminous Binder penetration method. The writer has no reliable data on the cost of machine application. Calcium Chloride. The cost of applying calcium chloride as a temporary dust layer on ten miles of road in Monroe County, N.Y., as given by Mr. Frank Bristow, First Assistant Engineer, New York State Department of Highways, is as follows: The material was applied by an ordinary agricultural drill. The force used was, 1 horse and driver, $0.30 per hour; 1 helper, $0.20 per hour. No preliminary work of sweeping was done; the material was spread on the middle 12 feet of macadam, using, approximately 0.75 lbs. to the sq. yd., the average speed being 0.5 miles, or 3,500 sq. yds., per day, at a cost of $0.0015 per sq. yd. Cost of calcium chloride at plant $13.00 net ton Freight 1 .60 per ' Unloading from cars, approximately. . . o. 15 Hauling three miles, . . 09° Total, delivered on road. ... $15 • 65 Total per sq. yd. delivered on road. . . . 0.0059 Labor of spreading 0.0015 Total per sq. yd. in place. . . $0.0074 Total per mile 12' wide, approximately $52.00 COST DATA ON RESHAPING ROAD 335 Cost of Applying Calcium Chloride Road No. 5507 Scottsville — Canawagus. Season 1915. W. G. Harger, Eng. in charge. 15 tons were applied at the rate of 1^ lbs. per sq. yd. on a 16' road for $22.00 or at the rate of $1.50 per ton. Force used, 1 team hauling agricultural plaster spreader. 2 laborers helping driver. Calcium Chloride in metal drums had been previously distributed along the road. Wages: Team, $5.00 per day; Laborers, $2.00 per day. Recapping The cost or recapping with any style of macadam is practically the same as original construction for that style of work except the item of scarifying and reshaping the old road. Scarifying. The cost of scarifying, as given by Mr. E. A. Bonney on the Erie County repair work for the season of 1907, is as follows: COST DATA ON RESHAPING ROAD Work was done on Main Street Road, No. 69, Erie County, N.Y., between July 15 and Sept. 13, 1907. The road had been built as a waterbound macadam. It was worn out, particularly in the center. There were few ruts, but the road was nearly level; in some stretches the center was lower than the sides. It was proposed to reshape the road and to lay a new top course treated with tarvia. The work of reshaping was done by loosening the old surface with spiked wheels of roller; this separated the crust into chunks of various sizes which were broken up by men with picks. The stone was then raked from the sides to the center, brought to the required crown, and rolled ready for the new course of stone. The cost of the complete operation included the number of men picking and the rollerman's salary. Labor $0. 175 per hour Rollerman 0.300 " " The roller was rented at a flat rate of $5.00 per day, and a portion of the time it was used on other parts of the work. This cost plus the coal and oil is not included. The data was compiled daily, and as the work was performed prac- tically every working day between the dates named an average of the square yard price should be nearly correct. The highest cost on any one day was $0.06 per sq. yd., the lowest cost $0,016, and the general average $0.03 per sq. yd. 1 Through the courtesy of Mr. Halbert P. Gillette, author of ''Handbook of Cost Data," we are able to publish the following: Cost of Resurfacing old Limestone Macadam. "In Engineer- ing News, June 6, 1901, I gave the following data to show that the intermittent method of repairing macadam is the most economic. 1 Gillette's Handbook of Cost Data, Myron C. Clark Publishing Company, edition of 1907, page 147. Pages 288 and 289, edition of 1910, in slightly different form. 336 COST DATA AND ESTIMATES The data were taken from my timebooks and can be relied upon as being well within the probable cost of similar work done by contract under a good foreman. It will be noted that the cost of operating the roller is estimated at $10.00 per day. This includes interest and depreciation as well as fuel and engineman's wages. "The road was worn unevenly, but^as it still had sufficient metal left, very little new metal was added. "The roller used was a 12-ton Buffalo Pitts, provided with steel picks on the rear wheels. It required eighty hours of rolling with the picks in to break up the crust of a surface 19,400 sq. yds. in area, 240 sq. yds. being loosened per hour. The crust was exceedingly hard, and, at times, the picks rode the surface without sinking in, so that a lighter roller would probably have been far less efficient. In fact, a ten-ton roller had been used a few years previous for the same purpose at more than double the expense per square yard, I am told. The picks simply open up cracks in the crust of a depth of about four inches, and it is necessary to follow the roller with a gang of laborers using hand picks to complete the loosening process. The labor of loosening and spreading anew the metal was 1.880 man-hours, or a trifle more than 10 sq. yds., per man-hour. About 60% of this time was spent in picking and 40% in respreading with shovels and potato hooks. "After the material had been respread, the short section was drenched with a sprinkling cart, water being put on in such abundance that when the roller came upon the metal the screenings which had settled at the bottom in the spreading process were floated up into the interstices. The roller and sprinkling cart were engaged only 63 hours in this process, 300 sq. yds. being rolled per hour; an excep- tionally fast rate. The rapidity of rolling was due to four factors: 1. The great abundance of water used, the water being a very short haul. 2. The unyielding foundation (telford) beneath. 3. The abundance of screenings and fine dust, the road not having been swept for some time. 4. The great weight of the roller, which was run at a high rate of speed. I am not prepared to say that longer roiling woule not have secured a harder surface, but I doubt very much whether it would. The metal, I should add, was hard lime- stone. Summing up, we have the cost of resurfacing the road per square yard to have been as follows: Cents per sq. yd. Picking with roller at $1 per hour $0.40 Picking by hand labor at $0.20 per hour . 1 . 20 Respreading by hand labor at $0.20 per hour 0.80 Rolling with roller at $1 per hour 0.33 Sprinkling with cart at $0.40 per hour 0.13 Foreman, 143 hours at $0.30 for 19,400 sq. yds 0.44 Total 3.30 "At this rate a macadam road sixteen feet wide can be resurfaced for a little more than $300 per mile. The frequency with which such resurfacing is necessary will, of course, depend upon several factors, COST DATA ON RESHAPING ROAD 337 chief of which are the amount of traffic and the quality of the road metal. I should say that five years would not be far from the aver- age for a country road built of hard limestone. Unless the road has had an excess of metal used in its construction, new metal should be added at the time of resurfacing to replace that worn out. "I am unable to see how any system of continuous repair with its puttering work here and there can be as economical as work done in the manner above described. I would not be understood, how r ever, as favoring an entire neglect of the road between repair periods. At times of heavy rains and snows, ditches and culverts need attention and there should be some one whose duty it is to look after such matters. What I do question is the economy of having a man con- tinuously at work putting in patches upon the road." 1 NEW YORK STATE PATROL MAINTENANCE, 1910 The standard Patrol distance is five miles. The standard Patrol distance, brick roads, is twelve miles. Patrolman's wages $78 per month, including horse and cart. Patrol is operated eight and one-half months in a year. The cost of this system of maintenance per mile for 19 10 was, approximately, $250 exclusive of administration charges. Patrolman's wages $125 .00 Materials 125.00 $250.00 These costs do not include surface treatments. Such a treat- ment of a road every two years would amount to about $375 a mile per year on waterbound roads. Automobile Truck Repair System. The tendency on minor repair maintenance work seems to be towards lengthening the patrol distance; confining the duties of the patrolman to cleaning culverts and ditches, trimming shoulders, and reporting the necessity of minor repairs. It is believed that these repairs can be handled more economically from a central point by the use of an automobile truck specially equipped for such work and which can operate within a radius of 20 to 30 miles. Special trucks have been devised with facilities for heating and applying bituminous materials as well as carrying materials. Conclusion. In conclusion the author desires to again call the attention of the reader to the fact that while cost data is valuable it must be used with discretion and not figured too closely. 1 Data obtained from Mr. Frank Bristow, Supt. of Repairs, N.Y.S. Dept. of Highways. 33& COST DATA AND ESTIMATES M ON H g P^ So < o g& Ph O o o 5 M § a I in o I— I m Pn o o I— I C/3 C/2 M o u B < H M O -t-> . d HI'S CO H (ON OO Ti- Nt^OOO HOO es 13 -4-J O OnvO CO O vo vd wT^ N On -^ • CO % M (N COO O %# m VO00 -fNN lO M 3 crj lONrONOOO M rO O M » vo co t& H M CO oo €© ■- 4 1>» h M >0 CO "T M M 00 H •3 rt OO O w rt i«h CO M CO O COOO ON %** ^ ^t tN CS CO CN CN €# £ o o o o o o q q q q q o O , , O H O •^j- H Ol CO M vO 4 H NH N 1^ COVO VO W % vo covO 1^ COOO <* a Ph OiOO CO o o O 00* CM c] CO a NvO 00 vo OO ^ o ^t- O o 1 On O co OO M 4©= ^ H vo co 3 CO 4©= P4 CO O O N vo H O to ^ n oq § co vo co 4 vd M to O Oi-io 0\vO co JLJ oq»H q h to vq § H OOO N O H 4 CO O VOOO > w d ,• d fi .3 t-l 3 o (J 1 a. er > c On vd d # o o u Oh a O ro ^ o £} ^J- omo o q moo +j o co O^ u cvi l O bo O *0 vovo vO ioOnj^ onO Tfoj On On ^ vo ro ^ 2" S (N On d .S G J? ^ bcV. ■<* o vovo «2x o c^^d w wi -^ d o p^ .S d^ — O £ll 0 179 186 238 244 Detail Quantities Sta. to Sta. | Exc. Emb. 123 123+50 575 225 Quantifies in cu. ft- 123+50 124 150 900 124 1 ■ 24+501 — 1450 /24+50 125 150 300 125 /25+50 320 20O 125+50 126 170 500 126 126+50 30 925 126+50 127 30 850 127 121+50 260 410 I27t50 128 350 250 128 128+50 635 160 /28+50 129 635 75 v., J ^ y Fig. 69 Cuts. For cuts over 3 feet deep slope stakes are placed and care taken that the slopes are properly carried down. If excavated beyond the finished lines it is practically impossible to make a back-fill that will hold and the resulting irregularities are unsightly. Fills. For fills slope stakes are set in the same manner as for cuts. 1 The earth should be deposited in thin layers, six to eight inches deep, extending from slope to slope, and each layer well compacted either with a roller or by driving over it with wagons in the process of building. Where the old surface has a steep slope it must be plowed to give a good bond with the new fill and prevent slide. It is bad practice to build the center of the fill and then shovel 1 Slope stakes can be located directly from the templet Cross Sections which is a much easier method than the railroad practice of rod and level computation. 342 NOTES ON CONSTRUCTION loose material off of the edge to widen the slopes, as this loose side-fill is not compacted and under the action of frost will nearly always slough away from the harder central portion. Old m ^ ^ £'^ Layers Surface. To get the full benefit of the teaming in compacting the dirt, a deep fill should be started at a point nearest the cut from which the material is hauled and each load driven over the loose layer. In this way nearly every fill can be better compacted than by the use of a roller alone. For long fills where there is considerable teaming over each layer a roller is not usually needed. Wet clay or heavy loam should never be placed in the bottom of a fill, as it dries slowly when not in contact with the air and keeps the fill " spongy." The writer has seen cases where fills not over 3 feet deep have remained soft for two months where wet material had been used and it was finally necessary to remove it. Transferring Grade from Stakes. A handy level for trans- ferring the grade from stakes to the center of the road is shown below. If well made it will transfer the grade elevation 50 feet with an error of less than 3 inches, which is close enough for this stage of the construction: Ditches. The ditches must always be dug out enough to protect the center grading before the fine grading (stone trench) is completed, and it is usually cheaper for the contractor, as well as better for the road, to dig them out before the fine grading begins. String * Rin 9 __ i " -=" — s String Level. Fig. 70 Regulation of Material in Fills. In fills, particularly shallow ones, the road can be greatly improved by a judicious selection of available materials. Material taken from two nearby cuts, or at different depths in the same cut, will often vary in character and the most experienced man on the job should indicate which materials to use in the center of the fill, under the metalling, and which on the sides. The soils in the order of value for fills are gravel, coarse sand, loam, and clay. For shallow fills on a good foundation clay should not be used under, the stone, as men- tioned on page 62, and a good material must be overhauled or borrowed. It is better to avoid overhaul if possible, as it is an item liable to be disputed as to the amount. Where it is neces- FINE GRADING FOR STONE TRENCH 343 sary, a good practical method of determining the amount of the small quantities of earth usually needed is to keep track of the number of wagon loads overhauled from station to station. Sod may be used in the sides of the fill, but should be kept at least eleven feet off center. It should NEVER be used as a shoulder close to the stone or in the center of the fill under the metalling. The author wishes to emphasize the importance of this regu- lation of material. At present the inspection of rough grading is often confined to keeping the sod from the center fill, and the center fill is made of the dirt just as it happens along. As a result, the subgrade will vary greatly in character and if a uni- form depth of stone is used over this " spotty " fill the results are often not satisfactory, while if the depth of stone is varied to meet the subgrade conditions an unnecessary amount of stone is used. In cases where there is no choice of earth materials the stone depth must be made thick enough to meet the require- ments of the grade. FINE GRADING FOR STONE TRENCH The fine grading includes the shaping and consolidation of the stone trench. , Grading ; P/'m^ Construction Shoulder Fig. 71. — Showing 3 Lines of Grading Pins The construction shoulder must be at least 2.5' and well consoli- dated in order to hold the stone solidly during rolling. This must be watched continually by the inspector as it is a point often slighted. Shaping the Grade. A simple guide for shaping the grade is shown in the accompanying sketch and consists of three strings (center and sides) stretched between pins driven at least every 50 feet and preferably every 25 feet. The pins should not be placed at intervals of more than 50 feet as this will cause objectionable sag in the lines and the grade will be undulating. The grade ele- vation is transferred and the lines carefully set at their proper elevation by means of a straight-edge, level and rod, or by stretch- ing a line between grade stakes on opposite sides of the road as previously described. The string level recommended for rough grading cannot be used, as it is not sufficiently accurate. The general level of the finished consolidated grade should be correct to within 1 inch. This leeway of 1 inch from the figured grade makes it possible to get satisfactory results without wast- 344 NOTES ON CONSTRUCTION ing time on finical work and does not appreciably affect the total amount of excavation, as the errors tend to balance. There should, however, be no short, small irregularities of grade notice- able to the eye. Continuous inspection on shaping the grade is not necessary. Consolidating the Grade. Most soils when slightly moist will consolidate readily if thoroughly rolled. Clay, heavy loams, or excessively fine sandy loams (quicksand) will not pack when wet. Continued rolling is injurious for these soils in this con- dition, as they will "work" under the roller. If they occur only in small pockets they can be removed and replaced with good material; if in stretches of any length the grade must dry out before placing the stone. Under drains are constructed at this time, where necessary, and the surface ditches are cleaned out and made effective. Where a hard shower has softened the surface only of a previously consolidated grade of this kind and the contractor wishes to lay stone, the surface can be hardened by spreading a thin layer of gravel or waste No. 2 stone and rolling it into the earth. This will help in preventing the stone teams from cutting up the grade. Gravels and finely pulverized clay, or clay loams (deep dust), will not consolidate when dry; such material must be thoroughly sprinkled to get a compact grade. It is not, however, customary to sprinkle coarse gravels, even if slightly loose, as no objectionable results follows from placing stone on such a grade; deep clay or loam dust is objectionable and must be sprinkled. Coarse sand makes an ideal foundation but is hard to keep in shape while placing the first layer of stone. In some cases sprinkling will harden it sufficiently; in others a layer of fine loam has been spread over the sand and flushed in with satisfac- tory results. Sometimes where loam is not available a cheap cheese-cloth has been spread over the top of shifting sand to prevent the stone from punching in too much under the roller. The author has never encountered any coarse sand that could not be successfully treated by sprinkling and covering with 1 inch or 2 inches of No. 2 stone; the blanket of No. 2 stone pre- vents the sand from squeezing up into the loose' bottom stone and spreading the fragments. While coarse sand makes a good foundation, a fine sand or sandy loam approaching quicksand is very treacherous; it is difficult to judge the degree of fineness at which a sand becomes treacherous, particularly when it is dry. A laboratory method is given on page 64, but a good practical method in the field is to saturate the material thoroughly with water; a satisfactory sand becomes more compact while an exceedingly fine sand gets " quaky." DETERMINATION OF DEPTH 345 DETERMINATION OF STONE DEPTHS AND CONSTRUCTION OF SUB-BASE Practically the only engineering problem that the constructing engineer has to solve is that of foundations. It is recognized by most designers and estimators that it is impossible from even a careful preliminary examination of the soil to specify exactly the amounts and depths of foundation stone. To meet this an extra quantity of sub-base or bottom stone is allowed the con- structor, to be used as he sees fit. During the progress of the rough and fine grading the exact limits of the different kinds of subgrade soil are determined and the stone depths varied ac- cording to his judgment. (See page 64.) Men that really under- stand this part of the work are hard to get, as it is only from ex- tended experience and intelligent study of their own failures and successes that a sound judgment is developed. A good con- structing engineer is much more difficult to find at present than a good technical designer. Where sub-base is used the subgrade is dug out to the required extra depth and rolled if it is in such shape that it will not " work." Peat, muck, wet fine sand, or wet clay cannot be rolled until the sub-base is placed and filled. Where it is possible, such soils should be drained and allowed to dry before placing the base, but is often not feasible to dry them enough to allow rolling, even though underdrainage is put in, which partially hardens them and successfully protects the road after the stone has been placed. This is particularly true on flats where it is hard to get an outlet for a drain or in the fine sands on which an under drain has little effect on account of the capillary action of the material. Where a soft subgrade of this kind is encountered, a stony gravel makes the best sub-base, as it contains no voids between the larger fragments and when rolled the soft underlying material cannot squeeze up through the course. In case boulder or quarry stone base is used on a soft grade, it is necessary to lay them in close contact by hand and then fill the voids completely with gravel or No. 2 stone before rolling; otherwise the subgrade material would squeeze up between the stones, separating them and par- tially destroying the efficiency of the base. In the Spring and Fall of the year it is common to find good material so saturated from long-continued rains that it acts badly under the roller and instead of waiting for the grade to dry out, when the normal thickness of stone would be sufficient, sub-base is often put in either to help the contractor so that he will not be delayed or because the engineer is misled as to the character of the material. This results in a. waste of money. On the other hand, clay, when thoroughly dry, is hard and firm, which often influences a new man to omit sub-base where it will surely be needed. The use of sub-base should not depend too much on the action of the grade under the roller unless the degree of saturation of 346 NOTES ON CONSTRUCTION the material is considered, although it serves as a guide in locating doubtful spots. The final determination should depend on test pits, which develop the character of the underlying material. The sub-base is constructed, as explained, in the chapter on Foundations, either of gravel, boulder or quarry stone. The depth is gauged by lines. The ratio of loose to rolled depth is given on page 272. Continuous imspection is not needed on sub-base; the depth of grading is checked before the stone is placed and the width, depth, and workmanship can be readily determined after the base is completed, and by an occasional inspection during the progress of the work. Bottom Stone. The earth subgrade must be firm and compact before the stone is spread. Bottom stone must NEVER be laid on a soft grade. One of the most common slips of inspection is to allow this to be done and the result is a " punky " bottom course that is never up to standard. The distributing power of this course depends largely on the stone fragments being firmly interlocked; if the stone is placed on a soft grade and rolled, the earth will squeeze up between the fragments and separate them. The depth of the loose stone is gauged by the lines or cubical wooden blocks placed on the subgrade. Blocks are more con- venient than lines except over sub-base of stone fills, where lines must be used to get a spread true to shape and grade. The ratio of loose to rolled depths is given on page 272. The loose stone is rolled until the stones are solidly inter- locked and there is no movement under the roller. A thin layer of satisfactory filler (see materials page 129.) is spread over the top, rolled and broomed in; the process is repeated until the stone is thoroughly filled. Continuous inspection on bottom course is not necessary. The widths and depths can be readily checked by occasional inspection. The two points to be care- fully watched during construction are: 1. That the grade is firm; 2. that the loose fragments are thoroughly rolled before the filler is applied. It is desirable to complete the bottom course well in advance of the top, in which case the contractor can work to advantage after rains, and the course will be better compacted by subjecting it to some traffic action. Where local stone is crushed on the job and the stone used ranges in size from 1 in. to tailings, care must be used in spreading that the sizes are well mixed, as pockets of fine or coarse stone are objectionable. The simplest method of mixing is to run the No. 3 and No. 4 and tailings into one bin at the crusher; if they are separated they can be well mixed by loading one end of the wagons with the No. 3 and the other end with No. 4 and when dumped on the grade they will run together. When dif- ficulty is experienced with these methods in obtaining a well- mixed stone spread the loose stone can be harrowed. Many specifications call for harrowing thoroughly where a large range APPROXIMATE AMOUNT OF FILLER 347 of crushed stone size are allowed in one course. If possible, tailings should be used as sub-base. When used in the bottom course having a rolled depth of 4 or 5 inches they should be placed in the lower part of the course, but for a 3-inch depth they should be placed on top and broken with a knapping hammer into frag- ments of less than 3^ inches. The filler should not be dumped directly on the stone unless absolutely necessary. Drawing the loads onto the unfilled stone loosens the course, and, also, at each pile of filler there is apt to be left an excess which is hard to clean off. Table 52 gives the approximate amount of filler required per 100 feet, and the spacing of ij-yard loads. The amount varies for the different materials used. Grading and foundations have been treated at some length, as they are the most difficult parts of the construction. Table 52. Giving the Approximate Amount of Filler P.e- quired per 100 Feet of Road for Crushed Stone Macadam Bottom Courses of Different Wldths and Depths, Using 0.35 Cubic Yards of Filler per Cubic Yard of Rolled Bottom Rolled Depth of Bottom Course .13-0 a & Tj O ££ 3 " 4" 5* 6" 10' 3.2 cu. yds. 4.3 cu. yds. 5.4 cu. yds. 6.6 cu. yds. 12' 3.8 ' i a 5-1 " " 6.5 " "* 7.6 " a i4 r 4-5 ' 1 a 6.0 " " 7-5 " " 9.0 " a 15' 4.9 ' 1 a 6.4 " " 8.0 " " 9.9 " a 16' 5- 2 ' c 11 6.9 " " 8.6 " " 10.4 " a 18' 5-9 ' ( it 7.9 " " 9-7 " " 11.8 " a 20' 6.4 ' i tt 8.6 " " 10.8 " " 12.8 " a 22 r 7.0 ' i (t 9-4 " " 11.8 " " 14.2 " a Table 5 2 A. Giving the Approximate Spacing of 1.5 Cubic Yard Loads of Filler for the Widths and Depths shown in Table 52 Width of Rolled Depth 01 Bottom Course Macadam in 4" 5" 6" 10' 46 feet 34 feet 27 feet 23 feet 12' 40 " 30 " 23 " 20 " 14' 33 " 25 " 20 " 17 " IS' 3i " 23 " 19 " 15 " i6 r 18' 29 M 25 " 22 " 19 " i7 " 16 " 13 " 12 • " 20' 23 " 18 " 13 " 11 " 22' 21 " 16 " 12 " 10 " 348 NOTES ON CONSTRUCTION TOP COURSES Waterbound Top. Waterbound top is constructed in the same way as the bottom course except that stone dust is used for a filler and the course is puddled as has been described. If the stone used is a local stone crushed on the job the output of the crusher must be carefully controlled, especially where selected boulders are used, as it is very important that the size and quality of such stone shall be uniform. Imported stone can be inspected on the cars. Aside from this, comparatively little inspection is required except at the stage when the loose stone has been rolled and before the binder is spread. At this time the inspector should examine the rolled course very carefully to see that it is true to shape and has no short depressions or humps. The smooth riding quality of the road depends on this inspection and too much care cannot be taken. This point is particularly emphasized, as many of the stone roads in New York State have been criticized as rough for automobile traffic. Any depressions are filled with stone of the same size as the body of the course and rolled, after which the course is again inspected and corrected until it is made true. The binder is then spread, broomed in dry, and puddled. In puddling use plenty of water and roll rapidly. If a pipe line and hose are used a pressure of ioo to 125 pounds at the pump should be maintained. The road can be conveniently puddled in stretches of 100 to 200 feet. After the road has dried out and been opened to traffic, if raveling occurs it can usually be remedied by light sprinkling and rolling. Where the top course is granite, gneiss, or trap, it is often necessary to use a certain percentage of limestone dust with the normal screenings. The limestone is more effective when spread last, filling the top voids of the course. Bituminous Top. Penetration Method. The same pro- cedure applies to the quality, size, and laying of the stone for a bituminous as for waterbound top, and does not require con- tinuous inspection. Just before pouring the bitumen the course should be care- fully examined and any pockets of fine stone, dirt, dirty or dusty stone removed, as fine stone or dirt prevents the pene- tration of the binder and the bitumen will not adhere properly to the stone unless it is clean and dry. The course is not rolled as firmly at this stage as for waterbound tops because excessive rolling tightens the stone too much and prevents the penetration of the bitumen. There should, however, be no creep in front of the roller. The bitumen is poured into the voids of this clean, dry, partially compacted course, usually by means of hand- sprinkling pots or hods. Pots having vertical slots are prefer- able to the fan-spout pots, as they give better penetration. Hods are to be preferred to pots. When hods are used, however, the bitumen should be poured across the road instead of in a TOP COURSES 349 longitudinal direction as this prevents overlap and minimizes the difficulty of preventing humps or waves. In placing the bitumen the following precautions must be observed: It must be hot enough to run freely; for each grade the temperature of applications is usually specified and it must not be overheated, for if charred it is useless. In applying, by whatever method, care must be taken not to overlap, as waves or humps will develop at these points. These defects do not appear for some time after the road is opened to travel, and an inexperienced inspector fails to realize the necessity of care in this particular. The stone must be clean and dry, and, in the writer's opinion, the air temperature should not be less than 50 F., as bitumen applied in cold weather is so chilled when it strikes the cold stone that an excessive amount is retained on the surface. As soon as the bitumen is applied a thin layer of No. 2 stone is spread over the surface and rolled lightly; continued rolling at this point is injurious, as freshly laid bituminous tops tend to shove under the roller and form waves. The road can be thor- oughly rolled and shaped to advantage only after the bitumen has had some time to harden. Good results have been obtained by rolling thoroughly the succeeding day after the binder is applied, unless in the meantime rain has saturated the course, in which case it must be allowed to dry before rolling. The amount of bitumen spread per square yard is usually con- trolled by spreading a given number of pots or hods in a given length of the road. These units of length can readily be marked off by the inspector with a stick or tape. This method will be satisfactory if checked up twice a day by the number of barrels used. When the binder is heated in small kettles it will some- times catch fire, but this is usually due to scale which has col- lected in the tank and if cleaned out it generally remedies the trouble. Where bituminous materials are heated by steam it is often con- venient to know the temperature of steam at different pressures; the following table is inserted for this purpose : Table 53 Pressure Gauge Lbs. per Sq. In. Temperature of Steam °F Pressure Lbs. per Sq. In. Temperature °F of steam Pressure Lbs. per Sq. In. Temperature °F of steam 'is 20 40 60 80 IOO 1 213 228 267 293 312 328 IOO I20 140 160 180 200 328 341 353 373 382 200 220 240 260 280 300 382 390 . 397 404 411 417 1 Fifteen pounds normal air pressure; to get ordinary steam gauge reading sub< tract 15 lbs. from the values given in this table. 350 NOTES ON CONSTRUCTION HASSAM CONCRETE PAVEMENT By E. E. Kidder The principal mechanical difficulty in laying a Hassam pavement is in getting a proper penetration of the grout. This requires stone free from small particles and a grout of the proper consistency. Stone. The stone should be sized ij" to 3^" uniformly mixed. Any pockets of fine stone should be shoveled out or if they occur in small areas raked over till the fine goes to the bottom of the course. The spreading is followed by rolling with a 10- ton road roller. Close attention should be given to obtaining as nearly a perfect surface as possible as it is practically impossible to add or deduct material once the stone is grouted. Grout. The grout should penetrate to the bottom but should not be so thin that separation occurs. The size of sand is important. Coarse sand will not penetrate well. Sand passing a 10" mesh and containing much that is finer works well. Manipulation. Each morning the end of the previous day's work should be cut down vertically and square across the road, shoveled out and replaced with new stone. This insures a vertical joint. Continuous Inspection is Necessary on both Stone and Grout. The grout will float a few of the top stone out of place but the 5 -ton tandem roller will smooth them down. The final finish is obtained by hand tamping and brooming the surface. The tamping is abso- lutely necessary to get the best results and it will be neglected unless insisted upon. Shoulders. It is desirable that the earth shoulders be left 1" higher than the finished pavement until the grouting is completed to prevent waste. During the rolling of the grout some water will flush to the surface and run to the edge; it should be let off by digging small trenches through the shoulder. The grading of the shoulders should be practically complete before laying stone in order that the Sand and Cement may be placed on one side of the road and the other shoulder may be used as a walk for the workmen and traffic. The gang organization is shown by a sketch (page 288 Cost Data). FIRST CLASS CONCRETE PAVEMENTS By F. W. Bristow The sub-grade should be formed true to alignment, elevation and shape and consolidated well in advance of the mixing machine to permit the delivery of materials both on the sub-grade and shoulders. The materials, stone or screened gravel, sand and cement should be distributed uniformly in the proper quantities to construct the pavement as planned. (For quality of materials see specifications. For amounts required see Cost Data.) The cement should be de- livered on the road only as required and covers provided for its protection in case of storm. Inspection of Manipulation. A diagram showing a typical mixing gang organization is given in the chapter on Cost Data. Two inspec- tors are necessary to properly supervise the work. The inspection must CONCRETE PAVEMENT 351 be continuous. The inspector ahead of the mixer sees that the sub- grade is correct; that the edging forms are properly set; that the fine and coarse aggregate conforms to the requirements, and that the proper amount of materials are placed in each batch of concrete. He also should keep a daily record of the amount of cement used, the amount of concrete laid and should figure the amount of cement per cu. yd. of concrete as a check on his batch inspection. He should be careful to observe that no empty cement sacks from the previous days run are counted the second time. The inspector back of the mixer first sees that the sub-grade is smoothed as the mixing machine moves ahead; that any muddy conditions is remedied by removal and that a dry dusty sub-grade is sprinkled to prevent rapid absorption of water from the concrete; that the concrete as delivered from the mixer has the proper consist- ency and is thoroughly mixed; that the transverse expansion joints are properly placed; that the striking of the concrete with the screed l or template is so done as to leave no projecting stone, or humps or hollows in the surface. Any surface irregularities must be imme- diately remedied and the mass restruck. The screeding is kept up closely to the mixer and is followed by the wood float finishers work- ing from a bridge that spans the concrete. "" /6 '- —>/Body of Screed 5 Pieces /"Planks Spiked Together* Wrought Iron Shoe^ F/atSubgrade ■Side Forms Sketch of Screed Fig. 71 a In case surface brooming is required the inspector determines when it shall be done; the best time is just after the initial set starts. Long-handled steel brooms are used and the brooming is done lightly transversely to the road. In hot weather the fresh concrete should be sprinkled to prevent sun checking. It is covered within 24 hours with a coat of earth 1" to 2" thick which is sprinkled and kept damp for 10 days when it is removed. Traffic must be barricaded from the road for this time. Before turning traffic on to the completed concrete the earth shoulders should be finished along the edge to prevent spauling. Sheet Asphalt, Topeka Mix, etc. The important points in any form of a mixed Bituminous surface are a proper grading of the aggre- gate and care not to char the binder in mixing. Two inspectors are required; one at the plant and one where the asphalt is being laid. Plant inspection should be continuous. The plant inspector is responsible for the proper proportions of the dif- ferent sizes of the aggregate and for the proper temperature of the mix. To insure the proper proportions he should test the measuring scales at short intervals and sift a sample of the dry mineral aggre- 1 The screed should be two feet wider than the finished pavement as it progresses with a see saw movement rather than a direct full. 352 NOTES ON CONSTRUCTION gate at least once a day. His most important duty is to prevent charring of the binder. It is not necessary to take the tempera- ture of each batmen as with a little experience any objectionable con- dition can be detected by the character of the smoke; a dense white smoke given off when mixing indicates a dangerously high tempera- ture. When this is observed the batch should be tested with a thermometer and immediately rejected if over the specified tempera- ture limit. (See specifications.) The temperature of the mineral aggregate will fluctuate very rapidly in the small plants generally used for road work and care should be observed in picking the plant inspector that he is a very conscientious man. He should also furnish the driver of each load of asphalt with a ticket giving the weight of the mix on that load. The Inspector on the road records the temperature of the mix as received, the weight of each load and indicates to the spreaders the number of sq. yds it should cover. (The surface mix weights approx. ioo lbs. per sq. yd. per inch of consolidated depth.) He also should take a sample of the mix as delivered once a day and ship to the laboratory for check analysis. The following sketches show a sample of a plant inspector's and road inspector's record book. Sept. 29, 191 5. Plant Record Load Record Bitumen Record Load Time Temperature of Mix. Weight of Load Time Temperature I 2 3 4 5 6 etc. 7.30 7-45 7.55 8.15 8.25 8.45 320° F. 310 300 300 290 310 50000 lbs. 5000 " 6000 " 6000 " 5000 " 5000 " 7.00 8.00 9.00 10.00 11.00 12.00 310 F. 345 34o 3io 305 320 Test & \mple No. 7. Sept 29, ic 115. Road Record Load No. Time Tempera- ture of Mix. Weight as per Ticket No.Yds. Covered Location on Road 1 2 3 4 5 etc. 8.00 8.20 8.30 8.45 9.00 3io°F. 300 295 295 280 5000 lbs. 5000 " 6000 " 6000 " 5000 " 25 25 30 30 25 Sta. 10 + 30 to 10 + 42 " IO+ 42 " IO+ 54 " 10 + 54 " 10 + 69 " 10 + 69 " 10 + 84 " 10 + 84 " 10 + 96 Proportions of Mix. The proportions of mix should be deter- mined by the engineer by screen analysis of the different materials that the contractor proposes to use. As an example assume that a mixture of cement, fine sand, coarse sand and buckwheat stone is proposed and it is desired to determine the relative amounts of the different materials to use in order to get the correct proportion of sizes specified. ASPHALT PAVEMENTS 353 For all ordinary purposes a size analysis can be safely made using the following screens: #200, #80, #40, #10, \" and V . The materials are thoroughly dried and the percentages expressed by weight. Fine Sand (Feeder Pit) Passing # 200 5 % 1 ' #80 retained on # 200 70 % " #40 " " # 80 Good Quality Coarse Sand (Bauerman Pit) Passing #200 " # 80 retained on #200 " #40 " " # 80 " #10 " "#40 Buckwheat Stone. (Commercial plant.) Passing #200 1 % " # 40 retained #200 " #10 " #40 25% 1% 2% 29% 68% A 1" 2 IO 1" 4 • 2% 7% 60% 30% The proportions can now be varied to produce practically any required mix. Tabulation Showing Method of Determining the Number of Pounds of Each Material to be Used in a 100 lb. Batch to Produce a Required Mix. Material No. lbs. Bitu- men #200 #40 #10 1 \ Bitumen Cement Fine Sand Coarse Sand . . Buckwheat Stone Totals IO 7 47 20 16 100 IO IO 7 2-5 9-5 . 44-5 6.0 5o.5 14.0 1.0 15.0 9.6 9.6 5-4 5-4 In this way the effect of varying any of the component parts of the mix can be readily seen and determined. The total size of the batch is of course varied to suit the capacity of the plant. The laboratory analysis of the daily sample taken on the road furnishes a check on the plant inspector. Rollers. The best results can be obtained by the use of two tandem rollers; a light roller not over 5 tons for first compression to anneal the surface while hot and a heavier 8 to 10 ton roller for final compres- sion and cross-rolling. This is more important where the asphalt is laid on a macadam base than when laid on concrete. 354 NOTES ON CONSTRUCTION BRICK ROADS To cover the points of construction of brick roads we cannot do better than to give " Instructions for Inspectors," by William C. Perkins, Resident Engineer, New York State Department of Highways. Mr. Perkins is well qualified to judge of this class of work. Grading. " Read your specifications carefully and follow them in every particular. " Do not let the contractor dig beyond the back slopes of your ditches. Your ditches should be straight, no sudden jogs; back slopes all true; no rubbish deposited back of the ditches, and be sure that your ditches drain. " Follow your cross-sections as closely as possible. Try to aid the contractor to take care of his dirt so that when the road is cleaned up there will not be a great amount of material to be moved. " Never make a shovel nil over 6 inches without rolling it. " In making a heavy fill with dump wagons begin to dump at the end toward your dirt supply. Have each pile of dump dirt spread thin and draw the next load over this, which will help to pack it. All should then be thoroughly rolled. " Examine your subgrade carefully, particularly when the roller is going over same, and if it waves or shakes under the roller, sub-base or drain should be put in, or the material dug out and the proper material put in. Do not make a fill with any old material found along the road. Use judgment in this particular. " Clearing and grubbing does not mean the grubbing of sod. It means the cutting down of bushes, trees, etc. Remember that the life of your pavement is the condition of your subgrade. The same should be inspected by the engineer in charge before any stone or concrete is placed. " Grade the full width of your macadam or concrete. Never deposit stone in the rut. Keep your sub-base free of ruts. " If your roller is not working on other work roll your subgrade. You cannot roll it too much. " Do not shift center line or grades until you have reported the necessity for it to headquarters, and if absolutely necessary give an estimate of the increase or decrease in quantities that such change would make. " Shoulders should not contain sod within 18 inches of the macadam. " Back slope all ditches i on i|. Be careful that your gutters are not too deep. Deep gutters where not necessary for drainage purposes make a road dangerous and must be avoided. " In trimming shoulders and ditches a good inspector should be put on the work, and instructed to see that the contractor sets proper stakes. A stake should be set out from the edge of the macadam, and also one in the ditch, and should be set at least every ioo feet. The bottom of the ditch must be a true BRICK ROADS 355 grade, no depression, and the ditch alignment must be good. These stakes can be easily set with a 1 6-foot level board. When approaching a culvert it is not necessary to deepen the gutters until you reach within 50 feet of same, when a straight grade can then be run to the invert. " In all cases be sure your ditches will carry water, and, I re- peat, be sure they are not ragged and the back slopes are well graded. In trimming shoulders be sure there is no ridge next to the macadam. " In setting your stakes for the shoulder work use the or- dinates and distances shown on the standard section. " Subgrade. Be sure that your subgrade has been properly graded so as to obtain 5 inches of concrete. If the contractor builds the curb first, a templet should be run over the curbing and test made to be sure that you have the correct depth. " Concrete Edging. Stakes for concrete edging can be placed every 50 feet for line and grading, with the exception at change of grades and curves, where they should be placed every 25 feet. " Be sure that your forms are properly set as to line and grade. " With stakes 50 feet apart be careful that there is no sag in the line when the forms are set. " If edging is set first it is better that the concrete be hand- mixed, as a machine turns out too large a quantity and cannot be placed in the proper time. " See that your forms are wet before the concrete is placed, and if steel forms are used they should be oiled. " Have a careful inspector on the mixing of the concrete for the edging and watch the mix. " Keep track of the number of bags of cement used and see that the proper proportion of cement to the lineal foot of edging is obtained. " Edging 6" X 10 \" will use 1 bag in 12.95 feet " Edging 8" X io|" will use 1 bag in 9.73 feet " Mixture, 1-2^-5. " Make the mixture rather wet and spade the same thoroughly, using a hoe straightened and punched full of holes, or some similar instrument, so as to get a good face next to the forms. " If you find you cannot get a good top surface keep the edging a couple of inches low, and about every third batch mix a batch of fine material and bring the edging up to the proper height, throughly working the same in. " Do not get a plaster effect, but get a good top surface. " Round both edges with a rounding tool, making the inner edge of a smaller radius than the outer edge. " When the forms are taken down all spots which are honey- combed, or rough, should be floated at once with cement. A rough edging should not be left on any road. " Have the contractor back up the edging as soon as possible. " In warm weather the edging should be kept wet for, at least, 356 NOTES ON CONSTRUCTION twenty-four hours. Have the contractor use care in delivering materials after the edging is built so that the edges of same are not broken by wagons, etc. " A good edging is often ruined by carelessness on the part of the contractor. " Concrete Base. Before laying base be sure that the founda- tion is in proper shape and of a proper depth. " Lay the concrete rather wet and drag same with a heavy templet. Have men back of the templet with tamping irons or blocks, tamping the concrete. This is important if you wish to get a smooth surface, and you must insist that the concrete be well tamped, " Be sure that you keep track of your bags, and, also, that the machine is working properly. " For a 16-ft. road nf bags will lay 10 ft. concrete base, mix- ture 1-2^-5. " After the day's run examine your base, and if there are any spots which are porous, grout same and check up your bags at the end of each day. " If the weather is very hot the base should be kept wet for twenty-four hours. " Sand Cushion. Sand for this cushion should be absolutely free of stones, and you must insist that the contractor screen same, if stones are in the sand delivered. No excuses will be taken for stones or pebbles in the cushion. Spread sand for a sufficient depth, then roll same with a small roller; then drag, roll again, and then drag with templet. " This should be sufficient to give a firm cushion. " The smoothness of the pavement depends on the proper form of the cushion. " Brick. Great care must be used in obtaining proper brick surface. " Be sure that your strips on the side expansion joints are in when the contractor starts to lay brick. " Allow no pinning in at the ends under 2§ inches. " Be sure that the expansion joint is not ragged. It must be uniform in width, otherwise you will have transverse cracks. " All bricks should be laid with lugs in the same direction. This is a point that the bricklayers very often do not do. The bricks should be laid by experienced bricklayers, not by amateurs. " After the brick are laid the contractor will start culling. Then you and your inspectors should carefully go over them, marking all soft 1 bricks to be taken out and rejected; all kiln- marked bricks to be turned over, and if not satisfactory to be taken out and used for pinning in; all overburned bricks, 2 J Soft brick are found by sprinkling the pavement lightly; the soft or under-burned brick will absorb the moisture, rapidly becoming dull , while the good brick still glisten with the water. 2 Over-burned brick are known by their color, which is much darker than the average. BRICK ROADS 357 which are burned to a cinder to be rejected. All underburned bricks, which, in your opinion, will not make a satisfactory pavement, to be rejected. All bronzed bricks (which have the appearance of overburned brick but this on one side only) to be turned over, and if satisfactory allowed to remain in the pave- ment. " Be sure that you have culled all of the bricks before the pavement is rolled, for after the pavement is rolled if much cull- ing is done you are liable to have a rough pavement. After the pavement is rolled go over same and mark all broken and spalled bricks, to be taken out or turned over. " Be careful of all high and low bricks in the pavement, for same will wear badly when the road is finished. " Be sure that your bricks are laid at right angles to the curb and are not wavy as to line. " In no case allow any ' Dutchman' 1 in your pavement except on curves where absolutely necessary. " Grouting. The grouting of the pavement is its life, and the greatest care must be used. Insist that all grout be placed on the pavement by the use of scoops from a box with unequal legs. " The grout should be mixed in small quantities and of the exact proportions. The sand should be sharp, not too coarse nor too fine. Care should be taken in using lake sand, as same is probably not sharp and too heavy for the grout. As soon as the grout reaches the pavement it should, at once, be pushed into the joints by means of brooms or squeegees. " It is best to use brooms on the first grouting and a squeegee on the second and third groutings. " Be sure that the joints are well filled in the first grouting, and do not let the grout escape over the edging and be lost. " Follow closely with the second grouting, otherwise the two groutings will not unite. " Be careful that the second grouting does not overlap the first. After the second grouting examine the pavement care- fully and, if necessary, put on a third grout to get flush joints. " The pavement should be completely covered with grout and all joints should be well filled before you pass on same. " Allow enough time for the grout to obtain initial set, and cover pavement with a layer of sand to protect same from the weather; and pavement should be kept wet for, at least, twenty- four hours. " In no case permit traffic on the pavement under ten days; longer, if possible. "Expansion Joints. Be careful in removing the expansion Joint boards that you do not disturb the pinning-in bricks and break the bond. We found it advisable to use two wedge-shaped boards to make the expansion joints and loosen up the back one as soon as grouting was started. 1 "Dutchman." Brick chipped to wedge shape to fill in between radial courses on curves. 358 . NOTES ON CONSTRUCTION " In pouring the asphalt filler be sure that the joints are abso- lutely clean the full depth. This is very important, or, other- wise, you will have cracks in the pavement. The joints are to be flushed with asphalt." CULVERTS Culverts are usually constructed before the road is graded. They should be completed well in advance of the macadam, because even though the back-fill is carefully tamped there is bound to be some additional settlement under traffic action, and if the macadam is laid over a fresh back-fill depressions are sure to develop which, if not repaired, make " thank-you-marms " in the road. Cast-iron Pipe. Trenches for pipe are dug the required depth, making the bottom wide enough to allow the joints to be properly calked. This requires a trench 18" to 24" wider than the pipe diameter, i.e., for a 12" pipe the trench is 30" to 36". Bell holes are dug as shown in Fig. 72, so that the pipe will have a uniform bearing its entire length. At no point should it rest directly on boulders or ledge rocks. If the foundation is soft the pipe should be laid on a concrete base. For ordinary soils the only precaution the inspector need take is to prevent back- fill under the pipe. Fig. 72 Unless the foreman is alert the trench is often excavated too much in some places, which are then back-filled. This is bad practice except where boulders are encountered which must be removed and the cavities back-filled with good material. Pipe. The pipe is inspected for flaws; it is then placed in the trench with the bell end upstream. At each joint the spigot end is placed in the bell and forced against the shoulder, making a tight joint. The pipe is then lined correctly and a gasket of jute or oakum driven into the joint with an iron calking tool having a 2" to 3" offset, as shown in Fig. 73- The balance of the joint is then filled with a 1 to 1 cement mortar. Fig. 73. — Steel Caulking Tool The trench is then back-filled, care being taken not to throw the pipe out of line; the back-fill must be well tamped in layers not exceeding 6", using heavy paver's rammers. A good work- ing rule is to use two of the best men on the job tamping and the laziest man on the force throwing dirt to them. CONCRETE CULVERTS 359 Head-walls for Culverts. The face of the head-wall should extend beyond the end of the pipe, as it is difficult to get a good- looking connection if it is flush with the end. -*_ Fig. 74 Figure 74 shows a convenient plug form for this extension. This plug is set into the end of the pipe and can be readily re- moved; the resulting head-wall being pleasing in appearance. The head-wall form can, also, be readily skewed (set at an angle with the pipe) if required. CONCRETE CULVERTS Excavation. The trench is dug to the required depth; if the material will stand vertically no back forms are necessary, and the width of the trench is made the width of the out to out di- mensions of the culverts. If back forms are needed the trench is usually made 2 feet wider. If running water is encountered which cannot be temporarily dammed, or diverted, the trench is made wide enough to flume the stream through on one side of the back forms for small culverts, or between the abutments for larger span structures. Back-fill. The back-fill is made as for cast-iron pipe except that it should not be deposited on the fresh top of a culvert within twenty-four hours of laying the concrete. Forms. Forms should be true to shape and constructed of planed tongue and groove lumber, for the exposed surfaces. They should be water-tight, as otherwise the fine material will run out of the face of the concrete and leave a rough " pop-corn " surface. They must be well braced to prevent bulging. Tri- angular or feather-edged grooved moldings are placed in the angles of the forms to shape them satisfactorily. Removal of Forms. The length of time that the forms should remain in place is a matter of judgment; it depends upon the cement and weather conditions. The author's practice is as follows: Head-walls or parapet forms are removed within thirty-six hours in dry weather or within forty-eight hours in damp, cold weather, in order to rub down the surfaces. Low side-wall forms for spans of 2' to 3', where the deck is constructed later, may be removed in 36 to 48 hours. Trunk forms for small culverts 2' to 3' span may be removed in from 3 to 7 days. 360 NOTES ON. CONSTRUCTION Trunk forms for medium culverts up to io' span 7 to 14 days. Deck forms for spans above io' may be removed in from 14 to 28 days. Any unusual load, such as a roller, should not be allowed over a new culvert of even a small span in less than seven days, unless precautions are taken to distribute the pressure by planking the back-fill, or otherwise, and on the larger structures a time limit of three to four weeks is advisable. Amount of Cement, Sand, and Stone required. Table 49, page 304 gives these amounts for one yard of concrete. The following table gives the amount of stone, sand, and cement required for culverts similar to Plate 6, assuming that no embedded boulders are used in the sides and bottom. If boulders are used see footnote, Table 49. MIXING AND PLACING CONCRETE The strength of the concrete depends largely upon the thoroughness of the mixing. The author's practice has been as follows: Hand-mixing. Cement and Sand. 3 turns dry. . . .3d class concrete (foundations and side walls) 4 " " .... 2d " " (decks and parapets) Add water and mix mortar. Drench stone and turn stone and mortar 3 times for 3d class concrete 4 " " 2d " Deposit in forms by dropping. Do not cast, as this separates the coarse and fine material. Use enough water to give a mixture that quakes like liver under the rammer. Deposit in layers not over 6" deep and ram each layer thor- oughly; spade the concrete thoroughly, and work an excess of the fine stuff to the face of the forms by prying the larger frag- ments back from the form with a narrow spade or broad-tined fork. Machine -mixing. Culverts generally contain such a small quantity of concrete that machine-mixing is rarely used. In case a batch-mixer is employed, the inspection is simplified to checking the quantities of cement, sand, and stone in each charge. If a continuous mixer is used it is well to keep watch of the cement hopper, as the cement is liable to run low, feeding only a portion of the worm, or a large lump of cement may ride on top of the worm and hinder the feed; or the worm may become coated with damp cement which reduces the capacity. If the inspector watches the cement hopper the contractor will tend to the sand and stone hoppers. Finishing Concrete. If a smooth, marble-like surface is de- sired it can be obtained by rubbing down the surface before it has fully set with a cement sand brick moistened with water. If CONCRETE CULVERTS Concrete Culverts 361 1. 5' high X 2.0' wide Length Feet Concrete Cubic Yards Paving Square Yards Ex. Met. Square Feet Cement Barrels Sand Cubic Yards Crushed Stone Cubic Second Third Yards 20 2.2 5.6 6.4 80 8.4 3-6 7.2 21 2.2 5-8 6.4 84 8.6 3-7 74 22 2-3 6.1 6.4 88 9.0 3-9 7.8 23 2.4 6-3 6.4 92 9-3 4.1 8.1 24 2.5 6.5 6.4 96 9-7 4.2 8-3 25 2.5 6. 7 6.4 IOO 9.9 4-3 8-5 26 2.6 6.9 6.4 I04 10.2 4.4 8.8 27 2.7 7.2 6.4 IO8 10.6 4.6 9.2 28 2.8 7-4 6.4 112 10.9 4.8 9-5 29 2.8 7.6 6.4 Il6 11. 1 4-9 9.6 30 2.9 7.8 6.4 I20 n-5 5.o 9.9 31 3-o 8.1 6.4 124 11.9 5-2 10.3 32 3-i 8-3 6.4 128 12.2 5-3 10.6 33 3-1 8-5 6.4 132 12.4 5-4 10.8 34 3-2 8.7 6.4 I36 12.7 5.6 11. 35 3-3 8.9 6.4 I40 13-1 5-7 "-3 36 3-4 9.2 6.4 144 !3-5 5-9 11. 7 37 3-4 9.4 6.4 I48 *3-7 6.0 11.9 38 3-5 9.6 6.4 152 14.0 6.1 12. 1 39 3.6 9.8 6.4 156 14-5 6.3 12.4 40 3.6 10. 1 6.4 l6o 14.8 6.4 12.7 4i 3-7 10.3 6.4 164 i5-i 6-5 13.0 42 3-8 io-5 6.4 168 15.4 6.7 T-3-3 43 3-9 10.7 6.4 172 15-7 6.8 13-5 44 3-9 10.9 6.4 176 15-9 6.9 *3-7 45 4.0 11. 2 6.4 l8o 16.4 7-i 14.1 46 4.1 11.4 6.4 184 16.7 7.2 14.4 47 4.2 11. 6 6.4 188 17.0 ■ 7-4 14-7 48 4.2 11.8 6.4 I92 17.2 7-5 14.8 49 4-3 12. 1 6.4 I96 17.6 7-7 15.2 5o 4.4 12.3 6.4 200 18.0 7.8 15.5 362 NOTES ON CONSTRUCTION Concrete Culverts. — Continued 2' high X 2' wide Length Feet Concrete Cubic Yards Expanded Metal Square Paving Square Yards Portland Cement Barrels Sand Cubic Yards Crushed Stone Cubic Second Third Feet Yards 20 2.4 7-1 80 9.8 IO. I 4-4 S.S 21 2.4 7-3 84 9.8 10.4 4-5 9.0 22 2-5 7.6 88 9.8 10.8 4-7 9.4 23 2.6 7-9 92 9.8 II. 2 4.9 9-7 24 2.7 8.1 96 9.8 us 5.o 10.0 25 2.7 8.4 100 9.8 11. 8 5-2 10.3 26 2.8 8.6 104 9.8 12.2 5-3 10.6 27 2.9 8.9 108 9.8 12.6 5.5 10.9 28 3.0 9.2 112 9.8 13.0 5-7 ii.3 29 3.o 9-4 116 9.8 13.2 5.8 11. 5 30 3-1 9-7 120 9.8 13.6 6.0 11.9 31 3-2 9.9 124 9.8 14.0 6.1 12. 1 32 3-3 10.2 128 9.8 14.4 6.3 12.5 33 3-3 10.5 132 9.8 14.7 6.4 12.8 34 3-4 10.7 136 9.8 i5-o 6.6 13.0 35 3-5 11.0 140 9.8 15-4 6.8 13-4 36 3-6 11. 2 144 9.8 15.8 6.9 13-7 37 3-6 "•5 148 9.8 16.1 7-1 14.0 38 3-7 11.8 152 9.8 16.5 7.2 14.4 39 3-8 12.0 156 9.8 16.8 7-4 14.7 40 3-9 12.3 160 9.8 17-3 7.6 15.0 4i 3-9 12.5 164 9.8 17.5 7.7 15-2 42 4.0 12.8 168 9.8 17.9 7-9 15.6 43 4.1 131 172 9.8 18.3 8.0 16.0 44 4.2 13-3 176 9.8 18.6 8.2 16.2 45 4.2 13.6 180 9.8 18.9 8-3 I6.S 46 4-3 13-9 184 9.8 19.4 8-5 16.9 47 4-4 ■ 14.1 188 9.8 19.7 8.6 17.2 48 4.4 14.4 192 9.8 20.0 8.8 17-4 49 4.5 14.6 196 9.8 20.4 8.9 17.7 5o 4.6 14.9 200 9.8 20.8 9.1 18.1 CONCRETE CULVERTS Concrete Culverts. — Continued 3£>3 2' high X 3' wide Length Feet Concrete Cubic Yards Expended Metal Square Steel Pounds Portland Cement Barrels Sand Cubic Yards Crushed Stone Cubic Second Third Feet Yards 20 2-3 7.6 IOO 78 I0.5 4.6 9.2 21 2-4 7.9 105 81 II. O 4-8 9.6 22 2-S 8.2 no 85 11.4 5-o 9.9 23 2.6 8-5 ii5 88 11.8 5-2 IO.3 24 2.6 8.8 120 9i 12. 1 5-3 IO.6 25 2.7 9.1 125 95 12.5 5-5 IO.9 26 2.8 9.4 130 98 13.0 5-7 II-3 27 2.9 9-7 135 101 13-4 5-9 11. 7 28 3.0 9.9 140 !05 13-7 6.0 12.0 29 3-i 10.2 145 108 14.1 6.2 12.3 30 3-2 10.5 150 112 14.6 6.4 12.7 31 3-3 10.8 155 115 i5-o 6.6 i3-i 32 3-4 11. 1 160 Il8 15-4 6.8 13-4 33 3-5 11.4 165 122 15-9 7.0 13.8 34 3-6 11. 7 170 125 16.3 7.2 14.2 35 3-7 12.0 175 128 16.7 7-3 14.6 36 3.8 12.2 180 132 17.0 7-5 14.8 37 3-9 12.5 185 135 17.5 7-7 15-2 38 3-9 12.8 190 139 17.8 7.8 15-5 39 4.0 13. 1 195 142 18.2 8.0 15-9 40 4-i 13-4 200 145 18.6 8.2 16.2 41 4.2 13-7 205 149 19.0 8.4 16.6 42 4-3 ' 14.0 210 152 19-5 8.6 17.0 43 4.4 14-3 215 156 19.9 8.7 17-3 44 4.5 14.5 220 159 20.2 8.9 17.6 45 4.6 14.8 225 l62 ' 20.7 9.1 18.0 46 4-7 i5-i 230 166 21. 1 9.2 18.4 47 4.8 15-4 235 169 21.5 9-4 18.7 48 4-9 15-7 240 172 21.9 9.6 19.1 49 5-o 16.0 245 176 22.4 9.8 19-5 5o 5-1 16.3 250 179 22.8 10. 19.8 364 NOTES ON CONSTRUCTION Concrete Culverts. — Continued 2' high X 4' wide Length Feet • Con Cubic crete Yards Expanded Metal Square Steel Pounds Portland Cement Barrels Sand Cubic Yards Crushed Stone Cubic Second Third Feet Yards 20 2.7 8.4 120 78 11.8 5-2 10.3 21 2.8 8.7 126 81 12.3 5-3 10.7 22 2.9 9.0 132 85 12.7 5-6 II. O 23 3-i 9-3 138 88 13.2 5-8 II.5 24 3-2 9-7 144 9i 13.8 6.0 I2.0 25 3-3 10. I50 95 14.2 6.2 12.3 26 3-4 10.3 156 98 14.6 6.4 12.7 27 3-5 10.6 162 101 15.0 6.6 13. 1 28 3-6 10.9 168 105 i5-5 6.8 13-4 29 3-7 11. 2 174 108 15-9 6.9 13.8 30 3-8 "•5 180 112 16.3 7-1 14.2 31 3-9 11.9 186 115 16.8 7-4 14.6 32 4.0 12.2 192 118 *7-3 7.6 15-0 33 4.2 12.5 198 122 17.8 7.8 15-5 34 4-3 12.8 204 125 18.3 8.0 15-9 35 4-4 13. 1 2IO 128 18.7 8.2 16.2 36 4-5 134 216 132 19.1 8.4 16.6 37 4.6 - 13.8 222 135 19.6 8.6 17. 1 38 4-7 14.1 228 139 20.1 8.7 17.4 39 4.8 14.4 234 142 20.5 9.0 17.8 40 4-9 14.7 240 145 20.9 9.1 18.2 4i 5.o 15.0 246 149 21.4 9.4 18.6 42 5-2 15-3 252 152 21.9 9.6 19.1 43 5-3 15.6 258 156 22.3 9.8 19.4 44 5-4 16.0 264 159 22.9 10. 19.9 45 5-5 16.3 270 162 23-3 10.2 20.2 46 5-6 16.6 276 166 23-7 10.4 20.6 47 5-7 16.9 282 169 24.1 10.6 21.0 48 5-8 17.2 288 172 24.6 10.8 21.3 49 5-9 i7-5 294 176 25.0 10.9 21.7 5o 6.0 17.8 3OO 179 25-4 11. 1 22.1 CONCRETE CULVERTS Concrete Culverts. — Continued 3°5 3' high X 3' wick ■ Length Concrete Cubic Yards Expanded Metal Steel Pounds Portland Cement Sand Cubic Crushed Stone Feet Square Barrels Yards Cubic Second Third Feet Yards 20 2-3 10.4 IOO 82 134 5-9 II.8 21 2.4 10.8 105 85 13-9 6.2 12.3 22 2-5 11. 2 no 88 14.4 6.4 12.7 23 2.6 n-5 115 92 14.9 6.6 i3-i 24 2.6 11.9 120 95 15-3 6.8 13.5 25 2.7 12.2 125 99 15-7 7.0 13.8 26 2.8 12.6 130 102 16.2 7.2 14.3 27 2.9 13.0 135 105 16.8 7-4 14.8 28 3-0 t-3-3 140 109 17.2 7.6 i5-i 29 3-i 13-7 145 112 17.7 7-9 15.6 30 3-2 14.0 150 116 18.2 8.1 16.0 3i 3-3 14.4 155 119 18.7 8-3 16.4 32 3-4 14.8 160 122 19.2 8-5 16.9 33 3-5 i5-i 165 126 19.6 8.7 17.3 34 3.6 15-5 170 129 20.2 8.9 17.6 35 3-7 15.8 175 133 20.6 9.1 18.1 36 3-8 16.2 180 136 21. 1 9-4 18.6 37 3-9 16.6 185 139 21.6 9.6 19.0 38 3-9 16.9 190 143 22.0 9-7 19-3 39 4.0 17-3 195 146 22.5 10. 19.8 40 4.1 17.6 200 150 22.9 10.2 20.1 4i 4.2 18.0 205 153 23-4 10.4 20.6 42 4.3 18.4 210 156 24.0 10.6 21. 1 43 4.4 18.7 215 160 24.4 10.8 21.4 44 4-5 19.1 220 163 24.9 n. 21.9 45 4.6 19.4 225 167 25-4 11. 2 22.3 46 4-7 19.8 23O 170 25.9 11.4 22.7 47 4.8 20.2 235 *73 26.4 n. 7 23.2 48 4-9 20.5 24O 177 26.8 11.9 23.6 49 5.o 20.9 245 180 27.4 12. 1 24.0 50 5-1 21.2 25O 184 27.8 12.3 24.4 3 66 NOTES ON CONSTRUCTION Concrete Culverts. — Continued 3' high X 4' wide Length Concrete Cubic Yards Expanded Metal Steel Portland Cement Sand Cubic Crushed Stone Feet Square Pounds Barrels Yards Cubic Second Third Feet Yards 20 2.7 n-3 I20 82 14.8 6.5 13.0 21 2.8 11. 7 126 85 15-3 6.8 13-5 22 2.9 12. 1 132 88 is-8 7.0 13-9 23 3-i 12.5 138 92 16.5 7-3 14-5 24 3-2 12.9 144 95 17.O 7-5 14.9 25 3-3 13.2 I50 99 17.4 7.7 15-3 26 3-4 13.6 156 102 18.0 7-9 15.8 27 3-5 14.0 162 105 18.5 8.2 16.3 28 3.6 14.4 168 109 19.0 8.4 16.7 29 3-7 14.8 174 112 19.6 8-7 17.2 30 3-8 15.2 180 116 20.I 8.9 17.6 31 3-9 15.6 186 119 20.6 9.1 18.1 32 4.0 16.0 192 122 21. 1 9-4 18.6 33 4.2 16.4 198 126 21.8 9.6 19.1 34 4-3 16.8 204 129 22.3 9.8 19.6 35 44 17. 1 210 133 22.8 IO.I 20.0 36 4-5 17-5 216 136 23-3 10.3 20.4 37 4.6 17.9 222 139 23.8 10.5 20.9 38 4-7 18.3 228 143 24-3 10.8 21.3 39 4.8 18.7 234 146 24.9 II.O 21.8 40 4.9 19. 1 240 150 25.4 II. 2 22.3 4i 5.o 19-5 246 153 25-9 II.4 22.7 42 5-i 19.9 252 156' 26.5 11. 7 23.2 43 5-3 .20.3 258 160 27.1 12.0 23-7 44 5-4 20.7 264 163 27.7 12.2 24.2 45 5-5 21.0 27O 167 28.1 12.4 24.6 46 5-6 21.4 276 170 28.6 12.6 25.0 47 5-7 21.8 282 173 29.1 12.9 25.5 48 5-8 22.2 288 177 29.7 i3-i 26.0 49 5-9 22.6 294 180 30.2 13-3 26.4 5o 6.0 23.0 3OO 184 30.7 13.6 26.9 CONCRETE CULVERTS Concrete Culverts. — Continued 367 4' high X 4' wide Length Concrete Cubic Yards Expanded Metal Steel Portland Cement Sand Cubic Crushed Stone Feet Square Pounds Barrels Yards Cubic Second Third Feet Yards 20 2.7 14-5 120 ^ 18.1 8.1 15-9 21 2.8 15.0 126 90 18.7 S-3 16.5 22 2.9 15-4 132 94 19.2 8.6 17.0 23 3-i 15-9 138 97 20.O 8.9 17.6 24 3-2 16.4 144 100 20.6 9.2 18.2 25 3-3 16.8 I50 104 21. 1 9.4 18.7 26 3-4 17-3 156 107 21.8 9-7 19.2 27 3-5 17.7 162 in 22.3 9.9 19.7 28 3-6 18.2 168 114 22.9 10.2 20.2 29 3-7 18.7 174 117 23-5 io-5 20.8 30 3-8 19.1 180 121 24.I 10.7 21.2 31 3-9 19.6 186 124 24.7 11.0 21.8 32 4.0 20.1 192 128 25-3 n-3 22.4 33 4.2 20.5 198 131 26.0 11.6 22.9 34 4-3 21.0 204 134 26.6 11.9 23-5 35 4.4 21.4 2IO 138 27.1 12. 1 24.0 36 4.5 21.9 216 141 27.8 12.4 24.5 37 4.6 . 22.4 222 145 28.4 12.6 25.1 38 4-7 22.8 228 148 28.9 12.9 25.5 39 4.8 23-3 234 151 29.6 13. 1 26.1 40 4-9 23.8 24O 155 30.2 13-4 26.6 4i 5.o 24.2 246 158 30.7 13-7 27.1 42 5-i 24.7 252 162 31-4 14.0 27.7 43 5-3 25.2 258 165 32.1 14.3 28.3 44 5-4 25.6 264 168 32.6 14.5 28.8 45 5-5 26.1 270 172 33-3 14.8 29-3 46 5-6 26.5 276 175 33-8 i5-o 29.8 47 5-7 27.0 282 179 34-4 15-3 30.3 48 5-8 27-5 288 182 35-i i5-6 30-9 49 5-9 27.9 294 185 35-6 15.8 31-4 5o 6.0 28.4 300 189 36.2 16.1 3i-9 368 NOTES ON CONSTRUCTION Concrete Culverts. — Continued 3 ; high X 5' wide Length Feet Concrete Cubic Yards Expanded Metal Steel Pounds Portland Cement Sand Cubic Crushed Stone Square Barrels Yards Cubic Second Third Feet Yards 20 4.0 12.4 140 ^3 17.5 7-7 15-2 21 4.2 12.8 147 86 18.1 7-9 15-7 22 4-4 ^3-3 154 90 18.9 ^3 16.4 23 4.6 13-7 161 93 19-5 8.6 17.0 24 4-7 14.1 168 96 20.1 8.8 17-4 25 4-9 14-5 175 100 20.7 9.1 18.0 26 5-i 14.9 182 103 21.4 9-3 18.5 27 5-3 15-4 189 106 22.1 9.6 19.2 28 5-4 15-8 196 no 22.6 9.9 19.7 29 5.6 16.2 203 113 23-3 10.2 20.2 30 5-8 16.6 2IO 117 23-9 10.5 20.8 31 5-9 17.0 217 120 24.5 10.7 21.2 32 6.1 17.4 224 123 25-1 n. 21.8 33 6-3 17.9 231 127 25-9 n-3 22.4 34 6-5 18.3 238 130 26.5 11.6 23.0 35 6.6 18.7 245 134 27.1 11.8 23-5 36 6.8 19. 1 252 137 27.7 12. 1 24.0 31 7.0 19-5 259 140 28.4 12.4 24.6 38 7.2 19.9 266 144 29.0 12.7 25.1 39 7-3 20.4 2 73 147 29.6 12.9 25-7 40 7-5 20.8 280 150 30-3 13.2 26.2 4i 7-7 21.2 287 154 30-9 13-5 26.8 42 7-8 21.6 294 157 3i-5 13-7 27-3 43 8.0 22.0 301 161 32.1 14.0 27.8 44 8.2 22.4 308 164 32.8 14-3 28.4 45 8.4 22.9 315 167 33-4 14.6 29.0 46 8-5 23-3 322 171 34-i 14.8 29-5 47 8-7 23-7 329 174 34-7 15.1 30.0 48 8.9 24.1 336 177' 35-3 15-3 30.6 49 9.1 24-5 343 181 36.0 15.6 31.2 50 9.2 24.9 35o 184 36.5 15.9 31.6 CONCRETE CULVERTS Concrete Culverts. — Continued 369 4' high X 5' wide Length Feet Concrete Cubic Yards Expanded Metal Square Steel Pounds Portland Cement Barrels Sand Cubic Yards Crushed Stone Cubic Second Third Feet Yards 20 4.0 15.8 140 88 2I.O 9.2 18.4 21 4.2 16.3 147 92 21.7 9.6 19.0 22 4.4 16.8 154 95 22.5 9.9 19.7 23 4.6 17.2 161 99 23.I I0.2 20.2 24 4-7 17.7 168 102 23-7 io-5 20.8 25 4.9 18.2 175 105 24.5 10.8 21.4 26 5-i 18.7 182 109 25.2 11. 1 22.1 27 5-3 19.2 189 112 26.O n-5 22.7 28 5-4 19.7 196 116 26.6 11. 7 23-3 29 5.6 20.2 203 119 27.4 12. 1 23-9 30 5-8 20.7 2IO 122 28.I 12.4 24.6 31 5-9 21.2 217 126 28.8 12.7 25.1 32 6.1 21.7 224 129 29-5 13.0 25.8 33 6.3 22.1 231 133 30.2 13-3 26.3 34 6-5 22.6 238 136 30.9 13.6 27.O 35 6.6 23.I 245 139 3i-5 13-9 27.6 36 6.8 23.6 252 143 32-3 14.2 28.2 37 7.0 24.I 259 146 33-o 14.5 28.8 38 7.2 24.6 266 150 33-8 14.9 29-5 39 7-3 25.I 273 153 34-4 i5-i 30.I 40 7-5 25.6 280 156 35-2 15-5 30.7 4i 7-7 26.I 287 160 35-9 15-8 31-3 42 7.8 26.6 294 163 36.6 16.1 3i-9 43 8.0 27.O 301 167 37-2 16.4 32.5 44 8.2 27.5 308 170 38.0 16.7 33-?- 45 8-4 28.O 315 173 38.7 17.0 33-8 46 8-5 28.5 322 177 39-3 17.3 34-3 47 8.7 29.O 329 180 40.1 17.6 35-0 48 8.9 29.5 336 184 40.9 18.0 35-6 49 9.1 30.0 343 187 41.6 18.3 36-3 5o 9.2 30-5 35o 190 42.2 18.6 36.8 37° NOTES ON CONSTRUCTION Concrete Culverts. — Continued 5' high X 5' wide • Length Concrete Cubic Yards Expanded Metal Steel Portland Cement Sand Cubic Crushed Stone Feet Square Pounds Barrels Yards Cubic Second Third Feet Yards 20 4.0 19-5 140 93 24.7 11. 21.8 21 4.2 20.0 147 •96 25-5 11.3 22.5 22 4.4 20.6 154 100 26.3 II. 7 23.2 23 4.6 21.2 161 103 27.2 12. 1 24.2 24 4-7 21.7 168 106 27.8 12.4 24.8 25 4.9 22.3 175 no 28.7 12.7 25-4 26 5-i 22.9 182 113 29.5. i3-i 26.2 27 5-3 23.4 189 117 30-3 13-4 26.8 28 5-4 24.0 196 120 31.0 13.8 27.6 29 5-6 24.6 203 123 3**9 14.1 28.2 30 5-8 25.1 210 127 32.6 14-5 29.0 31 5-9 25-7 217 130 33-4 14.8 29.6 32 6.1 26.2 224 134 34-i i5-i 30.2 33 6.3 26.8 231 *37 35-o 15-5 31.0 34 6-5 27.4 238 140 35-8 15-9 31.8 35 6.6 27.9 245 144 36.4 16.2 32.4 36 6.8 28.5 252 147 37-3 16.5 33-o 37 7.0 29.1 259 150 38.2 16.9 33-8 38 7.2 29.6 266 154 38.9 17.2 34-4 39 7-3 30.2 2 73 157 39-6 17.6 35-i 40 7-5 30.8 280 161 40.5 17.9 35-8 4i 7-7 3^-3 287 164 41.2 18.3 36.5 42 7.8 3i-9 294 167 42.0 18.6 37-2 43 8.0 32.5 301 171 42.8 19.0 37-9 44 8.2 33-o 308 174 43-6 19-3 38.6 45 8.4 33-6 315 178 44.4 19.7 39-3 46 8-5 34-2 322 181 45-2 20.0 40.0 47 8.7 34-7 329 184 45-9 20.3 40.6 48 8.9 35-2 336 188 46.7 20.6 41.2 49 9.1 35-9 343 191 47.6 21.0 42.0 5o 9.2 36.4 35o 195 48.3 21.4 42.8 CONCLUSION 371 a rough sandpaper-like finish is wanted it can be secured by rubbing with a wooden float moistened with water. This finish is not as apt to hair-check as the smooth finish. Freshly laid concrete should be protected from a hot sun by covering it with canvas, or blankets, and wetting it down fre- quently for four or five days. No plastering of surfaces should be allowed after the cement has set. If, however, it has been badly hair-checked from heat the defect can usually be remedied by rubbing with a carborundum brick. Freshly laid concrete must be protected from frost. A satisfactory method is to cover with canvas and a thick layer of manure or straw. If the concrete has been frost-pitted, on the surface only, bush hammer- ing will give a rough stone finish, pleasing in appearance. No culvert work should be allowed in continued cold weather, as it is difficult to get a good finish and in roadwork there is no neces- sity of doing this work in the winter. Concrete inspection must be continuous. CONCLUSION For obvious reasons the inspection of construction is generally the weak point in Municipal and State Engineering undertak- ings. It is often due to the employment of inferior inspectors, and frequently to the impossibility of even good inspectors con- trolling certain contractors. The work is rarely bad, but it will not be as strong nor as lasting as a first-class job, and if such conditions are foreseen, and cannot be avoided, it is, per- haps, best to design the work stronger than would otherwise be required, as this seems to be the only practical method of meeting a recognized evil. CHAPTER XII SPECIFICATIONS Under this heading are included extracts from the State speci- fications of New York and Washington covering "Materials" and the more common construction methods. It is difficult to write a specification that is definite and fair, and it is impossible to avoid criticism. The following clauses are examples of current practice. They are not ideal, but show the points to be considered. No attempt is made in this book to discuss methods of bidding or of forms of proposals. MATERIALS (New York State Specifications, 19 14) Materials of Construction All materials proposed to be used in construction shall have due examination and pass all required tests before acceptance. Those which are to be tested by the Bureau of Tests at Albany shall have samples taken and submitted in accordance with the commission's instructions to its employees. Samples are to be taken of all sand, gravel, cement, concrete, bituminous material, stone, and all other pavement ingredients, of which the engineer in charge has not been notified that satisfactory samples have already been taken. None of this material is to be used until the written notification of accept- ance is received by the engineer in charge of the contract, and then only so long as its quality remains equal to that of the accepted sample. Portland Cement 0.1. All the cement used in the work shall be true Portland cement of well-known brands which have been in successful use on large engineering works in America for not less than two years and which are manufactured at works which have been in successful operation for at least one year. 0.2. Tests will be made as follows: — first, for fineness; second, for constancy of volume; third, for time of setting; fourth, for tensile strength; fifth, for composition by chemical tests; sixth, for specific gravity. The average result of the separate samples shall be the test for tensile strength of any lot. The samples of each lot shall be required to show uniform results in tests. Marked deviations' from such results may be considered cause for rejection, even though test re- quirements may be otherwise fulfilled. The results of the tests may be expected in 12 days after shipment of samples. 372 CEMENT TESTS 373 Cement not satisfactory in the 7-day tests will be held awaiting the result of the 28-day tests before acceptance or rejection. 0.3. The cement shall meet the following requirements: It shall be ground to such fineness that not less than 92 per cent by weight shall pass through a No. 100 standard sieve of 10,000 meshes per square inch, and not less than 75 per cent by weight shall pass through a No. 200 standard sieve of 40,000 meshes per square inch. 0.4. Pats of neat cement about 3 inches by 4 inches in size, i inch thick at the center, and tapering to a thin edge, shall be kept in moist air for a period of 24 hours. Normal Tests Air Test. One of these pats is then kept in air at normal tempera- ture for 28 days. Water Test : Another pat is kept in water maintained as near 70 degrees Fahrenheit as practical for 28 days. Accelerated Test : A pat is exposed in any convenient way in an atmosphere of steam, above boiling water, in a loosely closed vessel for 5 hours. These pats are observed at intervals and, to satisfactorily pass the requirements, shall remain firm and hard and show no signs of distortion, checking, cracking or disintegration. 0.5. Cement shall not develop its initial set in less than 30 minutes, and shall develop a hard set in not less than 60 minutes nor more than 600 minutes, the determination being made with the Vicat needle apparatus from pastes of normal consistency, as follows: The paste is molded upon a glass in a conical hard rubber mold 4 centimeters high; this cake is to set in moist air and a Vicat needle with a wire 1 millimeter in diameter and loaded to 300 grammes shall be placed upon it. When the needle ceases to pass a point 5 millimeters above the upper surface of the glass plate the initial set has taken place. 0.6. Briquettes of neat cement mixed 1 minute, in an air tem- perature between 65 and 70 degrees Fahrenheit and using water of about the same temperature, and put into the molds with fingers and trowel and kept in moist air at this temperature for 1 day of 24 hours, shall show an average tensile strength of one hundred and seventy-five (175) pounds per square inch. Briquettes of neat cement mixed and molded as above and kept under above temperature for 1 day in moist air and 6 days in water shall show an average tensile strength of at least five hundred (500) pounds per square inch. Briquettes of neat cement mixed and molded as above and kept under above temperature for 1 day in moist air and 27 days in water shall show an average tensile strength of six hundred (600) pounds per square inch. Briquettes of 3 parts by weight of standard Ottawa sand and 1 part by weight of cement, mixed in the same manner as above and kept 7 days under the same conditions, shall show 374 SPECIFICATIONS an average tensile strength of at least two hundred pounds (200) per square inch. Briquettes of sand and cement mixed and molded as above and kept under above conditions for 28 days shall show an average tensile strength of at least two hundred and seventy-five (275) pounds per square inch. In the above tests for tensile strength the briquettes must not show any retrogression in strength within the periods specified. 0.7. The Commission of Highways may cause chemical tests, or analyses, of cement to be made, and may reject any cement which shows any adulteration, or excess of ingredients, which in its judgment would be detrimental to the work. The cement shall not contain more than 1.75 per cent of anhy- drous sulphuric acid (SO3) nor more than 4 per cent of magnesia (MgO). 0.8. The specific gravity of the cement after ignition to a low red heat shall not be less than 3.10; and the cement shall not show a loss in weight on ignition of more than 4 per cent. 0.9. The standard sand used in the tests shall be natural sand from Ottawa, 111., screened to pass a No. 20 standard sieve of 400 meshes per square inch and be retained on a No. 30 standard sieve of 900 meshes per square inch. Fine Aggregate for Concrete 0.10. Fine aggregate shall consist of sand free from organic matter; that which shows a coating on the grains shall not be used until satisfactorily washed. Sand shall be classified as No. 1, No. 2 and No. 3, and Grout Sand. No. 1 Sand 0.1 1. No. 1 sand shall be of the following gradation: 100 per cent shall pass a J-inch screen; not more than 20 per cent shall pass a No. 50 sieve; and not more than 6 per cent shall pass a No. 100 sieve. In special cases where more than 20 per cent of a sand passes a No. 50 sieve and the sand is well graded to give a low percentage of voids, written permission for use of the sand may be given by the first deputy commissioner. Sand may be rejected for this class if it contains more than 5 per cent of loam and silt. 0.12. Mortar in the proportion of 1 part of cement to 3 parts of the sand to be tested shall develop a compressive strength at least equal to the strength of a similar mortar of the same age composed of the same cement and standard Ottawa sand. No. 2 Sand 0.13. No. 2 sand shall fulfill all of the requirements for No. 1 sand except that restrictions on the percentage that will pass a No. 50 and No. 100 sieve shall be governed by the compressive strength of the mortar. The compressive strength of the mortar shall be at least equal to that obtained with the standard Ottawa sand. COARSE AGGREGATE FOR CONCRETE 375 No. 3 Sand 0.14. Sand may be rejected for this class if it contains more than 8 per cent of loam and silt. Mortar in the proportion of 1 part of cement to 3 parts of the sand when tested shall develop a compressive strength of at least 80 per cent of the strength of a similar mortar of the same age composed of the same cement and standard Ottawa sand. Screenings 0.15. Screenings shall not be used as fine aggregate except to the extent and under the restrictions given below. The division engineer shall submit samples to the Bureau of Tests; they must pass the required tests; their use must be approved in writing by the First Deputy Commissioner. 0.16. Screenings may be substituted for a portion of the No. 1, 2 and 3 sand under the following conditions : The screenings shall be free from dust coating and other dirt. 100 per cent shall pass a | inch screen and not more than 6 per cent shall pass a No. 100 sieve. The compressive strength of a mortar in which the screenings and sand are in the proportions intended for use, shall be at least equal to the standard strength obtained with sand of the given class. . Grout Sand 0.17. Grout sand shall be a sand of which 100 per cent passes a No. 20 sieve, and not over 30 per cent a No. 160 sieve. Sand may be rejected for this class if it contains more than 5 per cent of loam and silt. The grains shall be free from coating. Mortar in the proportion of 1 part of cement to 3 parts of the sand shall develop a compressive strength of at least 40 per cent of the strength of a similar mortar of the same age composed of the same cement and standard Ottawa sand. Cushion Sand 0.18. Cushion sand shall be a sand of which 100% passes a No. 6 sieve and 90% passes a No'. 20 sieve; an excessively fine sand will not be accepted in this class. Sand may be rejected for this class if it contains more than 10% of loam and silt. Coarse Aggregate for Concrete Stone 0.19. Crushed stone for concrete shall be of hard, durable stone, tested by the Bureau of Tests and satisfactory to the engineer. Stone for concrete shall be of an approved kind and quality of rock and shall be free, before being crushed, from soil, mud and dust. Crushed stone for first-class concrete shall be in fragments that will pass through a ij-inoh circular hole and that will not pass through a J-inch square hole. Crushed stone for second-class or third-class concrete shall be in fragments that will pass through a 2|-inch cir- cular hole and that will not pass through a J-inch square hole. 376 SPECIFICATIONS Gravel 0.20. Gravel shall not be used in concrete except when it has been submitted by the Division Engineer to the Bureau of Tests, has been approved by the Bureau of Tests, and its use has been approved by the First Deputy Commissioner in writing, — and then only under the restrictions given below. 0.21. Gravel for use in concrete pavement and first-class concrete shall be composed of hard, durable stone absolutely clean and free from coating. No gravel will be accepted that contains any disin- tegrated or soft stone or shale. Gravel containing any flat stone shall not be permitted. Gravel for use in second and third class concrete shall be composed of a sound, durable stone. It shall be clean and free from coating. It shall not contain more than 10 per cent of soft stone or shale. Gravel containing a larger percentage of flat stone shall not be permitted. 0.22. Gravel for first-class concrete shall be in particles that will pass through a ij-inch circular hole and that will not pass through a J inch square hole. Gravel for second and third-class concrete shall be in particles that will pass through a 2j-inch circular hole and that will not pass through a f-inch square hole. 0.23. Gravel mixed with mud, clay, dirt or quicksand shall be washed to the satisfaction of the engineer. Run of bank gravel shall not be permitted. All gravel shall be properly screened and the coarse and fine aggregate regularly proportioned thereafter. 0.24. All coarse aggregate used for concrete shall be uniformly graded from the minimum to the maximum sizes of stone or gravel specified above for the several types of concrete, thus producing an aggregate in which the voids will be a minimum. Stone, Gravel, etc., for Pavements 0.25. The sizes of all stone, gravel, etc., used under these specifi- cations shall be determined by the size of screen aperture through which the stone will pass when revolved in a rotary screen. They shall be designated as follows: Diameter of Aperture Min. Max. J-inch square Screenings J-inch square f-inch circular No. 1 f-inch circular ii-inch circular No. 2 1 1-inch circular 2f-inch circular No. 3 2 f-inch circular 3f-inch circular No. 4 0.26. Gravel shall consist of clean, sound, tough hard stone. Gravel shall be separated into five grades or sizes by means of a rotary screen having openings as specified above for broken stone. All the general specifications given below relating to broken stone shall apply to gravel, excepting that gravel may contain not more than 5 per cent, of loam, but must otherwise be free from BITUMINOUS MATERIALS 377 dirt or foreign matter and shall be washed if so directed by the engineer. 0.27. Broken slag shall be approved acid slag, clean, sound, tough, hard, sharp-angled and weigh not less than 1,800 pounds per cubic yard. If specified for use, it shall conform to all the general require- ments for broken stone as specified below. 0.28. Broken stone shall be clean and sharp angled, shall pass the standard tests for abrasion and toughness as adopted by the American Society for Testing Materials, and shall be approved by the Bureau of Tests and acceptable to the engineer before being used. 0.29. Field stones, boulders, or fence stones which are crushed for macadam purposes shall be 6 or more inches in diameter, if con- sisting of rounded cobbles. If of the flat variety, the minimum thickness shall be 2 inches, which latter requirement will also apply to laminated quarry stone. 0.30. If after trial it is found that partially developed quarries, ledges or other sources of supply do not furnish a uniform product, or if, for any reason, the product from any source, at any time, proves to be unsatisfactory to the engineer, said engineer may require the contractor to furnish stone from other sources of supply, and the contractor shall have no claim for increased payment on account of such requirement. 0.31. The contractor shall furnish one or more stone crushing plants of type, composition, and capacity satisfactory to the engineer. The rotary screens shall be provided with openings of size and shape given under "Stone Sizes," unless otherwise ordered by the engineer. All crushing plants installed on the work shall be fitted up with a tailing chute so that no stone will reach the bins other than that which passes through the proper screen. 0.32. All stone must be of the required size when placed in the roadway, and no breaking up of stone by hammers or otherwise will be permitted after the stone has been placed in the work. 0.33. In no case shall any constituent of macadam pavement be dumped into place in mass; the final placing shall be by shovel or by thin spreading such that no appreciable fall occurs. Filler or Binder. The filler for the bottom course shall be clean, coarse sand or stone screenings supplemented by product of the crusher not otherwise used in top or bottom courses. The filler and wearing surface for the top course shall be of top course stone screenings and when bituminous binder is used screenings must be dry, free from dust, and not larger than will pass a f-inch screen. BITUMINOUS MATERIALS Methods of Testing Bituminous Materials in the Laboratory of the Commission Preparing Laboratory Samples. Each laboratory sample is usually composed of several samples that have been taken to repre- sent one lot of material. The material in the separate samples is examined, and, if uniform in appearance, equal amounts are taken 378 SPECIFICATIONS from each and thoroughly blended to form a sample of about one- half pint on which the complete analysis is run. In case of mineral bitumen, the sample received is thrown on a large piece of paper, pieces which are evidently foreign to the material are rejected, and the whole " quartered down" to a sample of about 300 grams. This is ground in a mortar and the analysis run on this part of the original sample. Water Present. The presence of water in an oil, asphalt, or tar is determined by putting about 40 grams of the material into a deep, seamless 3-ounce tin box, a thermometer being suspended in the material. This is then heated to about 230 F. without stiring. If water is present, even in very small quantities, the material will froth when heated to about 212 F. The per cent of water present is determined by heating 20 grams of the material in a 2-ounce seam- less tin box in an oven maintained at a temperature of 212 F. for an hour. The per cent of water in mineral bitumen is determined in a similar manner. The loss in weight, while not absolutely cor- rect, is considered as moisture. Homogeneity. The homogeneity of the mixture is shown by its general appearance at a temperature of 77 F. when in a melted condition and when examined under the microscope. Gravity. The gravity is determined by taking a small test tube about f of an inch by 3^ inches, which is accurately weighed (weight A). The tube is then filled with distilled water at 77 F. and weighed (weight B). To get the gravity of the oil, asphalt, or tar the tube is filled with the material, cooled to a temperature of 77 F., cut off level with the top, and weighed (weight C) . The gravity is determined C — A as follows: ~ = gravity. Jj — A Penetration. The penetration test is made by putting the ma- terial to be tested in a 3-ounce deep, seamless tin box. Melting the material at the lowest possible temperature, cooling in air and then placing the material in a bath, for one hour, maintained at the tem- perature at which the test is to be made. The penetration is the distance expressed in hundredths of a centimeter which a standard needle under a stated load, and at a stated temperature, will pene- trate into the material. The factors usually employed are a No. 2 sewing needle, loaded with 100 grams, applied for five seconds at a temperature of 77 Fahrenheit. Residue having a Penetration of 10 Millimeters. This test is made as follows: 50 grams of the oil are placed in a 3-ounce deep, seamless tin box, the box placed in a sand bath and heated over a Bunsen Burner. A thermometer is suspended in the oil, the bulb not touching the bottom of the box. The temperature of the oil is kept at from 480 F. to 500 F. and the oil is stirred from time to time with the thermometer to prevent overheating in any part. Depending upon the nature of the oil, as usually indicated by its flash, consistency at 77 F. and gravity, the operator can tell about what per cent it will be necessary to evaporate before coolmg and taking a penetration as described under the test for penetration. _ It is sometimes necessary to make several trials before the desired BITUMINOUS MATERIALS 379 result is obtained. When the required penetration is reached, the residue left from evaporation is weighed and its per cent of the orig- inal sample taken is computed. Ductility. The ductility of an asphalt cement or bitumen is determined by the distance in centimeters that a briquette of the material will draw out before breaking. The briquette of the asphalt cement is molded in a Dow briquette mold having a central cross- section 1 centimeter square, a 2-square centimeter cross-section at mouth of clips, and a distance of 3 centimeters between clips. The molding of the briquette is done as follows: The mold is placed on a brass plate. To prevent the asphalt cement from adhering to this plate and the inner sides of the two pieces of the mold, they shall be well amalgamated. The asphalt cement to be tested is poured into the mold While in a molten state, a slight excess being added to allow for shrinkage on cooling. After the asphalt cement is nearly cooled, the briquette is smoothed off level by means of a hot spatula. When it is thoroughly cooled to the temperature at which it is desired to make the test, the clamp and the two side-pieces are removed, leaving the briquette of asphalt cement held at each end by the ends of the mold which serve as clips. The test is made by pulling the two clips apart at a uniform rate of 5 centimeters per minute by means of hooks inserted in the eyes, until rupture occurs. The briquette is kept in water at 77 F. for at least 30 minutes before testing, and the test is performed while the briquette is so immersed in the water at the above temperature, and at no time is the temperature of the water allowed to vary more than half a degree from the standard temperature. Toughness. The bitumen is heated until liquid; it is then poured into an amalgamated brass mold of such shape as to give a cylinder of the bitumen if inches in height by if inches in diameter. After cooling, the mold is removed and the cylinder of bitumen is placed in a mixture of finely crushed ice and water, giving a temperature of zero degrees centigrade. After remaining in the freezing mixture for about three hours the cylinders are broken in a Page impact machine (the standard machine of the American Society for Testing Materials for determining the toughness of macadam stone). When the cylinder to be broken is placed in the impact machine a piece of linen cloth about one inch square is placed on the end on which the plunger rests. This prevents the plunger from sticking to the bitumen and makes it easier to clean the machine. In making the test, the first drop of the hammer is from a height of five centimeters, and for each succeeding blow the height of the drop is increased five centimeters. The height from which the hammer falls when rupture occurs is given as the toughness of the material. Melting Point of Bitumen. The melting or softening point of bitumen is determined by filling a ring f inch in diameter by \ inch in depth, with the bitumen to be tested. After cooling, the bitumen is cut off level with the top of the ring. The ring containing the bitu- men is placed in water at 41 F. for 20 minutes before making the test. In performing the test the ring is put in a support so placed that the bottom of the ring is 1 inch above the bottom of an 800 cc. 380 SPECIFICATIONS beaker. On the center of the bitumen in the ring, is placed a f-inch steel ball, a thermometer being placed with its bulb on a level with the ring containing the bitumen. The beaker is nearly filled with water at a temperature of 41 F. and the temperature raised at the rate of 8° F. to io° F. per minute. The temperature recorded by the thermometer at the time the ball touches the bottom of the beaker is taken as the melting point of the bitumen. Evaporation. Fifty grams of the material are weighed into a flat-bottomed dish 2y-g in. in diameter by if in. in depth. This is placed in an oven maintained at a uniform temperature of 325 Fahrenheit for a period of five hours. At the end of this period the loss in weight or per cent of loss is found by reweighing. Flash. About 40 grams of the material to be tested are placed in a 3-ounce deep, seamless tin box. The box containing the ma- terial is placed on a sand bath over a Bunsen Burner, the bulb of a thermometer being placed in the material, but so adjusted as not to touch the bottom of the box. The flame of the Bunsen Burner is so adjusted that the temperature of the material being tested is raised at the rate of io° F. to 15 F. per minute. As soon as vapors are seen coming off, the small flame from a capillary tube is passed over the center of the liquid and about \ inch above it, and repeated for about every 5 F. rise in temperature until the slight explosion indi- cates the flash-point is reached. The temperature at this point is recorded as the open flash-point of the material being tested. Total Bitumen. The solubility in C S 2 is found by weighing approximately 1 gram of the material into an Erlenmeyer flask, adding 50 cc. of C 5 2 and allowing the solvent to act 12 hours at lab- oratory temperature, care being taken to break up all lumps before filtering. The filtration is made through a C. S. & S. 9-centimeter filter paper No. 589. The papers are first dried, and weighed imme- diately before using. The filtration is made in a valve funnel, a watch glass beieg placed on the funnel to prevent evaporation of the solvent. After washing until washings come clean, the filter and residue are placed in an oven at 212 F. for 30 minutes, cooled in a desiccator and weighed. The difference in weight gives the amount of material insoluble in C So from which the per cent of soluble bitumen is computed. The total bitumen in mineral bitumen is determined by weighing about 25 grams of the dried material into a dried and weighed C. S. & S. extraction cartridge and extracting in a continuous extraction apparatus, using C S 2 for a solvent; drying and weighing after extraction is completed. The loss gives the amount of bitumen soluble in C S 2 . Carbon Tetrachloride Solubility. This test is made in the same manner as determining the bitumen soluble C S 2 , except that C C I4 is used as solvent^ Naphtha Solubility. The amount of material soluble in 76 naphtha (boiling point 140 F. to 190 F.) is found by the same method that is used in getting the amount soluble in C S 2 , except that naphtha is used for a solvent in place of C 5 2 . The character of the filtrate is determined by placing about 10 cc. of the nitrate in the BITUMINOUS MATERIALS 381 tin covers of the 2-ounce boxes used in making the heating tests and allowing the nitrate to evaporate. The residue is noted to be sticky or oily by rubbing between the fingers. Water Soluble Materials. Water soluble materials in tar are determined by weighing about 2 grams into a casserole, adding 50 cc. of distilled water, and boiling for 1 hour. The solution is then filtered into a weighted porcelain evaporating dish, using hot distilled water for a wash and evaporated to dryness on a steam bath. The weight at the evaporating dish and contents after drying to a constant weight at 212 F., less the weight of the dish itself, gives the amount of water soluble materials in the tar, from which the per cent may be calculated. Free Carbon. The free carbon in tar is determined by extrac- tion at room temperature with C 5 2 . In extraction C S 2 is used in the same manner as making the determination for the amount of bitumen soluble in C S2 in asphalts. Determination as to whether extraction is complete is made by placing some of the carbon on white porcelain, moistening it with C S2, and if the porcelain is stained the extraction is not complete, and the carbon requires more washing. Paraffine. Fifty grams of the material are placed in a half -pint retort, E. & A. No. 4521, fitted with a tee condenser. To the 20- inch iron delivery tube of the retort is attached a 10-inch glass tube, and between the cover and the retort is placed a paper gasket cut from heavy wrapping-paper. The material is rapidly distilled to a dry coke from wTiich no further distillate can be obtained, not over 25 minutes being allowed from the time of placing flame under retort until distillation ceases. About 5 grams of the distillate are taken if the materials contain 2 per cent or less of paraffine and about 3 grams if the material contains over 2 per cent of paraffine. This amount of distillate is dissolved in 25 cc. of Squibbs Absolute Ether in a 2-ounce glass flask, after which 25 cc. of Squibbs Absolute Alco- hol are added. A one-to-one wash of 25 cc. each of similar ether and alcohol is made up, and the solution of oil and the wash are then frozen separately for 40 minutes in a salt and ice mixture, giving a temperature of o° F. The precipitate is filtered quickly by means of a suction pump by using a No. 575 C. S. & S. 9-centimeter hardened filter-paper; the paper being placed in a funnel packed in a freezing mixture of salt and ice. The paraffine caught on the filter-paper is washed with the cool one-to-one wash until the paraffine is white. The paraffine is then scraped into a weighted crystallizing dish and maintained at a temperature of 212 F. until a constant weight is obtained, after which it is weighed and the percentage of paraffine in the original material is computed by dividing the weight of the paraffine obtained by the number of grams of distillate taken for freezing, and multiply this result by the percentage distilled from the original sample (i.e., by 100 per cent less weight of coke expressed in percentage). The paraffine so determined to have a melting point of at least 120 F. The melting-point of paraffine is determined by covering the bulb of a thermometer with the paraffine; suspending the thermometer in a beaker of water at 65 F., and heating the water at the rate of 382 SPECIFICATIONS 8° to io° F. per minute. The temperature recorded by the ther- mometer at the time the paraffine melts from the bulb is taken as the melting-point of the paraffine. Distillation of Tar. The distillation test of tar is made by measuring 100 cubic centimeters of the tar into a 250 cc. Engler distilling flask with delivery tube at the middle of the neck. The thermometer is so placed that the mercury bulb is opposite the outlet of the flask. The thermometer used to have a nitrogen chamber to insure accurate reading at high temperatures. The flame is so regulated that approximately 1 cc. of distillate is caught per minute. The distillation is made continuous. The following fractions should be reported: Start of distillation to no° C. no° C. " 170 C. 170 C. " 235 C. 235 C. " 270 C. 270 C. " 300 C. Residue (pitch) Fixed Carbon and Mineral Matter. The fixed carbon is deter- mined by weighing approximately 1 gram of the material into a weighed platinum crucible with a tightly fitting cover. The crucible, with its cover in place, is then placed about 4 inches over a freely burning Bunsen Burner so as to be completely enveloped in the flame and exposed to the full heatof the burner for about 3 minutes or until the top of the crucible cover is burned free from the carbon; the under side of the cover being covered with the carbon. The flame is then withdrawn, the crucible cooled and weighed. The weight after burning, less the weight of the crucible, gives the amount of fixed carbon plus the mineral matter. The fixed carbon is then burned off in the open crucible until a constant weight is obtained; the crucible cooled and weighed. This weight is the crucible plus the mineral matter. The mineral matter subtracted from the com- bined weight of fixed carbon and mineral matter gives the fixed carbon. Items 64 to 74 Inclusive — Bituminous Materials 64.1. Under items 64 to 74 inclusive the Contractor shall furnish and deliver on the work at such points as the Engineer may direct, bituminous material of the kind shown on the proposal sheet as to be furnished under its respective item. 64.2. Bituminous material furnished shall be of approved quality and shall meet the requirements specified below for the kind of ma- terial furnished, and for any contract, the material furnished shall be of one brand and shall show a uniform test unless special permis- sion is given to furnish other brands of material. 64.3. The quantity to be paid for under this item shall be the number of gallons delivered on the work, unless the material is to be incorporated in the work by the same Contractor, and under the same contract. BITUMINOUS MATERIAL A 383 If the material is to be incorporated in the work by the same Con- tractor, and under the same contract, the quantity to be paid for under this item shall be the number of gallons incorporated in the work under directions of the Engineer. Bituminous material that has been rendered unfit for use by over- heating or by long-continued heating, shall not be paid for. For purposes of measurement, a gallon shall be a volume of 231 cubic inches and measurement shall be based on the volume of the bitu- minous material of a temperature of 60 degrees Fahrenheit. The price bid when the material is not to be incorporated in the work under this contract, shall include the furnishing of the material along the road as directed by the Engineer. Any material wasted through careless handling will not be paid for. The price bid shall include the furnishing of the bituminous material alongside the road at places designated by the Engineer. Where the material delivered is to be incorporated into the work under this contract, the cost of manipulating and incorporating this material shall be included in the price bid for the top course of the pavement being constructed. Item 64 — Specification for Bituminous Material A Mixing Method (Type i) This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) The various hydrocarbons composing it shall be present in a homogeneous solution. (3) It shall have a specific gravity at 77 degrees Fahrenheit of not less than 0.97. (4) The penetration shall be between 8 and 12 millimeters when tested for 5 seconds at 77 degrees Fahrenheit with a No. 2 needle, weighted with 100 grams. (5) Fifty grams of it upon being maintained at a uniform tempera- ture of 325 degrees Fahrenheit for 5 hours in a cylindrical vessel 5! centimeters in diameter by 3 J centimeters high shall not lose more than 4 per centum in weight. The penetration (5 seconds, 77 de- grees Fahrenheit, No. 2 needle, 100 grams weight) of this residue shall be at least 50 per centum of the original penetration. (6) Its solubility at air temperature in chemically pure carbon disulphide for the following named materials, or materials similar thereto, shall be at least 99.5 per centum for pure bitumen products, 96 per centum for Bermudez products, 81 per centum for Cuban products and 66 per centum for Trinidad products. (7) The solubility of the bitumen at air temperature, in 76 degrees Beaume paraffine petroleum naphtha distilling between 140 degrees and 190 degrees Fahrenheit, shall be between 68 and 88 per centum. (8) The bitumen shall show between 8 and 17 per centum fixed carbon. (9) It shall show an open flash point not less than 375 degrees Fahrenheit. 384 SPECIFICATIONS (10) It shall not contain more than 4.7 per centum paraffine scale. (11) It shall show a toughness at 32 degrees Fahrenheit not less than 10 centimeters. Toughness is determined by breaking a cylinder of the material if inches in diameter by if inches in height in a Page impact machine. (American Society of Testing Materials, August 15, 1908.) The first drop of the hammer is from a height of 5 centi- meters and each succeeding blow is increased by 5 centimeters. (12) It shall have a ductility at 77 degrees Fahrenheit of not less than 25 centimeters (Dow mould). (13) All bituminous material A. will be sampled by an Engineer of the Department of Highways and samples sent to the Bureau of Tests, Albany, N. Y. Item 65 — Specification for Bituminous Material A Mixing Method (Type No. 2) This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) The various hydrocarbons composing it shall be present in a homogeneous solution. (3) It shall have a specific gravity at 77 degrees Fahrenheit of not less than 0.97. (4) The penetration shall be between 6 and 8 millimeters when tested for 5 seconds at 77 degrees Fahrenheit with a No. 2 needle, weighted with 100 grams. (5) Fifty grams of it being upon maintained at a uniform tempera- ture of 325 degrees Fahrenheit for 5 hours in a cylindrical vessel, 5 1 centimeters in diameter by 3 J centimeters high shall not lose more than 4 per centum in weight. The penetration (5 seconds, 77 de- grees Fahrenheit, No. 2 needle, 100 grams weight) of this residue shall be at least 50 per centum of the original penetration. (6) Its solubility at air temperature in chemically pure carbon disulphide for the following named materials, or materials similar thereto, shall be at least 99.5 per centum for pure bitumen products, 96 per centum for Bermudez products, 81 per centum for Cuban products and 66 per centum for Trinidad products. (7) The solubility of the bitumen at air temperature, in 76 degrees Beaume paraffine petroleum naphtha distilling between 140 degrees and 190 degrees Fahrenheit shall be between 68 and 88 per centum. (8) The bitumen shall show between 8 and 17 per centum fixed carbon. (9) It shall show an open flash point not less than 375 degrees Fahrenheit. (10) It shall not contain more than 4.7 per centum paraffine scale, (n) It shall show a toughness at 32 degrees Fahrenheit not less than 5 centimeters. Toughness is determined by breaking a cylinder of the material if inches in diameter by if inches in height in a Page impact machine. (American Society of Testing Materials, August 15, 1908.) The first drop of the hammer is from a height of 5 centi- meters and each succeeding blow is increased by 5 centimeters. BITUMINOUS MATERIAL A 385 (12) It shall have a ductility at 77 degrees Fahrenheit of not less than 25 centimeters (Dow mould). (13) All bituminous material A. will be sampled by an Engineer of the Department of Highways and samples sent to the Bureau of Tests, Albany, N. Y. Item 66 — Specification for Bituminous Material A Penetration Method This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) The various hydrocarbons composing it shall be present in a homogeneous solution. (3) It shall have a specific gravity at 77 degrees Fahrenheit of not less than 0.97. (4) The penetration shall be between 14 and 19 millimeters when tested for 5 seconds at 77 degrees Fahrenheit with a No. 2 needle, weighted with 100 grams. (5) Fifty grams of it upon being maintained at a uniform tempera- ture of 325 degrees Fahrenheit for 5 hours in a cylindrical vessel 5 J centimeters in diameter by 3! centimeters high shall not lose more than 5 per centum in weight. The penetration (5 seconds, 77 degrees Fahrenheit, No. 2 needle, 100 grams weight) of this residue shall be at least 50 per centum of the original penetration. • (6) Its solubility at air temperature in chemically pure carbon disulphide for the following named materials, or materials similar thereto, shall be at least 99.5 per centum for pure bitumen products, 96 per centum for Bermudez products, 81 per centum for Cuban products and 66 per centum for Trinidad products. (7) The solubility of the bitumen at air temperature, in 76 degrees Beaume parafnne petroleum naphtha distilling between 140 degrees and 190 degrees Fahrenheit shall be between 70 and 88 per centum. (8) The bitumen shall show between 8 and 16 per centum fixed carbon. (9) It shall show an open flash point not less than 375 degrees Fahrenheit. (10) It shall not contain more than 4.7 per centum parafnne scale. (n) It shall show a toughness at 32 degrees Fahrenheit not less than 15 centimeters. Toughness is determined by breaking a cylinder of the material if inches in diameter by if inches in height, in a Page impact machine. (American Society of Testing Materials August 15, 1908.) The first drop of the hammer is from a height of 5 centimeters and each succeeding blow is increased by 5 centimeters. (12) It shall have a ductility at 77 degrees Fahrenheit of not less than 40 centimeters (Dow mould). (13) All bituminous material A. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N.Y. 386 SPECIFICATIONS Item 67 — Specification for Bituminous Material H. O. This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) The various hydrocarbons composing it shall be present in a homogeneous solution. (3) It shall have a specific gravity at 77 degrees Fahrenheit of not less than 0.96. (4) When evaporated in the open air at a temperature not exceed- ing 500 degrees Fahrenheit until the residue remaining has a pene- tration (5 seconds, 77 degrees Fahrenheit, No. 2 needle, 100 grams weight) of 10 millimeters the amount of such residue shall not be less than 85 per centum nor more than 95 per centum of the original oil. At a temperature of 77 degrees Fahrenheit such residue shall have a ductility of at least 25 centimeters (Dow mould). (5) Fifty grams of it upon being maintained at a uniform tempera- ture of 325 degrees Fahrenheit for five hours, in a cylindrical vessel 5I centimeters in diameter by 3! centimeters high, shall not lose more than 10 per centum in weight. (6) It shall be soluble in chemically pure carbon disulphide at air temperature to the extent of at least 99.5 per centum. (7) It shall be soluble at air temperature in j6 degrees Beaume paraffine petroleum naphtha distilling between 140 degrees and 190 degrees Fahrenheit to the extent of not less than 75 per centum and not more than 90 per centum. (8) It shall show between 6 and 14 per centum of fixed carbon. (9) It shall show an open flash point of not less than 325 degrees Fahrenheit. (10) It shall not contain more than 4.7 per centum paraffine scale. (11) It shall show a toughness at 32 degrees Fahrenheit not less than 20 centimeters. Toughness is determined by breaking a cylinder of the material 1 J inches in diameter by if inches in height in a Page impact machine. (American Society of Testing Materials, August 15, 1908.) The first drop of the hammer is from a height of 5 centimeters and each succeeding blow is increased by 5 centimeters. (12) All bituminous material H. O. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. Item 68 — Specification for Bituminous Material C. O. This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) The various hydrocarbons composing it shall be present in a homogeneous solution. (3) It shall have a specific gravity at 77 degrees Fahrenheit of not less than 0.93. (4) When evaporated in the open air at a temperature not exceed- ing 500 degrees Fahrenheit until the residue remaining has a pene- tration (5 seconds, 77 degrees Fahrenheit, No. 2 needle, 100 grams weight) of 10 millimeters the amount of residue shall not be less than 50 per centum nor more than 65 per centum of the original oil. At BITUMINOUS MATERIAL T 387 a temperature of 77 degrees Fahrenheit such residue shall have a ductility of at least 25 centimeters (Dow mould). (5) Fifty grams of it upon being maintained at a uniform tempera- ture of 325 degrees Fahrenheit for five hours, in a cylindrical vessel Si centimeters in diameter by 3 J centimeters high, shall not lose more than 30 per centum in weight. (6) It shall be soluble in chemically pure carbon disulphide at air temperature to the extent of at least 99.5 per centum. (7) It shall be soluble at air temperature in 76 degrees Beaume paraifine petroleum naphtha distilling between 140 and 190 degrees Fahrenheit to the extent of not less than 80 per centum and not more than 95 per centum. (8) It shall not show more than 10 per centum fixed carbon. (9) It shall show an open flash point of not less than 125 degrees Fahrenheit. (10) It shall not contain more than 4.0 per centum paraffine scale. (n) All bituminous material C. O. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. Item 69 — Specification for Bituminous Material T High Carbon — Binder This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) It shall be uniform in character, appearance and viscosity. (3) It shall have a specific gravity of not less than 1.20 at 25 degrees centigrade. (4) It shall contain not more than 25 per centum nor less than 12 per centum of free carbon. (5) When distilled by the method of the American Society for Testing Materials, it shall contain no body that distills at a lower temperature than 170 degrees centigrade; not over 3 per centum shall distill below 235 degrees centigrade; not over 12 per centum shall distill below 270 degrees centigrade, and not over 16 per cen- tum shall distill below 300 degrees centigrade. The specific gravity of the entire distillate shall not be less than 1.03 at 25 degrees centi- grade. The residue from the foregoing distillation shall have a melting point not greater than 75 degrees centigrade ball and ring method. (6) It shall have a melting point of not less than 27 degrees C, and not more than 34 degrees C., by ball and ring method. (7) .All bituminous material T. will be sampled by an Engineer of the Department of Highways and samples sent to the Bureau of Tests, Albany, N. Y. Item 70 — Specification for Bituminous Material T High Carbon — Hot Application This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) It shall be uniform in character, appearance and viscosity. $88 SPECIFICATIONS (3) It shaU have a specific gravity not less than 1.19 at 25 degrees centigrade. (4) It shall contain not more than 22 per centum nor less than 10 per centum of free carbon. (5) When distilled by the method of the American Society for Testing Materials, it shall contain no body that distills at a lower temperature than 170 degrees centigrade, not over 10 per centum shall distill below 235 degrees centigrade; not over 16 per centum shall distill below 270 degrees centigrade and not over 20 per cen- tum shall distill below 300 degrees centigrade. The specific gravity of the entire distillate shall not be less than 1.03 at 25 degrees centi- grade. The residue from the foregoing distillation shall have a melting point not greater than 75 degrees centigrade ball and ring method. (6) It shall have a float test (New York Testing Laboratory method) at 100 degrees centigrade between eighteen and twenty-eight seconds. (7) All bituminous material T. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. Item 71 — Specification for Bituminous Material T High Carbon — Cold Application This bituminous material shall have the following characteristics: (1) It shall have a specific gravity of 1.14 to 1.18 at 25 degrees centigrade. (2) It shall contain not more than 12 per centum nor less than 4 per centum of free carbon. (3) When distilled by the method of the American Society for Testing Materials, not over 5 per centum shall distill below 170 de- grees centigrade; not over 18 per centum shall distill below 235 degrees centigrade; not over 25 per centum shall distill below 270 degrees centigrade, and not over 32 per centum shall distill below 300 degrees centigrade. The specific gravity of the entire distillate shall not be less than 1.01 at 25 degrees centigrade. The residue from the foregoing distillation shall have a melting point not greater than 70 degrees centigrade ball and ring method. (4) The viscosity when tested by the standard Engler viscosim- eter shall not be more than 125 seconds at 60 degrees centigrade for the first 100 cubic centimeters. (5) All bituminous material T. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. Item 72 — Specification for Bituminous Material T Low Carbon — Binder This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) It shall be uniform in character, appearance and viscosity. (3) It shall have a specific gravity not less than 1.16 at 25 degrees centigrade. BITUMINOUS MATERIAL T 389 (4) It shall contain not more than 5 per centum free carbon. (5) When distilled by the method of the American Society for Testing Materials, it shall contain no body that distills at a lower temperature than 170 degrees centigrade; not over 5 per centum shall distill below 235 degrees centigrade; not over 15 per cen- tum shall distill below 270 degrees centigrade; not over 20 per centum shall distill below 300 degrees centigrade. The residue from the foregoing distillation shall have a melting point not greater than 75 degrees centigrade ball and ring method. (6) It shall have a melting point of not less than 27 degrees C, and not more than 34 degrees C, by ball and ring method. (7) All bituminous material T. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. Item 73 — Specification for Bituminous Material T Low Carbon — Hot Application This bituminous material shall have the following characteristics: (1) It shall be free from water. (2) It shall be uniform in character, appearance and viscosity. (3) It shall have a specific gravity of not less than 1.14 at 25 degrees centigrade. (4) It shall contain not more than 4 per centum of free carbon. (5) When distilled by the method of the American Society for Testing Materials, not over 1 per centum shall distill below 170 de- grees centigrade; not over 12 per centum shall distill below 235 degrees centigrade; not over 20 per centum shall distill below 270 degrees centigrade, and not over 25 per centum shall distill below 300 degrees centigrade. The residue from the foregoing distillation shall have a melting point not greater than 75 degrees centigrade ball and ring method. (6) It shall have a float test (New York Testing Laboratory method) at 100 degrees centigrade between fifteen and twenty-five seconds. (7) All bituminous material T. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Test, Albany, N. Y. Item 74 — Specification for Bituminous Material T Low Carbon — Cold Application This bituminous material shall have the following characteristics: (1) It shall have a specific gravity of 1.10 to 1.13 at 25 degrees centigrade. (2) It shall contain not more than 2 per centum of free carbon. (3) When distilled by the method of the American Society for Testing Materials, not over 5 per centum shall distill below 1 70 de- grees centigrade; not over 20 per centum shall distill below 235 degrees centigrade; not over 28 per centum shall distill below 270 degrees centigrade, and not over 35 per centum shall distill below 300 degrees centigrade. The residue from the foregoing distilla- 390 SPECIFICATIONS tion shall have a melting point not greater than 70 degrees centigrade ball and ring method. (4) The viscosity when tested by the standard Engler viscosim- eter shall not be more than 125 seconds at 60 degrees centigrade for the first 100 cubic centimeters. (5) All bituminous material T. will be sampled by an Engineer of the Department of Highways, and samples sent to the Bureau of Tests, Albany, N. Y. BRICK Paving brick shall be reasonably perfect in shape — shall be free from marked warping or distortion, and shall be uniform in size, so as to fit closely together and to make a smooth pavement. All brick shall be homogeneous in texture and free from laminations and seams. All brick shall be evenly burned and thoroughly vitrified. Soft, brittle, cracked, or spalled brick, or brick kiln-marked to a height or depth of over q\ parts of an inch will be rejected. If brick have rounded corners, the radius shall not be greater than X6 part of an inch. Brick must have not less than two nor more than four vertical lugs or projections not more than i inch wide, on one side of each brick, the total area of all lugs being not more than 3 square inches, so that when laid there shall be a separation between the bricks of at least | inch and not more than J inch. The imprint, or name of the brick, or maker, if used, shall be by means of recessed and not by raised letters. The two ends of the brick shall have a semi-circular groove, with a radius of not less than J of an inch and not more than \ of an inch. Grooves shall be so located that when the brick are laid together the grooves shall match perfectly; grooves shall be horizontal when brick is laid in pavement. All brick shall not be less than 3 J* X 3!" X 8§" nor more than 3i" X 4" X9" in size. All brick shall be subject to tests for abrasion and impact, for absorption, according to the standard methods prescribed by the National Brick Manufacturers' Association, as follows : THE RATTLER The machine shall be of good mechanical construction, self-con- tained, and shall conform to the following details of material and dimensions, and shall consist of barrel, frame and driving mechanism as herein described. THE BARREL The barrel of the machine shall be made up of the heads, head- liners and staves. The heads shall be cast with trunnions in one piece. The trunnion bearings shall not be less than two and one-half (2 \) inches in diam- eter or less than six (6) inches in length. BRICK RATTLER 391 V4/L- -j6 ••■U?h- ' i |J fES^§ ilBl S> >* Fig. 74 a 392 . SPECIFICATIONS The heads shall not be less than three-fourths (J) inch thick nor more than seven-eighths (f ) inch. In outline they shall be a regular fourteen-sided (14) polygon inscribed in a circle twenty-eight and three-eighths (28I) inches in diameter. The heads shall be provided with flanges not less than three-fourths (}) inch thick and extending outward two and one-half (2 J) inches from the inside face of head to afford a means of fastening the staves. The flanges shall be slotted on the outer edge, so as to provide for two (2) three-fourths (f) inch bolts at each end of each stave, said slots to be thirteen- sixteenths (i-f) inch wide and two and three-fourths (2!) inches center to center. Under each section of the flanges there shall be a brace three-eighths (f) inch thick and extending down the outside of the head not less than two (2) inches. Each slot shall be provided with recess for bolt head, which shall act to prevent the turning of the same. There shall be for each head a cast-iron headliner one (1) inch in thickness and conforming to the outline of the head, but inscribed in a circle twenty-eight and one-eighth (28!) inches in diameter. This liner or wear plate shall be fastened to the head by seven (7) five-eighths (f) inch cap screws, through the head from the outside. These wear plates, whenever they become worn down one- half (J) inch below their initial surface level, at any point of their surface, must be replaced with new. The metal of which these wear plates are to be composed shall be what is known as hard machinery iron, and must contain not less than one (1) per cent of combined carbon. The faces of the polygon must be smooth and give uniform bearing for the staves. To secure the desired uniform bearing the faces of the head may be ground or machined. THE STAVES The staves shall be made of six (6) inch medium steel structural channels twenty- seven and one-fourth (2 7 J) inches long and weighing fifteen and five-tenths (15.5) pounds per lineal foot. The channels shall be drilled with holes thirteen-sixteenths (y|) inch in diameter, two (2) in each end, for bolts to fasten same to head, the center line of the holes being one (1) inch from either end and one and three-eighths (if) inches either way from the longitudinal center line. The space between the staves will be determined by the accuracy of the heads, but must not exceed five-sixteenths ( T \) inch. The interior or flat side of each channel must be protected by a lining or wear plate three-eighths (f ) inch thick by five and one-half (si) inches wide by nineteen and three-fourths (19D inches long. The wear plate shall consist of medium steel plate, and shall be riveted to the channel by three (3) one-half (J) inch rivets, one of which shall be on the center line both ways and the other two on the longitudinal center line and spaced seven (7) inches from the center each way. The rivet holes shall be countersunk on the face of the wear plate and the rivets shall be driven hot and chipped off flush with the sur- face of the wear plate. These wear plates shall be inspected from time to time, and if found loose shall be at once reriveted, but no THE ABRASIVE CHARGE 393 wear plate shall be replaced by a new one except as the whole set is changed. No set of wear plates shall be used for more than one hundred and fifty (150) tests under any circumstances. The record must show the date when each set of wear plates goes into service and the number of tests made upon each set. The staves when bolted to the heads shall form a barrel twenty (20) inches long, inside measurement, between wear plates. The wear plates of the staves must be so placed as to drop between the wear plates of the heads. These staves shall be bolted tightly to the heads by four (4) three-fourths (f) inch bolts, and each bolt shall be provided with lock nuts, and shall be inspected at not less frequent intervals than every fifth (5th) test and all nuts kept tight. A record shall be made after each such inspection, showing in what condition the bolts were found. THE FRAME AND DRIVING MECHANISM The barrel should be mounted on a cast-iron frame of sufficient strength and rigidity to support same without undue vibration. It should rest on a rigid foundation and be fastened to same by bolts at not less than four (4) points. It should be driven by gearing whose ratio of driver to driven should not be less than one (1) to four (4). The counter shaft upon which the driving pinion is mounted should not be less than one and fifteen- sixteenths (if\) inches in diameter, with bearings not less than six (6) inches in length and belt driven, and the pulley should not be less than eighteen (18) inches in diameter and six and one-half (6|) inches in face. A belt of six (6) inch double-strength leather, properly adjusted, so as to avoid unnecessary slipping, should be used. (As a part of this publication will be found a complete working drawing of a machine which will meet the above specifications and requirements.) THE ABRASIVE CHARGE (a) The abrasive charge shall consist of two sizes of cast-iron spheres. The larger size shall be three and seventy-five-hundredths (3.75) inches in diameter when new and shall weigh when new approx- imately seven and five- tenths (7.5) pounds (3.40 kilos) each. Ten shall be used. These shall be weighed separately after each ten (10) tests, and if the weight of any large shot falls to seven (7) pounds (3.175 kilos) it shall be discarded and a new one substituted; provided, however, that all of the large shot shall not be discarded and substituted by new ones at any single time, and that so far as possible the large shots shall compose a graduated series in various stages of wear. The smaller size spheres shall be when new one and eight hundred seventy-five-thousandths (1.875) inches in diameter and shall weigh not to exceed ninety-five-hundredths (.95) pounds (0.430 kilos) each. Of these spheres so many shall be used as will bring the collective 394 SPECIFICATIONS weight of the large and small spheres most nearly to three hundred (300) pounds, provided that no small sphere shall be retained in use after it has been worn down so that it will pass a circular hole one and seventy-five-hundredths (1.75) inches in diameter, drilled in a cast-iron plate one-fourth (J) inch in thickness or weigh less than seventy-five-hundredths (.75) pounds (or .34 kilos). Further, the small spheres shall be tested by passing them over such an iron plate drilled with such holes, or shall be weighed after every ten (10) tests, and any which pass through or fall below specified weight, shall be replaced by new spheres, and provided, further, that all of the small spheres shall not be rejected and replaced by new ones at any one time, and that so far as possible the small spheres shall compose a graduated series in various stages of wear. At any time that any sphere is found to be broken or defective it shall at once be replaced. (b) The iron composing these spheres shall have a chemical com- position within the following limits: Combined carbon — Not less than 2.50%. Graphitic carbon — Not more than 0.10%. Silicon — Not more than 1 %. Manganese — Not more than 0.50%. Phosphorus — Not more than 0.25 %. Sulphur — Not more than 0.08 %. For each new batch of spheres used the chemical analysis must be furnished by the maker, or be obtained by the user, before intro- duction into the charge, and unless the analysis meets the above specifications, the batch of spheres shall be rejected. THE BRICK CHARGE The number of brick per charge shall be ten (10) for all bricks of the so-called " block size" whose dimensions fall between from eight (8) to nine (9) inches in length, three (3) and three and three-fourths (3D inches in breadth and three and three-fourths ($j) and four and one-fourth (4I) inches in thickness. No block should be selected for test that would be rejected by any other requirements for the specifications. The brick shall be clean and dried for at least three (3) hours in a temperature of one hundred (100) degrees Fahr. before testing. SPEED AND DURATION OF REVOLUTION The rattler shall be rotated at a uniform rate of not less than twenty-nine and one-half (29 J) nor more than thirty and one-half (30 J) revolutions per minute, and eighteen hundred (1,800) revolu- tions shall constitute the standard test. A counting machine shall be attached to the rattler for counting the revolutions. A margin of not to exceed ten (10) revolutions will be allowed for stopping. Only one (1) start and stop per test is regular and acceptable. RECORDS 395 THE RESULTS The loss shall be calculated in percentage of the original weight of the dried brick composing the charge. In weighing the rattled brick any piece weighing less than one (i) pound shall be rejected. RECORDS (a) The operator shall keep an official book, in which the alternate pages are perforated for removal. The record shall be kept in dupli- cate, by use of a carbon paper between the first and second sheets, REPORT OP Standard Rattler Test of Paving Bricks Identification Data Serial No. ( ) Name of the firm furnishing sample Name of the firm manufacturing sample Street or job which sample represents Brands or marks on the brick Quantity furnished Date received Length Breadth Standardization Data Number of charges tested since last inspection Drying treatment Date tested Thickness Weight of Charge (After Standardization', i of Locknuti on Staves Condition of Scale* 10 Large spheres Small spheres Total Number of charges tested since stave linings were renewed Repairs (Note any repairs affecting the condition of the barrel) Running Data Time Readings Revolution Counter Readings Running Notes, Stops, Etc. Hours Minutes Seconds Beginning of test . . • Final Reading .... Weights and Calculations Initial Weight of 10 Bricks . Final Weight of Same . . Loss of Weight Percentage Loss (Note.— The Calculation Must Appear) Number of broken bricks and remarks on same I certify that the foregoing test was made under the specifications of and is a true record. Date Location of Laboratory Fig. 74 B 396 SPECIFICATIONS and when all entries are made and calculations are completed the original record shall be removed and the carbon duplicate preserved in the book. All calculations must be made in the space left for that purpose in the record blank, and the actual figures must appear. The record must bear its serial number and be filled out completely for each test, and all data as to dates of inspection and weighing of shot and replacement of worn-out parts must be carefully entered, so that the records remaining in the book constitute a continuous one. In event of further copies of a record being needed, they may be furnished on separate sheets, but in no case shall the original carbon copy be removed from the record book. (b) The blank form upon which the record of all official brick tests is to be kept and reported is shown in Fig. 74B. Any brick which loses twenty-four (24) per cent or more in the rattler, or increases more than 3^ per cent in weight or less than § of 1 per cent in the absorption test, will be rejected. HILLSIDE BRICK On grades of five (5%) per cent or over the engineer may, if he deems advisable for the traffic, order the contractor to use special form of brick suitable for steep grades. Expansion Joint Paving Pitch. This cushion shall be composed of heavy pitch or asphaltum composition, having a melting point of not less than 120 F. nor more than 140 F., filling the allotted space. PREMOLDED EXPANSION JOINT The expansion joints shall be composed of a high grade asphalt, that shall pass the following tests: Specific gravity at 77 F., .98 to 1.05. Melting point, ball and ring method, 220 to 250 F. Loss on heating for 5 hours at 325 F., not over 1 %. Bitumen soluble in carbon disulphide, at least 98.5 %. Bitumen soluble in carbon tetrachloride, at least 99.8%. Bitumen soluble in 76 Beaume naphtha, 50 to 75 %. Penetration at 32 F. 200 grams, 1 min., at least 12. Penetration at 77 F. 100 grams. 5 sec. between 15 and 35. Penetration at 115 F. 50 grams, 5 sec. not more than 45. BLOCK STONE PAVEMENT (City of Rochester, N. Y., Specifications, 191 i) Paving blocks shall consist of the best quality of Medina sand- stone free from quarry checks or cracks, and shall be quarried from fine-grain live rock, showing a straight and even fracture. The material shall be of uniform quality and texture, free from seams or lines of clay or other substances which, in the opinion of the City Engineer, will be injurious to its use as paving material. SECOND QUALITY BLOCKS 397 Blocks shall measure not less than three (3) nor more than six (6) inches thick, and not less than six (6) nor more than six and one- half (6i) inches deep, and from seven (7) to twelve (12) inches in length. Stones to have parallel sides and ends, and right-angle joints. All roughness in joints of stone to be broken off, so that when set in place they shall have tight joints for a distance of at least two and one-half .(2 J) inches from the top down. The top to have a smooth even surface, with no projection or depression exceed- ing one-quarter (J) inch. When approved by the City Engineer, paving blocks of the fol- lowing dimensions may be used. Three to five inches in width; five inches in depth, with an allowable variation of one-quarter inch, more or less, in said depth, and seven to twelve inches in length. Paving blocks as here referred to shall be understood to mean blocks of Medina sandstone, prepared in the usual manner for dressed block paving by nicking and breaking the stone from larger blocks, as is done at the quarries where such blocks are usually prepared, and not made by re-dressing or selecting from common stone paving material. The stones will be carefully inspected after they are brought on the line of the work, and the blocks which, in quality and dimensions, do not conform strictly to these specifications, will be rejected and must be immediately removed from the line of the work. The con- tractor will be required to furnish such laborers as may be necessary to aid the inspector in the examination and the culling of the blocks. The stones brought upon the ground having been carefully and thoroughly inspected, as provided for herein, and all rejected stones removed from the line of the work, the contractor will then be re- quired to pile such stone as may have been approved, neatly, on the front of the sidewalk, and not within three (3) feet of any fire hydrant, and in such manner as will preserve sufficient passageway, on the line of the sidewalks, and also permit of free access from the roadway to each entrance on the line of the street. SECOND QUALITY BLOCKS (the following not in any specifications) Second quality block, known as pavers, are practically the same material as the first quality block, the only difference being a greater range of size and a less careful top and joint finish. They cost So . 50 per square yard less. These pavers can be furnished under a speci- fication allowing the following range of size and joint width : (City of Cleveland Specifications) "Common paving stones shall consist of the best quality of Medina sandstone, and shall be not less than three (3) nor more than five (5) inches thick, and not less than seven (7) nor more than eight (8) inches deep, and from eight (8) to thirteen (13) inches long. The stones to have parallel sides and ends, with right-angle joints, all 398 SPECIFICATIONS roughness and points of stone to be broken off so that when set in place they shall have tight joints for a distance of at least three inches from the top; the area of the bottom of any stone to be not less than three-quarters (f ) of the area of the top, the top of all stones to have a smooth even surface." CAST-IRON PIPE Cast-iron pipe shall be light weight and may be second quality, but it shall be free from all defects impairing its strength. The iron must be of good quality, uniform in thickness and of full strength, and the pipe shall be coated with coal pitch varnish mixed with lin- seed oil to form a firm, tough coating. The joint shall be formed by calking into the hub a gasket of jute or oakum and then filling with mortar formed of equal parts of Portland Cement and clean sharp sand. MESH REINFORCEMENT Mesh reinforcement shall be placed where called for on the plans or ordered by the engineer. It shall be of medium steel. If expanded metal is used it shall conform to the above require- ments, and the weight per square foot shall be as shown on the stand- ard structure sheet, and any reinforcement shall be of a character that it will distribute the loads evenly. DEFORMED BARS Deformed bars shall be placed where called for on the plans or ordered by the engineer. They shall be of medium steel and shall have a deformed cross-section, that is, the various cross-sections must be of different shape or their centers must not lie in the same axis. CAST IRON Cast iron shall be of full standard pattern for shapes or forms used, according to drawings or detailed specifications. All cast iron shall be of good gray iron, free from blows, sand holes, or other defects, and shall have a tensile strength of not less than 17,000 pounds per square inch of section. WROUGHT IRON Wrought iron shall be tough, fibrous, and uniform in quality and shall be manufactured by approved methods. Steel scrap shall not be used in its manufacture. Finished material shall be clean, smooth, straight, true to shape, of workmanlike finish and free from defects. Test pieces cut from finished material shall show an ultimate tensile strength of not less than 48,000 pounds per square inch, an elastic limit of not less than 25,000 pounds per square inch, and an elongation of not less than 20 per cent in 8 inches. STEEL 399 Wrought-iron test pieces cut from finished material when cold, or when heated to a bright, cherry-red, shall endure bending 180 degrees around a circle whose diameter is equal to twice the thickness of the test piece, without signs of cracking. Test pieces when nicked and broken shall show a fracture not less than 90 per cent fibrous, free from coarse, crystalline spots. Wrought iron when welded shall not show signs of red shortness. STEEL (1) Steel, except as otherwise provided by these specifications, shall be made by the acid or basic open-hearth process and shall be uniform in character; finished material shall be clean, smooth, straight, true to shape, of workmanlike finish, and free from defects. (2) Fractures must show a uniform fine grain of a blue, steel-gray color entirely free from a fiery luster or a blackish cast. (3) No work shall be put upon any steel at or near the blue tem- perature or between the temperature of boiling water and of the ignition of hardwood sawdust. (4) No sharp or unfilleted corners will be allowed in any piece of metal. (5) Annealing. Crimped stiffeners and buckled plates need not be annealed. All other steel that has been bent cold or partially heated and all forgings must be wholly annealed; exception may be made in unimportant cases and then only upon written permission from the Commission. (6) Tests of steel that is to be annealed shall be made after anneal- ing, or strips cut from such steel shall be annealed at the same time, before testing. (7) Tests of Medium Steel. Test pieces cut from finished material shall show an ultimate strength of not less than sixty thou- sand (60,000) pounds per square inch and not more than sixty-eight thousand' (68,000) pounds per square inch, an elastic limit of not less than thirty-five thousand (35,000) pounds per square inch, an elongation of not less than twenty-two (22) per cent in eight (8) inches, and a reduction of area at the fracture of not less than forty (40) per cent. (8) Medium steel shall not contain more than five one-hundredths (5-100) of one per cent of sulphur. (9) Acid steel shall not contain more than eight one-hundredths (8-100) of one per cent, and basic steel shall not contain more than four one-hundredths (4-100) of one per cent of phosphorus. (10) Medium steel shall endure bending cold or after quenching from a red heat in water at 8o° F., 180 around a circle whose di- ameter is equal to the thickness of the test piece, without signs of cracking. (11) Tests for Soft Steel. Test pieces cut from finished material shall show an ultimate strength of not less than fifty thou- sand (50,000) pounds per square inch and not more than fifty-eight thousand (58,000) pounds per square inch, an elastic limit of not less than thirty thousand (30,000) pounds per square inch, an elongation 400 SPECIFICATIONS of not less than twenty-eight per cent in eight inches, and a reduction in area at the fracture of not less than fifty (50) per cent. (12) Soft steel shall not contain more than four one-hundredths (4-100) of one per cent of sulphur. (13) Acid steel shall not contain more than six one-hundredths (6-100) of one per cent, and basic steel shall not contain more than four one-hundredths (4-100) of one per cent of phosphorus. (14) Soft steel shall endure bending flat upon itself without signs of cracking, when cold, or after quenching, from a red heat, in water at eighty (80) degrees F. VITRIFIED PIPE Vitrified pipe shall be double strength salt-glazed vitrified stone- ware sewer pipe of the first quality. The item will include the fur- nishing, delivering, handling, laying, and cementing of joints; also the operations of excavating the trench, bracing, sheeting, or other- wise supporting the sides, grading and preparing the bottom, back- filling and compacting to the original surface, and the removal of all surplus material. POROUS TILE Where called for on the plans, or ordered by the engineer, porous tile shall be laid true to line and grade, and firmly bedded in clean cinders, gravel, or crushed stone. The tile must be whole and free from cracks and other defects, and must be satisfactory to the engineer. TIMBER (Washington State Specifications) Quality of Timber and Plank. All timber and plank in cul- verts, trestlework, bridge abutments, and pile bridges shall be of good quality, of such kinds as the highway commissioner may direct, free from shakes, wanes, black and unsound knots, and all descrip- tions of decay, and shall be measured by the thousand feet, board measure; the price shall be understood to cover the expense of all labor (including all necessary digging and filling at the ends of bridges where grading is done before bridges are put in) and materials, pins, or treenails required in the performance of the work. All timber structures shall be built in conformity with plans to be furnished by the engineer. Piles and Pile-driving. Piles, whether used in foundations, trestlework, or pile bridges, shall be of good, sound quality of such timber as the Highway Commissioner may accept, not less than ten inches in diameter at the smaller end and of such lengths as the engineer may require. They shall be measured by the lineal foot after they are driven and cut off to receive the superstructure, and the price per lineal foot shall be understood to cover the expense of driving, cutting off, removing the bark from the part above the ground, and all other labor and material required in the performance EXCAVATION 401 of the work; but that portion of each pile cut off shall be estimated and paid for by the lineal foot as " piling cut off." Piles shall be driven of such lengths and to such depths as the engineer may re- quire. All piles shall be capped during the driving to prevent brooming. CLEARING AND GRUBBING Clearing. The right-of-way must be cleared to the width of feet on each side of the center line, or as shall be designated by the engineer; all trees, brush, and other vegetable matter within the space designated to be cut down, and the same, together with all other logs, brushwood, and fences already down, shall be burned or removed from the grounds, as the engineer may direct, so as not to injure the adjoining lands or to obstruct the line of the fences along the boundaries of the said right-of-way. When the embank- ments exceed two feet in height it will be required to cut the trees, brush, and stumps close to the ground. Light clearing shall include the removal of all standing trees of a size up to one foot in diameter, together with all other logs, brush, and other vegetable matter already down or lying loose on the ground. Heavy clearing shall include the removal of all standing trees over one foot in diameter, together with all other logs, brush, and other vegetable matter already down or lying loose on the ground. Grubbing. From the space required for the roadbed and neces- sary slopes and side drains, and whatever additional space may be required by the engineer, except where the excavations are three feet or more in depth, or embankments two feet or more in height, all stumps and other wood or vegetable matter embedded in the ground shall be grubbed up, and removed or disposed of as the engineer may direct, and only the area so grubbed shall be estimated. EXCAVATION Under the head of excavation shall be included all excavations required for the formation of the road-bed, the digging of all ditches, cutting new channels for streams, preparing foundations, the altering of all highway or private roads and all excavations in any way con- nected with or incidental to the construction of the road, and the expense of hauling and depositing same in embankments wherever required. Embankments. Under the head of embankments shall be included all embankments for any of the purposes mentioned not formed from excavations taken from the prism of the road or other necessary excavations. All grading shall be done and estimated by the cubic yard, meas- ured in the excavation, except material borrowed for embankment, which shall be measured in embankment, and shall be comprised under heads, viz.: Earth, Hard-pan, Loose Rock, Solid Rock, Shell Rock, and Solid Rock Borrow. 402 SPECIFICATIONS Earth. Earth will include clay, sand, loam, gravel, and all hard material that can, in the opinion of the chief engineer, be reasonably plowed, and all earthy matter or earth containing loose stones or boulders intermixed, and all other material that does not come under the classification of hard-pan, loose rock, solid rock, shell rock, and solid rock borrow. Hard-pan. Hard-pan will include material, not loose or solid rock, that cannot, in the opinion of the chief engineer, be reasonably plowed on account of its own inherent hardness. Loose Rock. Loose rock will include all stone and detached rock, found in separate masses, containing not less than one cubic foot, nor more than one-half cubic yard, and all slate or other rock, soft or loose enough to be removed without blasting, although blasting may occasionally be resorted to. Solid Rock. Solid rock will include all rock in place, and boulders measuring one-half cubic yard and upwards, in removing which it is necessary to resort to drilling and blasting. Shell-Rock Excavation. Shell-rock excavation will include all deposits composed entirely of rock in masses of less than one cubic foot which have broken off from the cliffs above the roadbed, but will only be estimated when in large deposits. Solid Rock Borrow. Solid rock borrow shall consist of solid rock, according to above classification, excavated outside of the regular cross-sections of the cuts for the roadbed, and placed and measured in embankment. EXCAVATION Item 2 — Earth Excavation Item 3 — Rock Excavation 2.1. Under these items the Contractor shall grade the entire length of roadway, ditches and side slopes to the required lines and grades; shall make all excavations for culverts, under-drains, catch basins, leaching basins, and other structures except posts; shall grade connecting public highways as directed and remove spongy material from the sub-grade to the depths required — all as shown on the plans or as directed by the Engineer. This item includes the excavation, filling and rolling necessary to complete the road and all structures connected therewith except as noted above, and includes the removal of all objectionable material for the full width of the improvement except as noted under section 1.1, and the filling to the required grade with acceptable material of all areas originally below the required grade, or excavated below grade under orders of the Engineer. Backfill for structures, old macadam excavated, and sod ordered removed from the site of a new embankment, shall be paid for as Earth Excavation. 2.2. All suitable materials from the excavation shall be used so far as practicable in making embankments, building up low places on the sub-grade or shoulders, and such other places as directed. EXCAVATION 403 2.3. Surplus material shall be placed in embankments, shall be used for extending the shoulders or shall be deposited in spoil banks, as directed by the engineer. All surplus materials shall be removed and disposed of as directed by the Engineer before the sub-grade or shoulder rolling is completed and beofre any stone is placed on the roadway. 2.4. If there is not sufficient suitable material to complete the grading and to bring the sub-grade to the required height, the con- tractor shall borrow additional material from the sides of the roadway or from other borrow pits as directed by the engineer so that the estab- lished grade for the road, embankments, etc. will be secured. All borrow pits outside the highway shall be acquired by the Contractor at his own expense, and any borrow pits in or adjacent to the high- way shall be left in a neat and satisfactory condition and shall be thoroughly drained. 2.5. The contractor shall remove boulders and all muck, quick- sand, soft clay and spongy material which will not consolidate under the roller, from the sub-grade to a depth to be determined by the engineer, and refill the space with acceptable materials from the excavations, or with stone or gravel, as directed. If stone or gravel is used, the same will be paid for at the contract price bid for item " Foundation Course." After all drains have been laid and the surface of the sub-grade has been properly shaped, it shall be thor- oughly rolled and compacted with an approved self-propelled roller weighing not less than 10 tons. Water puddling shall be resorted to in case the soil requires it. Care shall be taken not to roll clay foundations too much, thus developing a plastic condition. All hollows and depressions which develop shall be rilled with acceptable material, and the sub-grade shall again be rolled. This process of filling and rolling shall be repeated until no depressions develop. In places where the character of the material makes the use of such a roller impracticable, a lighter one may be permitted. The sub- grade shall not be muddy, or otherwise unsatisfactory when the foundation course is placed upon it. All culverts, ditches, and drains shall be satisfactorily completed to effectively drain the highway before the placing of any pavement will be permitted. The shoulders shall be rolled and left in a compact and satisfactory condition at the completion of the pavement. 2.6. Embankment shall be formed of suitable materials. If formed of stone, all reasonable precautions must be taken to ensure a solid embankment. The upper surface of the embankment shall be rolled and left in a satisfactory condition and approximately true to lines and grades. Large stone shall not project within 6 inches of the finished sub-grade, and all hollows and depressions shall be filled with the smaller stone from the excavation, with gravel or with other acceptable material. Stone in embankments shall not be used nearer than 6 inches to the surface of shoulders. Where the filling is less than 2 feet in depth all vegetable matter shall be removed from the original surface. Where necessitated by the existing slope, the original surface shall be trenched or other- wise broken up before placing new embankment thereon. 404 SPECIFICATIONS Embankment shall be constructed in successive horizontal layers not exceeding 12 inches in thickness; when concrete is to be placed thereon, these layers shall not exceed 6 inches in thickness. Each layer shall extend across the entire fill and shall be thoroughly rolled and compacted by approved methods. If impracticable to use a heavy roller for this work a grooved roller shall be used. 2.7. At all intersecting public highways the contractor shall grade back to a sufficient distance with acceptable materials, as directed by the engineer so that a smooth riding and satisfactory junction will be produced. 2.8. The quantity of excavation to be paid for under items 2 and 3 shall be the number of cubic yards of material, measured in its original position, excavated and disposed of as directed by the En- gineer, and the limits shall not exceed those shown upon the plans or fixed by the Engineer. The price bid for Earth Excavation shall include the removal of all materials, as specified under section 2.1 — except as provided below for "Rock Excavation," — the placing of same in embank- ment or spoil, the rolling, compacting, grading and all other work incidental thereto. No direct payment shall be made under Items 2 or 3 for work in connection with contractor's plant, nor for his other requirements in carrying out the provisions of this contract, but compensation therefor shall be considered as having been included in the prices stipulated for the various items of the contract. The price bid for Item 3 shall include the removal of all boulders of more than 13 cubic feet and all hard ledge rock and the placing of same in embankment or spoil if not used under other items of the contract, and rolling, compacting, grading and all other work inci- dental thereto. Boulders of less than 13 cubic feet, and all soft or disintegrated rock which can be removed with pick and shovel, shall not be paid for under Rock Excavation, but under "Earth Excava- tion." The price bid for the Items shall include all labor, materials, supplies, and plant and incidentals necessary to complete the work. Item 4 — Overhaul 4.1. If the haul on any material either from cuts or borrow pits made in accordance with directions from the Engineer exceeds 2,000 feet it shall be classified as overhaul. For each 100 feet of haul greater than 2,000 feet the Contractor shall receive the price bid for Overhaul per cubic yard of all material so moved, measured in its original position. The price bid shall include all labor, appliances, and incidentals necessary to complete the work. Item 5 — Vitrified Clay Pipe 5.1. Under this item the Contractor shall furnish and place vitri- fied pipe where directed by the Engineer. 5.2. Pipe shall be first quality, double strength, salt glazed, sound, vitrified, stoneware sewer pipe with bell joints. CONCRETE LEACHING BASINS 405 5.3. All pipe shall be laid true to line and grade with bells up- stream, and shall have a full, firm and even bearing. The joints shall be filled with jute and mortar consisting of one part Portland cement and two parts sand. 5.4. The quantity to be paid for under this item shall be the number of linear feet of pipe incorporated in the work under the directions and to the satisfaction of the Engineer. The price bid shall include the furnishing and laying and all mate- rials and incidentals necessary thereto, except that all excavation in connection therewith will be paid for under item " Excavation. " Item 6 — Vitrified Clay Underdrains 6.1. Under this item the Contractor shall furnish and lay 6-inch salt glazed vitrified pipe wherever required for drainage. 6.2. The pipe shall be laid true to line and grade with the bells up grade. A strap of burlap at least 6 inches wide and long enough to reach around the pipe and lap at least 1 foot shall be wrapped around each joint of pipe to give double thickness on the top and to act as a strainer. The pipe shall be covered as laid with clean gravel or broken stone of No. 2 or No. 3 size placed around and above it to the surface of the sub-grade. 6.3. The amount to be paid for under this item shall be the number of linear feet of pipe furnished and incorporated in the work. The price bid shall include all labor, materials, and incidentals necessary to complete the work, except that the necessary excava- tion will be paid for under item " Excavation," and the necessary broken stone will be paid for under item "Broken Stone, Loose Measurement." Item 7 — Porous Tile Underdrain 7.1. Under this item the Contractor shall furnish and lay 6-inch porous tile wherever required for drainage. 7.2. The tile must be whole and free from cracks and other defects, and must be satisfactory to the Engineer. 7.3. The tile shall be laid true to line and grade and shall be covered as laid with clean gravel or broken stone placed around and above it to the surface of the sub-grade. 7.4. The amount to be paid for under this item shall be the number of linear feet of pipe furnished and incorporated in the work. The price bid shall include all labor, materials, and incidentals necessary to complete the work, except that the necessary excava- tion will be paid for under item "Excavation," and the necessary broken stone will be paid for under item "Broken Stone, Loose Measurement." Item 8 — Concrete Leaching Basins 8.1. Under this item the Contractor shall build at places indicated on the plan or ordered by the Engineer, concrete leaching basins of a type shown on the detail plans. 406 SPECIFICATIONS 8.2. The concrete used in these basins shall be second-class con- crete. 8.3. The grating shall be of cast iron of the quality specified in item "Miscellaneous Iron and Steel." 8.4. For each basin completed, the Contractor shall receive the price bid. The price bid shall include all concrete, stone, grating, and all material, labor and incidentals necessary to complete the work, except that the excavation will be paid for under item "Excavation." Item 9 — Vitrified Leaching Basins 9.1. Under this item the Contractor shall build at places indicated on the plans or ordered by the Engineer, leaching basins of a type shown on the detail plans. 9.2. Vitrified pipe shall be of double thickness, sound, and thor- oughly tamped in place. 9.3. The broken stone used for filling shall be No. 4 broken stone or gravel. 9.4. The grating shall be of cast iron of the quality specified in item "Miscellaneous Iron and Steel." 9.5. For each basin completed in accordance with plans and under orders of the Engineer, the Contractor shall receive the price bid. The price bid shall include all materials, labor and appliances, and all expenses incidental to completing the work, except the excava- tion — which last will be paid for under item "Excavation." Item 10 — Catch Basins 10. 1. Under this item the Contractor shall build catch basins as shown on the plans, as directed by the Engineer. 10.2. The catch basins may be built of second-class concrete or of acceptable brick at the option of the Contractor. If bricks are used they shall be sound, hard burned brick of acceptable quality, and shall be laid by a competent mason and in a workmanlike manner. Mortar of one part Portland cement and two parts sand shall be used. 10.3. For each catch basin complete with cast iron top, as shown on plans and ordered by the Engineer, the Contractor shall receive the price bid. The price bid shall include all labor, materials and incidentals required to complete each basin, except that the excavation will be paid for under item " Excavation." drop inlets Drop inlets shall be constructed where shown upon the plans, or directed by the Engineer. The details of construction shall be such as he may direct. Payment for drop inlets will be made under appropriate items at the contract price for the materials entering into their construction; that is, payment will be made , for the various amounts of excava- RELAYING OLD PIPE 407 tion, concrete, cast iron, cast-iron pipe, etc. Payment under these items shall include all labor and materials necessary to complete the work. Item 11 — Changing Elevation of Manholes and Catch Basins 1 1. 1. Under this item the Contractor shall raise or lower to the grades given all existing covers of catch basins or manholes. 1 1.2. All changes shall be made with acceptable brick laid in Portland cement mortar of one part cement and two parts sand. All work shall be done in a workmanlike manner by competent masons. 1 1.3. For each manhole or catch basin raised or lowered as directed by the Engineer, the Contractor shall receive the unit price bid. The price bid shall include all labor, materials and incidentals necessary to complete the work. If any manhole or catch basin heads or covers are broken through carelessness on the part of the Contractor, they shall be replaced at his expense. Item 12 — Cast-iron Pipe 1 2. 1. Under this item the Contractor shall furnish and place cast-iron pipe as directed for culverts, drains and other necessary uses, and of the sizes and weights ordered. 12.2. Pipe shall be of class A unless otherwise called for by the plans or ordered in writing by the Division Engineer, and may be second^ quality; but it shall be free from all defects impairing its strength or utility. The iron must be of good quality, uniform in thickness and of full strength. The pipe shall be coated with coal- pitch varnish mixed with linseed oil to form a firm, tough coating. Joints shall be formed by caulking into the hubs a gasket of jute or oakum and then rilling with mortar composed of equal parts of Port- land cement and clean, sharp sand. It shall be laid true to line and grade and shall have a full, firm, even bearing. 12.3. The number of tons of cast-iron pipe to be paid for under this item shall be the actual weight in place in the work as directed by the Engineer when of class A, or an equal weight when of heavier class; except that when a heavier weight is used under written order of the Division Engineer, such weight shall be paid for. The price bid shall include the furnishing, delivering, handling, laying, cutting and all work and materials necessary to complete the work. Item 13 — Relaying Old Pipe 1 3. 1. Under this item the Contractor shall as directed carefully remove, preserve and relay old pipe found in existing culverts. 13.2. The old pipe when relaid shall be true to line and grade and have a full, firm, even bearing, and the work shall be in every way the same as if new pipe were being laid. 13.3. Any old pipe in good condition which is damaged in removing, due to the carelessness of the Contractor, shall be replaced with new pipe at the Contractor's expense. 408 SPECIFICATIONS Any old pipe which is, in the Engineer's judgment, unfit for relay- ing may be destroyed before removing. 13.4. The amount to be paid for under this item shall be the number of linear feet incorporated in the work. New pipe furnished to re- place old pipe which is destroyed through the carelessness of the Contractor shall be paid for as if the old pipe had been preserved and relaid. The price bid shall include all labor, materials and incidentals necessary to complete the work, except that the excavation neces- sary will be paid for under the item " Excavation." Item 14 — Stone Filling 14. 1. Under this item the Contractor shall furnish and place acceptable stone of either quarry, field or cobble stone for filling crib work, and similar work as required. 14.2. Stone filling shall be of acceptable quarry, field or cobble stone. The larger stones shall be properly embedded at the bottom of the fill; all stones shall be so placed as to make a fill of maximum stability. 14.3. The quantity to be paid for under this item shall be the number of cubic yards measured in its final position and incorporated in the work as directed by the Engineer. The price stipulated shall include the cost of obtaining the stone, placing, and all materials and expenses incidental thereto. Item 15 — Piles 1 5. 1. Under this item the Contractor shall furnish and drive piles of acceptable material and lengths for foundations, revetment and elsewhere as required. 15.2. Piles shall be furnished to fit the localities. The Contractor shall, when required, drive preliminary test piles, each of which will be paid for at the contract price therefor. After the test piles are driven a statement will be furnished the Contractor by the Engineer, showing for the information of the Contractor the probable number of piles of the different kinds required, grouped between certain lengths in feet. 15.3. Piles, shall be driven by hammer or combination of hammer and water jet methods, and the driving shall be satisfactory to the Engineer in every case. In driving piles the heads shall be protected from injury by a cap or shall be banded if required. The fall of the hammer shall not exceed 20 feet, and shall be regulated so as not to injure or shatter the pile. Driving shall continue until £he penetra- tion and bearing values are satisfactory to the Engineer. 15.4. The tops of all piles shall be sawed level and true to the ele- vation fixed by the Engineer. 15.5. Broken, split or misplaced piles shall be drawn and properly replaced. Piles driven below the grade fixed by the Engineer shall be drawn and replaced by new, and if necessary, longer, piles. No payment will be made for driving or withdrawing piles so injured or misplaced. RIP-RAP 409 15.6. The number of linear feet paid for under this item shall be the total length of piles driven in accordance with plans or orders of the Engineer. The price bid shall include the furnishing and delivering upon the work, the peeling, banding, tenoning, framing, driving, painting and all other labor and incidentals necessary to complete the work. Item 16 — Timber and Lumber 16. 1. Under this item the Contractor shall furnish timber and lumber of various sizes as may be ordered for sills or platforms beneath the road, for culverts, bridges, reinforcing existing struc- tures and for other similar purposes as ordered by the Engineer. 16.2. Timber and lumber shall be of short leaf yellow pine or spruce or other acceptable kind, sound, square-edged, free from shakes, loose knots or decay, and shall be planed, and tongued-and- grooved if required. 16.3. No payments will be made under this item for timber or lumber for forms, moulds, or centers, for sheeting or bracing, scaffolds, fences, guard rails or any part of the contractor's temporary bridges, roads, or plant; but payment for timber and lumber used in the above cases shall be included under the appropriate items covering the same. 16.4. The quantity of timber and lumber to be paid for shall be the number of thousand feet, board measure, actually placed in accordance with orders of the Engineer. If any round timber is used it shall be estimated as square timber of the largest size, omit- ting fractions of an inch, which can be inscribed in the small end of the log. Xo second hand timber shall be used except with the approval of the Engineer. The price bid shall include all bolts, spikes and other fastenings and all other material expenses incidental to fur- nishing, framing and placing the timber and lumber satisfactorily. Item 17 — Rip-Rap 1 7. 1. Under this item the Contractor shall furnish and place rip- rap for slope protection where shown upon the plans or ordered by the Engineer. 17.2. Rip-rap shall consist of field stone or rough, unhewn quarry stones as nearly cubical in form as is practicable, placed upon a slope not steeper than the angle of repose, and so laid that the weight of the large stones is carried by the soil and not by the stones adjacent. Fifty per centum of the mass shall be large stones of two cubic feet or more. The largest stones shall be placed first, roughly arranged and in close contact; the stones shall rest upon a 6-inch bed of stone chips or gravel or other acceptable porous material, where ordered by the Engineer. The spaces between the larger stones shall be filled with spalls of suitable size. 17.3. The quantity of rip-rap to be paid for under this item shall be the number of cubic yards placed in accordance with the plans or as directed by the Engineer. When a porous bed is placed in 410 SPECIFICATIONS accordance with the directions of the Engineer, the quantity of the same shall be included in the quantity of rip-rap and paid for as such. The price bid shall include all labor, materials and incidental expenses necessary to satisfactorily complete the work. CONCRETE MASONRY Item 18 — First-Class Concrete Item 19 — Second-Class Concrete Item 20 — Third-Class Concrete 18. 1. Under Items 18, 19 and 20 the Contractor shall place con- crete of the class indicated on the plans or ordered by the Engineer, for culverts, abutments, wing walls and in other structures as directed by the Engineer. This item shall not include concrete used in curbs, catch-basins, edging, sign posts, guard railing, resetting old curb, concrete pave- ment foundations or " Concrete Pavements" or in other structures for which there is a contract item, unless it is specifically stated under that item that such shall be. the case. All concrete placed in the work, whether included under Items 18, 19, or 20 or under other items, shall conform to the requirements for concrete of the class specified. 18.2. Concrete shall consist of approved Portland cement, a fine aggregate of sand, and a coarse aggregate of broken stone or gravel, mixed in the proportions specified for the various classes given below. Samples of all these ingredients shall be submitted to and approved by the Bureau of Tests, and shall be acceptable to the Engineer before being used in the work. 18.3. Concrete will be classified as follows: First-class concrete shall be made of one part Portland cement, two parts of No. 1 or No. 2 sand (see page 374) and four parts of coarse aggregate. Second-class concrete shall be made of one part Portland cement, two and one-half parts of No. 2 or No. 3 sand, and five parts of coarse aggregate. Third-class concrete shall be made of one part of Portland cement, three parts of No. 3 sand, and six parts of coarse aggregate. Cement, fine and coarse aggregate shall be proportioned by loose volumes. For this purpose one bag of cement shall be considered as T 9 o 5 o of a cubic foot. The fine and coarse aggregate shall be meas- ured separately. PORTLAND CEMENT 18.4. All the cement used in the work shall conform to the require- ments given under "Materials of Construction," pages 372-374; it shall be subject to rigid inspection, shall be sampled by the Engineer at once on delivery, and shall conform to the prescribed tests made at the testing laboratories of the Bureau of Tests. All cement which is rejected because of failure to stand the required tests shall be imme- diately removed at the expense of the contractor. CONCRETE 411 18.5. Cement barrels shall contain 376 pounds of Portland cement. Each bag of Portland cement shall contain 94 pounds net. 18.6. Provision shall be made by the Contractor for storing cement in a dry place. FINE AGGREGATE 18.7. Fine aggregate shall conform in all respects to the require- ments given under " Materials of Construction," page 374. Sand which contains foreign matter shall be satisfactorily washed before using. Screenings shall not be used except when they have been submitted by the Division Engineer to the Bureau of Tests, have been accepted by the Bureau of Tests, and their use has been approved by the First Deputy Commissioner in writing, and then only under the restrictions laid down under "Materials of Construction," page 375. COARSE AGGREGATE 18.8. Coarse aggregate shall conform in all respects to the require- ments given under "Materials of Construction," page 375. Materials which contain foreign matter shall be satisfactorily washed before using. MIXING, DEPOSITING AND FINISHING CONCRETE 18.9. Approved batch mixers shall be used in all cases where required by the Engineer. No continuous mixer shall «be used. Mixing shall continue through at least twelve revolutions of the mixer, and until every face of every, particle of stone of gravel is completely coated with mortar. In all machine mixing the batches of concrete shall be proportioned to the size of the mixer to produce the best results, 18.10. If hand mixing is permitted the following method shall be used: The sand and cement shall be thoroughly mixed dry and made into a thin mortar. After the mortar has been brought to the proper consistency, the broken stone or gravel, having been just previously drenched with water, shall be added, and the whole thoroughly mixed to the satisfaction of the Engineer. The mixing shall be done upon water-tight platforms, in a satisfactory manner; after the materials are wet, the work shall proceed rapidly until the concrete is in place, and is so thoroughly manipulated that water flushes to the surface and all the interstices between the stones are entirely filled with mortar. 1 8.1 1. All mortar and concrete shall be used while fresh and before the initial set has begun. Any mortar or concrete in which the initial set has begun shall be removed from the mixing boards or receptacle and not used in the work. No retempering of mortar or concrete shall be allowed. 18.12. The quantity of water to be used in making concrete shall be determined by the Engineer, but in general a wet mixture shall be used as tending to produce a uniform, dense and impervious concrete. 412 SPECIFICATIONS 18.13. When required by the Engineer, concrete shall be deposited in. layers averaging not more than six inches in thickness before compacting. In joining new concrete to old, or to concrete that has already set, the work already in place shall have its surface cut over thoroughly with picks to remove all laitance, loose and foreign ma- terial; this surface shall then be washed and be scrubbed with wire brooms before the new concrete is placed. In order to bond the successive courses, horizontal keys shall be formed at the top of the upper layer of each day's work and at such other levels as work is interrupted until the concrete has taken its initial set. Rough stone may, at the discretion of the Engineer, be embedded instead of using the keys. Whenever concreting is suspended on any section for more than one hour, all edges which will be exposed in the finished work shall be brought to a level. In any given layer the separate batches shall follow each other so closely that each one shall be placed and compacted before the pre- ceding one has set, so that there will be no line of separation between the batches. After the concrete has begun to set, it shall not be walked upon in less than twelve hours. The operation of compacting the concrete shall be conducted so as to form a compact, dense, impervious artificial stone which shall show a smooth face on exposed surfaces. The weight of rammers, if used, shall be satisfactory to the Engineer. If any monolith, the concrete of which is found porous, has been plastered or is otherwise defective, it shall be removed and replaced in whole or in part, as directed by the Engineer, entirely at the Contractor's expense. 18.14. The Contractor shall construct suitable forms, the cost of which shall be included in the contract price per cubic yard for the concrete, the interior shape and dimensions of which shall be such that the finished concrete shall be of the form and dimensions shown on the plans. Lagging for faces shall not be less than two (2) inches in thickness before being dressed, except where used for curved or special surfaces. Especial attention must be paid to bracing, and where the forms appear to be insufficiently braced, or unsatisfactorily built, either before or during concreting, the Engineer shall order work to be stopped until the defects have been corrected to his sat- isfaction. If desired, small rods to hold the forms may be embedded in the concrete, but in all such cases provision must be made by sleeve nuts or other satisfactory methods for the removal of the two inches nearest the surface. All holes thus left shall be immediately and completely filled with cement mortar and the surface left smooth and even. All forms shall be set and maintained true to the lines desig- nated until the concrete is sufficiently hardened. All forms shall be satisfactory to the Engineer and shall remain in place as long as he deems necessary. The interior surfaces of the forms which come in contact with the surfaces of the concrete which will be exposed in the finished work shall be of lumber dressed on both faces and both edges and having water-tight joints, and shall be so constructed as to leave all such exposed surfaces of the concrete with a smooth even CONCRETE 413 finish. Forms reused shall be maintained at all times in good con- dition as to accuracy of shape, strength, rigidity, watertightness and smoothness of surface. Forms unsatisfactory in any respect shall not be used, and if condemned shall be removed immediately from the work. 18.15. Boulders and fragments of rock may be bedded in a large mass of third-class concrete. Each stone before being bedded or placed shall be thoroughly washed and scrubbed, if necessary, to free it from all dirt. Stones bedded in concrete shall be at least three inches apart at all points, and no stones shall be placed within three inches of any face of the concrete. Stones shall be laid on their largest bed and worked down into the concrete by bars so as to exclude the air from any pockets in the lower surface of the stone. 18.16. The Contractor shall construct weep holes in all retaining walls at such points as are indicated on the plans or designated by the Engineer. Selected stones shall be placed by hand at the inner end of the holes to assist drainage in escaping and to prevent the overflow of earth. Payment for all labor and materials required to construct and protect these weep holes will be included in the contract price for concrete. 18.17. Whenever directed by the Engineer, newly laid masonry shall be protected to prevent freezing, and the protection shall be in all respects satisfactory to him. The Contractor shall be responsible for all damage to concrete by freezing, and any concrete so damaged shall be cut out and re- placed at the Contractor's expense as directed by and to the satis- faction of the Engineer. When the temperature falls below 35 degrees Fahrenheit the fine aggregate, water and stone shall be heated, and the newly laid con- crete shall be covered with canvas or otherwise protected from freezing. No concrete foundation for pavement or concrete pave- ment shall be laid when the temperature falls below 35 degrees Fahrenheit. 18.18. All damage to or disfigurement of concrete of any kind occurring prior to the final acceptance of the work shall be remedied by the Contractor at his own expense and to the satisfaction of the Engineer. 18.19. No piece of stone shall be left within one inch of any face, a broad-tined fork or other implement, if approved, being thrust between the form and the concrete to pry the fragments of stone back from the face. The top surface of concrete shall be formed immediately after the underlying course is completed and before this course takes its initial set. The top surface shall be formed by cutting off the excess with a template and shall then be rubbed smooth and hard with a wooden float by skilled men. As soon as the concrete has sufficiently set and the Engineer shall so direct, the forms shall be removed and all exposed faces immediately finished by being rubbed smooth with a mortar block and water. No plastering of any surface will be allowed, the required finish being obtained by rubbing down the irregularities of the face. All exposed surfaces shall be smooth, 414 SPECIFICATIONS dense, without pits, irregularities, blow holes or bubbles. The surface of all finished and unfinished work shall be kept wet for a period of six days unless otherwise directed by the Engineer. All edges, joints of sections and angles which will be exposed in the finished structure shall be rounded. A radius of one inch shall be used unless otherwise designated on the plans or directed by the Engineer. 18.20. Concrete shall not be laid in water nor exposed to the action of the water before setting, except by written permission of the Engineer, and then in such manner as he may specially direct. 18.21. Where concrete is to rest on any excavated surface other than rock, special care shall be taken not to disturb the bottom of the excavation, and the final removal of material to grade shall not be made until just before the concrete is laid, except in concrete foundations for pavement. The excavation lines and bases of structures shown on the plans shall be considered as only approximate; and they may be ordered in writing by the Engineer, to be placed at any elevation or of any dimensions that will give a satisfactory foundation. Any additional concrete that may be required by the Engineer below or beyond the lines shown on the plans will be paid for at the contract price. No structure shall be commenced without the Engineer's approval. All rock or hardpan foundation surfaces shall be freed from loose pieces, cut to firm surfaces and cleaned to the satisfaction of the Engineer, before laying concrete. All seams shall be cleaned out and filled with concrete or mortar; and payment for such cleaning out and filling shall be made at the contract price for the class of concrete used. 18.22. The quantity to be paid for under Items 18, 19 and 20 shall be the number of cubic yards of the various classes measured in place in the finished structures placed in accordance with the plans or as ordered by the Engineer. No payment will be made for any concrete outside of these limits, nor for any concrete whose placing is rendered necessary owing to lack of proper care. The price bid for Items 18, 19 and 20, respectively, shall include all materials, forms, labor and other incidental expenses necessary to satisfactorily complete the work as specified in the foregoing paragraphs for first-class concrete, second-class concrete and third- class concrete respectively. Item 21 — Stone Masonry 2 1. 1. Under this item the Contractor shall furnish and build all stone masonry in structures or elsewhere, as shown upon the plans or ordered by the Engineer. 21.2. Stone masonry shall be built of clean stone, free from structural defects, laid in full cement mortar beds. Selected stone, roughly squared and pitched to line, shall be used at all angles and ends of walls. 21.3. The stone shall be laid on its natural bed to form substan- tial masonry, presenting a neat and finished appearance. Spalls and pinners shall not be allowed to show on the face of the wall, and STONE CURBING AND HEADERS 415 shall be used only where necessary. The length of stretchers shall not exceed three times their rise; the width of stretchers shall in no case be less than their rise. At least one-fourth of the stone in the face shall be headers, and these shall be evenly distributed; the length of headers shall not be less than the thickness of the wall, where the wall is four feet or less in thickness; where the wall is more than four feet in thickness, the length of the headers shall not be less than two feet and eight inches, and not more than two-thirds of the thickness of the wall; the width of the headers shall not be less than their rise. All stones shall be laid to break joints six inches or more and to thoroughly bond the work. No joint of the face shall be over one inch in width. Backing shall consist of good- sized, well-shaped stone so laid as to break joints. All spaces between the stone shall be filled with spalls set in mortar. The rear faces shall present approximately plain surfaces. 21.4. End walls of culverts and retaining walls shall be capped with concrete or with stone, roughly squared, extending across the entire width of the wall, and on steps of wing walls the coping shall extend under the step next above it at least eight inches. 21.5. On all exposed faces, the joints shall be raked out and cleaned to a depth of two inches and then pointed with Portland cement mortar mixed in a proportion of one to one. 21.6. The quantity of stone masonry to be paid for under this item shall be the number of cubic yards measured in the completed work, and the limits shall not exceed those shown upon the plans or fixed by the Engineer. The price bid shall include all labor, materials and incidental expenses necessary to satisfactorily complete the work. Item 22 — Stone Curbing and Headers 22.1. Under this item the Contractor shall furnish and place stone curbing and headers where shown on the plans or ordered by the Engineer. 22.2. Stone curbing and headers shall be of approved bluestone, sandstone or granite, sound, uniform, free from seams or other imper- fections, and shall be nowhere less than 5 inches thick, 15 inches deep, and 3 feet long. The upper face shall be evenly cut and the front face shall be dressed for the full depth to an even surface with no projections or depressions exceeding one-quarter inch. The bottom shall be roughed off parallel to the top so that there will be no projections exceeding 2 inches beyond the required depth. The ends shall be squared and dressed to form joints 'not exceed- ing one-eighth inch for a depth of at least 2 inches from top and front face. The backs shall be rough dressed for full depth and dressed the same as the face for a depth of 2 inches from the top. The joints of circular curbing shall be cut on radial lines. 22.3. The curb or header shall be set in third-class concrete, as shown on the plans. It shall be true to line and grade and settled so as to have a firm and uniform bearing. 416 SPECIFICATIONS 22.4. If required by the plans, porous drain- tile shall be placed under stone curbing and firmly embedded and covered with cinders, gravel or broken stone. 22.5. After the curb or header has been set the trenches shall be filled with earth and thoroughly tamped. 22.6. The quantity to be paid for under this item shall be the number of linear feet of curbing or headers set in accordance with plans and directions of the Engineer. The price bid for this item shall include the furnishing and setting of the curb or header, all concrete, tile, broken stone or gravel, and all labor, materials and incidental expenses necessary to complete the work. Item 23 — Resetting Old Curbing 23.1. Under this item the Contractor shall remove and reset old curbing, as shown upon the plans or ordered by the Engineer. 23.2. Care shall be taken in removing old curbing so that there shall be no unnecessary breakage, and any curbing damaged in removing, hauling, or storing, due to the carelessness of the Contractor, shall be replaced with new curbing at his own expense. 23.3. All joints and tops shall be redressed, if directed by the Engineer, to obtain a smooth top surface and to obtain joints of the same class as specified for new curbing. 23.4. The quantity to be paid for under this item shall be the number of linear feet removed, stored, hauled, and reset in accordance with the plans and as directed by the Engineer. The price bid shall include all concrete, tile, removing, redressing, hauling, storing, resetting, and all materials, labor and incidental expenses necessary to complete the work. Item 24 — Concrete Curbing 24.1. Under this item the Contractor shall place concrete curbing, of the type shown on the plans, where shown on the plans or ordered by the Engineer. 24.2. All curbing shall be constructed of first-class concrete. The concrete shall be of such consistency, and be so spaded and worked, that a smooth mortar face will be produced. The coarse aggregate for concrete curbing shall be approved No. 2 stone or gravel. 24.3. Curbing shall be moulded in place in sections 6 feet long and provision made at each joint for expansion of one-sixteenth inch. 24.4. All forms shall be set true to line and grade and held rigidly in position. They shall be either of metal or of acceptable planed and matched lumber, and of such construction that a smooth surface will be provided. The forms shall be left in place until the concrete has set suffi- ciently so that they can in the opinion of the Engineer be removed without injury to the curbing. The curbing shall immediately upon the removal of the forms be rubbed down to a smooth and uniform surface, but no plastering will be allowed. For this work a competent and skillful finisher shall be employed. COBBLE GUTTERS 417 24.5. The Contractor shall protect the curbing and keep it in first-class condition until the completion of the contract. Any curb- ing which is damaged at any time previous to the final acceptance of the work shall be removed and replaced with satisfactory curbing at the Contractor's expense. (Also see section 18. 1.) 24.6. The quantity to be paid for under this item shall be the number of linear feet placed in accordance with the plans or directions of the Engineer. The price bid for concrete curbing shall include the furnishing and placing of all concrete, tile, porous filling, forms, and all other ma- terials, labor and incidental expenses necessary to complete the work. Item 25 — Concrete Edging 25.1. Under this item the Contractor shall furnish and mould in place concrete edging of the type shown on the plans and where designated on the plans or ordered by the Engineer. 25.2. The concrete edging shall be composed of second-class con- crete. The top shall be troweled to an even surface and the material shall be rammed and spaded so that a dense concrete and a smooth surface will result. (Also see section 18.1.) 25.3. The forms shall be set and held true to line and grade, and shall not be removed until the concrete has set sufficiently, in the judgment of the Engineer, so that no harm will result therefrom. The edging shall be protected from injury until the completion of the contract. After the removal of the forms, the trenches shall be back-filled with earth and thoroughly tamped. 25.4. The quantity to be paid for under this item shall be the number of linear feet of concrete edging completed as shown on the plans or ordered by the Engineer. The price bid shall include the furnishing and placing of concrete and forms, and all other materials, labor and incidentals necessary to complete the work. Item 26 — Cobble Gutters 26.1. Under this item the Contractor shall furnish and place cobble gutters where shown on the plans or ordered by the Engineer. 26.2. Cobble gutters shall consist of rounded "hardheads," quarry or field stone, and shall be laid on edge. If hardheads are used they shall be 4 inches to 8 inches in diameter. The largest stones shall be selected and set along the edges of the gutter. All stones except where embedded in mortar shall be set in sand, and shall be laid to line and grade with close joints by skilled workmen using regular paving tools. The whole shall then be thoroughly rammed in place and brought to a uniform surface. All joints shall be swept full of sand. On grades exceeding 6 per centum, and elsewhere if called for by the plans or ordered by the Engineer, cobble gutters shall be laid in Portland cement mortar, mixed one to three, as shown upon the plans. 4 i 8 SPECIFICATIONS 26.3. The quantity of cobble gutter to be paid for under this item will be the number of square yards of exposed surface laid in accord- ance with the plans and as directed by the Engineer. The price bid shall include the furnishing and placing of all stones, sand, mortar, and all other materials, labor and incidental expenses necessary to complete the work. Item 27 — Concrete Gutters 27.1. Under this item the Contractor shall furnish and place concrete gutters where shown upon the plans or ordered by the Engineer. 27.2. Concrete gutters shall be of first-class concrete and shall conform to all requirements therefor as elsewhere specified. They shall be of the shape and length shown upon the plans, and shall be placed true to line and grade as directed. (See section 18. 1.) 27.3. The quantity for which the Contractor will be paid shall be the number of square yards of concrete gutters placed in accordance with the plans and ordered by the Engineer. The price bid shall include the furnishing and placing of all con- crete, the preparation of foundation, together with all other labor and incidental expenses necessary to satisfactorily complete the work. Item 28 — Brick Gutters 28.1. Under this item the Contractor shall furnish and place brick gutters where shown upon the plans or ordered by the Engineer. 28.2. Brick gutters shall be constructed of approved brick, shall conform to the dimensions shown upon the plans, and shall be laid true to lines and grades upon a suitable bed of sand. 28. 3. Where brick gutters are to be laid next to a curbing in con- nection with a pavement having a concrete foundation, they shall be constructed in full conformity to the specifications for brick pavement, and shall be paid for as such. 28.4. The quantity for which the Contractor will be paid shall be the number of square yards of brick gutters placed in accordance with the plans and ordered by the Engineer. The price bid shall include the furnishing and placing of all ma- terials and the preparation of bed, together with all other labor and incidental expenses necessary to satisfactorily complete the work. Item 29 — Metal Reinforcement 29.1. Under this item the Contractor shall furnish and place metal bar and metal mesh reinforcing material where shown upon the plans or directed by the Engineer. 29.2. All metal reinforcement shall, when embedded, be free from mill scale, grease, injurious rust, dirt or other foreign substance. 29.3. All metal reinforcement shall be securely held in place so that it will be in the prescribed position after the concrete has been thoroughly compacted. MISCELLANEOUS IRON AND STEEL 419 29.4. Unless otherwise designated upon the plans, all bar rein- forcement shall be of open hearth steel, and shall consist of approved "deformed" bars or rods which shall have an elastic limit of not less than 30,000 nor more than 45,000 pounds per square inch, and an elongation of not less than 20 per centum in a length of 8 inches. Deformed bars shall not contain more than T f q of one per centum of sulphur nor more than r J T of one per centum of phosphorus. In small culverts and other structures of minor importance standard commercial deformed bars acceptable to the Engineer may be used. All deformed bars shall be uniform in quality, and shall endure bending 180 degrees, when cold, around a circle whose diameter is equal to the diameter or thickness of the test piece, without fracture on the outside of the bent portion. Bars shall overlap each other by 30 diameters. 29.5. Unless otherwise designated upon the plans, all metal mesh reinforcement shall be of an approved kind and quality, and of the cross-section shown upon the plans and acceptable to the Engineer, and equal in all respects to the best standard commercial products. Sheets of metal mesh shall overlap each other as directed by the Engineer or as shown upon the plans. 29.6. The quantity of metal reinforcement for which the Contractor will be paid shall be the number of pounds incorporated in the work in accordance with the plans or directions of the Engineer. The bid price shall include all labor, materials, and other expenses necessary to satisfactorily complete the work. Metal reinforcement used in rails and posts shall not be included in this item, but shall be considered as being included in the price bid for appropriate items. Item 30 — Miscellaneous Iron and Steel 30.1. Under this item the Contractor shall furnish and place all cast iron, wrought iron and steel not especially included in other items as shown on the plans and for miscellaneous structures as ordered by the Engineer. This item shall include beams, channels, and other structural shapes, as well as miscellaneous iron castings, wrought iron, etc. 30.2. All structural steel, bolts, etc., shown on the plans may be of stock steel. Stock steel shall be subjected only to surface in- spection and cold bending tests. Test pieces cut from finished materials shall endure bending cold, without signs of cracking, 180 degrees around a circle whose diameter is equal to the thickness of the test piece. Iron castings shall be made of the best tough gray iron of uniform quality and shall be free from defects and uneven shrinkage. No mill cinder iron, white or burnt iron or scrap of any kind shall be used. They shall be clean, out of wind, and true to dimensions. Castings having blow holes plugged or filled with putty or crust shall not be used. Wrought iron shall be tough, fibrous and uniform in quality and shall be manufactured by approved methods. Steel scrap shall 4 20 SPE CIFIC ATIONS not be used in its manufacture. Finished material shall be clean, smooth, true to shape and free from defects. All iron and steel except cast iron shall be given a shop coat of red lead and oil, and after being placed shall be given two coats of approved paint. 30.3. The quantity of iron, wrought iron and steel to be paid for under this item shall be the number of pounds furnished and placed in accordance with the plans or instructions of the Engineer. The price bid shall include the furnishing, placing, painting and all other labor, materials and incidental expenses necessary to satisfactorily complete the work. Item 31 — Wooden Guard Railing 3 1 . 1 . Under this item the Contractor shall furnish and erect wooden guard railing of the type indicated, where shown on the plans or ordered by the Engineer. 31.2. The posts shall be of seasoned white oak, cedar, locust, tamarack, white pine, or chestnut. They shall be at least 6 inches square, or if round they shall be 6 inches in diameter at the smaller end after the bark is removed, and 7 feet long. Round posts shall be shaved to even surfaces free from bark or skin. The lower part of the posts to a point 3 feet from the top shall be dipped while dry in suitable bituminous material heated to a temperature of 300 degrees Fahrenheit, or shall be charred as directed. The posts if dipped shall be thoroughly dry before being set in the ground. 31.3. Rails shall be of seasoned, planed spruce or other satis- factory wood, and be properly secured to the posts, all in a workman- like manner. 31.4. The joints of the rails and posts shall be given one coat of white lead and linseed oil before being put together; the beveled tops of posts shall receive two heavy coats of the same. The entire surface exposed above the ground shall be painted with three coats of white lead and linseed oil. 31.5. The white lead and the linseed oil shall be delivered sep- arately on the road in original containers; before being mixed and used a pint sample of each, covering each lot, shall be forwarded to the Bureau of Tests, and neither ingredient shall be used until accepted by the Commission. The mixing of the ingredients shall be as directed by the Engineer. This specification shall apply to all paint used under this contract. 31.6. The quantity of wooden guard railing to be paid for under this item shall be the number of linear feet completed in place. The price bid shall include the furnishing and erecting of all posts and rail, the excavation, painting, dipping, hardware and all expenses and incidentals necessary to complete the work. Item 32 — Special Guard Railing 32.1. Under this item the Contractor shall furnish and erect, true to line and grade, guard railing of the special design shown upon the plans, at the places indicated by the plans or ordered by the Engineer. GUIDE SIGNS 421 32.2. Except as otherwise provided by the plans, each class of work necessitated under this item shall be governed by the clauses of other items which are specially applicable thereto. 32.3. The quantity of guard railing to be paid for under this item shall be the number of linear feet placed in accordance with the plans and ordered by the Engineer. The price bid shall include all excavation, concrete, metal rein- forcement, hardware, backfilling and all other materials, labor and incidental expenses necessary to satisfactorily complete the work. Item 33 — Pipe Railing 33.1. Under this item the Contractor shall furnish and erect pipe railing of the type indicated where shown upon the plans or ordered by the Engineer. 33.2. Pipe railing shall consist of wrought iron pipe, rails, posts and pipe rail fittings of the sizes shown on the plans. All threaded joints shall be coated with lead and oil before being assembled. All parts shall be painted, after being put in place, with two coats of white lead and linseed oil. 2,^-3' The quantity of pipe railing to be paid for under this item shall be the number of linear feet placed in accordance with the plans and ordered by the Engineer. The price bid shall include the furnishing and erecting of all materials, the painting and all expenses and incidentals necessary to complete the work. Item 34 — Guide Signs 34.1. Under this item the Contractor shall furnish and erect guide signs of the type indicated where shown upon the plans or ordered by the Engineer. 34.2. Permanent guide signs shall be for the purpose of furnishing permanent directions to traffic after the completion of the contract. Permanent guide signs shall be constructed of kiln dried white pine and of the dimensions shown on the plans. They shall first be given four coats of white lead mixed with linseed oil. After the last coat has become thoroughly dried the letters shall be painted with black enamel paint, and when this is thoroughly dried they shall be given one coat of the finest white shellac. 34.3. Temporary guide signs shall be for the purpose of guiding traffic along a detour during construction. Temporary guide signs shall be constructed of kiln dried white pine and of the dimen- sions shown on the plans. They shall first be given three coats of white lead mixed with linseed oil. After the last coat has become thoroughly dried the letters shall be painted with black enamel paint. 34.4. The number of guide signs to be paid for under this item shall be the number of signs placed in accordance with the plans and ordered by the Engineer. All signs become the property of the State upon payment for this item. 422 SPECIFICATIONS The price bid shall include the furnishing of all labor and materials necessary to satisfactorily erect permanent guide signs on sign posts and temporary guide signs including sign posts, each guide sign com- plete in place. Item 35 — Highway Number Signs 35.1. Under this item the Contractor shall paint on the concrete sign posts highway number signs of the type indicated where shown upon the plans or ordered by the Engineer. 35.2. Highway number signs shall be painted on all concrete sign posts with letters which shall first be formed of two coats of flat black mixed in oil and afterward retraced with black enamel. 35.3. The number of highway number signs to be paid for under this item shall be the number placed in accordance with the plans and ordered by the Engineer. The price bid shall include the furnishing of all labor and materials to satisfactorily complete the work. Item 36 — Danger Signs 36.1. Under this item the Contractor shall furnish and erect danger signs where shown upon the plans or ordered by the Engineer. These shall be of the type called for by the plans. 36.2. Danger signs shall be constructed of a material and painted similar to that specified for guide signs and shall be of the dimensions and lettered as shown on the standard plans. These signs shall be placed on the standard concrete sign posts and set at an angle of forty-five degrees to the center line. When the standard sign is used the arrow shall point in the direction of the danger. 36.3. The number of completed danger signs for which the Con- tractor will receive payment will be the number placed in accordance with the plans and ordered by the Engineer. The price bid shall include the furnishing of all labor and materials necessary to complete each danger sign in a satisfactory manner. Item 37 — Concrete Sign Posts 37.1. Under this item the Contractor shall furnish and erect concrete sign posts of the type indicated, where shown upon the plans or ordered by the Engineer. 37.2. Concrete sign posts shall be made of first-class concrete and of the dimensions and materials shown on the standard plans. To these posts shall be securely fastened guide boards and signs. 37.3. The number of completed concrete sign posts to be paid for under this item shall be the number erected in accordance with the plans and ordered by the Engineer. The price bid shall include all concrete, reinforcement, forms, excavation and backfill, and the furnishing of all other labor and materials necessary to complete each concrete sign post in a satis- factory manner. FOUNDATION COURSE 423 LOOSE STONE Item 38 — Screened Gravel — Loose Measure Item 39 — Broken Stone — Loose Measure 38.1. Under these items the Contractor shall furnish and place upon the road, as directed by the Engineer, broken stone and gravel of the sizes designated on the Itemized Proposal. This stone and gravel will be used for general repair work and for miscellaneous work. 38.2. The stone or gravel delivered shall be of approved quality and shall conform to the general requirements for broken stone and gravel, and they shall be of the sizes ordered. 35.3. The quantity to be paid for under Items 38 and 39 respec- tively shall be the quantity of broken stone or gravel furnished and delivered on the work at the places and in the condition specified by the Engineer. When the material is produced by the contractor on the work, it shall be measured in cubic yards; it shall be measured in tons of 2,000 lbs. when the material is imported and the weight is obtainable from reliable sources such as certified quarry or railroad figures. The price bid shall include furnishing and delivering the stone or gravel as directed by the Engineer and all labor, appliances and ex- penses incidental thereto; also the spreading, rolling or incorporating of the stone or gravel in the work, when required by the Engineer. FOUNDATION COURSE Item 40 — Foundation Course — "Run of Bank" Gravel Item 41 — Foundation Course — Field of Quarry Stone 40.1. Under these items the Contractor shall furnish and place a foundation course of stone or gravel of the depth and in the places called for by the plans, or as ordered by the Engineer in accordance with section 2.5, " Preparation of Subgrade" of item " Excavation." 40.2. No stone or gravel shall be placed on the road until the culverts are completed and proper drainage provided. 40.3. When field or quarry stone is used for constructing the foundation course it shall be of a hard, sound and durable quality, acceptable to the Engineer; the stones shall be placed by hand so as to bring them in as close contact as possible. When quarry stones are used they shall be placed on edge. The depth of the stone shall in no case be greater than the depth specified for the course, the width shall not be greater than the depth, nor more than six inches; and the length shall not be greater than one and one-half times the depth, nor more than 12 inches. The distribution of the stone shall be of a uniformity satisfactory to the Engineer. The long dimension shall always be placed crosswise the road. After laying, this course shall be thoroughly rolled with an approved roller weighing not less than ten tons, and shall then be filled with stone or gravel as directed and again rolled until the stones are bound together and 424 SPECIFICATIONS thoroughly compacted; but no gravel shall be used for rilling except under written permission of the Engineer. All holes or depresssions found in rolling shall be filled with material of the same quality and the surface shall be re-rolled until it conforms to the lines and grades shown on the plans. When field stone is used approved tailings may be used for filling. In all cases a sufficient amount of fine material shall be used to fill all voids. In limited areas where the use of a roller is impracticable heavy tampers may be used to consolidate the material. 40.4. Wherever gravel is used for the foundation course it shall conform in all particulars to the gravel specified in section 2 of Bottom Course "Run of Bank" Gravel. 40.5. The quantity to be paid for under this item shall be the number of cubic yards of compacted material in place. The amount to be estimated shall be computed by multiplying the finished cross- section of the foundation course as shown upon the plans or ordered by the Engineer, by the length of the foundation course measured along the axis of the pavement. The price bid shall include the furnishing, placing, filling, rolling of the material and all labor and incidental expenses necessary to complete the work. Item 42 — Foundation Course — Telford Base 42.1. Under this item the Contractor shall furnish and place a foundation course of field or quarry stone laid on edge, in accordance with the plans or as ordered by the Engineer. 42.2. No stone shall be placed on the road until the culverts are completed and proper drainage has been provided. 42.3. Field or quarry stone of approximate rectangular shape shall be used. The stone shall be not less than one and one-half inches thick, in depth equal to the depth of the course, and in length not more than one and one-half times the depth. 42.4. The pieces shall be placed on edge by hand in as close con- tact as possible with long dimension crosswise of the road. After being placed, all pieces projecting more than one inch above the established plane of the surface shall be broken off flush so as to obtain a true and uniform surface. This course shall then be rolled with an approved self-propelled roller weighing not less than ten tons, and shall then be filled with approved screenings and again rolled until the course is thoroughly compacted/ Material other than screen- ings for filling this course shall not be used except under the written order of the Engineer. 42.5. The quantity to be paid for under this item shall be the number of cubic yards of compacted material in the completed course. The amount to be estimated shall be computed by multiply- ing the finished cross-section of the foundation course as shown upon the plans or ordered by the Engineer, by the length of the founda- tion course measured along the axis of the pavement. The price bid shall include the furnishing, placing, filling, rolling of the material and all labor and incidental expenses necessary to complete the work. BOTTOM COURSE 425 BOTTOM COURSE Item 43 — " Run of Bank" Gravel 43.1. Under this item the Contractor shall furnish and place approved "Run of Bank" gravel either upon the properly prepared subgrade or upon the foundation course. The work shall be per- formed in full conformity to the specifications given under sections 44.2 to 44.9 inclusive, so far as same are not inconsistent with the use of such gravel. 43.2. All gravel shall be of hard, durable stone satisfactory to the Engineer. The particles shall be of such size as will pass through a 3J-inch circular hole, and shall be well graded. Gravel shall be of such nature that the material passing a J-inch screen shall not be more than five per x centum in excess of the voids in the remaining material after its separation therefrom. Before using "Run of Bank" gravel in the work the same shall be tested to determine its suitability. Should at any time during the work and for any reason the gravel fail to maintain suitable proportions of the coarse and fine particles, the Contractor shall by the addition of selected material and satisfactory manipulation produce a material meeting the above requirements. 43.3. The depth of loose stone or gravel in all cases, whether in foundation, bottom or top courses, shall be gauged by the use of cubical blocks of suitable size. (See page 272.) 43.4. The spreading of any layer or course of broken stone, gravel or filler, whether in foundation, bottom or top courses, shall be done from suitable spreader wagons or from piles dumped along the road as directed by the Engineer. No segregation of large or fine particles will be allowed, but the stone as spread shall be well graded with no pockets of fine material. 43.5. The quantity to be paid for under this item shall be the number of cubic yards of compacted material in place in the completed course.. The amount to be estimated shall be computed by multi- plying the finished cross-section of the bottom course as shown on the plans or ordered by the Engineer, by the length of the bottom course measured along the axis of the pavement. The price bid shall include the furnishing, placing, rolling and filling the material, and all other labor, materials and incidental expenses necessary to satisfactorily complete the work. Item 44 — Bottom Course — Screened Gravel Item 45 — Bottom Course — Broken Stone 44.1. Under these items the Contractor shall furnish and place stone or gravel, conforming to the general requirements for same, either upon the properly prepared subgrade or upon the foundation course. This stone or gravel shall be of sizes specified below. 44.2. After the subgrade or foundation course shall have been properly prepared and proper drainage provided, a course of broken 1 Not feasible, see page 6g. 426 SPECIFICATIONS stone or gravel of graded No. 3 or No. 4 or a uniform mixture of same shall be spread evenly so that it will have after rolling the required thickness. If specifically allowed by the Engineer a lim- ited amount of No. 2 stone may be used in the bottom course. In cases where the finished thickness of the bottom course is to be more than 5 inches, the broken stone or gravel for it shall be spread, rolled and filled in two separate layers neither of which shall be of a greater depth than 6 inches measured loose. 44.3. The depth of loose sotne or gravel in all cases, whether in foundation, bottom or top courses, shall be gauged by the use of cubical blocks of suitable size. (See page 272.) 44.4. The spreading of any layer or course of broken stone, gravel or filler, whether in foundation, bottom or top courses, shall be done from suitable spreader wagons or from piles dumped along the road as directed by the Engineer. No segregation of large or fine particles will be allowed, but the stone spread shall be well graded with no pockets of fine material. 44.5. After the bottom course of stone or gravel has been laid loose it shall be thoroughly rolled with an approved roller weighing not less than ten tons. This rolling must begin at the sides and continue toward the center and shall continue until there is no disturbance of the stone ahead of the roller. After the stone is thoroughly compacted No 1. stone or gravel, and screenings or sand, or a mixture of these, shall be uni- formly spread upon the surface and swept in with rattan or steel brooms and rolled dry. After the completion of the rolling no team- ing other than that necessary for bringing material for the next course shall be allowed over the rolled material. It is the intention to bind this course with the small stone, but not to use so much that a good bond will not be secured between the bottom and top courses. 44.6. When two courses of bottom stone are laid each course shall be treated by rolling and adding fine material as described above. 44.7. If the subgrade material shall become churned up into or mixed with the bottom or sub-bottom courses through the Contractor's hauling over it or working on it when the subgrade is in a wet con- dition, the Contractor shall at his own expense remove such mixture of subgrade material and broken stone and replace it with clean broken stone of the proper size, and shall roll or otherwise compact the material so as to produce a uniform, firm and even bottom course. If the above condition occurs through no fault of the Contractor, the Contractor shall be paid both for excavating and replacing under the items " Excavation" and " Bottom Course" respectively. 44.8. All filler for top and bottom courses shall be delivered and piled alongside the road before the course in which it is to be used is placed. 44.9. The quantity to be paid for under these items respectively shall be the number of cubic yards of compacted material in place in the completed course. The amount to be estimated shall be computed by multiplying the finished cross-section of bottom course as shown upon the plans or ordered by the Engineer, by the length of the bottom course measured along the axis of the pavement. CONCRETE FOUNDATION FOR PAVEMENT 427 The price bid for the respective items shall include the furnishing, placing, filling, rolling of the material and all labor and incidental expenses necessary to complete the work. Item 46 — Concrete Foundation for pavement 46.1. Under this item the Contractor shall furnish and place upon a properly prepared subgrade, concrete foundation for pavement of the thickness shown upon the plans or ordered by the Engineer. 46.2. Concrete foundation shall not be placed on any subgrade until the subgrade has been properly drained, thoroughly rolled and compacted, and is true to line and grade in horizontal and transverse cross-section. 46.3. Concrete shall consist of a mixture of Portland cement, No. 2 or No. 3 sand and broken stone or gravel. The coarse aggregate shall consist of a well mixed product of No. 2 and No. 3 stone or No. 2 and No. 3 gravel. The fine aggregate shall consist of No. 2 or No. 3 sand. All of these materials shall conform in all respects to the requirements given under "Materials of Construction," pages 372 to 377. All specifications relating to second-class concrete shall apply to work done under this item, in so far as they are not inconsistent with the special specifications given below. 46.4. The concrete shall be mixed in the proportions of one volume of cement to two and one half volumes of sand and five volumes of broken stone or gravel. The relative proportions of fine and coarse aggregate may be varied slightly, as a result of tests for voids by the Engineer, to the end that the resulting concrete shall be as dense as possible. The concrete shall in all cases approximate a 1: 2 J: 5 mix. 46.5. The concrete shall be mixed in approved mechanical batch mixers. Mixing shall be continued through at least 12 revolutions and until every particle is coated with mortar and until the batch is of uniform color and consistency. After the materials are once wetted the work shall proceed rapidly until the concrete is in place. The quantity of water used shall be as directed by the Engineer and suitable measuring tanks shall be provided by the Contractor so that the same amount of water may be used in the separate batches. 46.6. Before any concrete is placed, the subgrade shall be sprinkled sufficiently to dampen it, but a muddy condition shall not be allowed. As soon as possible after mixing, the concrete shall be deposited in place and thoroughly spaded and rammed so as to bring the mortar flush to the surface. Especial care shall be taken to keep the con- crete uniform and to prevent pockets of stone or mortar. 46.7. The surface, when completed, shall conform to the lines and grades shown upon the plans, and shall be free from depressions or irregularities. No stone shall project above the general surface. All ramming and shaping shall be done before the concrete has taken its initial set. 46.8. When the work is stopped for any reason a vertical joint shall be put in and the work completed up to this joint. 428 SPECIFICATIONS 46.9. No concrete foundation for pavement shall be laid when the temperature falls below 35 F. 46.10. As soon as the concrete has taken its initial set the surface shall be covered with a one-inch layer of suitable material and this shall be kept moist for a period of at least seven days. For covering concrete foundations on which a sand cushion is called for, the sand cushion may be used for the cover coat if the contractor so elects; in case this is done the sand cushion shall be put in acceptable con- dition before preparing for laying the blocks; any portions which have become excessively dirty shall be removed and replaced with acceptable material to the satisfaction of the Engineer. In those cases where material other than sand cushion is used as a cover coat it shall be cleaned off after a period of ten days. 46.11. The quantity to be paid for under this item shall be the number of cubic yards of concrete foundation for pavement incor- porated in the work in accordance with the plans or as directed by the Engineer. The price bid shall include the furnishing and placing of all ma- terials; all mixing, tamping, finishing, and all labor, appliances and incidental expenses necessary to complete the work. The amount to be estimated shall be computed by multiplying the cross-section of concrete foundation as shown upon the plans or ordered by the Engineer by the total length of concrete foundation measured along the axis of the pavement. TOP COURSE Item 47 — Top Course — Water Bound Macadam — Gravel 47.1 . Under this item the Contractor shall furnish and place upon the bottom course, gravel of an approved character to form the top course. 47.2. The top course shall consist of approved gravel of the char- acter hereinbefore specified and of the thickness shown on the plans, together with the binder necessary to properly fill and bind the course. For this purpose gravel of No. 3 size with, when approved by the Engineer, a certain amount of No. 2 size, may be used. Run of bank gravel shall not be used except by written permission of the Division Engineer; this permit must be given in advance, shall specify the locality from which it is to be taken, and contain a proviso that if the material should at any time become unsatisfactory its use shall at once cease and proper material be furnished without additional recompense even if it has to be imported. A copy of any such permit must be filed with the State Highway Commission, and on this permit must be the written and signed acceptance of all the conditions by the Contractor. 47.3. The gravel shall be spread evenly upon the bottom course, using cubical blocks for gauging, to such a depth as to insure the required thickness after it shall have been thoroughly rolled and compacted with an approved roller weighing 10 to 12 tons. Care shall be exercised to prevent any depressions or surface irregularities after rolling the gravel and binder. WATER BOUND MACADAM 429 47.4. When the gravel consists of screened material the binder shall consist of a mixture of the sand screened out and the No. 1 size, with enough clay added when necessary to make a percentage of 12 to 17, but not to exceed 17 per cent. The binder shall be added as directed by the Engineer and thoroughly swept into interstices thereof until they are filled. After sprinkling the surface it shall be thoroughly rolled. The adding of binder where necessary and the sweeping, sprinkling and rolling shall continue until the course is compacted. The pavement shall then be opened to traffic and shall be in a first-class and satisfactory condition at the time of its acceptance. 47.5. When the gravel consists of run of the bank the binder shall be the fine particles contained in the material in its natural state except that when so ordered in writing by the division Engineer a small percentage of clay or loam may be added, when necessary to properly bind the course. The particles shall be of such size as will pass through a 3^-inch circular hole, and shall be well graded. Gravel shall be of such nature that the material passing a i-inch screen shall not be more than five per centum in excess of the voids in the remaining material after its separation therefrom. Should at any time during the work and for any reason the gravel fail to maintain suitable proportions of the coarse and fine particles, the Contractor shall by the addition of selected material and satisfactory manipulation produce a top course meeting the above requirements. Care shall be taken to keep the large stone away from the surface. After sprinkling the surface it shall be thoroughly rolled. Addi- tional material forbinder shall be added where necessary and the sprinkling and rolling shall continue until the course is compacted. The pavement shall then be opened to traffic and shall be in a first- class and satisfactory condition at the time of its acceptance. 47.6. The quantity to be paid for under this item shall be the number of cubic yards of compacted material in place in the completed course. The amount will be computed by multiplying the finished cross-section of the top course as shown on the plans or ordered by the Engineer, by the length of the top course measured along the axis of the pavement. The price bid under this item shall include the furnishing, placing, roiling, filling, and puddling of the material, and all labor, material and incidental expenses necessary to complete the work. Gravel or screenings remaining loose on the surface after the work is completed shall not be estimated as a part of the depth of the top course, but payment therefor shall be included in the price bid for this item. Item 48 — Top Course — Water Bound Macadam — Broken Stone 48.1. Under this item the Contractor shall furnish and place upon the bottom course broken stone to form the top course. 48.2. The top course shall, except as noted below, consist of No. 3 broken stone as shown on the plans and of the thickness shown 43° SPECIFICATIONS thereon, together with the binder necessary to properly fill and bind the course. The binder shall consist of screenings and No. i stone mixed. 48.3. The No. 3 stone shall be spread evenly upon the bottom course, using cubical blocks for gauging, to such a depth as to insure the required thickness after it shall have been thoroughly rolled and compacted. Care shall be used in the reading of the stone that no irregularities in the contour shall develop in the rolling; every such irregularity that does occur the Contractor shall remove before adding the smaller material. The rolling shall be done with a 10 to 12 ton self-propelled roller of approved pattern, and shall be con- tinued until the layer of stone does not creep or wave ahead of the roller. After the stone has been compacted to the satisfaction of the Engineer, a light coating of binder shall be spread on dry by shoveling from piles previously placed alongside the pavement, and imme- diately swept in and thoroughly rolled. Care must be taken through- out to add the binder only in light coatings and to thoroughly sweep each coating in order that the maximum amount of binder may be worked in to fill the voids. The spreading and sweeping and rolling shall be continued until no more binder will go in dry, after which the macadam shall be sprinkled until saturated, the sprinkler being followed by the roller. If the subgrade should become wet to such an extent that the pavement becomes unstable and waves under the roller, the roller shall be taken off and this portion left to dry out before puddling is resumed. More screenings shall be added where necessary, and the sweeping, sprinkling, and rolling shall continue until a grout has been formed that shall fill all the voids and be pushed into a wave by the wheels of the roller. After the wave of grout has been produced over the whole section of the macadam this portion shall be left to dry out, after which it shall be opened to traffic. The macadam shall be repuddled and back-rolled on succeeding days as much as may be necessary to secure satisfactory results. The macadam shall then be covered with a wearing carpet of screenings at least three-eighths of an inch thick; this wearing carpet shall be maintained and renewed until the whole road is accepted. During all the working hours when the roller is not needed for rohing the fills, subgrade, shoulders, and unfinished courses of the pavement, it shall be employed in back- rolling the earlier portions of the macadam. 48.4. The quantity to be paid for under this time shall be the number of cubic yards of compacted material in place in the completed course. The amount to be estimated shall be computed by multi- plying the finished cross-section of the top course as shown on the plans or ordered by the Engineer, by the length of the top course measured along the axis of the pavement. The price bid shall include the furnishing, placing, rolling^ filling and puddling of the material, and all labor, material and incidental expenses, necessary to complete the work. No. 1 stone or gravel, chips or screenings remaining loose on the surface after the work is completed shall not be estimated as a part CLEANING OLD PAVEMENT 431 of the depth of the top course, but payment therefor shall be included in the price bid for this item. Item 49 — Cleaning Old Pavement 49.1. The purpose of the work called for under this item is to prepare an old macadam or old concrete surface for the application of a new top course or a wearing carpet. 49.2. Under this item the Contractor shall clean the old macadam or concrete surface by the use of seal hand brooms or by the use of mechanical sweepers of approved type, as directed by the Engineer, so as to completely uncover but not dislodge the embedded stones of the pavement. All mud, dust, and other dirt so swept off shall then be removed and deposited in such places and in such manner as the Engineer may direct. 49.3. Ruts and depressions of a greater depth than one inch below the general surface of the pavement shall be completely swept out by hand brooms until all loose material has been removed and the embedded stones are fully uncovered. This operation of cleaning out the ruts and depressions and filling them with thoroughly compacted stone and binder to the general level of the surface, shall precede the general operation of cleaning the macadam surface. 49.4. The amount to be paid for under this item shall be the actual number of square yards of old macadam or concrete, including ruts and depressions, cleaned in accordance with the above sections and to the satisfaction of the Engineer. The price bid shall include all labor, tools, appliances, the removal of material cleaned from the surface, and all other expenses incidental thereto. Item 50 — Scarifying and Reshaping Old Macadam 50.1. The purpose of the work under this item is to prepare old macadam pavement for the application of a top course. 50.2. Under this item the Contractor shall thoroughly scarify the old macadam by hand picking or by means of a mechanical scarifier of approved type. Unless specifically authorized by the Engineer, the use of a roller with spiked wheels will not be permitted. The loosened stones shall then be forked or raked over as directed by the Engineer, after which the macadam shall be compacted by rolling with a self-propelled roller weighing not less than 10 tons until an even and firm surface is produced. If necessary in order to satisfactorily compact the stones, the macadam shall be sprinkled during the process of rolling. 50.3. The quantity to be paid for under this item shall be the actual number of square yards, scarified, reshaped, rolled and com- pacted to the satisfaction of the Engineer, and the price stipulated shall include all labor, appliances and expenses incidental thereto. 43 2 SPECIFICATIONS Item 51 — Surface Treatments with Bituminous Material 5 1. 1. Under this item the Contractor shall apply bituminous mate- rial and shall apply broken stone or gravel of specified sizes as a wearing carpet to a new or old pavement of macadam, concrete, or any other substance or type, as shown on the plans or ordered by the Engineer. 51.2. If the pavement to be treated is a newly built macadam or concrete, after it shall have become thoroughly dried and hardened, it shall be swept clean of all dust, dirt or other loose material; if ordered by the Engineer, the sweeping of the macadam shall be con- tinued until the voids are exposed in the surface to a satisfactory depth, not exceeding one-half inch. The price bid, under this item, shall include the aforesaid cleaning of the pavements. If the pavement to be treated is an old macadam or old concrete, the cleaning shall be paid for under Item " Cleaning Old Pavement." 51.3. After the pavement shall have been cleaned to the satis- faction of the Engineer, and when dry, the bituminous material shall be uniformly sprayed over the surface by means of an approved pressure distributor. The bituminous material for hot application shall be heated to a temperature between 250 degrees and 350 degrees F. as required, and when tar is used, it shall be heated to a tempera- ture between 200 degrees and 250 degrees F. as required. The amount of bituminous material to be used in any one applica- tion shall not be less than one-sixth nor more than one-half gallon per square yard, the precise quantity depending upon the character of the pavement, the materials and the local conditions. The Con- tractor shall, therefore, be subject entirely to the direction of the Engineer in this respect. 51.4. The bituminous material applied as above specified shall then be immediately covered, while soft, with a uniform layer of approved broken stone of No. 2. or No. 1 size, after which the stone shall be rolled with a self-propelled roller of approved weight. If ordered by the Engineer another application of bituminous material shall then be made to be followed by an application of approved No. 2 stone or approved No. 1 stone, and again rolled to the satis- faction of the Engineer. The quantity of No. 2 stone and of No. 1 stone to be used shall be sufficient to completely cover the bituminous- material and shall be spread in two or more thin applications, the roller being used after each spreading. The total amount of stone to be used after each application of the bituminous material being that which will become imbedded under the pressure of the roller. The final appli- cation of the stone shall be of No. 1 size. 51.5. Gravel, which has been tested and approved for use, may be substituted for broken stone if screened to produce particles cor- responding with No. 2 and No. 1 sizes. 51.6. No bituminous material for surface treatment shall be placed between October 15th and May 15th, except by written permission of the Engineer, nor when the air temperature on the work is below 50 degrees F., nor when the pavement is damp or in an otherwise unsatisfactory condition. BITUMINOUS MACADAM 433 51.7. Under this item the Contractor shall be paid for the number of gallons of bituminous material furnished in and incorporated in the work in accordance with these specifications and the orders of the Engineer. Bituminous material, that has been wasted or that has been rendered unfit for use by over-heating or by long-continued heating, shall not be paid for. For purposes of measurement, a gallon shall be a volume of 231 cubic inches and measurement shall be based on the volume of the bituminous material at a temperature of 60 degrees F. The price bid shall include the furnishing, hauling, heating and applying the bituminous material, and shall also include the spread- ing, rolling and incorporation of the stone into the wearing carpet. This item shall not include the furnishing of the No. 1 and No. 2 stone or gravel, nor the delivery of same along the side of the road; these will be paid for under Items Screened Gravel or Broken Stone Loose Measure, respectively. Item 52 — Top Course Bituminous Macadam — Penetration Method 52.1. Under this item the Contractor shall furnish and lay a broken stone top course composed of fragments of the specified sizes, and incorporate therewith bituminous material introduced from the surface by means of an approved pressure distributor. 1 52.2. After the bottom course shall have been completed to the satisfaction of the Engineer, a course of approved No. 3 broken stone shall be evenly spread thereon in such quantity that after the application of the bituminous material and broken stone of smaller sizes, hereafter specified, the final compacted thickness of the top course shall be as called for on the plans or ordered by the Engineer. The No. 3 stone shall then be smoothed out by passing over it a few times a self-propelled roller weighing approximately 10 tons, after which bituminous material of the kind specified in the proposal, heated to a temperature between 250 degrees and 350 degrees Fah- renheit if asphalt is used, and between 200 degrees and 250 degrees if tar is used, shall be evenly spreading over the surface. The quan- tity of bituminous material to be used in the first application shall be the amount ordered by the Engineer, which will approximate if gallons per square yard for a top course 3 inches thick, with a pro- portional reduction in the quantity for thinner courses. The surface shall then be immediately covered with a layer of approved No. 2 broken stone, after which it shall be compacted with a self-propelled roller weighing approximately 10 tons; during the rolling process, additional No. 2 broken stone shall be applied and broomed about until the voids in the No. 3 stone are entirely filled. 2 The rolling shall be continued until the course of stone is thoroughly 1 Hand spreading from pots or hods is more satisfactory for the first coat but not for the flush coat. 2 Too much rolling is injurious while the oil is hot; better results are obtained by- waiting till the next day to compact; the course should be rolled early in the morning for 10 days and gradually hardened down. 434 SPECIFICATIONS ■ compacted and its surface is true and even to the established grade and conforms in all respects to the requirements specified for finish- ing and testing the surface of "Top Course Bituminous Macadam, Mixing Method — Type i." 52.3. After this portion of the work shall have been completed to the satisfaction of the Engineer, all loose stone shall be swept from the surface and a sealing coat of one-half gallon of bituminous material per square yard shall be applied by means of an approved pressure distributor. After this it shall be immediately covered with approved No. 1 broken stone, spread and broomed about by experienced workmen, and again rolled; the rolling shall be continued and additional No. 1 stone shall be applied until a smooth, uniform surface is produced. 52.4. Before being opened to traffic a layer of No. 1 broken stone approximately one-half inch thick shall be spread loose on the surface for wearing course. 52.5. The quantity to be paid for under this item shall be the number of cubic yards of compacted material in place in the com- pleted course. The amount to be estimated shall be computed by multiplying the finished cross-section of the top course, as shown upon the plans or ordered by the Engineer, by the length of the top course measured along the axis of the pavement. The price bid shall include the heating and placing of the bituminous material, the furnishing, placing, rolling and filling of the broken stone, and all labor, materials, and incidental expenses necessary to complete the top course. This item shall not include the furnishing and delivery of the bi- tuminous material; such furnishing and delivering will be paid for under the item covering such material. No. 1 broken stone or gravel, chips or screenings remaining loose on the surface after the work is completed shall not be estimated as a part of the depth of the top course, but payment for these shall be included in the price bid for item 52. Item 53 — Top Course — Bituminous Macadam — Mixing Method. Type 1 53.1. Under this item the Contractor shall construct a top course of broken stone mixed with a bituminous material, upon a previously constructed bottom course of concrete, broken stone, quarry or field stone, or gravel. The surface laid shall be in conformity with the lines and grades shown upon the plans or ordered by the Engineer. A smooth finished surface will be insisted on free from irregularities and waviness. The entire top course shall consist of a wearing course finished over with a flush or sealing coat. 53.2. The broken stone used in this course shall be of approved material. When the top course is to be 2 inches or less in thickness in the completed work, the stone shall be of No. 2 size. When the top course is to be over 2 inches in thickness in the completed work, No. 2 and No. 3 stone shall be used, proportioned as directed by the Engineer. The sealing coat therefor shall be of stone of No. 1 size. BITUMINOUS MACADAM 435 53.3. Gravel of approved quality and corresponding to the same sizes as broken stone may be used in the top course. If used, it shall conform to the general requirements for broken stone and gravel for water-bound and bituminous macadam work. 53.4. The broken stone for the wearing course shall be heated, before entering the mixer, to between 225 degrees Fahrenheit and 300 degrees Fahrenheit in revolving dryers of an approved type. The stone shall be continuously agitated during the heating. 53.5. The bituminous material to be used in this course shall conform with the specifications for "Bituminous Material A for Mixing Method — Type I." 53.6. The bituminous material shall be heated in kettles so designed as to admit of even heating of the entire mass, with an efficient and positive control of the heat at all times. Bituminous material "A" shall be heated as directed to a temperature between 275 degrees Fahrenheit and 350 degrees Fahrenheit; all material heated beyond 350 degrees Fahrenheit, either before or during mixing with the broken stone, shall be rejected. Bituminous material "T" shall be heated as directed to a temperature between 200 degrees Fahrenheit and 275 degrees Fahrenheit; all bituminous material "T" heated beyond 275 degrees Fahrenheit either before or during mixing with the broken stone, shall be rejected. No tar shall be heated in kettles containing any asphalt cement, and no asphalt cement shall be heated in kettles containing any tar; before changing from one type of material to the other, kettles shall be scrupulously cleaned in order to avoid mixtures of the two; any such Mixtures shall be rejected. 53.7. When thoroughly heated to the proper temperatures, the bituminous material and the broken stone for the wearing course shall be mixed as directed from time to time, using approximately 18 gallons of bituminous material per cubic yard of loose stone. (The amount of bitumen in the completed work shall be from 5 to 7 J per centum of the total weight of the completed course.) The • Contractor shall provide approved means for accurately propor- tioning the mixture. Excess of bituminous material shall be pre- vented; and any such excess shall be cause for rejection of the course unless satisfactorily corrected previous to laying. The mixer used shall be of satisfactory design, having revolving blades and efficient means for keeping the temperature at the desired point without burn- ing the liquid. The mixing shall continue until every particle of the broken stone is thoroughly coated with the bituminous material and a uniform mixture has been obtained, which when discharged shall have a temperature between 200 degrees and 300 degrees Fahrenheit for bituminous material "A" and between 200 degrees and 250 degrees Fahrenheit for bituminous material "T." The use of batteries of small batch mixers will not be allowed. 53.8. The bituminous mixture, heated and prepared as specified above, shall be delivered from the mixer to the point of deposition in the pavement, if at a considerable distance, in suitable trucks or wagons, provided with canvas covers for retaining the heat. To prevent undue compacting on long hauls, partitions may be required 436 SPECIFICATIONS for large truck loads. As delivered, the mixture shall have a tempera- ture of at least 150 degrees Fahrenheit. Material having a lower temperature than this shall not be laid in the pavement. The mix- ture shall be immediately spread over the foundation course by men experienced in such work, so that when rolled it shall have the required thickness and shall be free from surface depressions and irregularities. The paving shall be done as continuously as practicable, to reduce to a minimum the number of joints between hot and cold materials. Such joints shall be constructed in an approved manner. The hot paving mixture shall not be dumped in large masses on the bottom course. It shall be dumped upon platforms and shoveled with hot shovels into position in the pavement. 53.9. The wearing course, placed as above specified, shall be rolled at once, while the material is still warm and pliable, beginning at the edges and working toward the center. Acceptable means shall be provided to prevent the asphalt from sticking to the roller. Rolling shall continue without interruption until all roller marks disappear and the surface shows no further compressibility. Places which the roller cannot effectively reach shall be compressed with hot iron tamps. 53.10. As soon as possible after the compacting of the wearing course, when the surface is clean and dry, a sealing coat of hot bitu- minous material "A" shall be evenly spread over the wearing course by means of approved pressure distributors. The bituminous material "A" shall be applied at a temperature not less than 275 degrees Fahrenheit nor more than 350 degrees Fahrenheit, at a rate of \ to f gallon per square yard, as directed. A thin and uniform layer of dry, clean No. 1 stone shall be immediately spread over the bituminous material "A" by machines or skilled workmen, sufficient to more than take up all' the excess bituminous material "A." The spreading of the No. 1 stone shall not lag more than 20 feet behind the placing of the asphalt coating. The pavement shall then be again thoroughly rolled. The surface of the wearing course shall be kept scrupulously clean until the sealing coat is applied. The Con- tractor shall not permit any hauling over the surface before the com- pletion of the sealing coat. 53.11. Before placing the sealing coat, the pavement shall be tested with a ten foot straight edge laid parallel with the center line of the pavement, and any depressions exceeding one-half inch shall be satis- factorily eliminated or the pavement relaid. 53.12. Rollers used for the bituminous wearing course and the sealing coat shall be well balanced, self-propelled rollers of satis- factory design, weighing between eight and ten tons. They shall give a compression under the rear roller of between 200 and 350 pounds per linear inch of roll, and shall be provided with an ash pan which shall prevent ashes from dropping upon the pavement. 53.13. No top course bituminous material shall be mixed or placed between October 15 and May 15 except by written permission of the Engineer, nor when the air temperature in the shade is below 50 degrees Fahrenheit, nor when the foundation is damp or otherwise unsatisfactory. BITUMINOUS MACADAM 437 53.14. The Contractor shall provide a sufficient number of accu- rate, efficient thermometers for determining the temperatures of the bituminous material and the broken stone at all stages of the work. 53.15. The quantity to be paid for under this item shall be the number of square yards of compacted material in place in the com- pleted course. The amount to be estimated shall be computed by multiplying the finished width of the top course as shown upon the plans or ordered by the Engineer, by the length of the top course measured along the axis of the pavement. The price bid shall include the furnishing (bituminous material excepted), the heating, placing, rolling and compacting of all ma- terials, together with all other labor and incidental expenses neces- sary to satisfactorily complete the work. The furnishing of the bituminous material will be paid for under the appropriate item therefor as shown on the proposal sheet. Item 54 — Top Course — Bituminous Macadam — Mixing Method. Type 2 54.1. Under this item the Contractor will be required to construct a top course consisting of a compacted mixture of broken stone, sand and bituminous material "A" laid to conform to the required grade and cross-section, as shown on the plans and ordered by the Engineer. 54.2. Broken stone for this course shall be of the character specified all of which shall pass a one-half-inch screen and shall be so graded that when combined in a bituminous mixture containing not less than thirty (30) per centum of the sand specified in section 54.3 it shall produce a bituminous mixture coming within the limits speci- fied in section 54.8. 54.3. The sand shall be clean, hard grained and sharp. It shall all pass a ten (10) mesh screen, and shall contain at least fifteen (15) per centum of material retained on a forty (40) mesh screen and at least twenty (20) per centum of material that will pass an eighty (80) mesh screen except as hereinafter provided for. If the sand does not contain the required amount of fine material, approved stone dust may be added to make up the deficiency. 54.4. The bituminous material to be used in this course shall conform with the specifications for bituminous material "A" for Type 2. 54.5. The broken stone and sand shall be heated as directed, before entering the mixer, to between 225 degrees Fahrenheit and 325 degrees Fahrenheit in revolving dryers of an approved type. The broken stone and sand shall be continuously agitated during the heating. 54.6. The bituminous material shall be heated in kettles so designed as to produce an even heating of the entire mass, with an efficient and positive control of the heat at all times. It shall be heated as directed to a temperature between 275 degrees Fahrenheit and 350 degrees Fahrenheit. If heated beyond 350 degrees Fahrenheit either before or during the mixing with the broken stone it shall be rejected. 438 SPECIFICATIONS 54.7. The Contractor shall provide a sufficient number of accurate, efficient, stationary thermometers for determining the temperature of the asphalt cement in the kettles. 54.8. When thoroughly heated to the temperature directed, the bituminous material and the broken stone and sand shall be mixed in the following proportions by weight : Bitumen from 7 to n % Mineral aggregate, passing 200 mesh from 7 to n % Mineral aggregate, passing 40 mesh from 45 to 55 % Mineral aggregate, passing 10 mesh from 16 to 25 % Mineral aggregate, passing 4 mesh from 8 to 15 % Mineral aggregate, passing 2 mesh less than 10% the sieves being used in the order named. A mixer shall be used, having revolving blades, and so designed and operated as to produce and discharge a thoroughly coated and uniform mixture of non- segregated broken stone, sand and bituminous material. When discharged the mixture shall have a temperature not more than 325 degrees Fahrenheit and not less than 225 degrees Fahrenheit as directed. 54.9. All defective areas in the cement concrete foundation shall be repaired as directed at least ten (10) days in advance of lay- ing the bituminous concrete. Before laying the bituminous concrete the surface of the foundation shall be dry and thoroughly cleaned. 54.10. The mixture heated and prepared as specified in section 54.8, shall be delivered direct from the mixer to the point of deposi- tion on the foundation in trucks or wagons, provided with canvas covers for retaining the heat. As delivered, the bituminous concrete shall have a temperature of at least 200 degrees Fahrenheit: material having a lower temperature than this shall not be laid upon the foundation. Before the mixture is placed, all contact surfaces of curbs, edgings, manholes, etc., shall be well painted with hot asphalt cement. The hot mixture shall be dumped upon platforms, con- structed as directed, and shoveled with hot shovels into position on the foundation. It shall be immediately spread as directed over the foundation course by men experienced in such work, so that when rolled it shall have at no place less than the required thickness and shall be free from surface depressions and irregularities. Joints between hot and cold materials shall be constructed as di- rected. The paving shall be done as continuously as practicable, to reduce to a minimum the number of such joints. 54.11. Rollers used on the bituminous concrete shall be well balanced, self-propelled, tandem rollers, weighing between seven (7) and eight (8) tons each. Each shall have a compression under the rear roller of between two hundred (200) and three hundred (300) pounds per linear inch of roll, and shall be provided with an ash pan which shall prevent ashes from dropping into the bituminous con- crete or sealing coat. 54.12. The surface of the top course shall be tested with a ten (10) foot straight edge laid parallel with the center line of the road BITULITHIC PAVEMENT 439 upon any portion of the surface, and any depression or other irregu- larity exceeding one-half (i) inch \_\" is a better limit] shall be satis- factorily eliminated as directed. 54.13. After the pavement has been satisfactorily finished and has thoroughly dried out, Portland cement shall be dusted over the sur- face in a quantity sufficient to form a complete film over all parts of the pavement. This film shall remain undisturbed by rain or other- wise until it has set; in case of disturbance before setting, it shall be renewed. 54.14. No top course material shall be mixed or placed between October 15th and May 15th, except by written permission of the Engineer, nor when the air temperature in the shade is below 50 degrees Fahrenheit nor when the foundation is damp or otherwise unsatisfactory. 54.15. The quantity to be paid for under this- item shall be the number of square yards of compacted material in place in the com- pleted pavement. The amount to be estimated shall be computed by multiplying the width of top course as shown on the plans or ordered by the Engineer, by the length of the top course measured along the axis of the road. The price bid for this item shall include the furnishing of the sand; the furnishing, crushing and screening of the broken stone; the heat- ing, mixing, placing and rolling of the broken stone, sand and bitu- minous material, and the cement film and all work and expense in- cidental to the completion of the work except the furnishing of the bituminous material, which shall be paid for under the item Bitu- minous Material "A" for Mixing Method, Type 2. Item 55 — Bitulithic Pavement 55.1. Under this item the Contractor shall furnish the necessary stone, bituminous material, machinery, labor and other equipment, and shall construct upon a properly prepared foundation a bitulithic pavement composed of an accurately proportioned aggregate of carefully graded broken stone properly heated and mixed with sep- arately heated Warren's Puritan Brand bitulithic cement, placed and rolled and covered with Warren's quick drying bituminous flush coat composition, followed by two coats of hot stone chips thoroughly rolled into the surface. 55.2. The several grades and sizes of mineral aggregate shall be accurately measured in proportions previously determined by labora- tory tests to give the best results; that is, the most dense mixture of mineral aggregate and one having inherent stability; heated in a rotary mechanical heater so designed as to keep each batch by itself until heated, or after heating the stone in a rotary mechanical heater to a temperature of about 250 degrees Fahrenheit, it shall be ele- vated and passed through a rotary screen, having sections with various sized openings. The difference in the width of openings in successive sections shall not exceed one-fourth (\) inch in sections having open- ings smaller than one-half (|) inch, and shall not exceed one-half (i) inch, in sections having openings greater than one-half (J) inch. 440 SPECIFICATIONS The several sizes of stone thus separated by the screen sections shall pass into a bin containing sections or compartments corresponding to screen sections. From these compartments the stone shall be drawn into a weighing-box, resting on a scale having seven beams. The stone from these compartments shall be accurately weighed, using the proportions which have been previously determined by laboratory tests to give the best results; that is, the most dense mixture of mineral aggregate, and one having inherent stability. If the crushed stone in the wearing course does not provide the best proportions of fine-grained particles, such deficiency must be sup- plied by the use of not to exceed 25 per centum hydraulic cement, pulverized stone, or very fine sand. 55.3. The mineral aggregate, composed of differing sizes accurately measured or weighed as above, shall pass into a "twin pug," or other approved form of- mixer. In this mixer shall be added a sufficient quantity of Warren's Purilan Brand, bituminous water-proof cement, or bitulithic cement, to thoroughly coat all the particles of stone and to fill all voids in the mixture. The bituminous cement shall, before mixing with the stone, be heated to between 200 degrees and 250 degrees Fahrenheit, and the amount used in each batch shall be accurately weighed and used in such proportions as have been pre- viously determined by laboratory tests to give the best results — that is, to coat all particles of stone and fill the voids in the mineral aggregate. The mixing shall be continued until the combination is a uniform bituminous concrete. In this condition it shall be hauled to the street, and there spread on the prepared foundation to such a depth that, after thorough compression with a steam road roller, it shall have a thickness of two (2) inches. The proportioning of the varying sizes of stone and bituminous cement shall be such, that the compressed mixture shall, as closely as practicable, have the density of solid stone. 55.4. After rolling the wearing surface, there shall be spread over it, while it is still warm, a thin coating of Warren's Quick Drying Bituminous Flush Coat Composition, by means of a suitable flush coat spreading machine, so designed as to spread quickly over the surface a uniform thickness of flush coat composition. This spread- ing machine shall be provided with a flexible spreading band and an adjustable device for regulating, to any desired amount, the quantity and uniformity of flush coat composition to be spread. There shall be spread over the flush coat composition, in at least two coats, fine particles of hot crushed stone, in sufficient quantity to completely cover the surface of the pavement. These stone chips shall be spread by means of a suitable stone spreading machine, so designed as to provide a storage receptacle of at least five (5) cubic feet capacity and to rapidly and uniformly cover the surface of the pavement with the desired quantity of stone. This spreading machine shall be provided with an adjustable attachment for regu- lating uniformly the quantity of stone spread at each operation. The hot stone chips shall be immediately and thoroughly rolled into the surface until it has become cool. The purposes of the flush coat composition and the fine particles of hot crushed stone are to not AMIESITE PAVEMENT 441 only fill any unevenness in the surface, but also to make the surface waterproof and gritty, thus providing a good foothold for horses. On grades a mineral flush coat may be used in place of the liquid flush coat. 55.5. Warren Bros., owners of the patents used in the construc- tion of Bitulithic pavement, have filed with the State Commission of Highways a properly executed binding agreement to furnish any contractor to bid for the work all the necessary surface material mixed and ready for use, and bituminous flush coating cement necessary for coating the wearing surface, delivered on wagons of the Contractor at the mixing plant (which will be located within three miles of the point of use) at a stipulated price per square yard for each contract. Such price for Bitulithic pavement mixture and flush coating composition will include a license to use all the patents required in the construction of the pavement as herein specified. The filing of a bid under these specifications will be construed as an acceptance of the terms of the license agreement filed by the Warren Bros. Company, at the price fixed in said agreement, which is on file with the secretary of the Commission. 55.6. The quantity of pavement to be paid for under this item shall be the number of square yards of Bitulithic pavement placed in accordance with the plans, or as directed by the Engineer. The bid price shall include the furnishing and placing of all materials, the mixing, spreading, rolling and all labor and incidental expenses necessary to complete the work. GENERAL SPECIFICATIONS FOR AMIESITE PAVEMENT Foundation. The excavation, filling or embankment, drainage and rolling of the sub-base shall be in full accordance with the stand- ard specifications for street or highway paving before placing the foundation, the depth and nature of which is governed by existing conditions of the sub-grade. The foundation, whether in re-surfacing or new work, shall be, before applying the Amiesite, even and compact and swept clean of all loose dirt and foreign material. New stone, if put on, shall be thoroughly bonded with screenings, sprinkled and rolled hard and uniformly. The foundation must be uniform and be brought up to a true and even grade, parallel to and inches below the ele- vation of the finished surface of the street or road. Application. The bottom course shall be spread in a uniform layer, using blocks or strips to insure an even distribution, then rolled. (Size) Stone used in Amiesite Bottom Course-Graded — |" to ii". If any depressions appear, due to foundation not being firm or any other reason, they must be filled with Amiesite and rolled until surface is even and to grade desired. After the preliminary rolling, the Amiesite top course, made of stone graded from \" to \" in size, shall be applied, not less than one inch (1") deep, loose measurement, and raked to an even depth so as to cover the underlying Amiesite and fill the voids. In no case 442 SPECIFICATIONS shall the bottom course be spread over 300' in advance of the top course, nor shall over 50' be left uncovered during the night. The compressed depth of finished Amiesite surface shall be: Pref- erably not less than 2\ inches. Rolling. After the top course has been evenly spread to a true grade, the surface shall be rolled with a standard ten ton road roller until the material is thoroughly compacted and ceases to creep in front of the roller. In rolling the roller must start from the side- lines of the street or road and work towards the center. Care must be taken that the shoulders are firm and solid, as otherwise the surface will iron out to a feather edge and crack. No rolling shall be done unless the Amiesite is free from water. Surface Finish. After rolling as called for above, clean, sharp sand or stone dust (Lines tone where obtainable) shall be spread in a thin layer and the road may then be immediately thrown open to traffic. General. No Amiesite shall be spread when the road-bed con- tains depressions holding water. The Amiesite must at all times be kept clean. Dirt or other foreign material must not be allowed to mix with, under, or on the Amiesite while being unloaded from cars, spread and rolled. Should the bottom course become coated or partly coated with dust or dirt before the top course can be applied, the part thus coated must be swept and then given a light application of bituminous cement, that can be applied in a thin coating from a sprinkling pot so constructed that a thin and uniform application can easily be applied. The Amiesite may be steamed to facilitate its unloading from the cars. Steam pressure shall not exceed fifteen pounds to the square inch. The Amiesite should not be steamed more than fifteen minutes in any one place. This shall be done under the supervision of the inspector in charge. The Amiesite shall be unloaded from wagons upon iron sheets or boards, so as to insure the material being kept clean and being spread uniformly. Cross rolling shall be done, when ordered, to equalize the bond and prevent waves in the surface. Care must be taken that the bottom course is not rolled down hard before the top course is applied. The finished surface of the Amiesite after rolling shall be kept \" higher than any permanent elevation, depending upon the traffic to ultimately compress or pound it down to grade. Grading slopes or shoulders shall not be carried on after the Amie- site course is started until completion of roadway. The street or road shall be closed to traffic when the Amiesite surface is being applied. Item 56 — Hassam Compressed Concrete Pavement 56.1. Under this item the Contractor shall furnish all materials for and place upon a properly prepared sub-grade or sub-bottom course Hassam Compressed Concrete Pavement of the thickness shown upon the plans or ordered by the Engineer. 56.2. Hassam Compressed Concrete Pavement will be placed on the AMIESITE PAVEMENT 443 sub-grade or on the sub-bottom course and shall not be placed until these are in first-class condition as required for macadam pavement. 56.3. Hassam Compressed Concrete Pavement shall consist of a graded No. 3 and No. 4 stone, of an approved quality, spread evenly and gauged by the use of cubical blocks; after rolling and thoroughly compacting with a 10-ton roller, it shall have the required depth and shall conform to the established lines, grades and cross-sections. Where any depressions or irregularities develop in rolling the surface shall be forked over and material added or taken away to the end that a smooth surface shall be provided after re-rolling. 56.4. After the rolling has been satisfactorily completed and the surface of the broken stone has been brought to the required uni- form surface, and before there is any displacement of the stone, the voids shall be filled with a grout consisting of one part Portland cement and two parts of approved sand. The sand shall be of such sizes that it will not separate readily from the cement, when placing the grout, and any batch of grout, when being placed, shall at all times be of a uniform product and of such consistency that it will flow readily but shall not be so wet as to cause a separation of the cement and sand. The rolling shall be continued during the process of grouting and until all the voids are filled. 56.5. The grout shall be mixed in a Hassam Grout Mixer or other mechanical mixer which will properly mix the ingredients and from which they can be deposited without a separation of the cement and sand. 56.6. Immediately after the voids shall have been filled with grout, a thin layer of No. 1 broken stone or fine aggregate shall be spread over the entire surface and rolled until the grout flushes to the surface. 56.7. After placing the surface stone the surface shall not be worked upon or disturbed for a period of ten days, during which time the surface shall be kept thoroughly wet. 56.8. Any cracks either longitudinal or transverse which develop before the acceptance of the work shall be thoroughly cleaned out and filled with acceptable bituminous material. 56.9. The quantity to be paid for, under this item, shall be the number of cubic yards of Hassam Compressed Concrete Pavement incorporated in the work in accordance with the plans or as directed by the Engineer. The price bid shall include the furnishing and placing of all materials, all grouting, rolling, forms and all labor, appliances, royalties and incidental expenses necessary to complete the work. The amount to be estimated shall be computed by multiplying the cross-section of concrete pavement as shown upon the plans or ordered by the Engineer, by the total length of pavement measured along the axis of the pavement. Item 57 — Cement Concrete Pavement 57.1. Under this item the Contractor shall furnish and place upon a properly prepared sub-grade or sub-bottom course, concrete pave- ment of the thickness shown upon the plans or ordered by the Engineer. 444 SPECIFICATIONS 57.2. Concrete pavement will be placed on the sub-grade or on the sub-bottom course, and shall not be placed until these are in first- class condition, as required for macadam pavement. 57.3. Concrete shall consist of a mixture of Portland cement, No. 1 sand, and broken stone or gravel. All these materials shall conform in all respects to the requirements given under "Materials of Construction," and all the specifications relating to first-class concrete shall apply to work done under this item, in so far as same are not inconsistent with the special specifications given below. 57.4. The concrete shall be mixed in the proportions of one volume of cement to four and one-half volumes of sand and broken stone or gravel. The volumes of sand and broken stone or gravel, shall be measured separately in approved hoppers. The relative proportions of fine and coarse aggregate will be varied slightly, as a result of tests for voids by the Engineer, to the end that resulting concrete shall be as dense as possible. The concrete shall in all cases approxi- mate at 1: 1 J: 3 mix. 57.5. The coarse aggregate shall consist of a well-mixed product of No. 2 and No. 3 stone or No. 1 and No. 2 gravel. Gravel shall not be used except when it has been submitted by the Division Engineer to the Bureau of Tests, has been approved by the Bureau of Tests, and its use has been approved by the First Deputy Com- missioner in writing, — and then only under the restrictions laid down under "Materials of Construction." The fine aggregate shall consist of No. 1 sand. 57.6. The concrete shall be mixed in approved mechanical batch mixers. Mixing shall be continued through at least 12 revolutions and until every particle is coated with mortar and until the batch is of uniform color and consistency. After the materials are once wetted the work shall proceed rapidly until the concrete is in place. The quantity of water used shall be as directed by the Engineer and suitable measuring tanks shall be provided by the Contractor so that the same amount of water may be used in the separate batches. No concrete pavement shall be laid when the temperature falls below 35 F. 57.7. Substantial forms shall be placed along the edge of the con- crete pavement and shall be set and held true to line and grade. 57.8. Before any concrete is placed, the sub-grade shall be sprinkled sufficiently to dampen it but a muddy condition shall not be allowed. As soon as possible after mixing, the concrete shall be deposited in place and thoroughly spaded and screeded so as to bring the mortar flush to the surface. Especial care shall be taken to keep the con- crete uniform and to prevent pockets of stone or mortar. 57.9. Heavy screeds cut to the lines required for the finished surface and resting upon the side forms shall be used for consolidat- ing and screeding the concrete, and the surface, when completed, shall conform to the lines and grades shown upon the plans, and shall be free from depressions or irregularities. No stone shall pro- ject above the general surface. All shaping and screeding shall be done before the concrete has taken its initial set. Any concrete CONCRETE PAVEMENT 445 which has not been shaped and finished previous to the time of initial set, shall be removed for the full depth of the roadway and replaced with satisfactory concrete. 57.10. If a satisfactory finish cannot be obtained with the screed, the screeding shall be immediately followed, and before the cement has taken its initial set, by rubbing down with a wooden float. The men employed for this work shall be competent and experienced and shall work from a platform which rests on the forms or shoulders. The surface, when finished, shall be such that no water will stand on the finished pavement. It shall then be slightly roughened by brooming. 57.11. As soon as the concrete has taken its initial set the surface shall be covered with a one-inch layer of sand or other suitable material of which the shoulders are to be constructed and this shall be thoroughly sprinkled every morning and night, and more often if necessary, so that it will be kept moist for a period of ten days after placing; the material shall then be cleaned from the surface and the road may be opened to traffic if so directed by the Engineer. 57.12. The concrete shall be deposited in sections 30 feet in length, and at the end of each section expansion joints of the type shown on the plans shall be placed. After starting any section, an effort shall be made to complete it at one operation. If for any reason this cannot be done, a vertical joint shall be made when the work is stopped and the work completed up to this joint. 57.13. Any cracks, either longitudinal or transverse, which develop before the acceptance of the work, shall be thoroughly cleaned out and filled with acceptable bituminous material. 57.14. The quantity to be paid for under this item shall be the number of cubic yards of concrete pavement incorporated in the work, in accordance with the plans or as directed by the Engineer. The price bid shall include the furnishing and placing of all materials; all mixing, screeding, finishing, forms, expansion joints and all labor, appliances and incidental expenses necessary to com- plete the work. The amount to be estimated shall be computed by multiplying the cross-section of concrete pavement as shown upon the plans or ordered by the Engineer by the total length of pavement measured along the axis of the pavement. Item 58 — Lignin or Sulphite Liquor 58.1. Under this item the Contractor shall furnish and apply lignin or sulphite binder at the rate of one-half gallon of binder (not of the mixture) to the square yard. 58.2. A quart sample, from each carload of the material to be used, shall be submitted to the Bureau of Tests for acceptance before it may be used. This material shall be a neutral or basic liquor secured by the extraction of lignin from organic matter. It shall be concentrated by evaporation at a temperature not exceeding 210 degrees Fahren- heit until it has a specific gravity at 77 degrees Fahrenheit of not less than 1.23. When concentrated to a constant weight at 212 446 SPECIFICATIONS degrees Fahrenheit, it shall have a residue of not less than 45 per cent. It shall contain not more than 9 per cent of ash. It shall be 99.5 per cent soluble in cold water. 58.3. After the road has been thoroughly filled and brought to a puddle with water, the application of lignin or sulphite binder shall commence and the puddle continued, using a mixture of one part binder to not less than three parts water. This puddling shall con- tinue until the road has received at treatment for its full width of one-quarter gallon of the lignin (not of the mixture) to the square yard. After the roadway has set, but not entirely dried out, the balance of one-quarter gallon of the lignin to the square yard shall be applied to 80 per cent of the width of roadway, using a mixture of one part lignin to two parts water. 58.4. Lignin or sulphite binder shall be applied to the roadway by means of an improved sprinkler which can be regulated so that a uniform distribution is obtained and so that not over one-half of the required amount of binder shall be spread to the square yard on each trip of the sprinkler. The sprinkler shall be equipped with necessary brooms so arranged as to sweep forward any excess material that does not immediately penetrate into the surface. 58.5. The quantity of material to be paid for under this item shall be the actual number of gallons of lignin binder, measured before dilution, actually applied and incorporated in the work to the satisfaction of the Engineer. Binder that has been wasted or that has been applied not in accordance with the requirements of this specification or the orders of the Engineer, shall not be included in this item for payment. The price bid shall include the cost of furnishing, hauling, applying and all necessary appliances and expenses incidental thereto. Item 59 — Wood Block Pavement 59.1. Under this item the Contractor shall furnish and place upon a properly prepared foundation wood block of the quality specified where shown upon the plans or ordered by the Engineer. This pavement shall be placed upon the old macadam, old concrete pavement, new concrete foundation or on other foundation as shown on the plans and ordered by the Engineer. 59.2. The blocks shall be from 6 to 9 inches long and shall average 8 inches; they shall be 3 inches in depth and from 3 to 4 inches in width; but all blocks in one piece of pavement shall be of uniform width. No variation greater than -j^-inch shall be allowed in the depth and |-inch in the width of the blocks. 59.3. Blocks shall be made from Southern yellow pine, North Carolina pine, Norway pine, black gum or tamarack; only one kind of wood, however, shall be used in one piece of pavement. Yellow pine block shall be made from what is known as Southern yellow pine, well manufactured, full size, saw butted, all square edges, and shall be free from the following defects: Unsound, loose and hollow knots, worm holes and knot holes, through shakes and round shakes that show on the surface. In WOOD BLOCK 447 yellow pine timber the annular rings shall average not less than six to the inch and shall be in no case less than four to the inch, measured radially. Norway pine, gum, North Carolina pine and tamarack block shall be cut from timber that is first-class in every respect, and shall be of the same grade as that defined for the Southern yellow pine. 59.4. The creosote oil with which the blocks shall be treated shall conform to either of the following specifications, designated as "A" and "B." The preservative to be used under this specification shall be a product of coal gas, water gas or coke oven tar, which shall be free from all adulterations and contain no raw or unfiltered tars, petro- leum compounds, or tar products obtained from processes other than those stated. Specification "A" The specific gravity shall not be less than one and eight-hundredths (1.08) nor more than one and fourteen hundredths (1.14) at a tem- perature of thirty-eight (38) degrees centigrade. Not more than three and one-half (3^) per centum shall be insol- uble by continuous hot extraction with benzol and chloroform. On distillation, which shall be made exactly as described in Bul- letin 65 of the Railway Engineering and Maintenance of Way As- sociation, the distillate, based on water free oil, shall not exceed one-half (J) of one (1) per centum at one hundred and fifty (150) degrees centigrade, and shall not be less than thirty (30) nor more than forty (40) per centum at three hundred and fifteen (315) degrees centigrade. The oil shall contain not more than three (3) per centum of water. Specification "B" It shall be completely liquid at thirty-eight (38) degrees centigrade, and shall have a specific gravity at that temperature of not less than one and three hundredths (1.03) nor more than one and eight hun- dredths (1.08). It shall contain not more than two (2) per centum of matter in- soluble by hot extraction with benzol and chloroform. On distillation, which shall be made exactly as described in Bulle- tin No. 65 of the American Railway Engineering and Maintenance of Way Association, the distillate based on water free oil shall be within the following limits: At 210 degrees centigrade, not more than 5 per centum. At 235 degrees centigrade, not more than 35 per centum. At 315 degrees centigrade, not more than 85 per centum. The oil shall yield a coke residue not exceeding three (3) per centum. The distillate, between 210 degrees centigrade and 235 degrees centigrade, shall yield solids on cooling to 15 degrees centigrade. The preservative shall contain not more than 3 per centum of water. 59.5. The manufacturer of the oil shall permit full and complete inspection and sampling at the factory at which the oil is produced, 448 SPECIFICATIONS of all materials either crude or refined, entering into the manufacture of the finished product itself, in order that the materials used can be determined to be in accordance with the foregoing requirements. He shall also submit satisfactory proof of the origin of all materials entering into the composition of the finished product. Samples of the preservative taken by the inspector from the treat- ing tank during the progress of the work shall at no time show an accumulation of more than 2 per centum of foreign matter, such as sawdust or dirt. 59.6. The blocks shall be treated with the preservative above described, so that they shall contain at least sixteen pounds of the same per cubic foot of timber. The manufacturer of the block shall equip his plant with all neces- sary gauges, appliances and facilities to enable the inspector to satisfy himself that the requirements of the specifications are fulfilled. 59.7. Upon the foundation shall be spread a bed of cement mortar at no place less than one-half inch in thickness, composed of one part Portland cement and four parts sand thoroughly mixed dry. This mortar bed shall be struck with a template to a true surface exactly parallel to the top of the proposed pavement surface and three inches below it. This bed shall be sprinkled, immediately in advance of the block laying, with hand sprinklers. 59.8. On the mortar surface prepared as described, the blocks shall be laid with grain vertical and at such angles with the curb as the Engineer may direct. The blocks shall be laid in straight and parallel courses and set snugly together but not driven together. Each course of blocks shall be of uniform width and depth, with end joints broken by a lap of not less than two and one-half inches. Only whole blocks shall be used except in starting courses, cutting closures, or where specially permitted by the Engineer. Closures shall be carefully cut and trimmed by experienced men, the portion of the blocks used shall be free from defects and the cut end shall have a surface perpendicular to the top of the block and cut at a proper angle to give a close joint. In laying block the pavers must stand on the block previously laid. After the laying is completed, defective blocks shall be carefully culled out, low blocks raised, the courses carefully aligned and the blocks spaced up. The pavement shall then be rolled by a steam tandem roller weighing not less than two and one-half tons nor more than five tons; the pavement being at the same time lightly sprinkled and the rolling continued until a uniform surface is obtained. Upon the completion of the rolling any defective blocks shall be removed and be replaced with sound blocks, and displaced blocks shall be re- aligned. The joints in the pavement shall then be immediately filled in the manner hereinafter described. If deemed advisable by the Engineer, portions of pavement laid with blocks which have become "dried out" shall be sprinkled with water at frequent inter- vals before joints of same are filled. 59.9. After rolling, the blocks shall be flushed with an approved bituminous filler heated to at least 300 degrees Fahrenheit, which shall be poured over the whole surface and well forced into the joints ASPHALT BLOCK PAVEMENT 449 by rubber squeegees. While the bituminous filler is still hot it shall be immediately followed with a thin coating of clean dry sand. Before turning traffic onto the pavement a coating of one-half inch in thickness of dry screened sand shall be spread over the entire surface. 59.10. The quantity to be paid for under this item shall be the number of square yards, including expansion joints, of pavement laid in accordance with the plans and as directed by the Engineer. The price bid shall include the furnishing and placing of the mortar bed, wood block, bituminous filier and sand surfacing and all other labor and incidental expenses necessary to complete the work. Item 60 — Asphalt Block Pavement 60.1. Under this item the Contractor shall furnish and place upon a properly prepared foundation asphalt block of the quality specified where shown upon the plans or ordered by the Engineer. This pavement shall be placed upon the old macadam, old concrete pavement, new concrete foundation or on other foundations as ordered by the Engineer and shown upon the plans. 60.2. The blocks shall be five inches in width, by twelve inches in length, by two inches in depth, and a variation of more than one- fourth of an inch in length and one-eighth of an inch in width or depth from these dimensions will be sufficient ground for rejecting any block. 60.3. The blocks shall consist of the following materials: Asphaltic cement. Crushed trap rock or other approved crushed rock. Inorganic dust. The rock used in the blocks must be crushed so that every particle will pass a screen of one-fourth of an inch mesh. The blocks shall receive a compression in the moulds of not less than one ton per cubic inch of material in the blocks, and must weigh not less than ten and one-half pounds per block. The blocks shall have a specific gravity of not less than 2.40, and after having been dried for twenty- four hours at a temperature, of 150 degrees Fahrenheit, they shall not absorb more than one per centum of moisture when immersed in water for seven days. Whatever the character of the asphalt used, the block shall yield not less than six and one-half per centum of bitumen, when extracted with carbon bisulphide. The inorganic dust, or filler, shall be produced from sound lime- stone or other approved material, and shall be powdered to such a fineness that all of it shall pass a thirty mesh sieve and not less than fifty per centum of it shall pass a 200 mesh sieve. Sufficient inor- ganic dust shall be used to give a minimum percentage of voids in the block, and provide a sufficient medium for absorbing the asphaltic cement. 60.4. The asphaltic cement shall be composed of natural or oil asphalt, and asphaltic oil, as approved. This asphaltic cement shall be of acceptable consistency and quality. 45° SPECIFICATIONS The material shall have a specific gravity of at least 0.98 at 77 degrees Fahrenheit. Its penetration shall be not more than ten mm. when tested for five seconds at 77 degrees Fahrenheit, with a No. 2 needle weighted with 100 grams. When twenty grams are heated in a hot air oven in a flat-bottom dish two and one-half inches in diameter at 325 degrees Fahrenheit for five hours, the loss in weight shall not be more than eight per centum. It shall show an open flash point not less than 325 degrees Fahrenheit. Its solubility at air temperature in chemically pure carbon disulphide shall be at least sixty-six per centum. 60.5. Upon the foundation shall be spread a bed of the thickness shown upon the plans, composed of one part Portland cement and four parts sand, thoroughly mixed. This mortar bed shall be struck with a template to a true surface, exactly parallel to the top of the proposed pavement surface and two inches below it. This bed shall be sprinkled immediately in advance of the block laying with hand sprinklers. The blocks shall be laid while the mortar is fresh and before it has taken its initial set. All depressions and other irregularities in the surface shall be corrected by the Contractor immediately. The blocks shall be laid by the pavers standing upon the blocks already laid and not upon the bed of mortar. The blocks shall be laid at right angles with the line of the street, and in such a manner that all longitudinal joints shall be broken by a lap of at least four inches. The blocks shall be so laid as to make the lateral joints as tight as possible, consistent with keeping a good alignment of the courses across the street. When thus laid the blocks shall be immediately covered with clean, fine sand, per- fectly dry, and screened through a one-eighth inch screen. This sand shall be spread over the surface and swept into the joints and be allowed to remain on the pavement not less than thirty days or for such time as the action of the traffic on the street shall have thoroughly ground the sand into all the joints. 60.6. The materials incorporated into blocks shall be approved by the Engineer, and samples of all materials shall be sent to the Bureau of Tests and they shall pass the tests required by this Bureau for these materials. 60.7. The methods of work and materials used shall at all times be subject to the inspection and supervision of the Engineer or his representative upon the work. 60.8. The quantity to be paid for under this item shall be the number of square yards of asphalt block laid in accordance with the plans or as directed by the Engineer. The price bid shall include the furnishing and placing of all materials, mortar bed, and all labor and incidental expenses necessary to complete the work. Where placed upon old concrete foundation or upon old macadam the preparation of the foundation to receive the mortar bed will be paid for under item "Cleaning Old Pavement" or item " Scarifying and Reshaping Old Macadam." BRICK PAVEMENT 451 Item 61 — Brick Pavement 6 1. 1. Under this item the Contractor shall furnish and place the number of square yards of brick pavement required in accordance with the plans or as ordered by the Engineer. The item will include the furnishing and placing of all the block, sand cushion, grout, expansion joints and all material, labor and other expenses incidental thereto but will not include the concrete foundation, edging, curbing, manholes, catch basins, etc., which will be paid for under the espe- cially designated items therefor. 61.2. All bricks or blocks used must be vitrified and especially burned for street paving and of the very best quality as regards hardness, dimensions, toughness, straight lines and non-absorption of water. 61.3. The paving bricks shall be subjected to modulus of rupture test and to abrasion tests conducted by the Commission in the manner and with rattlers of the type adopted February 7, 191 1, by the National Paving Brick Manufacturers Association. One sample shall be tested for every two hundred thousand (200,000) bricks and less than this when conditions warrant. An average loss in weight in a rattler test exceeding twenty-four (24) per centum, or an average absorption of three and one-half (3J) per centum of water shall cause the rejection of the total quantity that the test represents, provided, however, that if permitted the bricks may be carefully reculled, and new samples taken and tested. If this second test passes the requirements, the bricks represented by it may be used. If this second test fails, no further test shall be permitted but the entire lot shall be rejected. To ensure the furnishing of bricks of uniformly acceptable quality, any " brand" of brick shall be rejected and shall not be further considered if three lots, each of ten thousand (10,000) bricks or more, offered consecutively for acceptance tests, fail to meet the requirements for this section without reculling them. Modulus of Rupture. When tested on edge as laid on the pave- ment, the modulus of rupture shall be not less than two thousand WL (2,000) pounds per square inch. Computed by formula R = 3-7—^ in which R is the modulus of rupture in pounds per square inch, L the length between supports in inches ( = 6 inches), b and d the breadth and depth in inches, and W the load in pounds, which pro- duces rupture. All the above tests will be made by the Bureau of Tests of the State Commission of Highways. 61.4. On grades of 5 per centum or over an approved special form of block suitable for steep grades shall be used. 61.5. The size of the brick shall be 3 \ inches in width by four inches in depth by 8J inches in length, and shall not vary from these dimensions more than one-eighth inch in width or depth nor more than one-half inch in length. Bricks of a given brand shall not vary among themselves more than J-inch in depth nor more than J-inch in width nor more than J-inch in length in any one shipment. If the edges 452 SPECIFICATIONS are rounded the radius shall not be greater than ft of an inch. One side shall contain lugs of such dimensions that transverse joints will not be less than -f^ of an inch nor more than J-inch in width. Each end shall contain a semi-circular groove of J to f-inch radius, or a bulge or at least -£6 inch. The grooves shall be horizontal, and shall match perfectly when the bricks are laid in the finished pavement. Bricks in any course shall not vary in width by more than J-inch. 61.6. Not less than ten days after the concrete foundation has been completed, there shall be laid a bed of clean Cushion Sand as described under " Materials of Construction," which shall be one and one-half inches thick after being rolled with a roller weighing 150 pounds per foot of width. Before being rolled this bed of sand shall be brought to the proper elevation and crown as shown on plans by a template of a shape and size satisfactory to the Engineer. After being rolled all irregularities of the surface shall be eliminated and the sand cushion shall be brought to the exact form and section by the use of lutes or hand templates. 61.7. Longitudinal expansion joints shall be placed alongside each curb or edging, and shall be one of the following types: Premolded Type, requiring no heating or pouring at the place of insertion. These expansion joints shall be of the proper thickness and width, as specified, made in convenient lengths ready for use. The joints shall be placed as the paving progresses, and shall rest directly on the sand cushion. The expansion joints shall be composed entirely of a high grade asphalt, and shall pass the following tests: Specific gravity . . . . 985 to 1 . 002 Melting point 235 to 265 F. Loss on heating for 5 hours at 325 F .0 % Bitumen soluble in carbon disulphide 99 . 5 % to 99 . 9 % Bitumen soluble in carbon tetra-chloride 99.4% to 99.8% Bitumen soluble in Be. Naphtha to % Penetration at 32 F 25 to 35 Penetration at 77 F 40 to 50 Penetration at 115 F 65 to 75 Poured Type. This type shall be provided for by placing along- side each curb or edging wooden strips with metal wedge shape pieces dropped over the top of the boards and between the board and the curb every three feet apart to facilitate the removal of the boards, or, by using two planed wedge-shaped strips so cut that when placed together in reverse positions their total section shall be rectangular and of a thickness and depth equal to the thickness and depth of the required expansion joint. The strip placed next to the curb shall be set with the wide edge up. These expansion joint forms shall be set next to the curb on a true grade with all end joints tight, and be pressed into sand so that their tops shall be one-quarter inch below the top surface of the pavement blocks before rolling. The two strips comprising the joint form shall break joints. The thickness of longitudinal expansion joints shall be as called for by the plan. BRICK PAVEMENT 453 61.8. On the sand cushion prepared as in section 61.6 the blocks shall be carefully set on edge with the best edge up, shall be laid straight at and right angles to the edging line, except at road inter- sections, where they shall be laid at such angles as directed by the Engineer. All block shall be laid with the lugs in the same direction, joints shall be close and at right angles to the tops and sides. Each alternate course shall be commenced with a half brick. No half bricks or bats shall be used except at the ends of courses. All joints shall be broken with a lap of not less than three (3) inches. All brick shall be clean when placed in the pavement. Brick which in the opinion of the Engineer are not satisfactorily clean, shall be washed before being placed. In no case shall the sand cushion in front of the pavement be disturbed or walked on during the laying of the blocks. 61.9. After a sufficient number of blocks have been laid, all soft, broken or badly misshapen blocks shall be marked by the inspector and removed. Any blocks slightly spalled or kiln-marked shall be turned over, and should the opposite face be acceptable, it may be replaced in the pavement, otherwise, it must be removed. In laying block pavement, the inspector shall keep the blocks culled, and the Contractor shall make the necessary changes and replacements so that the work shall at all times be ready for grouting within 300 feet from the block-laying. 61.10. After all objectionable blocks have been removed from the pavement and all replacements have been made, the pavement shall be swept clean and thoroughly rolled with a self-propelled tandem roller weighing not over five tons and not less than three tons. Horse rolling shall not be permitted. This rolling shall start along the outside edges and progress toward the center. It shall then be re- rolled diagonally both ways until the surface is even. After final rolling the pavement shall be tested with a ten-foot straight edge laid parallel with the curb, and any depression exceeding one-quarter inch shall be corrected and brought to the proper grade. All blocks disturbed in making replacements or correcting depressions shall be settled into place by ramming or by rerolling. Each section of pavement must be acceptable to the Engineer before the grouting on that section may be commenced. 6 1.1 1. Grout for filling the joints of brick or block pavements shall be composed of one part Portland cement and one part Grout Sand. 61.12. The box for mixing this grout shall be about four feet eight inches long, two feet six inches wide and one foot two inches deep, supported on legs of different lengths in order that the mixture shall readily flow to the lowest corner, which shall not be more than six inches above the pavement. Approved mechanical grout mixers may be used. 61.13. The mixture, not exceeding one sack of cement together with a like amount of sand, shall be placed in the box and mixed dry, until the mass assumes a uniform color. Water shall then be added, forming a liquid mixture of the consistency of thin cream for the first coat and slightly thicker for each succeeding coat. From 454 SPECIFICATIONS the time the water is applied until the last drop is removed and floated into the joints of the pavement the mixture must be constantly agitated. 61.14. The brick shall be wet to the satisfaction of the Engineer before any grout is placed. The grout shall be removed from the box to the street surface with a scoop shovel and immediately swept into the joints, the mixture in the box being constantly agitated while this is being done. The work of grouting shall proceed for the entire width of the pavement. When sufficient time has elapsed for the grout to thor- oughly penetrate all the joints, but before the cement has attained its initial set, the section treated shall be gone over a second time in the same manner, care being taken to thoroughly fill all joints from the bottom flush with the top of the block. If necessary to secure flush joints, a third, fourth or fifth coat of the grout shall be swept in and smoothed off with a suitable squeegee. Care shall be taken to so conduct the grouting that no part of any joint will receive an application of the second grout until the first is satisfactorily completed, nor of the third until the second is completed, etc. To insure this result metal strips 1-16 in. by 6 in. by 3 ft. must be inserted, for the full length of the joint, at work intervals; all of the several applications of grout must be completed up to this joint before any grouting is begun on the other side of it. 61.15. After the joints are thus filled flush with the top of the blocks and sufficient time for hardening has taken place, so that the cover coat will not absorb any moisture from the grout, one inch of suitable material shall be spread evenly over the entire surface, and be kept moist for a period of at least ten days and until the grout has thor- oughly set. During this period the section grouted must remain absolutely free from disturbance or traffic of any kind. After 30 days from the spreading, this cover coat shall be completely removed. 61.16. In case the poured type of expansion joint is used, after the grout has set but within thirty-six hours after its application the expansion joint forms shall be withdrawn and the space thus formed thoroughly cleaned and a bituminous filler having a melting point not less than 120 degrees Fahrenheit nor more than 140 degrees Fahrenheit shall be immediately poured into place at a temperature not less than 200 degrees Fahrenheit. 61.17. If required, transverse expansion joints shall be constructed of the materials and in the manner prescribed by the Engineer. 61.18. The quantity of pavement to be paid for under this item shall be the number of square yards placed in accordance with the plans or directions of the Engineer. The price bid per square yard shall include the sand cushion, paving block, grout, material for expansion joint, sand covering, sprinkling, and all other labor, materials and incidentals necessary to satisfactorily complete the work. The amount to be estimated under this item shall be computed by multiplying the actual width of pavement, including expansion joints, by the total length of pavement measured along the axis of the road and parallel to the surface. STONE BLOCK PAVEMENT 455 Item 62 — Stone Block Pavement 62.1. Under this item the Contractor shall furnish and place upon a properly prepared foundation Stone Block pavement of the quality specified below, where shown upon the plans or directed by the Engineer. The item will include the furnishing and placing of all the block, sand cushion, grout, expansion joints and all material, labor and other expenses incidental thereto but will not include the concrete foundation, edging, curbing, manholes, catch basins, etc., which will be paid for under the especially designated items therefor. 62.2. The dimensions of the blocks shall be as follows: Not less than six inches nor more than twelve inches long on top, not less than three and one-half inches nor more than four and one-half inches wide on top, and not less than four inches nor more than five inches deep. They shall be dressed so that after laying, no measure- ment of any joint shall show a width of more than one-half inch for a depth of one inch, or a width of more than one inch in any part of the joint. The head of the block shall be so cut that it shall not have a depression in it more than three-eighths inch deep, and the edges and corners must be full unchipped and unbroken. All blocks shall be sorted and laid in straight courses of uniform width and depth. 62.3. The blocks shall be of stone of medium sized grain showing an even distribution of constituent material. They shall be of uniform quality and texture, without seams, scales or disintegration, and free from an excess of mica or feldspar. They shall be made from rock which when tested in the Deval Rattler will show a "co- efficient of wear" of more than 7 and less than 14. All blocks for any one contract shall be from the same quarry unless otherwise directed. 62.4. On the prepared foundation, sufficient clean Cushion Sand as described under "Materials of Construction" on page 375, shall be spread to such a thickness that after the pavement has been thoroughly rammed or settled the sand under the block shall be nowhere less than one inch thick. On the sand cushion above specified, the blocks shall be set ver- tically on edge in close contact with each other, and in straight rows across the road at right angles to the curb, except at intersections, where the angle of the rows with the curb shall be varied to meet the conditions. Blocks in adjoining rows shall be set to break joints not less than three inches. All blocks shall be set so that when thoroughly rammed or settled to a firm, unyielding bearing, they will then be true to lines, grades and cross sections, and have no joints greater than the maximum allowable. All depressions or irregularities in the surface shall be corrected to the satisfaction of the Engineer. Only practiced and competent pavers shall be em- ployed in laying the blocks. After the blocks are laid, sufficient approved clean gravel shall be spread over the surface and swept into the joints so as to fill the latter to a depth of about two inches from the bottom. The blocks shall 456 SPECIFICATIONS then be thoroughly rammed or rolled until firm, even and true to the lines, grades and cross sections. Approved expansion joints shall be provided along the curb as may be required and shall be filled with the same quality of filler as is specified for expansion joints in brick pavements. Portland cement grout mixed in proportions of one part cement and one part sand shall then be poured into the joints until the grout flushes to the surface of the pavement. The grout shall be broomed when required, and the pouring and brooming shall be continued until all the joints are thoroughly filled, and the grout is even with the highest part of any and all blocks. Sprinkling or otherwise wetting the blocks before grouting shall be done when atmospheric or other conditions require this precaution to be taken. 62.5. After grouting shall have been completed and the grout shall have sufficiently hardened, a coating of suitable material about one inch deep shall be spread over the whole surface of the grouted pavement, and the road shall then be sprinkled with water. This covering shall be kept wet, and no travel of any kind shall be allowed on the completed pavement for at least seven days thereafter, nor until the grout shall have thoroughly set, when the covering shall be completely removed. 62.6. The quantity to be paid for under this item shall be the number of square yards, including expansion joints, of pavement laid in accordance with the plans and as directed by the Engineer. The price bid shall include the furnishing and placing of all materials, the spreading of sand cushion, the laying, ramming or rolling, grouting, surfacing and all labor and incidental expenses necessary to complete ^he work. MEDINA SANDSTONE BLOCK PAVEMENT (City of Rochester, N.Y., Specifications, 191 1) The grading, subwork, and curbs having been completed as herein specified under the proper headings, the work of laying the concrete foundation and paving will then proceed. A concrete foundation six (6) inches thick, of Portland cement, as specified in the bidding sheet and shown in plans, will be laid in accordance with the specifications herein contained. The surface will be eight (8) inches below the finished pavement and parallel thereto, or seven (7) inches if a five (5) inch block is specified. The surface to be kept wet until covered with sand, and, at least, thirty-six (36) hours shall be allowed for the concrete to set before the pavement is laid. When connection is to be made with any layer set, or partially set, the edge of such layer must be broken down, shall be free from dust and properly wet, so as to make the joints fresh and close. On this concrete foundation shall be laid a bed of clean, sharp sand, perfectly free from moisture (made so by artificial heat if deemed necessary), not less than one (1) inch thick, to the depth necessary to bring the pavement and crosswalks to the proper grade when thoroughly rammed. MEDINA SANDSTONE BLOCK PAVEMENT 457 Upon this bed of sand, the stone blocks and crosswalks must be laid. The stone blocks are to be laid in straight courses at right angles with the line of the street, except in intersections of streets, where the courses shall be laid diagonally, and except in special cases, when they shall be laid at such angle, with such crown and at such grade as the city engineer may direct. Each course of blocks shall be uniform in width and depth, and shall be gauged and selected for the pavers on the sidewalks, and so laid that all longitudinal joints or end joints shall be close joints and shall be broken by a lap of at least three inches, and that joints between courses shall not be more than one-half inch in width. The blocks shall then be thoroughly rammed by courses at least three times by a rammer weighing not less than eighty (80) pounds — no iron of any kind being allowed on its lower face to come in contact with the paving, and until brought to an unyielding bearing, with a uniform surface, true to the roadway on the established grade. The surface of the pavement thus com- pleted must be even and smooth throughout and molded to conform to the wells of the surface sewers, street and alley intersections, drainage details, and the grade lines established by the city engineer. During the final ramming the pavement shall be tested with a straight- edge and templet, and any unevenness must be taken out and made true to the required grade, level, and cross-section. If a paving pitch filler is used, the joints shall be filled with clean, dry, hot gravel of proper size as herein specified, heated in pans especially provided for that purpose, and poured from cans having small spouts and thoroughly settled in place with wire picks until the level of the gravel is at least two inches below the top of the pavement. The gravel used between the blocks shall be of such size as will pass through a sieve having four meshes per square inch, and be retained on a sieve of sixty-four meshes per square inch, and must be screened when dry. There shall be immediately poured into the joints, while the gravel is hot, boiling paving cement as hereinafter described, heated to a temperature of 300 F. until the joints and all interstices of gravel filling are full and will take no more, and are filled flush with the top of the blocks. Dry, hot gravel must then be poured along the joints, filled with paving cement, as above described. The paving cement to be used in filling the joints as herein provided shall be a paving pitch of the best quality, of a brand that has been proved by actual use in pavements known to the city engineer to be best adapted to the purpose. It shall be delivered on the work in lots at least one week before using, in order that the necessary analysis and examination may be made by the city engineer. In addition to this the contractor must furnish the city engineer with the certificate of the manufacturer or refiner that the materials are of the kind specified. The city engineer may direct that a Portland cement grout filler may be used in the joints instead of a paving pitch, in which case the pavement shall be thoroughly sprinkled or washed with water before grouting. The grout shall be mixed with clean, sharp sand 458 SPECIFICATIONS of approved quality, in the proportion of one to one, the cement and sand to be thoroughly mixed together dry, in a box, and then only a sufficient amount of water added to make the grout of the proper fluidity when thoroughly stirred. The grout shall- be prepared only in small quantities at a time, and shall be stirred rapidly and constantly in the box and while being applied to the pavement, and no settlings or residue will be allowed to be used. The grout shall be transferred to the pavement in such a way as the engineer may think most advantageous and best for the work, and shall then be rapidly swept into the joints of the pavement with proper brooms. The stones shall be well wet as directed before the grout is applied, and the pouring must be continued until the joints remain full. All teams and traffic of any kind, except on planks, shall be rigidly prohibited on the pavement for ten days after the grout is applied, or until, in the opinion of the engineer, it has become thoroughly set and hardened, so that the bond will not be broken by traffic over the pavement. CONVERSION TABLE 459 Linear Units Conversion Table 55 Old Surveyors' Units 1 link = 7.92 in. 100 links = 1 chain = 66 ft. 25 links = 1 rod = 16.5 ft. Ordinary Measure 12 in. = 1 ft. 3 ft. = 1 yd. 5280 ft. = 1 mile Square Units 1 sq.ft. 1 sq. yd. 1 acre 1 sq. mile = 144 sq. in. = 9 sq. ft. = 1296 sq. in, = 43>5°° sq. ft. = 4840 sq. yds. = 27,878,400 sq. ft. = 3,097,600 sq. yds. = 640 acres Volume Units 1 cu. ft. 1 cu. yd. 1 ordinary gal. 1 Imperial gal. 1 barrel = 1728 cu. in. = 7.4805 ordinary gal. = 6.232 Imperial gal. = 27 cu. ft. = 46,656 cu. in. = 231 cu. in. = 277 cu. in. = 3i-5 gal. = 4.21 cu. ft. Weight Units 1 pound 1 ordinary 1 long ton = 16 ounces ton = 2000 pounds = 2240 pounds Temperature Units Freezing point of water = 32° Fahrenheit = o° Centigrade Boiling point of water at normal air pressure =212° Fahrenheit = ioo° Centigrade 1 degree Fahrenheit 1 degree Centigrade — °-5556 degree Centigrade = 1.8 degrees Fahrenheit 460 GENERAL REFERENCE TABLES Table 56 Equivalents of Inches and Fractions of Inches in Decimals of a Foot In. In. 1 In. 2 In. 3 In. 4 In. . 5 In. •0 8 33 .1667 .2500 -3333 .4167 A .0026 .0859 .1693 .2526 •3359 •4193 ft .0052 .0885 .1719 •2552 .3385 .4219 3 32 .0078 .0911 •1745 .2578 •34i 1 •4245 i .0104 .0938 .1771 .2604 •3438 .4271 5 32 .0130 .0964 .1797 .2630 •3464 .4297 A .0156 .0990 .1823 .26^6 •349o •4323 T2 .0182 .1016 .1849 .2682 •35i6 •4349 1 .0208 .1042 .1875 .2708 •3542 •4375 9 32 .0234 .1068 .1901 .2734 .3568 .4401 5 16 .0260 .1094 .1927 .2760 •3594 .4427 11 32 .0286 .1120 •1953 .2786 .3620 •4453 I •°3I3 .1146 .1979 .2813 .3646 •4479 if •°339 .1172 .2005 .2839 .3672 •4505 ft •0365 .1198 .2031 .2865 .3698 •4531 II .0391 .1224 .2057 .2891 •3724 •4557 1 2 .0417 •1253 .2083 .2917 •375o 4583 H .0443 .1276 .2091 •2943 •3776 .4609 ft .0469 .1302 • 2I 35 .2969 .3802 •4635 U •0495 .1328 .2161 •2995 .3828 .4661 f .0521 -1354 .2188 .3021 .3854 .4688 21 32 •°547 .1380 .2214 •3°47 .3880 .4714 1 1 T6 •°573 .1406 .2240 •3°73 .3906 .4740 £3 32 •°599 .1432 .2266 •3°99 •393 2 .4766 3 4 .0625 .1458 .2292 •3 I2 5 •3958 •4792 25 3 2 .0651 .1484 .2318 •3 I 5 I •3984 .4818 13 "16 .0677 .1510 .2344 •3i77 .4010 .4844 21 32 .0703 •1536 .2370 •3203 .4036 .4870 1 8 .0729 •1563 .2396 .3229 .4063 .4896 2.9 32 •o755 .1589 .2422 •3255 .4089 .4922 15 16 .0781 .1615 .2448 .3281 •4115 .4948 31 32 .0807 .1641 .2474 -33°7 .4141 •4974 EQUIVALENTS OF DECIMALS OF A FOOT 461 Equivalents of Inches and Fractions of Inches in Decimals of a Foot In. 6 In. 7 In. 8 In. 9 In. 10 In. 11 In. .5000 .5833 .6667 .7500 •8333 .9167 1 "32 .5026 .5859 .6693 .7526 .8359 •9193 1 1 6 •5°5 2 .5885 .6719 •7552 .8385 .9219 A .5078 •5911 •6745 .7578 .8411 •9245 1 8 .5104 .5938 .6771 .7604 .8438 .9271 5 "32 •513° .5964 .6797 .7630 .8464 .9297 A •5!56 •599° .6823 .7656 .S490 •9323 J7 32 .5182 .6016 .6849 .7682 .8516 •9349 i .5208 .6042 .6875 .7708 .8542 •9375 9 32 •5 2 34 .6068 .6901 •7734 .8568 .9401 5 Te- .5260 .6094 .6927 .7760 •8594 .9427 ll 32 .5286 .6120 • 6 953 .7786 .8620 •9453 1 •53 J 3 .6146 .6979 .7813 .8646 •9479 13 32 •5339 .6172 .7005 .7839 .8672 •9505 A •5365 .6198 •703 1 .7865 .8698 •953i it •539i .6224 •7057 .7891 .8724 •9557 4 .5417 .6250 .7083 .7917 .87^0 .9583 11 32 •5443 .6276 .7109 •7943 .8776 .9609 9 16 •5469 .6302 •7i35 .7969 .8802 •9635 19 32 •5495 .6328 .7161 •7995 .8828 .9661 5 S •552i •6354 .7188 .8021 •8854 .9688 21 ¥2 •5547 .6380 .7214 .8047 .8880 .9714 11 16 •5573 .6406 .7240 .8073 .8906 .9740 2 3 32 •5599 .6432 .7266 .8099 •8932 .9766 3 4 •5625 .6458 •7292 .8125 .8958 .9792 23 32 •5651 .6484 .73i8 .81^1 .8984 .9818 13 1 6 •5677 .6510 •7344 .8i77 .9010 •9844 27 32 •57°3 .6536 •737° .8203 .9036 .9870 7 8 •5729 •6563 •7396 .8229 .9063 .9896 2.9 32 •5755 .6589 .7422 •8255 .9089 .9922 1 5 16 .578i .6615 .7448 .8281 •9115 .9948 3 1 32 .5807 .6641 •7474 .8307 .9141 •9974 462 GENERAL REFERENCE TABLES Table 57. Areas and Volumes Areas ' 7T h <--b-->l y k — -b — -> -b— >i/ Squares, Rectangles, and Parallelograms. Area = bh h b h ____*_ h-b->| K- h b *i b -—x K-b-H Triangles Area = ^ bh Trapezoids Area = // Circles Area = n R 2 = - D Circumference of Circle = 2 n R = tt D Commonly used value of n = 3.1416 Sector of Circle Area = n R 2 A° 3 6o c .A Segment of a Circle Area ° ( - R2 w) ' ( (i? sin t> (i? cos 1) ) Volumes Cubes, Rectangular Prisms, Parallelopipeds, Cylinders, etc. All solids having parallel bases and a constant cross-section. Volume = area of base X perpendicular height between the planes of the bases. i < area of section parA top+base + fA 61 tc \ and midway) between base and top ' Prismoidal Formula Trautwine defines a prismoid as a solid having for its ends two parallel plane figures connected by other plane figures on which and through every point of which a straight line may be drawn from one of the two parallel ends to the other. These connecting planes may be parallelograms or not and parallel to each other or not. This includes cubes, all parallelopipeds, prisms, cylinders, pyramids, cones, and their frustums, provided the top and base are parallel and wedges. The prismoidal formula applies to all these solids either alone or to any form that can be separated into units of the above forms. Prismoidal formula Volume = h X A + a + \M h = perpendicular distance between the parallel ends A = area of one of the parallel ends a = area of the other parallel end M = area of a cross-section midway between and parallel to the two parallel ends Sphere Volume = % n R 3 = 4- 1888 R s = in D* = O.5236 D 3 In which R = radius of sphere D = diameter of sphere 464 GENERAL REFERENCE TABLES Table 58 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 1 I I I .OOOO I .OOOO 3.142 O.7854 2 4 8 I.4142 1.2599 6.283 3.1416 3 9 27 I-732I 1.4422 9.425 7.0686 4 16 64 2 .OOOO I-5874 12.566 12.5664 5 25 125 2.2361 1. 7100 15.708 19.6350 6 36 216 2.4495 1.8171 18.850 28.2743 7 49 343 2.6458 1. 91 29 21.991 38.4845 8 64 512 2.8284 2 .OOOO 25-I33 50.2655 9 81 729 3.OOOO 2.0801 28.274 63.6173 10 100 1000 3.1623 2.1544 31.416 78.5398 11 121 I33 1 3.3166 2.2240 34.558 95-°33 12 144 1728 3.464I 2.2894 37-699 113.097 1 3 169 2197 3.6056 2.3513 40.841 132.732 14 196 2 744 3-7417 2.4101 43.982 I53-938 15 225 3375 3-8730 2.4662 47.124 176.715 16 256 4096 4.0000 2.5198 50.265 201.062 17 289 49i3 4-I23I 2.5713 53-407 226.980 18 324 5832 4.2426 2.6207 56.549 254.469 19 361 6859 4.3589 2.6684 59.690 283.529 20 400 8000 44721 2.7144 62.832 3*4-159 21 441 9261 4.5826 2.7589 65^73 346.361 22 484 10648 4.6904 2.8020 69.115 380.133 2 3 529 12167 4.7958 2.8439 72.257 1I5-476 24 576 13824 4.899O 2.8845 75.398 452.389 25 625 15625 5.OOOO 2.9240 78.540 490.874 26 676 17576 5.O99O 2.9625 81.681 530.929 27 729 19683 5.I962 3.0000 84.823 572.555 28 784 21952 5-29I5 3.0366 87.965 6i5752 29 841 24389 5-3852 3-0723 91.106 660.520 3° 900 27000 5-4772 3.1072 94.248 706.858 3 1 961 29791 5.5678 3.1414 90.389 754.768 3 2 1024 32768 5-6569 3-1748 100.531 804.248 33 1089 35937 5-7446 3-2075 103.673 855-299 -34 1156 39304 5-83IO 3.2396 106.814 907.920 35 1225 42875 5.9161 3-2711 109.956 962.113 36 1296 46656 6.0000 3-3 OI 9 113.097 1017.88 37 1369 50653 6.0828 3-3322 116.239 1075.21 38 1444 54872 6.1644 3.3620 119.381 1134.11 39 1521 59319 6.2450 3-3912 122.522 1194.59 40 1600 64000 6.3246 3.4200 125.660 1256.64 SQUARES, CUBES AND ROOTS 465 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root CnViP Rnnt Circle V.UUC J\UUl Circum. Area 41 1681 68921 6.4031 3.4482 128.81 1320.25 42 1764 74088 6.4807 3.4760 I3I-95 I385-44 43 l849 795°7 6-5574 3-5°34 I35-09 1452.20 44 1936 85184 6.6332 3-5303 138.23 !52o.53 45 2025 91125 6.7082 3-5569 141-37 i59o-43 46 2Il6 9733 6 6.7823 3-5830 I44oI 1661.90 47 2209 103823 6.8557 3.6088 I47-65 1734.94 48 2304 110592 6.9282 3-6342 150.80 1809.56 49 2401 1 1 7649 7.0000 3-6593 153-94 1885.74 5° 2500 125000 7.0711 3.6840 157.08 1963.50 5 1 2601 132651 7.1414 3.7084 160.22 2042.82 52 2704 140608 7.2111 3-7325 163.36 2123.72 53 2809 148877 7.2801 3-7563 166.50 2206.18 54 2916 157464 7-3485 3-7798 169.65 2290.22 55 3° 2 5 166375 7.4162 3.8030 172.79 2375-83 56 3*36 175616 74833 3-8259 175-93 2463.01 57 3249 185193 7-5498 3.8485 179.07 255I-76 58 33 6 4 195112 7.6158 3.8709 182.21 2642.08 59 348i 205379 7.6811 3-8930 185.35 2733-97 60 3600 216000 7.7460 3-9149 188.50 2827.43 61 372i 226981 7.8102 3-9365 191.64 2922.47 62 3844 238328 7.8740 3-9579 194.78 3019.07 63 3969 250047 7-9373 3-9791 197.92 3II7-25 64 4096 262144 8.0000 4.0000 201.06 3216.99 65 4225 274625 8.0623 4.0207 204.20 3318.31 66 4356 287496 8.1240 4.0412 207.35 3421.19 67 4489 300763 8.1854 4.0615 210.49 3525-65 68 4624 3 J 443 2 8.2462 4.0817 213.63 3631.68 69 4761 328509 8.3066 4.1016 216.77 3739-28 70 4900 343000 8.3666 4.1213 219.91 384845 7i 5°4i 3579" 8.4261 4.1408 223.05 3959.19 72 5184 373248 8.4853 4.1602 226.19 4071.50 73 5329 389017 8.5440 4.1793 229.34 4185.39 74 54/6 405224 8.6023 4.1983 232.48 4300.84 75 5625 421875 8.6603 4.2172 235.62 4417.86 76 5776 438976 8.7178 4-2358 238.76 453646 77 5929 450533 8.775o 4.2543 241.90 4656.63 78 6084 474552 8.8318 4.2727 245.04 4778.36 ■ 79 6241 493°39 8.8882 4.2908 248.19 4901.67 80 6400 512000 8-9443 4.3089 25I-33 5026.55 466 GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 8l 6561 53I44I 9 .OOOO 4.3267 2 5447 5I53.00 82 6724 551368 9-°554 4-3445 257.61 5281.02 83 6889 571787 9.IIO4 4.3621 260.75 5410.61 84 7056 592704 9.1652 4.3795 263.89 5541.77 85 7225 614125 9- 2I 95 4.3968 267.04 5674.50 86 7396 636056 9.2736 4.4140 270.18 5808.80 87 7569 658503 9-3 2 74 4.4310 273.32 5944.68 88 7744 681472 9.3808 4.4480 276.46 6082.12 89 7921 . 704969 9.4340 4.4647 279.60 6221.14 90 8100 729000 9.4868 4.4814 282.74 6361.73 9i 8281 753571 9-5394 4.4979 285.88 6503.88 92 8464 778688 9-59I7 4-5 x 44 289.03 6647.61 93 8649 804357 9.6437 4.5307 292.17 6792.91 94 8836 830584 9.6954 4.5468 295.31 6939.78 95 9025 857375 9.7468 4.5629 298.45 7088.22 96 9216 884736 9.7980 4.5789 3 OI -59 7238.23 97 9409 912673 9.8489 4.5947 304.73 7389.81 98 9604 941192 9.8995 4.6104 307.88 7542.96 99 9801 970299 9.9499 4.6261 311.02 7697.69 100 1 0000 I OOOOOO 10.0000 4.6416 314.16 7853.98 IOI 10201 1030301 10.0499 4.6570 317.30 8011.85 102 10404 1061208 10.0905 4.6723 320.44 8171.28 103 10609 1092727 10.1489 4.6875 323.58 8332.29 104 10816 I I 24864 10.1980 4.7027 326.73 8494.87 105 11025 1157625 10.2470 47177 329.87 8659.OI 106 11236 1191016 10.2956 4.7326 333-OI 8824.73 107 1 1 449 1225043 10.3441 4.7475 336.15 8992.02 108 1 1 664 1259712 10.3923 4.7622 339.29 9160.88 109 11881 1295029 10.4403 4.7769 342.43 9331.32 no 12100 I 33 I OOO 10.4881 4.7914 345.58 9503.32 III 12321 1367631 IO-5357 4.8059 348.72 9676.89 112 12544 1404928 10.5830 4.8203 351.86 9852.03 I1 3 12769 1442897 10.6301 4.8346 355.oo IOO28.7 114 12996 I 48 1 544 10.6771 4.8488 358.14 IO207.O 115 13225 1520875 10.7238 4.8629 361.28 10386.9 116 13456 1560896 10.7703 4.8770 364.42 10568.3 117 13689 1601613 10.8167 4.8910 367.57 I075L3 118 i39 2 4 1643032 10.8628 4.9049 37o.7i I0935.9 119 14161 1 685 1 59 10.9087 4.9187 373^5 III22.0 120 14400 1728000 io.9545 4.9324 376.99 II309-7 SQUARES, CUBES AND ROOTS 467 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 ClR<"T TT No. Square Cube Sq. Root Cube Root Circum. Area 121 14641 1771561 1 1 .OOOO 4.9461 380.13 1 1 499.O 122 14884 1815848 11.0454 4-9597 383.2 7 1 1 689 .9 123 15129 1860867 11.0905 4.9732 386.42 11882.3 124 15376 1906624 H-I355 4.9866 389.56 12076.3 125 15625 i953 I2 5 1 1. 1803 5 .OOOO 392.70 12271.8 126 15876 2000376 11.2250 5- OI 33 395.84 12469.O 127 16129 2048383 11.2694 5.0265 398.98 12667.7 128 16384 2097152 ^-3^37 5-°397 402.12 12868.0 129 1 664 1 2146689 ii-3578 5.0528 405.27 13069.8 130 16900 2197000 1 1. 401 8 5.0658 408.41 13273.2 131 17161 2248091 n-4455 5.0788 411.55 13478.2 132 17424 2299968 1 1. 489 1 5.0916 414.69 13684.8 *33 17689 2352637 11.5326 5- io 45 4I7.83 13892.9 134 17956 2406104 n.5758 5.1172 420.97 14102.6 135 18225 2460375 1 1. 6190 5.1299 424.12 I43I3-9 136 18496 2515456 1 1. 6619 5.1426 427.26 14526.7 137 18769 2571353 11.7047 5.i55i 430.40 14741.I 138 19044 2628072 n-7473 5.1676 433-54 I4957-I 139 19321 2685619 11.7898 5.1801 436.68 I5I747 140 19600 2744000 11.8322 5-!9 2 5 439.82 15393.8 141 19881 2803221 11.8743 5.2048 442.96 I56I4.5 142 20164 2863288 1 1 .9 1 64 5-2171 446.II 15836.8 143 20449 2924207 ii-9583 5-2293 449.25 16060.6 144 20736 2985984 12.0000 5-2415 452.39 16286.O 145 21025 3048625 12.0416 5.2536 455-53 16513.O 146 21316 3112136 12.0830 5-2656 458.67 16741.5 147 21609 3176523 12.1244 5.2776 461.81 16971.7 148 21904 3241792 12.1655 5.2896 464.96 17203.4 149 22201 33°7949 12.2066 5-3 OI 5 468.10 17436.6 150 22500 3375°°° 12.2474 5-3I33 471.24 17671.5 151 22801 344295 1 12.2882 5-325 1 474.38 17907.9 J 5 2 23104 3511808 12.3288 5.3368 477.52 18145.8 i53 23409 358i577 12.3693 5.3485 480.66 18385.4 i54 23716 3652264 12.4097 5-36oi 483.81 18626.5 i55 24025 3723875 12.4499 5.3717 486.95 18869.2 156 24336 3796416 12.4900 5-3832 490.09 19113.4 157 24649 3869893 12.5300 5-3947 493.23 19359-3 158 24964 39443 1 2 12.5698 5.4061 496.37 19606.7 i59 25281 4019679 12.6095 5-4175 499.51 I9855-7 160 25600 4096000 1 2 .649 1 5.4288 502.65 20106.2 468 GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 161 25921 4I7328I 12.6886 5.4401 505.8o 20358.3 162 26244 4251528 12.7279 5-4514 508.94 20612.0 163 26569 433°747 12.7671 5.4626 5 1 2 .08 20867.2 164 26896 4410944 12.8062 5-4737 5 J 5-22 21124.1 165 27225 4492125 12.8452 5.4848 518.36 21382.5 166 27556 4574296 12.8841 5-4959 521.50 21642.4 167 27889 4657463 12.9228 •5-5069 524.65 21904.0 168 28224 4741632 12.9,615 5.5178 527-79 22167. 1 169 28561 4826809 13.0000 5.5288 530.93 22431.8 170 28900 4913000 13.0384 5-5397 534.07 22698.0 171 29241 500021 1 13.0767 5-5505 537-21 22965.8 172 29584 5088448 13.1149 5-56i3 540.35 23235.2 173 29929 5I777I7 i3- I 5 2 9 5-5721 543-5° 23506.2 174 30276 5268024 13.1909 5.5828 546.64 23778.7 175 30625 5359375 13.2288 5-5934 549.78 24052.8 176 30976 5451776 13.2665 5.6041 552.92 24328.5 177 3 J 3 2 9 5545233 13-3041 5-6i47 556.o6 24605.7 178 31684 5639752 I3-34I7 5-6252 559-20 24884.6 179 32041 5735339 I3-379 1 5-6357 562.35 25164.9 180 32400 5832000 13.4164 5.6462 56549 25446.9 181 32761 5929741 I3-4536 5-6567 568.63 25730.4 182 33 I2 4 6028568 13.4907 5.6671 571.77 26015.5 183 33489 6128487 I3-5277 5-6774 574-9 1 26302.2 184 33856 6229504 I3-5647 5-6877 578.05 26^90.4 185 34225 6331625 13.6015 5.6980 581.19 26880.3 186 34596 6434856 13.6382 5-7083 584.34 271 7 1. 6 187 34969 6539 20 3 13.6748 5-7i85 587.48 27464.6 188 35344 6644672 i3-7ii3 5-7287 590.62 27759.1 189 35721 6751269 13-7477 5.7388 593.76 28055.2 190 36100 6859000 15.7840 5-7489 596.90 28352.9 191 36481 6967871 13.8203 5-759o 600.04 28652.1 192 36864 7077888 13.8564 5.7690 603.19 28952.9 193 37249 7189057 13.8924 5-779° 606.33 29255-3 194 37636 73 OI 3 8 4 13.9284 5.7890 609.47 29559-2 195 38025 7414875 13.9642 5-7989 612.61 29864.8 196 38416 7529536 14.0000 5.8088 6I5.75 301 7 1. 9 197 38809 7645373 14.0357 5.8186 618.89 30480.5 198 39204 7762392 14.0712 5.8285 622.04 30790.7 199 39601 7880599 14.1067 5.8383 625.18 31102.6 200 40000 8000000 14.1421 5.8480 628.32 3 r 4i5-9 SQUARES, CUBES AND ROOTS 469 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 201 40401 8120601 14.1774 5.8578 631.46 31730.9 202 40804 8242408 14.2127 5.8675 634.60 32047.4 203 41209 8365427 14.2478 5.8771 63774 32365.5 204 41616 8489664 14.2829 5.8868 640.89 32685.1 205 42025 8615125 14.3178 5.8964 644.03 33006.4 206 42436 8741816 14.3527 5.9059 647.17 33329.2 207 42849 8869743 14.3875 5.9I55 650.31 33653-5 208 432 64 8998912 14.4222 5.9250 65345 33979-5 209 43681 9129329 14.4568 5-9345 656.59 34307.0 210 44100 9261000 14.4914 5-9439 65973 34636.1 211 44521 9393931 14.5258 5-9533 662.88 34966.7 212 44944 9528128 14.5602 5.9627 666.02 35298.9 213 45369 9663597 14.5945 5.9721 669.16 35632.7 214 45796 9800344 14.6287 5.9814 672.30 35968.1 215 46225 9938375 14.6629 5.9907 67544 36305.0 2l6 46656 10077696 14.6969 6.0000 678.58 36643.5 217 47089 10218313 I47309 6.00Q2 681.73 36983.6 2l8 47524 10360232 14.7648 6.O185 684.87 37325.3 219 47961 10503459 14.7986 6.O277 688.01 37668.5 220 48400 10648000 • 14.8324 6.O368 691.15 38013.3 221 48841 10793861 14.8661 6.O459 694.29 38359-6 222 49284 10941048 14.8997 6.O55O 69743 38707.6 223 49729 I10S9567 I4.9332 6.064I 700.58 39057-I 2 24 50176 11239424 14.9666 6.O732 703.72 ■ 394o8.i 225 50625 11390625 15.0000 6.0822 706.86 3976o.8 226 51076 ii543!76 *5-°333 6.O912 710.OO 401 15.0 227 51529 1 1 69 7083 15.0665 6.I002 7 I 3-i4 40470.8 228 51984 11852352 15.0997 6.IOQI 716.28 40828.1 229 52441 12008989 i5-!327 6.Il8o 719.42 41187.1 23O 52900 1 2 167000 15.1658 6.I26C) 722.57 41547.6 23I 5336l 12326391 15.1987 6.I358 725.71 41909.6 232 53 8 24 12487168 15-2315 6.I446 728.85 42273.3 2 33 54289 I2 649337 15.2643 6.1534 73L99 42638.5 234 54756 12812904 15.2971 6.l622 735.13 43005-3 235 55225 12977875 J 5-3297 6.I7IO 738.27 43373-6 236 55696 13144256 15.3623 6.1797 741.42 43743-5 237 56169 !33i2053 15.3948 6.1885 744.56 44ii5-o 238 56644 13481272 15.4272 6.I972 747.7o 44488.1 239 57121 13651919 154596 6.2058 750.84 44862.7 240 57600 13824000 15.4919 6.2145 753.98 45238.9 47° GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 "NTr» Square Cube Sq. Root Cube Root Circle lNO. Circum. Area 241 58081 I399752I 15.5242 6.2231 757-12 45616.7 242 58564 14172488 I5-5563 6.2317 760.27 45996.1 243 59°49 14348907 15.5885 6.2403 76341 46377.0 244 59536 14526784 15.6205 6.2488 766.55 46759.5 245 60025 14706125 I5-6525 6.2573 769.69 47!43-5 246 60516 14886936 15.6844 6.2658 772.83 47529.2 247 61009 15069223 15.7162 6.2743 775-97 47916.4 248 61504 15252992 15.7480 6.2828 779.12 48305.1 249 62001 15438249 15.7797 6.2912 782.26 48695.5 250 62500 15625000 15,8114 6.2996 78540 49087.4 251 63001 15813251 15.8430 6.3080 788.54 49480.9 252 635 4 16003008 I5-8745 6.3164 791.68 49875-9 253 64009 16194277 15.9060 6.3247 794.82 50272.6 254 64516 16387064 15.9374 6 -333° 797.96 50670.7 255 65025 16581375 15.9687 6.3413 801. 11 5 io 7o-5 256 65536 16777216 16.0000 6.3496 804.25 5I47L9 257 66049 16974593 16.0312 6-3579 807.39 51874.8 258 66564 i7i735 J 2 16.0624 6.3661 810.53 52279.2 259 67081 17373979 16.0935 6-3743 813.67 52685.3 260 67600 17576000 16.1245 6.3825 816.81 53092.9 26l 68121 17779581 I6.I555 6.3907 819.96 53502.1 262 68644 17984728 16.1864 6.3988 823.10 53912.9 2 63 69169 18191447 16.2173 6.4070 826.24 54325.2 264 69696 18399744 16.2481 6.4151 829.38 54739-1 265 70225 18609625 16.2788 6.4232 832.52 55154.6 266 70756 18821096 16.3095 6.4312 835.66 55571.6 267 71289 19034163 16.3401 6-4393 838.81 55990.3 268 71824 19248832 16.3707 6-4473 841.95 56410.4 269 72361 19465109 16.4012 6-4553 845.09 56832.2 270 72900 19683000 16.4317 6.4633 848.23 57255.5 271 73441 19902511 16.4621 6.47J3 85I-37 57680.4 272 73984 20123648 16.4924 6.4792 854.51 58106.9 273 74529 20346417 16.5227 6.4872 857.66 58534.9 274 75076 20570824 16.5529 6.4951 860.80 58964.6 275 75625 20796875 16.5831 6.5030 863.94 59395-7 276 76176 21024576 16.6132 6.5108 867.08 59828.5 277 76729 21253933 16.6433 6.5187 870.22 60262.8 278 77284 21484952 16.6733 6.5265 873-36 60698.7 279 77841 21 71 7639 16.7033 6.5343 876.50 61136.2 280 78400 21952000 16.7332 6.5421 879.65 6i575-2 SQUARES, CUBES AND ROOTS 471 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 281 78961 22188041 16.7631 6.5499 882.79 62015.8 282 79524 22425768 16.7929 6-5577 885.93 62458.0 283 80089 22665187 16.8226 6.5654 889.07 62901.8 284 80656 22906304 16.8523 6-573 1 892.21 63347.1 285 81225 23149125 16.8819 6.5808 895.35 63794.0 286 81796 23393656 16.91 15 6.5885 898.50 64242.4 287 82369 236399 3 16.9411 6.5962 901.64 64692.5 288 82944 23887872 16.9706 6.6039 904.78 65144. 1 289 83521 24137569 1 7 .OOOO 6.6115 907.92 65597.2 290 84100 24389000 17.0294 6.6191 911.06 66052.O 291 84681 24642171 17.0587 6.6267 914.20 66508.3 292 85264 24897088 17.0880 6.6343 917.35 66966.2 293 85849 25153757 17.1172 6.6419 920.49 67425.6 294 86436 25412184 17.1464 6.6494 923.63 67886.7 295 87025 25672375 17.1756 6.6569 926.77 68349-3 296 87616 25934336 17.2047 6.6644 929.91 68813.5 297 88209 26198073 17.2337 6.6719 933.05 69279.2 298 88804 26463592 17.2627 6.6794 936.19 69746.5 299 89401 26730899 17.2916 6.6869 939-34 70215.4 300 90000 27000000 17.3205 6.6943 942.48 70685.8 301 90601 27270901 17.3494 6.7018 945.62 7II57.9 302 91204 27543608 I7.378I 6.7092 948.76 7I63I.5 3°3 91809 27818127 17.4069 6.7166 951.90 72106.6 3°4 92416 28094464 17.4356 6.7240 955.04 725834 3°5 93025 28372625 17.4642 6.73 J 3 958.19 73061.7 306 93636 28652616 17.4929 6.7387 96i.33 7354L5 3°7 94249 28934443 17.5214 6.7460 964.47 74023.0 308 94864 29218112 17-5499 6.7533 967.61 74506.0 3°9 9548i 29503629 I7.5784 6.7606 97°-75 74990.6 310 96100 29791000 17.6068 6:7679 973-89 75476.8 3 11 96721 30080231 17.6352 6.7752 977.04 75964.5 312 97344 30371328 17.6635 6.7824 980.18 76453.8 3*3 97969 30664297 17.6918 6.7897 983-32 76944.7 314 98596 30959144 17.7200 6.7969 986.46 77437-1 3i5 99225 31255875 17.7482 6.8041 989.60 7793I- 1 316 99856 31554496 17.7764 6.8113 992.74 78426.7 3i7 100489 31855013 17.8045 6.8185 995.88 78923.9 3i8 101124 32157432 17.8326 6.8256 999.03 79422.6 3i9 101761 32461759 17.8606 6.8328 1002.20 79922.9 320 102400 32768000 17.8885 6.8399 1005.30 80424.8 472 GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 321 103041 33076161 17.9165 6.8470 1008.5 80928.2 322 103684 33386248 17.9444 6.8^41 1011.6 81433.2 323 104329 33698267 17.9722 6.8612 1014.7 81939.8 3 2 4 104976 34012224 18.0000 6.8683 1017.9 82448.O 325 105625 34328125 18.0278 6-8753 1021.0 82957.7 326 106276 34645976 18.0555 6.8824 1024.2 83469.0 32 7 106929 34965783 18.0831 6.8894 1027.3 83981.8 328 107584 35287552 18.1108 6.8964 1030.4 84496.3 329 108241 35611289 18.1384 6.9034 1033.6 85012.3 33° 108900 35937000 18.1659 6.9104 1036.7 85529.9 33i 109561 36264691 18.1934 6.9174 1039.9 86049.0 332 110224 36594368 18.2209 6.9244 1043.0 86569.7 333 I 10889 36926037 18.2483 6.9313 1046.2 87092.O 334 III556 37259704 18.2757 6.9382 1049.3 87615.9 335 112225 37595375 18.3030 6.9451 1052.4 88141,3 336 112896 37933056 I8.3303 6.9521 I055-6 88668.3 337 H35 6 9 38272753 18.3576 6.9589 105.8.7 89196.9 33& I 14244 38614472 18.3848 6.9658 1061.9 89727.O 339 114921 38958219 18.4120 6.9727 1065.0 90258.7 34o 115600 39304000 18.4391 6.9795 1068. 1 ■ 90792.O 34i 116281 39651821 18.4662 6.9864 1071.3 91326.9 342 I 16964 40001688 18.4932 6.9932 1074.4 91863.3 343 1 1 7649 40353607 18.5203 7.0000 1077.6 92401.3 344 118336 40707584 18.5472 7.0068 1080.7 92940.9 345 H9 25 41063625 18.5742 7.0136 1083.8 93482.O 346 119716 41421736 18.6011 7.0203 1087.0 94024.7 347 120409 41781923 18.6279 7.0271 1090. 1 94569.O 348 121104 421.44192 18.6548 7-0338 I093-3 95114.9 349 121801 42508549 18.6815 7.0406 1096.4 95662.3 35o 122500 42875000 18.7083 7-°473 1099.6 96211.3 35i 123201 43243551 I8.7350 7.0540 1102.7 96761.8 352 123904 43614208 18.7617 7.0607 1105.8 97314.O 353 124609 43986977 18.7883 7.0674 1 109.0 978677 354 125316 44361864 18.8149 7.0740 1112.1 98423.O 355 126025 44738875 18.8414 7.0807 IH5-3 98979.8 356 126736 45118016 18.8680 7.0873 1118.4 99538.2 357 127449 45499293 18.8944 7.0940 1121.5 IOO098 358 128164 45882712 18.9209 7.1006 1124.7. 100660 359 128881 46268279 18.9473 7.1072 1 127.8 IOI223 360 129600 46656000 18.9737 7.1138 1131.0 IO1788 SQUARES, CUBES AND ROOTS 473 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 ClRft-t! No. Square Cube Sq. Root Cube Root Circum. Area 361 130321 47045881 19.0000 7.1204 1134*1 IO2354 362 131044 47437928 19.0263 7.1269 H37-3 IO2922 363 131769 47832147 19.0526 7-1335 1140.4 103491 364 132496 48228544 19.0788 7.1400 11430 IO4062 365 133225 48627125 19.1050 7.1466 1146.7 IO4635 366 133956 49027896 19.1311 7-I53I 1149.8 105209 367 134689 49430863 19.1572 7.1596 1153-0 105785 368 135424 49836032 19-1833 7.1661 1156.1 106362 369 136161 50243409 19.2094 7.1726 1159-2 106941 370 136900 50653000 I9-2354 7-I79I 1 1 62 .4 107521 371 137641 51064811 19.2614 7.1855 1165.5 108103 372 138384 51478848 19.2873 7.1920 1 1 68.7 108687 373 139129 5I895II7 19.3132 7.1984 1171.8 IO9272 374 139876 52313624 I9-339I 7.2048 1175.0 109858 375 140625 52734375 19.3649 7.2 1 12 1178.1 I 10447 376 141376 53157376 19.3907 7.2177 1181.2 IIIO36 377 142129 53582633 19.4165 7.224O 1 184.4 111628 378 142884 54010152 19.4422 7.2304 1187.5 II222I 379 143641 54439939 19.4679 7.2368 1190.7 II28I5 380 144400 54872000 19.4936 7.2432 H93-8 II34II 381 145161 553 634i 19.5192 7-2495 1 196.9 I I 4OO9 382 145924 55742968 19.5448 7.2558 1200. 1 I I4608 3*3 146689 56181887 19.5704 7.2622 1203.2 II5209 384 147456 56623104 19-5959 7.2685 1206.4 II5812 385 148225 57066625 19.6214 7.2748 1209.5 H6416 386 148996 57512456 19.6469 7.28ll 1212.7 I I 702 I 3*7 149769 57960603 19.6723 7.2874 1215.8 II7628 388 I5 544 58411072 19.6977 7.2936 1218.9 H8237 389 *5 1 3 21 58863869 19.7231 7.2999 1222. 1 I 18847 39o 152100 59319000 19.7484 7.3061 1225.2 I 19459 39i 152881 59776471 19-7737 7-3 I2 4 1228.4 120072 392 153664 60236288 19.7990 7.3186 1231-5 I20687 393 154449 60698457 19.8242 7.3248 1234.6 I21304 394 155236 61162984 19.8494 7.33io 1237.8 I2I922 395 156025 61629875 19.8746 7.3372 1240.9 122542 396 156816 62099136 19.8997 7-3434 1244. 1 I23163 397 157609 62570773 19.9249 7.3496 1247.2 I23786 398 158404 63044792 19.9499 7.3558 1250.4 I244IO 399 159201 63521199 19.9750 7.3619 1253.5 I25036 400 160000 64000000 20.0000 7.3684 1256.6 I25664 474 GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots. Circumferences, and Circular Areas of Nos. from i to 520 No. Square Cube Sq. Root Cube Root Circle Circum. Area 401 I 60801 64481 201 20.0250 7-3742 1259.8 126293 402 I 61 604 64964808 20.0499 7-3803 1262.9 126923 403 162409 6545 o8 27 20.0749 7.3864 1266.1 127556 404 163216 65939264 20.0998 7-3925 1269.2 128190 405 164025 66430125 20.1246 7.3986 1272.3 128825 406 164836 66923416 20.1494 7.4047 1275.5 129462 407 165649 67419143 20.1742 7.4108 1278.6 130100 408 166464 67917312 20.1990 7.4169 1281.8 I3°74I 409 167281 68417929 20.2237 7.4229 1284.9 131382 4.10 168100 68921000 20.2485 7.4290 1288. 1 132025 411 I 6892 I 69426531 20.2731 7-435° 1291.2 132670 412 169744 699345 2 8 20.2978 7.4410 1294.3 *333*7 413 170569 70444997 20.3224 7.4470 1297.5 133965 414 171396 70957944 20.3470 7-453° 1300.6 134614 415 172225 71473375 20.3715 7-459° l3°3-8 135265 416 I73 56 71991296 20.3961 7.4650 1306.9 i359i8 417 173889 72511713 20.4206 7.4710 1310.0 136572 418 174724 73034632 20.4450 7.4770 i3 I 3- 2 137228 419 I7556I 73560059 20.4695 7.4829 i3 l6 -3 137885 420 176400 74088000 20.4939 7.4889 i3 J 9-5 138544 421 177241 74618461 20.5183 7.4948 1322.6 139205 422 178084 75151448 20.5426 7-5°°7 1325-8 139867 423 178929 75686967 20.5670 7.5067 1328.9 i4°53 x 424. 179776 76225024 20.5913 7.5126 1332.0 141196 425 180625 76765625 20.6155 7-5i85 1335-2 141863 426 181476 77308776 20.6398 7.5244 1338.3 14253 1 427 182329 77854483 20.6640 7-53° 2 I34I-5 143201 428 183184 78402752 20.6882 7-536i 1344.6 143872 429 184041 78953589 20.7123 7.5420 1347-7 144545 43° 184900 79507000 20.7364 7-5478 i35°-9 145220 43i 185761 80062991 20.7605 7-5537 I354-0 145896 432 186624 80621568 20.7846 7-5595 1357-2 146574 433 187489 81182737 20.8087 7-5654 1360.3 147254 434 188356 81746504 20.8327 7-5712 1363-5 147934 435 189225 82312875 20.8567 7-577° 1366.6 148617 436 190096 82881856 20.8806 7.5828 1369.7 149301 437 190969 83453453 20.9045 7.5886 1372.9 149987 438 191844 84027672 20.9284 7-5944 1376.0 150674 439 192721 84604519 20.9523 7.6001 1379.2 *5 1 3 6 3 440 193600 85184000 20.9762 7.6059 1382.3 i52°53 SQUARES, CUBES AND ROOTS 475 Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 ClRPTTT No. Square 194481 Cube Sq. Root Cube Root Circum. | Area 441 8^766121 2 1 .OOOO 7.6117 I385-4 J 5 2 745 442 195364 86350888 21.0238 7.6174 1388.6 153439 443 196249 86938307 21.0476 7.6232 I39I-7 I54I34 444 197136 87528384 21.0713 7.6289 1394.9 154830 445 198025 88121125 21.0950 7.6346 1398.O i555 2 8 446 198916 88716536 21.1187 7.6403 1401.2 156228 447 199809 89314623 21.1424 7.6460 1404.3 156930 448 200704 89915392 21.1660 7-65I7 1407.4 157633 449 201601 90518849 21.1896 7-6574 1410.6 158337 45° 202500 91125000 21.2132 7.6631 I4I3-7 159043 45i 203401 9i733 8 5i 21.2368 7.6688 1416.9 i5975i 45 2 204304 92345408 21.2603 7.6744 1420.O 160460 453 205209 92959677 21.2838 7.6801 I423-I 161171 454 206116 93576664 21.3073 7.6857 1426.3 161883 455 207025 94I9 6 375 21.3307 7.6914 1429.4 162597 456 207936 94818816 21.3542 7.6970 1432.6 163313 457 208849 95443993 21.3776 7.7026 1435-7 164030 458 209764 96071912 2 1 .4OO9 7.7082 1438.9 164748 459 210681 96702579 2I.4243 7-7I38 1442.0 165468 460 21 l6oO 97336000 2I.4476 7-7 J 94 I445- 1 166190 461 2I252I 97972181 2I.4709 7.7250 1448.3 166914 462 213444 98611128 2I.4942 7.7306 i45 x -4 167639 463 214369 99252847 21.5174 7.7362 1454.6 168365 464 215296 99897344 2I.5407 7.7418 1457-7 169093 465 216225 100544625 2I.5639 7-7473 1460.8 169823 466 217156 101194696 2I.587O 7-7529 1464.0 170554 467 218089 101847563 2I.6I02 7-7584 1467. 1 171287 468 219024 102503232 21.6333 7-7639 i47o-3 172021 469 219961 103161709 2I.6564 7-7695 1473-4 172757 470 2209OO 103823000 2I.6795 7-775o 1476.5 173494 47i 22184I 104487111 21.7025 7-7805 1479-7 174234 472 222784 105154048 21.7256 7.7860 1482.8 174974 473 223729 105823817 2I.7486 7-7915 1486.0 175716 474 224676 106496424 2I.7715 7.7970 1489. 1 176460 475 225625 107171875 21-7945 7.8025 1492.3 177205 476 226576 107850176 2I.8174 7.8079 1495-4 177952 477 227529 108531333 21.8403 7-8i34 1498.5 178701 478 228484 I0 92i535 2 2I.8632 7.8188 1501.7 i7945i 479 22944I 109902239 2I.886I 7.8243 1504.8 180203 480 2304OO 110592000 2I.9089 7.8297 1508.0 180956 476 GENERAL REFERENCE TABLES Squares, Cubes, Square Roots, Cube Roots, Circumferences and Circular Areas of Nos. from i to 520 No Square Cube Sq. Root Cube Root Circle J.NV.J. Circum. Area 48l 231361 111284641 21.9317 7.8352 1511.I 181711 482 2323 2 4 111980168 2L9545 7.8406 I.5I4.3 182467 483 233289 112678587 21.9773 7.8460 I5I74 183225 484 234256 H33799°4 22.0000 7.85I4 1520.5 183984 48S 235225 114084125 22.0227 " 7.8568 15237 184745 486 236196 114791256 22.0454 7.8622 1526.8 185508 487 237169 II 55oi3°3 22.0681 7.8676 I530.0 186272 488 238144 116214272 22.0907 7.8730 I 533-i 187038 489 239121 1 1 69301 69 22.1133 7.8784 1536.2 187805 490 240100 1 1 7649000 22.1359 7.8837 15394 188574 491 241081 118370771 22.1585 7.8891 1542.5 189345 492 242064 1 19095488 22.1811 7.8944 1545.7 190117 493 243049 119823157 22.2036 7.8998 1548.8 190890 494 244036 120553784 22.2261 7.905 1 I55L9 191665 495 245025 121287375 22.2486 7.9I05 I555-I 192442 496 246016 122023936 22.2711 7.9158 1558.2 193221 497 247009 122763473 22.2935 7.921 1 1561.4 194000 498 248004 I2 35°5992 22.3159 7.9264 1564.5 194782 499 249001 124251499 22.3383 7.9317 1567.7 195565 500 250000 125000000 22.3607 7.9370 1570.8 196350 5oi 251001 125751501 22.3830 7.9423 1573-9 197136 502 252004 126506008 22.4054 7.9476 I577-I 197923 503 253009 127263527 22.4277 7.9528 1580.2 198713 5°4 254016 128024064 22.4499 7.958i 1583-4 199504 505 255025 128787625 22.4722 7.9634 1586.5 200296 506 256036 129554216 22.4944 7.9686 1589.7 201090 5o7 257049 130323843 22.5167 7-9739 1592.8 201886 508 258064 131096512 22.5389 7.9791 1595-9 202683 5o9 259081 131872229 22.5610 7-9843 I599-I 203482 5io 260100 13 265 1000 22.5832 7.9896 1602.2 204282 5ii 261121 13343283 1 22.6053 7.9948 1605.4 205084 512 262144 134217728 22.6274 8.0000 1608.5 205887 5i3 263169 i35 00 5 6 97 22.6495 8.0052 1611.6 206692 5i4 264196 135790744 22.6716 8.0104 1614.8 207499 5i5 265225 136590875 22.6936 8.0156 1617.9 208307 5i6 266256 137388096 22.7156 8.0208 1621.1 209117 5i7 267289 138188413 22.7376 8.0260 1624.2 209928 5i8 268324 138991832 22.7596 8.03 1 1 1627.3 210741 21155° 5i9 269361 139798359 22.7816 8.0363 1630.5 520 270400 140608000 22.8035 8.0415 1633.6 212372 TRIGONOMETRIC FUNCTIONS 477 Table 59. Trigonometric Functions and the Solution of Triangles In the accompanying figure the trig- onometric functions of the angle A between the lines B A and A C are as follows; sin A cos A tan cot sec cosec ex-sec = BC = AC = EF = G H = A E = A H = B E In the right-angled triangle ABC let a equal the side B C opposite the angle A; let b equal the side A C opposite the angle B; let equal A B, the side opposite the angle C. Let C = oo° The following formulae apply to right-angled triangles: Angles. A+B +C = 180 A + B = oo° A = 90 .- B B = go°~A Sides. sin A cos A tan A A rea a b area 2 --c sin A = b tan A V(c+b)(c-b$ c cos. A = tan A b = V (c + a) (c — a) _ a b sin A cos A c = V 7 a 2 + b 2 B 1 Oblique Triangles. Note. Where an angle is more than 90 its sine, cosine, and tangent are equal to that of the angle (180 — the angle in question); that is, if the sine of 120 is desired take the sine of (180 - 120 ) = 6o°. B / Let K=Area \ 478 GENERAL REFERENCE TABLES Given A,B, a Desired A, a, b C, a, b C,b c, K B, C i(A+B) i(A-B) A B K Formulae C = 1 80 — (A + B); b a sin B c = a sin A sin (.4 +£);#= sin .4 a 2 sin 5 sin C 2 sin ^4 sin 5= ^-^ 6; C = 180 - (A + 5) c = sin A sin C Two solutions are possible with B' as an acute angle and B as an obtuse angle i{A+B) =90° - \C tan \{A-B) = a + b A=l(A+B) + i(A-B) B = -\{A+B) - Hi ~B) sin i(A+B) tan i (A+B) c = (a — b) sin i (A - B) K = h ab sin C a, by c B K In the following formula s = \ (a + b + c) sin \ B - a) - c) sin B 2 y/ s (s — a) (s — b) (s — c) ac K = V s -a) (s - b) (s - c) NATURAL TANGENTS AND CO-TANGENTS 479 Table 60 Tan. Co-tan. .00000 Infinite. .00029 3437-750 .00058 1718.870 .00087 1145.920 .00116 859-436 .00145 687.549 .00175 572.957 .00204 491.106 .00233 429.718 .00262 381.971 .00291 343-774 .00320 312.521 .00349 286.478 •00378 264.441 .00407 245-552 .00436 229.182 .00465 214.858 .00495 202.219 .00524 190.984 •00553 180.932 .00582 171.885 .00611 163.700 .00640 156.259 .00669 149.465 .00698 143-237 .00727 137-507 .00756 132.219 .00785 127.321 .00814 122.774 .00844 118.540 .00873 114-589 .00902 110.892 •00931 107.426 .00960 104. 171 .00989 101.107 .01018 98.2179 .01047 95-4895 .01076 92.9085 .01105 90-4633 .01135 88.1436 .01164 85-9398 .01193 83.8435 .01222 81.8470 .01251 79-9434 .01280 78.1263 •01309 76.3900 •01338 74.7292 •01367 73-I390 .01396 71-6151 .01425 70.I533 •01455 68.7501 .01484 67.4019 •01513 66.1055 .01542 64.8580 .01571 63.6567 .01600 62.4992 .01629 61.3829 .01658 60.3058 .01687 592659 .01716 58.2612 .01746 57.2900 Co-tan. Tan. 1 $9° 1 2 O 3 Tan. Co-tax. Tan. CO -TAN. Tan. Co-tan. .01746 57.2900 .03492 28.6363 .05241 19.0811 .01775 56-3506 .03521 28.3994 .05270 18.9755 .01804 55-4415 •03550 28.1664 .05299 18.8711 .01833 54-5613 •03579 27.9372 .05328 18.7678 .01862 53.7086 .03609 27.7117 .05357 18.6656 .01891 52.8821 .03638 27.4899 •05387 18.5645 .01920 52.0807 .03667 27.2715 .05416 18.4645 .01949 51-3032 .03696 27.0566 •05445 18.3655 .01978 50.5485 .03725 26.8450 •05474 18.2677 .02007 49.8157 •03754 26.6367 •05503 18.1708 .02036 49.1039 •03783 26.4316 •05533 18.0750 .02066 48.4121 .03812 26.2296 .05562 17.9802 .02095 47-7395 .03842 26.0307 •05591 17.8863 .02124 47-0853 .03871 25.8348 .05620 17-7934 .02153 46.4489 .03900 25.6418 .05649 17.7015 .02182 45.8294 .03929 25.45I7 .05678 17.6106 .02211 45.2261 .03958 25.2644 •05708 17-5205 .02240 44.6386 .03987 25.0798 •05737 I7-43I4 .02269 44.0661 .04016 24.8978 .05766 17-3432 .02298 43-5o8i .04046 24.7I85 •05795 17-2558 .02328 42.9641 .04075 24.5418 .05824 17.1693 •02357 42-4335 .04104 24.3675 •05854 17.0837 .02386 41.9158 •04133 24.1957 .05883 16.9990 .02415 41.4106 .04162 24.0263 .05912 16.9150 .02444 40.9174 .04191 23-8593 .05941 16.8319 .02473 40.4358 .04220 23-6945 .05970 16.7496 .02502 39-9655 .04250 23-5321 .05999 16.6681 •02531 39.5059 .04279 23-37I8 .06029 16.5874 .02560 39.0568 .04308 23.2137 .06058 16.5075 .02589 38.6177 •04337 23-0577 .06087 16.4283 .02619 38.1885 •04366 22.9038 .06116 16.3499 .02648 37.7686 •04395 22.7519 .06145 16.2722 .02677 37-3579 .04424 22.6020 .06175 16.1952 .02706 36.9560 •04454 22.4541 .06204 16.1190 •02735 36.5627 •04483 22.3081 •06233 16.0435 .02764 36.1776 .04512 22.1640 .06262 15-9687 •02793 35.8006 .04541 22.0217 .06291 I5-8945 .02822 35.43I3 •04570 21.8813 .06321 15.8211 .02851 35-0695 .04599 21.7426 .06350 I5-7483 .02881 34.7I5I .04628 21.6056 .06379 15.6762 .02910 34-3678 .04658 21.4704 .06408 15.6048 .02939 34-0273 .04687 21.3369 .06437 I5-5340 .02968 33-6935 .04716 21.2049 .06467 15-4638 .02997 33.3662 •04745 21.0747 .06496 15-3943 .03026 33-0452 •04774 20.9460 •06525 I5.3254 .03055 32.7303 .04803 20.8188 •06554 I5.257I .03084 32.4213 .04832 20.6932 .06584 I5-I893 .03114 32.1181 .04862 20.5691 .06613 15.1222 •03143 31.8205 .04891 20.4465 .06642 I5-0557 .03172 31.5284 .04920 20.3253 .06671 14.9898 .03201 31.2416 .04949 20.2056 .06700 14.9244 •03230 30.9599 .04978 20.0872 •06730 14.8596 •03259 30.6833 .05007 19.9702 •06759 14-7954 .03288 30.4116 .05037 19.8546 .06788 14.7317 •03317 30.1446 .05066 19.7403 .06817 14.6685 •03346 29.8823 •05095 19.6273 .06847 14-6059 •03376 29.6245 .05124 I9-5I56 .06876 I4.5438 .03405 29.3711 •05153 19.4051 .06905 14-4823 •03434 29.1220 .05182 19.2959 .06934 14.4212 •03463 28.8771 .05212 19.1879 .06963 14.3607 .03492 28.6363 .05241 19.0811 .06993 14.3007 Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. I s 8° 1 8 7° 8 6° 480 NATURAL TANGENTS AND CO-TANGENTS 4 Tan. CO-TAN. .06993 I4.3007 .07022 14.24H .07051 14.1821 .07080 14-1235 .07110 14-0655 .07139 14.OO79 .07168 I3-9507 .07197 13.8940 .07227 13.8378 .07256 13.7821 .07285 13.7267 .073M 13.6719 -07344 13.6174 •07373 I3.5634 .07402 13.5098 .07431 13.4566 .07461 I3-4039 .07490 I3.35I5 .07519 13.2996 .07548 13.2480 .07578 13.1969 .07607 I3.I46I .07636 13.0958 .07665 13.0458 .07695 12.9962 .07724 12.9469 .07753 12.8981 .07782 12.8496 .07812 12.8014 .07841 12.7536 .07870 12.7062 .07899 I2.659I .07929 12.6124 .07958 12.5660 .07987 12.5199 .08017 12.4742 .08046 12.4288 .08075 12.3838 .08104 12.3390 .08134 12.2946 .08163 12.2505 .08192 12.2067 .08221 12.1632 .08251 I2.I20I .08280 I2.0772 .08309 I2.O346 •08339 II.9923 .08368 II.9504 •08397 II.9087 .08427 H.8673 .08456 II.8262 .08485 H.7853 .08514 11.7448 •08544 II.7045 .08573 II.6645 .08602 II.6248 .08632 11.5853 .08661 II- 546l .08690 H.5072 .08720 II.4685 .08749 1 1. 43OI Co-tan. Tan. 8 5° 5 6 7 Tan. 1 Co-tan. Tan. Co-tan. Tan. Co-tan. .08749 1 1. 4301 .10510 9-51436 .12278 8.14435 .08778 11-3919 .10540 9.48781 .12308 8.12481 .08807 11-3540 •10569 9.46141 .12338 8.10536 .08837 11-3163 •10599 9-43515 .12367 8.08600 .08866 11.2789 .10628 9.40904 .12397 8.06674 .08895 11. 2417 .10657 9-38307 .12426 8.04756 .08925 11.2048 .10687 9-35724 .12456 8.02848 .08954 11.1681 .10716 9-33154 .12485 8.00948 .08983 11.1316 .10746 9-30599 •12515 7.99058 .09013 11.0954 •10775 9.28058 •12544 7.97176 .09042 11.0594 .10805 9-25530 •12574 7-95302 .09071 11.0237 .10834 9.23016 .12603 7-93438 .09101 10.9882 .10863 9.20516 .12633 7-91582 7-89734 .09130 10.9529 .10893 9.18028 .12662 .09159 10.9178 .10922 9-15554 .12692 7.87895 .09189 10.8829 .10952 9-I3093 .12722 7.86064 .09218 10.8483 .10981 9.10646 .12751 7.84242 .09247 10.8139 .IIOII 9.08211 .12781 7.82428 .09277 10.7797 .11040 9.05789 .12810 7.80622 .09306 io.7457 .11070 9-Q3379 .12840 7.78825 •09335 10.7119 . 1 1099 9.00983 .12869 7-77035 •09365 10.6783 .11128 8.98598 .12899 7-75254 •09394 10.6450 .11158 8.96227 .12929 7.73480 .09423 10.6118 .11187 8.93867 .12958 7-71715 •09453 10.5789 .11217 8.91520 .12988 7-69957 .09482 10.5462 .11246 8.89185 .13017 7.68208 .09511 10.5136 .11276 8.86862 .13047 7.66466 •09541 10.4813 •11305 8.84551 .13076 7.64732 •09570 10.4491 •H335 8.82252 .13106 7.63005 .09600 10.4172 .11364 8.79964 •13136 7.61287 .09629 10.3854 •II394 8.77689 .13165 7.59575 .09658 10.3538 .11423 8.75425 •13195 7.57872 .09688 10-3224 .11452 8.73172 .13224 7.56176 .09717 10.2913 .11482 8.70931 .13254 7-54487 .09746 10.2602 .11511 8.68701 •13284 7.52806 .09776 10.2294 .11541 8.66482 .13313 7-51132 .09805 10.1988 .11570 8.64275 •13343 7-49465 .09834 10.1683 .11600 8.62078 .13372 7.47806 .09864 10.1381 .11629 8.59893 .13402 7.46154 .09893 10.1080 .11659 8.57718 •13432 7-44509 •09923 10.0780 .11688 8-55555 .13461 7.42871 .09952 10.0483 .11718 8.53402 •13491 7.41240 .09981 10.0187 .11747 8.51259 •13521 7.39616 .10011 9.98931 .11777 8.49128 •13550 7-37999 .10040 9.96007 .11806 8.47007 , .13580 7.36389 .10069 9.93101 .11836 8.44896 1 .13609 7.34786 .10099 9.902 1 1 .11865 8.42795 1 .13639 7-33I90 .10128 9-87338 .11895 8.40705 .13669 7.31600 .10158 9.84482 .11924 8.38625 .13698 7.30018 .10187 9.81641 •II954 8.36555 1 .13728 7.28442 .10216 9.78817 •1 1983 8.34496 .13758 7.26873 .10246 9.76009 .12013 8.32446 .13787 7.25310 .10275 9.73217 .12042 8.30406 .13817 7-23754 .10305 9.70441 .12072 8.28376 .13846 7.22204 .10334 9.67680 .12101 8.26355 .13876 7.20661 •10363 964935 .12131 8-24345 .13906 7-I9I25 .10393 9.62205 .12160 8.22344 -13935 7-17594 .10422 9.59490 .12190 8.20352 .13965 7.16071 .10452 9.56791 .12219 8.18370 .13995 7-14553 .10481 9.54106 .12249 8.16398 .14024 7.13042 .10510 9-51436 .12278 8.14435 -14054 7.H537 Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. 1 84° 8 3° 8^ 1° NATURAL TANGENTS AND CO-TANGENTS 481 8 2> 9 3 10° 11° / Tan. CO-TAN. Tan. CO-TAN. Tan. Co-tan. Tan. Co-tan. / .14054 7-H537 .15838 6.51375 •17633 5.67128 .19438 5-14455 60 I .14084 7.10038 .15868 6.30189 .17663 5-66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5-65205 .19498 5.12862 58 3 .14143 7.07059 .15928 6.27829 •17723 5.64248 .19529 5.12069 57 4 .14173 7-Q5579 .15958 6.26655 •17753 5-63295 .19559 5-11279 56 5 .14202 7.04105 .15988 6.25486 .17783 5-62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24321 .17813 5-6i397 .19619 5.09704 54 7 .14262 7-01174 .16047 6.23160 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5-595H .19680 5.08139 52 9 .14321 6.98268 .16107 6.20851 .17903 5-58573 .19710 5.07360 51 10 •14351 6.96823 .16137 6.19703 .17933 5-57638 .19740 5.06584 50 11 .14381 6.95385 -16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 6.I74I9 •17993 5-55777 .19801 5-05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.04267 47 14 .14470 6.91104 .16256 6.I5I5I .18053 5-53927 .19861 5-03499 46 15 .14499 6.8*9688 .16286 6.14023 .18083 5-53007 .19891 5-02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6-86874 .16346 6.II779 .18143 5-51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.840S2 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5-48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4-97438 38 23 • 14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 .14767 6.77199 •16555 6.O4051 .18353 5.44857 .20164 4-95945 36 25 .14796 6.75838 .16585 6.02962 .18383 5.43966 .20194 4-952QI 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.OO797 .18444 5.42192 .20254 4-93721 33 28 .14886 6.71789 .16674 5.99720 .18474 5-41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 •20315 4.92249 31 30 •14945 6.69116 •16734 5.97576 .18534 5-39552 •20345 4-9i5i6 30 3i .14975 6.67787 .16764 5.965IO .18564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5-37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5-36936 .20436 4.89330 27 34 • 15064 6.63831 .16854 5-93335 .18654 5.36070 .20466 4.83605 26 35 .15094 6.62523 .16884 ' 5-92283 .18684 5-35206 •20497 4.87882 25 36 .15124 6.61219 .16914 5.91235 .18714 5-34345 .20527 4.87162 24 37 •15153 6.59921 .16944 5-90191 .18745 5-33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5-89151 .18775 5-32631 .20588 4.85727 22 39 .15213 6-57339 .17004 5.88114 .18805 5.3I778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 4i .15272 6-54777 .17063 5.86051 .18865 5.30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5-29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5-28393 •20739 4-82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5-27553 .20770 4.81471 16 45 • 15391 6.49710 .17183 5.81966 .18986 5-26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5-79944 .19046 5.25048 .20861 4-79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 •15511 6.44720 •17303 5.77936 .19106 5-23391 .20921 4.77978 11 50 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 10 51 .15570 6.42253 .17363 5-75941 .19166 5-21744 .20982 4.76595 9 52 .15600 6.41026 • 17393 5-74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5-7396o .19227 5.20107 .21043 4.75219 7 54 .15660 6.38587 •17453 5.72974 .19257 5.19293 .21073 4-74534 6 55 .15689 6-37374 .17483 5.71992 .19287 5.18480 .21104 4-73851 5 56 .15719 6.36165 •17513 5-71013 .19317 5-17671 .21134 4.73I70 4 57 •15749 6.34961 • 17543 5-70037 .19347 5-16863 .21164 4.72490 3 58 •15779 6.33761 • 17573 5.69064 •19378 5.16058 .21195 4-71813 2 59 .15809 6.32566 .17603 5.68094 .19408 5.15256 .21225 4.7II37 1 60 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 / Co-tan. Tan. Co-tan . Tan. Co-tan. Tan. Co~tan. Tan. / 8 1° 8 0° i 7 9° I 7 3° J 482 NATURAL TANGENTS AND CO-TANGENTS 12° 13° 14° 15° / Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. .21256 4.70463 .23087 4-33148 •24933 4.01078 .26795 3-73205 I .21286 4.69791 .23117 4.32573 .24964 4.00582 .26826 3.72771 2 .21316 4.69121 .23148 4.32001 .24995 4.00086 .26857 3-7233^ 3 .21347 4.68452 .23179 4-31430 .25026 3-99592 .26888 3.71907 4 .21377 4.67786 .23209 4.30860 •25056 3.99099 .26920 3-71476 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3-71046 6 .21438 4.66458 .23271 4.29724 .25118 3-98ii7 .26982 3.70616 7 .21469 4-65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 8 .21499 4-65138 ■23332 4-28595 .25180 3-97139 .27044 3-69761 9 .21529 4.64480 -23363 4.28032 .25211 3.96651 .27076 3-69335 10 .21560 4-63825 •23393 4.27471 .25242 3.96165 .27107 3.68909 11 .21590 4.631 71 .23424 4.26911 •25273 3.95680 .27138 3.68485 12 .21621 4.62518 .23455 4.26352 •25304 3-95I96 .27169 3.68061 13 .21651 4.61868 .23485 4-25795 •25335 3-94713 .27201 3.67638 14 .21682 4.61219 .23516 4-25239 .25366 3-94232 .27232 3.67217 15 .21712 4.60572 •23547 4.24685 •25397 3-93751 .27263 3.66796 16 .21743 4-59927 ■23578 4.24132 .25428 3-93271 .27294 3.66376 17 •21773 4.59283 .23608 4.23580 •25459 3-92793 .27326 3-65957 18 .21804 4.58641 •23639 4.23030 .25490 3.92316 .27357 3-65538 19 .21834 4.58001 .23670 4.22481 .25521 3-91839 .27388 3-65121 20 .21864 4-57363 .23700 4-21933 •25552 3.91364 .27419 3.64705 21 .21895 4.56726 .23731 4.21387 •25583 3.90890 .27451 3.64289 22 •21925 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 23 .21956 4-55458 •23793 4.20298 •25645 3-89945 , -27513 3.63461 24 .21986 4.54826 .23823 4.19756 .25676 3-89474 1 -27545 3.63048 25 .22017 4.54196 •23854 4.19215 .25707 3.89004 1 .27576 3.62636 26 .22047 4.53568 .23885 4.18675 .25738 3.88536 1 .27607 3.62224 27 .22078 4.52941 .23916 4-18137 .25769 3.88068 .27638 3.61814 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 29 .22139 4-51693 .23977 4.17064 .25831 3-87136 .27701 3.60996 30 .22169 4-51071 .24008 4.16530 .25862 3.86671 .27732 3-60588 3i .22200 4-50451 .24039 4-15997 •25893 3.86208 •27764 3.60181 32 .22231 4.49832 .24069 4-I5465 .25924 3-85745 •27795 3-59775 33 .22261 4.49215 .24100 4-14934 -25955 3-85284 .27826 3.59370 34 .22292 4.48600 .24131 4.14405 .25986 .3.84824 .27858 3.58966 35 .22322 4.47986 .24162 4.13877 .26017 3-84364 .27889 3-58562 36 .22353 4-47374 .24193 4-I3350 .26048 3.83906 .27920 3.58160 37 .22383 4.46764 .24223 4.12825 .26079 3-83449 .27952 3-57758 38 .22414 4-46155 .24254 4.12301 .26110 3.82992 .27983 3-57357 39 .22444 4-45548 .24285 4-II778 .26141 3-82537 .28015 3-56957 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3-56557 4i .22505 4-44338 •24347 4.10736 .26203 3.81630 .28077 3-56i59 42 .22536 4-43735 •24377 4.10216 .26235 3.8ii77 .28109 3-5576i 43 .22567 4-43134 .24408 4.09699 .26266 3.80726 .28140 3-55364 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3-54968 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3-54573 46 .22658 4.41340 .24501 4.08152 •26359 3-79378 .28234 3-54179 47 .22689 4-40745 •24532 4.07639 .26390 3-78931 .28266 3-53785 *8 .22719 4.40152 .24562 4.07127 .26421 3-78485 .28297 3-53393 \9 .22750 4-3956o .24593 4.06616 .26452 3.78040 .28329 3-53QOI 50 .22781 4.38969 .24624 4.06107 .26483 3-77595 .28360 3.52609 ;i .22811 4.38381 .24655 4-05599 •26515 3-77I52 .28391 3.52219 52 .22842 4-37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 53 .22872 4.37207 .24717 4.04586 .26577 3.76268 .28454 3-5I44I 54 .22903 4.36623 .24747 4.04081 .26608 3-75828 .28486 3-51053 >5 -22934 4.36040 .24778 4-03578 .26639 3.75388 .28517 3.50666 ;6 .22964 4-35459 .24809 4-03075 .26670 3-7495Q .28549 3-50279 57 .22995 4.34879 .24840 4.02574 .26701 3-74512 .28580 3.49894 ;8 .23026 4-34300 .24871 4.02074 .26733 3-74075 .28612 3-49509 J9 .23056 4-33723 .24902 4-OI576 .26764 3-73640 .28643 3-49125 )0 .23087 4-33I48 •24933 4.01078 .26795 3-73205 .28675 3.48741 / Co-tan. Tan. CO-TAN. Tan. Co-tan. Tan. Co-tan. Tan. Ti TO 76 ;° 71 )° 7< t° NATURAL TANGENTS AND CO-TANGENTS 483 16° 17° | 18° 19° / Tan. CO -TAN. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. / .28675 3.48741 .30573 3.27085 .32492 3-07768 •34433 2.90421 6 I .28706 3-48359 .30605 3.26745 •32524 3.07464 •34465 2.90147 5 2 .28738 3-47977 .30637 3.26406 .32556 3.07160 •34498 2.89873 5 3 .28769 3-47596 .30669 3.26067 .32588 3.06857 .34530 2.89600 5 4 .28800 3.47216 .30700 3-25729 .32621 3-06554 •34563 2.89327 5 5 .28832 3-46837 •30732 3-25392 •32653 3.06252 •34596 2.89055 5 6 .28864 3-46458 •30764 3-25055 .32685 3-05950 .34628 2.88783 5 7 .28895 3.46080 .30796 3.24719 •32717 3-05649 •34661 2.88511 5 8 .28927 3-45703 .30828 3-24383 •32749 3-Q5349 •34693 2.88240 5 9 .28958 3-45327 .30860 3.24049 •32782 3-05049 .34726 2.87970 5 10 .28990 3-44951 .30891 3-23714 •32814 3-04749 •34758 2.87700 5 11 .29021 3-44576 .30923 3.23381 .32846 3-04450 •34791 2.87430 4 12 •29053 3.44202 •30955 3.23048 .32878 3.04152 .34824 2.87161 4 13 .29084 3-43829 •30987 3.22715 .32911 3-03854 .34856 2.86892 4 14 .29116 3-43456 .31019 3.22384 •32943 3-03556 .34889 2.86624 4 15 .29147 3-43084 .31051 3-22053 •32975 3.03260 .34922 2.86356 4 16 .29179 3-427I3 .31083 3.21722 •33007 3.02963 •34954 2.86089 4 17 .29210 3-42343 •31115 3-21392 •33040 3.02667 •34987 2.85822 4 18 .29242 3-41973 •3H47 3.21063 •33072 3.02372 •35019 2.85555 4 19 .29274 3.41604 .31178 3-20734 .33104 3.02077 •35052 2.85289 4 20 .29305 3-41236 .31210 3.20406 •33136 3-01783 .35085 2.85023 4< 21 •29337 3.40869 .31242 3.20079 .33160 3.01489 .35H7 2.84758 3< 22 .29368 3.40502 •31274 3-19752 •33201 3.01196 •35i5o 2.84494 3^ 23 .29400 3.40136 .31306 3.19426 -33233 3.00903 .35183 2.84229 3 24 .29432 3-39771 .31338 3.19100 .33266 3.0061 1 .35216 2.83965 3< 25 .29463 3-394o6 •31370 3-18775 •33298 3.00319 •35248 2.83702 3. 26 .29495 3-39042 .31402 3-18451 -33330 3.00028 .35281 2.83439 3- 27 .29526 3-38679 •31434 3-18127 •33363 2.99738 .35314 2.83176 3. 28 •29558 3-383I7 .31466 3-17804 •33395 2-99447 •35346 2.82914 3- 29 .29590 3-37955 .31498 3-17481 •33427 2.99158 •35379 2.82653 3^ 30 .29621 3-37594 .31530 3-17159 •3346o 2.98868 •35412 2.82391 3C 31 .29653 3-37234 .31562 3-16838 •33492 2.98580 •35445 2.82130 2C 32 .29685 3-36875 •31594 3-16517 •33524 2.98292 •35477 2.81870 2* 33 .29716 3-36516 .31626 3-16197 •33557 2.98004 •355IO 2.81610 2; 34 .29748 3-36158 .31658 3-15877 •33589 2.97717 •35543 2.81350 2t 35 .29780 3-358oo .31690 3-I5558 •33621 2.97430 •35576 2.81091 2« 36 .29811 3-35443 .31722 3.15240 •33654 2.97144 .35608 2.80833 2A 37 .29843 3-35o87 .31754 3.14922 •33686 2.96858 •35641 2.80574 22 38 .29875 3-34732 .31786 3.14605 .33718 2-96573 •35674 2.80316 22 39 .29906 3-34377 .31818 3.14288 •33751 2.96288 •35707 2.80059 21 40 .29938 3-34023 .31850 3-13972 •33783 2.96004 .35740 2.79802 2C 4i .29970 3-33670 .31882 3-13656 .33816 2.95721 •35772 2-79545 IS 42 .30001 3-33317 .31914 3-i334i •33848 2-95437 .35805 2.79289 18 43 •30033 3-32965 .31946 3.13027 .33881 2-95155 .35838 2.79033 17 44 .30065 3-32614 •31978 3-12713 .33913 2.94872 .35871 2.78778 16 45 •30097 3-32264 .32010 3.12400 •33945 2.94590 .35904 2.78523 15 46 .30128 3-3I9I4 .32042 3.12087 •33978 2.94309 •35937 2.78269 14 47 .30160 3-3I565 •32074 3.II775 •34010 2.94028 •35969 2.78014 13 48 •30192 3-3i2i6 .32106 3-11464 •34043 2.93748 .36002 2.77761 12 49 .30224 3-30868 •32139 3-IH53 •34075 2.93468 •36035 2.77507 II 5o •30255 3-3052I .32171 3.10842 .34108 2.93189 .36068 2.77254 10 5i .30287 3-30I74 .32203 3-10532 .34140 2.92910 .36101 2.77002 9 52 •30319 3.29829 •32235 3.10223 •34173 2.92632 •36134 2.76750 S 53 •30351 3-29483 .32267 3.09914 •34205 2.92354 .36167 2.76498 7 54 .30382 3-29I39 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3-28795 •32331 3.09298 .34270 2.91799 •36232 2.75996 5 56 .30446 3.28452 .32363 3.08991 •34303 2.91523 .36265 2.75746 A 57 .30478 3.28109 .32396 3-08685 •34335 2.91246 .36298 2.75496 3 58 •30509 3.27767 .32428 3-08379 .34368 2.90971 •36331 2.75246 2 59 •30541 3.27426 .32460 3.08073 •34400 2.90696 •36364 2.74997 I 60 •30573 3-27085 .32492 3.07768 •34433 2.90421 •36397 2.74748 c / Co-tan- Tan. Co-tan. Tan. CO-TAN. Tan. Co-tanJ Tan. / 7 3° 7 2° 7 1° 1 7 0° 484 NATURAL TANGENTS AND CO-TANGENTS 20° 21° 22° 23° Tan. Co-tan. Tan. CO-TAN. Tan. Co-tan. Tan. Co-tan. / .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2-35585 6c •36430 2-74499 .38420 2.60283 .40436 2.47302 .42482 2-35395 59 .36463 2.74251 .38453 2.60057 •40470 2.47095 .42516 2.35205 58 .36496 2.74004 .38487 2.59831 .40504 2.46888 •42551 2.35015 57 •36529 2.73756 .38520 2.59606 •40538 2.46682 •42585 2.34825 56 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 •36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2-34447 54 .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.34258 53 .36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 52 .36694 2.72526 .38687 2.58484 •40707 2.45655 •42757 2.33881 5i .36727 2.72281 .38721 2.58261 •40741 2.45451 .42791 2.33693 50 .36760 2.72036 •38754 2.58038 .40775 2.45246 .42826 2.33505 4Q •36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 48 .36826 2.71548 .38S21 2-57593 •40843 2.44839 .42894 2.33130 47 .36859 2.71305 .38854 2-57371 .40877 2.44636 .42929 2-32943 46 .36892 2.71062 .38888 2.57I50 .40911 2-44433 .42963 2.32756 45 .36925 2.70819 .38921 2.56928 •40945 2.44230 .42998 2.32570 44 .36958 2.70577 •38955 2.56707 •40979 2.44027 •43032 2.32383 43 .36991 2.70335 .38988 2.56487 •41013 2.43825 •43067 2.32197 42 .37024 2.70094 .39022 2.56266 .41047 2.43623 .43101 2.32012 4i •37057 2.69853 •39055 2.56046 .41081 2.43422 •43136 2.31826 40 .37090 2.69612 .39089 2.55827 •41115 2.43220 •43170 2.31641 39 •37124 2.69371 .39122 2.55608 .41149 2.43019 ■43205 2.31456 S^ .37157 2.69131 •39156 2.55389 •41183 2.42819 •43239 2.31271 2.31086 37 .37190 2.68892 .39190 2.55I70 .4*217 2.42618 •43274 36 •37223 2.68653 .39223 2.54952 .41251 2.42418 •43308 2.30902 35 •37256 2.68414 ! •39257 2.54734 .41285 2.42218 •43343 2.30718 34 .37289 2.S8i75 ! .39290 2.54516 •41319 2.42019 .43378 2.30534 33 •37322 2.67937 •39324 2.54299 •4*353 2.41819 .43412 2.30351 32 •37355 2.67700 •39357 2.54082 •41387 2.41620 •43447 2.30167 3i .37388 2.67462 •39391 2.53865 .41421 2.41421 .43481 2.29984 30 .37422 2.67225 .39425 2.53648 •41455 2.41223 •43516 2.29801 29 •37455 2.66989 .39458 2.53432 .41490 2.41025 •43550 2.29619 28 .37488 2.66752 .39492 2.53217 .41524 2.40827 •43585 2.29437 27 •37521 2.66516 •39526 2.5300I .41558 2.40629 .43620 2.29254 26 •37554 2.66281 •39559 2.52786 .41592 2.40432 •43654 2.29073 25 .37588 2.66046 •39593 2.52571 .41626 2.40235 •43689 2.28891 24 .37621 2.65811 .39626 2.52357 .41660 2.40038 •43724 2.28710 23 •37654 2.65576 .39660 2.52142 .41694 2.39841 •43758 2.28528 22 ■37687 2.65342 •39694 2.51929 .41728 2.39645 •43793 2.28348 21 .37720 2.65109 .39727 2.5I7I5 •41763 2.39449 .43828 2.28167 20 •37754 2.64875 .39761 2.51502 •41797 2.39253 .43862 2.27987 19 .37787 2.64642 •39795 2.51289 .41831 2.39058 •43897 2.27806 18 •37820 2.64410 .39829 2.51076 .41865 2.38862 •43932 2.27626 17 •37853 2.64T77 i .39862 2.50864 .41899 2.38668 .43966 2.27447 16 •3738 7 2.63945 .39896 2.50652 •41933 2-38473 .44001 2.27267 15 .37920 2.63714 ■39930 2.50440 .41968 2.38279 .44036 2.27088 14 •37953 2.63483 •39963 2.50229 .42002 2.38084 .44071 2.26909 13 .37986 2.63252 •39997 2.50018 .42036 2.37891 .44105 2.26730 12 .38020 2.63021 .40031 2.49807 .42070 2.37697 .44140 2.26552 11 .38053 2.62791 .40065 2-49597 .42105 2.37504 •44175 2.26374 10 .38086 2.62561 .40098 2.49386 .42139 237311 .44210 2.26196 9 .38120 2.62332 .40132 2.49177 •42173 2.37118 •44244 2.26018 8 .38153 2.62103 .40166 2.48967 .42207 2.36925 •44279 2.25840 7 .38186 2.61874 .40200 2.48758 .42242 2.36733 •44314 2.25663 6 .38220 2.61646 .40234 2.48549 .42276 2.36541 •44349 2.25486 5 .38253 2.61418 .40267 2.48340 .42310 2.36349 •44384 2.25309 4 .38286 2.61190 .40301 2.48132 •42345 2.36158 .44418 2.25132 3 .38320 2.60963 .40335 2.47924 .42379 2.35967 •44453 2.24956 2 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44488 2.24780 1 .38386 2.60509 .40403 2.47509 .42447 2.35585 •44523 2.24604 CO-TAN. Tan. CO-TAN. Tan. Co-tan. Tan. Co-tan. Tan. / 6< 2° 6, 3° 6' 7° 6< 30 NATURAL TANGENTS AND CO-TANGENTS 485 1 24° 2 5° 26° 2 7° ' Tan. Co-tan. Tan. Co-tan. Tan. CO-TAN. Tan. CO-TAN. - •44523 2.24604 .46631 2.14451 •48773 2.05030 •50953 I.96261 60 I •44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 I.96120 59 2 •44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 1-95979 58 3 .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 1.95698 56 5 .44697 2.23727 .46808 2.13639 •48953 2.04276 •51 136 1-95557 55 6 •44732 2.23553 •46843 2.13477 .48989 2.04125 .51173 I-954I7 54 7 •44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 1.952-77 53 8 .44802 2.23204 .46914 2.13154 .49062 2.03825 .51246 I-95I37 52 9 •44837 2.23030 •46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12832 •49134 2.03526 •51319 1.94858 50 11 .44907 2.22683 .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 •47056 2.12511 .49206 2.03227 •51393 1-94579 48 13 •44977 2.22337 .47092 2.12350 .49242 2.03078 •51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 •51467 1. 94301 46 15 •45047 2.21992 •47163 2.12030 •49315 2.02780 •51503 1.94162 45 16 .45082 2.21819 .47199 2.11871 •49351 2.02631 •51540 1.94023 44 17 .45H7 2.21647 •47234 2.11711 •49387 2.02483 .51577 1.93885 43 18 ■45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 1.93746 42 19 •45i87 2.21304 •47305 2.11392 •49459 2.02187 •51651 1.93608 41 20 .45222 2.21132 •47341 2.11233 •49495 2.02039 .51688 1.93470 40 21 •45257 2.20961 •47577 2.11075 •49532 2.01891 •51724 1-93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.OI743 .51761 I-93I95 38 23 •45327 2.20619 .47448 2.10758 .49604 2.01596 •51798 I-93057 31 24 •45302 2.20449 •47483 2.10600 .49640 2.01449 .51835 1.92920 36 25 •45397 2.20278 •47519 2.10442 .49677 2.01302 .51872 1.92782 35 26 •45432 2.20108 •47555 2.10284 •49713 2.OI155 .51909 1.92645 34 27 •45467 2.19938 •47590 2.10126 •49749 2.OIO08 .51946 1.92508 33 28 •45502 2.19769 .47626 2.09969 .49786 2.00862 -51983 1.92371 32 29 •45537 2.19599 .47662 2.0981 1 .49822 2.00715 .52020 1.92235 3i 30 •45573 2.19430 .47698 2.09654 .49858 2.00569 •52057 1.92098 30 3i .45608 2.19261 •47733 2.09498 •49894 2.OO423 .52094 1.91962 29 32 .45643 2.19092 •47769 2.09341 •49931 2.OO277 •52131 1. 9 1 826 28 33 .45678 2.18923 •47805 2.09184 •49967 2.OOI31 .52168 1. 91690 27 34 •45713 2.18755 .47840 2.09028 .50004 I.99986 .52205 I-9I554 26 35 •45748 2.18587 .47876 2.08872 .50040 1. 99841 .52242 1.91418 25 36 •45784 2.18419 .47912 2.08716 .50076 I.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 •50113 I -99550 .52316 I-9H47 23 38 •45854 2.18084 .47984 2.08405 •50149 I.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 • 50185 I.99261 •52390 1.90876 21 40 .45924 2.17749 •48055 2.08094 .50222 I.99116 .52427 1. 90741 20 4i .45960 2.17582 .48091 2.07939 .50258 I.98972 .52464 1.90607 19 42 •45995 2.17416 .48127 2.07785 •50295 I.98828 .52501 1.90472 18 43 .46030 2.17249 ■48163 2.07630 •50331 I.98684 .52538 I-90337 17 44 .46065 2.17083 .48198 2.07476 .50368 I.9854O •52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 I.98396 .52613 1 .90069 15 46 .46136 2. 167=51 .48270 2.07167 .50441 1.98253 .52650 1-89935 14 47 .46171 2.16585 .48306 2.07014 •50477 I.98110 .52687 1. 8980 1 13 48 .46206 2.16420 .48342 2.06860 •50514 I.97966 .52724 1.89667 12 49 .46242 2.16255 •48378 2.06706 •50550 I.97823 .52761 1-89533 11 50 .46277 2.16090 .48414 2.06553 •50587 I.97680 .52798 1 .89400 10 5i .46312 2.15925 •48450 2.06400 •50623 I.97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 1-97395 .52873 1-89133 8 53 .46383 2.15596 .48521 2.06094 .50696 1-97253 .52910 1 .89000 7 54 .46418 2.15432 •48557 2.05942 •50733 1.97m 1 •52947 1.88867 6 55 .46454 2.15268 •48593 2.05790 .50769 1.96969 .52984 1.88734 5 56 .46489 2.15104 .48629 2.05637 ! .50806 1.96827 .53022 1.88602 4 57 •46525 2.14940 .48665 2.05485 ! -50843 1.96685 .53059 1.88469 3 58 .46560 2.14777 .48701 2-05333 1 .50879 1.96544 •53096 1-88337 2 59 •46595 2.14614 •48737 2.051S2 j 50916 1.96402 •53134 1. 8820 s 1 60 •46631 2.14451 •48773 2.05030 •50953 1. 96261 •53171 1.88073 / CO-TAN . Tan. Co-tan. Tan. Co-tan. Tan. CO -TAN. Tan. / 6 5° 6 40 i 6 3° 6 2° 486 NATURAL TANGENTS AND CO-TANGENTS O Ad 8° 29° 30° 31° Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. 1.66428 o .53171 1.88073 •5543i 1.80405 •57735 1-73205 .60086 I .53208 1. 87941 •55469 1. 8028 1 •57774 1.73089 .60126 1. 663 18 2 .53246 1.87809 •55507 1. 80 1 58 .57813 1.72973 .60165 1. 6209 3 .53283 1.87677 •55545 1.80034 .57851 1.72857 .60205 1.66099 4 .53320 1.87546 •55583 1.79911 .57890 1.72741 .60245 1.65990 5 .53358 1.874^5 .55621 1.79788 •57929 1.72625 .60284 1. 6588 1 6 •53395 1.87283 •55659 1.79665 .57968 1.72509 .60324 1.65772 7 -53432 1.87152 •55697 1.79542 .58007 1-72393 .60364 1.65663 8 .53470 1.87021 •55736 1. 79419 .58046 1.72278 .60403 1-65534 9 .53507 1. 8689 1 •55774 1.79296 .58085 1. 72163 .60443 1.65445 IO •53545 1.86760 .55812 I-79I74 .58124 1.72047 .60483 1-65337 ii .53582 1.86630 .55850 1-79051 .58162 1. 71932 .60522 1.65228 12 .53620 1.86499 .55888 1.78929 .58201 1.71817 .60562 1.65120 13 .53057 1.86369 •55926 1.78807 .58240 1. 71702 .60602 1.65011 14 •53694 1.86239 •55964 1.78685 .58279 1.71588 .60642 1.64903 15 •53732 1.86109 •56003 1.78563 .58318 i-7i473 .60681 1.64795 16 .53769 1.85979 .56041 1. 78441 •58357 i-7i358 .60721 1.64687 17 .53807 1.85850 .56079 1. 78319 .58396 1.71244 .60761 1.64579 18 •53844 1.85720 .56117 1.78198 .58435 1.71129 .60801 1. 6447 1 19 .53882 1.85591 .56156 1.78077 •58474 1-71015 .60841 1.64363 20 •53920 1.85462 .56194 1.77955 .58513 1. 70901 .60881 1.64256 21 •53957 1-85333 .56232 1.77834 .58552 1.70787 .60921 1. 64148 22 •53995 1.85204 •56270 I-777I3 .58591 1.70673 .60960 1. 64041 23 •54032 1-85075 •56309 1.77592 .58631 1.70560 .61000 1-63934 24 .54070 1.84946 •56347 1. 77471 •58670 1.70446 .61040 1.63826 25 .54107 1. 84818 .56385 I-7735I .58709 1.70332 .61080 1.63719 26 •54145 1.84689 .56424 1.77230 .58748 1. 70219 .61120 1.63612 27 .54183 1. 84561 .56462 1.77110 .58787 1. 70106 .61160 1-63505 28 .54220 1-84433 •56500 1.76990 .58826 1.69992 .61200 1.63398 29 •54258 1.84305 .56539 1.76869 .58865 1.69879 .61240 1.63292 30 .54296 1-84177 •56577 1.76749 .58904 1.69766 .61280 1.63185 31 •54333 1.84049 .56616 1.76630 •58944 1.69653 .61320 1.63079 32 •54371 1.83922 •56654 1. 76510 .58983 1. 69541 .61360 1.62972 33 •54409 1.83794 •56693 1.76390 .59022 1.69428 .61400 1.62866 34 •54446 1.83667 .56731 1. 76271 .59061 1.69316 .61440 1.62760 35 .54484 1.83540 .56769 1.76151 .59101 1.69203 .61480 1.62654 36 •54522 1-83413 .56808 1*76032 .59140 1. 69091 .61520 1.62548 37 •5456o 1.83286 .56846 I.759I3 .59179 1.68979 .61561 1.62442 38 •54597 1-83159 .56885 1.75794 .59218 1.68866 .61601 1.62336 39 •54635 1.83033 .56923 1.75675 •59258 1.68754 .61641 1.62230 40 •54673 1.82906 .56962 1-75556 •59297 1.68643 .61681 1.62125 4i •547" 1.82780 •57ooo 1-75437 ■5933^ 1. 68531 .61721 1.62019 42 •54748 1.82654 •57039 1. 75319 •59376 1. 68419 .61761 1.61914 43 .54786 1.82528 .57078 1.75200 •59415 1.68308 .61801 1. 61808 44 .54824 1.82402 .57116 1.75082 •59454 1.68196 .61842 1.61703 45 .54862 1.82276 •57155 1.74964 •59494 1.68085 .61882 1. 6 1 598 46 .54900 1. 82 1 50 •57193 1.74846 •59533 1.67974 .61922 1. 61493 47 •54938 1.82025 •57232 1.74728 •59573 1.67863 .61962 1.61388 48 •54975 1.81899 •57271 1. 74610 .59612 1.67752 .62003 1.61283 49 •55013 1-81774 •57309 1.74492 •59651 1.67641 .62043 1.61179 50 •55051 1. 8 1 649 .57348 1-74375 .59691 1.67530 .62083 1.61074 5i •55089 1.81524 .57386 1-74257 •59730 1.67419 .62124 1.60970 52 •55127 1.81399 .57425 1. 74140 •59770 1.67309 .62164 1.60865 53 •55i65 1.81274 .57464 1.74022 .59809 1.67198 .62204 1. 60761 54 •55203 1.81150 •5V503 I-73905 •59849 1.67088 .62245 1.60657 55 •55241 1.81025 •57541 1.73788 .59888 1.66978 .62285 1.60553 56 •55279 1. 8090 1 .5758o 1.73671 •59928 1.66867 •62325 1.60449 57 •55317 1.80777 .57619 1-73555 •59967 1.66757 .62366 1.60345 58 •55355 1.80653 •57657 I-73438 .60037 1.66647 .62406 1. 60241 59 •55393 1.80529 •57696 1.73321 .60046 1.66538 .62446 1. 60137 60 •55431 1.80405 •57735 1.73205 .60086 1.66428 .62487 1 .60033 Co-tan. Tan. Co-tan. Tan. Co-tan . Tan. Co-tan. Tan. 6 1° 6 0° 5 9° 5 8° NATURAL TANGENTS AND CO-TANGENTS 487 32° 33° 34° 35° Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. / .62487 1.60033 .64941 1.53986 .67451 1.48256 ; .70021 1.42815 60 .62527 1.59930 .64982 1.53888 •67493 1.48163 1 .70064 1.42726 59 .62568 1.59826 .65023 1.53791 •67536 1.48070 .70107 1.42638 58 .62608 1-59723 .65065 1-53693 •67578 1.47977 .70151 1.42550 57 .62649 1.59620 .65106 1-53595 .67620 1.47885 .70194 1.42462 56 .62689 I-595I7 .65148 1-53497 .67663 1.47792 •70238 1.42374 55 .62730 1. 59414 .65189 1.53400 : -67705 1.47699 .70281 1.42286 54 .62770 I-593H •65231 I-533Q2 .67748 1.47607 •70325 1. 42 198 53 .62811 1.59208 '65272 1-53205 .67790 i-475i4 .70368 1.42110 52 .62852 i-59io5 •65314 i-53io7 ■67832 1.47422 .70412 1.42022 5i .62892 1.59002 .65355 1. 53010 .67875 1-47330 .70455 i-4i934 5o .62933 1 .58900 •65397 1-52913 .67917 1.47238 .70499 1.41847 49 •62973 1.58797 .65438 1.52816 .67960 1.47146 •70542 I-4I759 48 .63014 1.58695 .65480 I-527I9 .68002 1.47053 .70586 1.41672 47 •63055 I-58593 •65521 1.52622 .68045 1.46962 .70629 1.41584 46 •63095 1.58490 •65563 1-52525 .68088 1.46870 .70673 i-4i497 45 .63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1. 4 1 409 44 •63177 1.58286 .65646 1-52332 .68173 1.46686 .70760 1.41322 43 .63217 1.58184 .65688 1.52235 .68215 1-46595 .70804 1-41235 42 .63258 1.58083 .65729 I-52I39 .68258 1.46503 .70848 1. 41 148 4i •63299 i.5798i •65771 1.52043 .68301 1. 4641 1 .70891 1.41061 40 •63340 I-57879 .65813 1. 51946 .68343 1.46320 •70935 1.40974 39 .63380 1.57778 •65854 1.51850 .68386 1.46229 •70979 1.40887 38 .63421 1.57676 .65896 I-5I754 .68429 1-46137 .71023 1 .40800 37 .63462 1-57575 .65938 1.51658 .68471 1.46046 .71066 1.40714 36 •63503 1-57474 .65980 1-51562 .68514 1-45955 .71110 1.40627 35 •63544 1-57372 .66021 1.51466 .68557 1.45864 •7ii54 1.40540 34 •63584 1.57271 .66063 i-5i37o .68600 1-45773 .71198 1.40454 33 •63625 1.57170 .66105 I-5I275 .68642 1.45682 .71242 1.40367 32 .63666 1.57069 .66147 I-5II79 .68685 1-45592 .71285 1.40281 3i .63707 1.56969 .66189 1.51084 .68728 i-455oi .71329 1. 40 1 95 30 .63748 1.56868 .66230 1.50988 .68771 I.454IO •71373 1. 40 109 29 .63789 1-56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 .63830 1.56667 .66314 1.50797 .68857 1.45229 .71461 I.39936 27 .63071 1.56566 .66356 1.50702 .68900 I-45I39 .71505 1.39850 26 .63912 1.56466 .66398 1.50607 .68942 1.45049 •71549 1.39764 25 •63953 1.56366 .66440 1.50512 .68985 1.44958 •71593 1.39679 24 .63994 1.56265 .66482 1-50417 .69028 1.44868 •71637 1-39593 23 •6403s 1.56165 .66524 1.50322 .69071 1.44778 .71681 I-39507 22 .64076 1.56065 .66566 1.50228 .69114 1.44688 .71725 1. 3942 1 21 .64117 1.55966 .66608 1-50133 •69157 1.44598 .71769 1.39336 20 •64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 .64199 1.55766 '66692 1.49944 .69243 1. 44418 .71857 1-39165 18 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 .64281 I-55567 .66776 1-49755 •69329 1.44239 .71946 1.38994 16 .64322 I-55467 .66818 1. 4966 1 .69372 1.44149 .7.1990 1.38909 15 •64363 1-55368 .66860 1.49566 .69416 1.44060 •72034 1.38824 14 .64404 1.55269 .66902 1.49472 •69459 1.43970 .72078 1.38738 13 .64446 i-55i7o .66944 1.49378 .69502 1. 43881 .72122 1.38653 12 .64487 1-55071 .66986 1.49284 •69545 1.43792 .72166 1.38568 11 .64528 1.54972 .67028 1. 49 1 90 .69588 I-43703 .72211 1.38484 10 •64569 1.54873 .67071 1.49097 .69631 1. 43614 .72255 I-38399 9 .64610 1-54774 .67113 1.49003 •69675 1-43525 .72299 1-38314 8 .64652 1.54675 •67155 1.48909 .69718 I-43436 •72344 1.38229 7 .64693 1-54576 .67197 1. 488 1 6 .69761 1-43347 .72388 I-38I45 6 •64734 1.54478 •67239 1.48722 .69804 1.43258 •72432 1.38060 5 •64775 1-54379 .67282 1.48629 ■69847 1.43169 .72477 1.37976 4 .64817 1.54281 .67324 1.48536 .69891 1.43080 •72521 I-3789I 3 .64858 i-54i83 •67366 1.48442 .69934 1.42992 •72565 1.37807 2 .64899 1.54085 .67409 1.48349 •69977 1-42903 .72610 1.37722 1 .64941 1.53986 •67451 1.48256 .70021 1.42815 .72654 1.37638 Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. CO -TAN. Tan. / 5 7° 5 6° 5 5° 5 40 488 NATURAL TANGENTS AND CO-TANGENTS 36° 37° 38° 39° / Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. / o .72654 1-37638 -75355 1.32704 .78129 1.27994 .80978 1-23490 60 I .72699 1-37554 .75401 1.32624 .78175 1.27917 .81027 1. 23416 59 2 •72743 i-3747o •75447 1.32544 .78222 1.27841 .81075 1-23343 58 3 .72788 1.37386 •75492 1.32464 .78269 1.27764 .81123 1.23270 57 4 .72832 1.37302 .75538 1.32384 .78316 1.27688 .81171 1.23196 56 5 .72877 i-372i8 .75584 1.32304 .78363 1.27611 .81220 1-23123 55 6 .72921 I-37I34 .75629 1.32224 .78410 1-27535 .81268 1.23050 54 7 .72966 1.37050 .75675 1.32144 •78457 1.27458 .81316 1.22977 53 8 .73010 1.36967 •75721 1.32064 .78504 1.27382 •81364 1.22904 52 9 .73055 1.36883 •75767 1-31984 .78551 1.27306 .81413 1. 22831 51 TO .73100 1 .36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 50 ii •73144 1.36716 .75858 1.31825 .78645 I-27I53 .81510 1.22685 49 12 .73189 1-36633 .75904 1. 3 1 745 .78692 1.27077 .81558 1. 22612 48 13 •73234 I-36549 •75950 1. 31666 .78739 1. 2 700 1 .81606 1.22539 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1-36383 .76042 1-31507 .78834 1.26849 .81703 1.22394 45 16 .73368 1.36300 .76088 1-31427 .78881 1.26774 .81752 1. 22321 44 17 .73413 1.36217 .76134 I-3I348 .78928 1.26698 .81800 1.22249 43 18 •73457 1.36133 .76180 1.31269 .78975 1.26622 .81849 1. 22176 42 19 .73502 1-36051 .76226 1-31190 .79022 1.26546 .81898 1. 22104 4i 20 •73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 1. 22031 40 21 •73592 1-35885 .76318 1-31031 .79117 1.26395 .81995 1.21959 39 22 •73637 1.35802 .76364 1.30952 .79164 1. 26319 .82044 1.21886 38 23 .73681 I-357-I9 .76410 1.30873 .79212 1.26244 .82092 1.21814 37 24 .73726 1-35637 .76456 1-30795 •79259 1. 26169 .82141 1. 21742 36 25 .73771 1-35554 .76502 1. 30716 •79306 1.26093 .82190 1. 21670 35 26 .73816 1-35472 .76548 1.30637 •79354 1.26018 .82238 1. 21598 34 27 .73861 I-35389 .76594 1-30558 •79401 1-25943 .82287 1. 21526 33 28 •739o6 I-35307 .76640 1.30480 •79449 1.25S67 .82336 1.21454 32 29 •73951 1.35224 .76686 1. 3040 1 •79496 1.25792 .82385 1. 21382 3i 30 •73996 I-35I42 .76733 1.30323 •79544 I.257I7 •82434 1.21310 30 3* .74041 1.35060 .76779 1.30244 •79591 1.25642 .82483 1.21238 29 32 .74086 1.34978 • 76325 1.30166 •79639 1.25567 .82531 1.21166 28 33 .74131 1 .34896 .76871 1.30087 .79686 1.25492 .82580 1. 21094 27 34 .74176 1-34814 .76918 1.30009 •79734 I.254I7 .82629 1.21023 26 35 .74221 1-34732 .76964 1-29931 .7978i 1.25343 .82678 1. 20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 .82727 1.20879 24 37 •74312 1.34568 '77057 1.29775 •79877 I-25I93 .82776 1.20808 23 38 •74357 1.34487 •77103 1.29696 •79924 1.25118 .82825 1.20736 22 39 .74402 I-34405 .77149 1. 29618 •79972 1.25044 .82874 1.20665 21 40 •74447 1-34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 •74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1. 34160 •77289 1.29385 .80115 1.24820 .83022 1. 20451 18 43 •74583 1.34079 •77335 1.29307 .80163 1.24746 .83071 1.20379 17 44 .74628 I-33998 .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 •74674 1.33916 .77428 1.-9152 .80258 1.24597 .83169 1.20237 15 46 .74719 I-33835 •77475 1.29074 .80306 1.24523 .83218 1. 20166 14 47 .74764 1-33754 •77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74810 I-33673 .77568 1. 28919 .80402 1-24375 -83317 1.20024 12 49 •74855 1-33592 .77615 1.28842 .80450 1. 24301 .83366 I-I9953 11 SO .74900 I-335H .77661 1.28764 .80498 1.24227 •83415 1. 1 9882 10 5i .74946 I-3343Q .777o8 1.28687 .80546 I-24I53 •83465 1.19811 9 52 .74991 1-33349 •77754 1. 28610 .80594 1.24079 .83514 1. 19740 8 53 •75037 1.33268 .77801 1-28533 .80642 1.24005 .83564 1. 1 9669 7 54 • 75082 i.33i87 .77848 1.28456 .80690 I-2393I .83613 1. 1 9599 6 55 .75128 I-33I07 .77895 1.28379 .80738 1.23858 .83662 1. 19528 5 56 •75173 1.33026 .77941 1.28302 .80786 1.23784 .83712 1. 19457 4 57 .75219 1.32946 .77988 1.28225 .80834 1. 23710 .83761 1. 19387 3 58 .75264 1.32865 .7S035 1. 28148 .80882 1.23637 .83811 1.19316 2 59 •753io 1-32785 .78082 1. 28071 .80930 1.23563 .83860 1. 19246 1 60 •75355 1.32704 .78129 1.27994 .80978 1.23400 .83910 I-I9I75 t Co-tan. Tan. Co-tan. Tan. I Co-tan Tan. Co-tan. Tan t 5; 3° & 2° 5 1° 5( 3° NATURAL TANGENTS AND CO-TANGENTS 489 40° 41° 42° 43° 1 Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. / .83910 1.19175 .86929 1.15037 .90040 1.11061 •93252 1.07237 60 I .83960 1.19105 .86980 1. 1 4969 •90093 1. 10996 •93306 1-07174 59 2 .84009 1-19035 .87031 1. 14902 .90146 1.10931 •93360 1.07112 58 3 .84059 1. 1 8964 .87082 1. 14834 .90199 1. 10867 •93415 1.07049 57 4 .84108 1. 18894 .87133 1. 14767 .90251 1. 10802 .93469 1.06987 56 5 .84158 1. 18824 .87184 1. 1 4699 .93304 1. 10737 •93524 1.06925 55 6 .84208 1. 18754 .87236 1. 14632 •90357 1. 10673 •93578 1.06862 54 7 .84258 1. 1 8684 .87287 1-14565 .90410 1. 10607 •93633 1 .06800 53 8 .84307 1.18614 .87338 1. 1 4498 •90463 i- 10543 .93688 1.06738 52 9 •84357 1. 18544 .87389 1. 14430 .90516 1. 10478 •93742 1.06676 Si 10 .84407 1. 18474 .87441 1-14363 | .90569 1.10414 •93797 1.06613 50 11 •84457 1. 18404 .87492 1. 14296 .90621 1. 10349 •93852 1-06551 49 12 .84507 1-18334 •87543 1. 14229 .90674 1. 10285 •939o6 1.06489 48 13 .84556 1. 18264 •87595 1.14162 .90727 1. 10220 .93961 1.06427 47 14 .84606 1.18194 .87646 1. 14095 .90781 1.10156 1 .94016 1.06365 46 15 .84656 1.18125 .87698 1. 14028 | •90834 1.10091 1 -94071 1.06303 45 16 .84706 1. 18055 .87749 1.13961 1 .90887 1. 10027 ! -94125 1. 06241 44 17 .84756 1. 17986 .87801 1. 13894 .90940 1.09963 .94180 1.06179 43 18 .84806 1.17016 .87852 1.1382S .90993 1.09899 •94235 1. 061 1 7 42 19 .84856 1. 1 7846 .87904 1.13761 .91046 1 .09834 .94290 1.06056 4i 20 .84906 1. 17777 •87955 1. 13694 .91099 1.09770 •94345 1.05994 40 21 •84956 1*17708 ; .88007 1. 13627 J •9H53 1.09706 .94400 1.05932 39 22 .85006 1. 17638 .88059 1-13561 | .91206 1.09642 •94455 1.05870 38 23 .85057 1. 17569 .88110 1 .13494 .91259 1.09578 .94510 1.05809 37 24 .85107 1-17500 .88162 1. 13428 •91313 1. 09514 •94565 I-05747 36 25 .85157 1-17430 .88214 1-13361 .91366 1.09450 1 .94620 1.05685 35 26 .85207 1-17361 .88265 1-13295 .91419 1.09386 ' .94676 1.05624 34 27 .85257 1. 17292 .88317 1. 13228 •91473 1.09322 •94731 1.05562 33 28 .85307 1. 17223 .88369 1.13162 .91526 1.09258 .94786 1. 05501 32 29 .85358 1.17154 .88421 1. 1 3096 .91580 1. 09195 .94841 1 -Q5439 3i 30 .85408 1. 17085 .88473 1. 13029 •91633 1-09131 .94896 1-05378 30 3i .85458 1.17016 .88524 1. 12963 .91687 1.09067 •94952 1-05317 29 32 •85509 1. 16947 .88576 1. 12897 .91740 1 .09003 .95007 1-05255 28 33 .85559 1. 16878 .88628 1.12831 .91794 1 .08940 .95062 1-05194 27 34 .85609 1. 1 6809 .88680 1. 12765 .91847 1.08876 .95118 1-05133 26 35 .85660 1.16741 .88732 1. 12699 .91901 1. 088 1 3 •95173 1.05072 25 36 .85710 1. 16672 .88784 1. 12633 .91955 1.08749 .95229 1. 05010 24 37 .85761 1. 1 6603 .88836 1. 12567 .92008 1.08686 .95284 1.04949 23 38 .85811 1-16535 .88888 1.12501 .92062 1.08622 •95340 1.04888 22 39 .85862 1. 16466 .88940 1.1243s .92116 1.08559 •95395 1.04827 21 40 .85912 1. 16398 .88992 1. 12369 .92170 1 .08496 •95451 1.04766 20 41 .85963 1. 16329 .89045 1. 12303 .92224 1.08432 •95506 1.04705 19 42 .86014 1.16261 .89097 1. 12238 .92277 1.08369 •95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 •92331 1.08306 •95618 1.04583 17 44 .86115 1.16124 .89201 1.12106 •92385 1.08243 •95673 1.04522 16 45 .86166 1. 16056 •89253 1.12041 •92439 1.08179 •95729 1. 0446 1 15 46 .86216 1.15987 .89306 I-H975 .92493 1.08116 •95785 1. 0440 1 14 47 .86267 1. 15919 •89358 1.11909 •92547 1.08053 •95841 1.04340 13 48 .86318 1.15851 .89410 1.11844 .92601 1.07990 •95897 1.04279 12 49 .86368 1-15783 .89463 1.11778 .92655 1.07927 •95952 1.04218 11 50 .86419 1.15715 •89515 1.11713 .92709 1.07864 .96008 1.04158 10 5i .86470 1. 15647 .89567 1.11648 •92763 1. 0780 1 .96064 1.04097 9 52 .86521 I-I5579 .89620 1.11582 .92817 1.07738 .96120 1.04036 8 53 .86572 I.I55H .89672 1.11517 .92872 1.07676 .96176 1.03976 7 54 .86623 1. 1 5443 .89725 1.1 1452 .92926 1.07613 .96232 1-03915 6 55 .86674 I-I5375 .89777 1.11387 .92980 1-07550 .96288 1-03855 5 56 .86725 1-15308 .89830 1.11321 •93034 1.07487 •96344 1.03794 4 57 .86776 1. 15240 .89883 1.11256 .93088 1.07425 .96400 I-Q3734 3 58 .86827 1.15172 •89935 1.11191 •93143 1.07362 •96457 1.03674 2 59 .86878 1.15104 .89988 1.11126 •93197 1.07299 .96513 1-03613 1 60 .86929 1-15037 .90040 i.tio6i •93252 1.07237 .96569 I-Q3553 / Co-tan Tan. Co-tan. Tan. Co-tan. Tan. Co-tan. Tan. *• 4 3° 4 3° 4 7° 4 6° 490 NATURAL TANGENTS AND CO-TANGENTS 44° Tan. Co-tan. .96569 .96625 .96681 .96738 .96794 .96850 .96907 .96963 .97020 .97076 •97133 .97189 .97246 .97302 •97359 .97416 .97472 •97529 .97586 •97643 .97700 I-Q3553 1 -03493 1 -03433 1.03372 1-03312 1.03252 1.03192 1.03132 1.03072 1.03012 1.02952 1.02892 1.02832 1.02772 1 .02713 1.02653 1.02593 1.02533 1.02474 1.02414 1-02355 Co-tan. Tan. 45° 440 Tan. Co-tan. 97756 97813 97870 97927 97984 98041 98098 98155 98213 98270 98327 98384 98441 98499 98556 98613 98671 98728 98786 98843 1.02295 1.02236 1. 02 1 76 1. 021 17 1.02057 1. 01 998 1.01939 1.01879 1.01820 1.01761 1. 01 702 1.01642 1-01583 1.01524 1. 01 465 1. 01 406 1.01347 1. 01 288 1.01229 1.01170 Co-tan. I Tan. 45° / r 39 38 4i 42 37 36 43 44 35 34 45 46 33 32 47 48 31 49 30 50 29 28 5i 52 27 26 53 54 25 24 55 56 23 22 57 58 21 20 59 60 ' / Tan. 44° Co-tan. .98901 .98958 .99016 .99073 •99131 .99189 .99247 .99304 .99362 .99420 •99478 •99536 •99594 .99652 .99710 .99768 .99826 .99884 .99942 1.01112 1.01053 1.00994 1.00935 1.00876 1. 008 1 8 1.00759 1. 00 70 1 1.00642 1.00583 1.00525 1.00467 1 .00408 1.00350 1.00291 1.00233 1.00175 1. 001 16 1.00058 Co-tan. Tan. 45° NATURAL SINES AND COSINES o I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ^9 20 0° Sine Cosine .00000 .00029 .00058 .00087 .00116 •00145 .00175 .00204 •00233 .00262 .00291 .00320 .00349 •00378 .00407 .00436 .00465 .00495 .00524 •00553 .00582 .99999 •99999 •99999 •99999 •99999 ■99999 •99999 •99999 .99998 •99998 Cosine Sine 89° c >° c )° 60 r 21 Sine Cosine 39 / 4i Sine Cosine .00611 •99998 .01193 •99993 59 22 .00640 •99998 3^ 42 .01222 •99993 58 23 .00669 •99998 37 43 .01251 .99992 57 24 .00698 •99998 36 44 .01280 .99992 56 25 .00727 •99997 35 45 •01309 .99991 55 26 .00756 •99997 34 46 •01338 .99991 54 27 .00785 •99997 33 47 .01367 .99991 53 28 .00814 •99997 32 48 .01396 •99990 52 20 .00844 .99996 3i 49 .01425 •99990 5i 30 .00873 •99996 30 50 .01454 .99989 50 31 .00902 .99996 29 51 .01483 •99989 49 32 .00931 .99996 28 52 •01513 .99989 48 33 .00960 •99995 27 53 .01542 .99988 47 34 .00989 •99995 26 54 .01571 .99988 46 35 .01018 •99995 25 55 .01600 •99987 45 36 .01047 •99995 24 56 .01629 •99987 44 37 .01076 •99994 23 57 .oi6s8 .99986 43 38 .01105 •99994 22 58 .01687 .99986 42 39 .01134 •99994 21 59 .01716 •99985 4i 40 r 40 / .01164 •99993 20 / 60 •oi745 •99985 Cosine Sine Cosine Sine 8 3° 8< y> 17 TO 15 14 13 12 i r TO 9 8 7 6 5 4 3 2 1 o NATURAL SINES AND COSINES 491 2° e }° 40 Sine Cosine | Sine Cosine Sine Cosine Sine Cosine r .01745 .99985 .03490 •99939 •05234 .99863 .06976 •99756 60 •01774 .99984 ■03519 •99938 .05263 .99861 .07005 •99754 59 .01803 .99984 .03548 •99937 .05292 .99860 •07034 •99752 58 .01832 .99983 •03577 •99936 .05321 .99858 .07063 •9975o 57 .01862 .99983 .03606 •99935 •05350 •99857 .07092 •99748 56 .01891 .99982 •03635 •99934 •05379 •99855 .07121 •99746 55 .01920 .99982 .03664 •99933 .05408 .99854 .07150 •99744 54 .01949 .99981 •03693 .99932 •05437 .99852 .07179 •99742 53 .01978 .99980 •03723 •99931 •05466 .99851 .07208 •99740 52 .02007 .99980 •03752 .99930 •05495 •99849 .07237 •99738 5i .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 50 .02065 .99979 .03810 .99927 •05553 .99846 .07295 •99734 49 .02094 .99978 .03839 .99926 .05582 •99844 .07324 •99731 48 .02123 .99977 .03868 •99925 .05611 .99842 •07353 .99729 47 .02152 .99977 .03897 .99924 .05640 .99841 .07382 •99727 46 .02181 .99976 .03926 .99923 .05669 •99839 .07411 •99725 45 .02211 .99976 •03955 .99922 .05698 .99838 .07440 •99723 44 .02240 •99975 .03984 .99921 •05727 .99836 .07469 .99721 43 .02269 •99974 •04013 .99919 •05756 .99834 .07498 .99719 42 .02298 •99974 .04042 .99918 .05785 •99833 •07527 .99716 4i .02327 •99973 .04071 .99917 .05814 .99831 .07556 .99714 40 .02356 .99972 .04100 .99916 .05844 .99829 .07585 .99712 39 .02385 .99972 .04129 •99915 •05873 .99827 .07614 .99710 38 .02414 .99971 .04159 •99913 .05902 .99826 .07643 .99708 37 .02443 .99970 .04188 .99912 •05931 .99824 .07672 •99705 36 .02472 .99969 .04217 .99911 .05960 .99822 .07701 .99703 35 .02501 .99969 .04246 .99910 .05989 .99821 .07730 •99701 34 .02530 .99968 .04275 .99909 .06018 .99819 .07759 •99699 33 .02560 .99967 •04304 •9990 7 .06047 .99817 .07788 .99696 32 .02589 .99966 •04333 .99906 .06076 .99815 .07817 •99694 31 .02618 .99966 .04362 •99905 .06105 .99813 .07846 .99692 30 .02647 .99965 .04391 •99904 •06134 .99812 .07875 .99689 29 .02676 .99964 .04420 .99902 .06163 .99810 .07904 .99687 28 .02705 .99963 •04449 .99901 .06192 .99808 .07933 .99685 27 •02734 .99963 .04478 .99900 .06221 .99806 .07962 .99683 26 .02763 .99962 .04507 .99898 .06250 .99804 .07991 .99680 25 .02792 .99961 •04536 .99897 .06279 •99803 .08020 .99678 24 .02821 .99960 •04565 .99896 .06308 .99801 .08049 .99676 23 .02850 •99959 •04594 •99894 •06337 •99799 .08078 .99673 22 .02879 •99959 .04623 •99893 .06366 •99797 .08107 .99671 21 .02908 .99958 •04653 .99892 •06395 ■99795 .08136 .99668 20 .02938 •99957 .04682 .99890 .06424 •99793 .08165 .99666 19 .02967 .99956 .04711 .99889 •06453 •99792 .08194 .99664 18 .02996 •99955 .04740 .99S88 .06482 •99790 .08223 .99661 17 •03025 •99954 .04769 .99886 .06511 .99788 .08252 •99659 16 •03054 •99953 .04798 .99885 .06540 .99786 .08281 •99657 15 .03083 .99952 .04827 .99883 .06569 •99784 .08310 .99654 14 .03112 •99952 .04856 .99882 .06598 .99782 •08339 .99652 13 .03141 •99951 .04885 .99881 .06627 .99780 .08368 -99649 12 .03170 •99950 .04914 .99879 .06656 •99778 .08397 •99647 11 .03199 •99949 •04943 .99878 .06685 .99776 .08426 .99644 10 .03228 .99948 .04972 .99876 .06714 •99774 .08455 .99642 9 •03257 •99947 .05001 •99875 •06743 .99772 .08484 •99639 8 .03286 .99946 .05030 •99873 .06773 •9977o .08513 •99637 7 .03516 •99945 .05059 .99872 .06802 .99768 .08542 •99635 6 •03345 •99944 .05088 .99870 .06831 .99766 .08571 •99632 5 •03374 •99943 .05117 .99869 .06860 .99764 .08600 .99630 4 •03403 .99942 .05146 .99867 .06889 .99762 .08629 .99627 3 •03432 .99941 .05175 .99866 .06918 .99760 .08658 .99625 2 •03461 •99940 .05205 .99864 .06947 •99758 .08687 .99622 1 •03490 •99939 •05234 •99863 .06976 •99756 .08716 .99619 Cosine Sine Cosine Sine Cosine Sine Cosine Sine / Si i° 8' 7° 81 3° 8 5° 49 2 NATURAL SINES AND COSINES 5° ( 3° 7° 8° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine o .08716 .99619 •10453 .99452 .12187 .99255 .13917 .99027 I .08745 .99617 .10482 .99449 .12216 .99251 .13946 .99023 2 .08774 .99614 .10511 •99446 .12245 .99248 •13975 .99019 3 .08803 .99612 .10540 •99443 .12274 .99244 .14004 .99015 4 .08831 .99609 .10569 •99440 .12302 .99240 •14033 .99011 5 .08860 .99607 •10597 •99437 .12331 •99237 .14061 .99006 6 .08889 .99604 .10626 •99434 .12360 •99233 .14090 .99002 7 .08918 .99602 .10655 •99431 .12389 .99230 .14119 .98998 8 .08947 •99599 . .10684 .99428 .12418 .99226 .14148 .98994 9 .08976 .99596 .10713 •99424 .12447 .99222 .14177 .98990 IO .09005 •99594 .10742 .99421 .12476 .99219 .14205 .98986 ii .09034 •99591 .10771 .99418 .12504 .99215 .14234 .98982 12 .09063 .99588 .10800 •99415 •12533 .99211 .14263 .98978 13 .09092 .99586 .10829 .99412 .12562 .99208 .14292 •98973 14 .09121 •99583 .10858 .99409 .12591 .99204 .14320 .98969 15 .09150 .99580 .10887 .99406 .12620 .99200 •14349 •98965 16 .09179 •99578 .10916 .99402 .12649 .99197 •14378 .98961 17 .09208 •99575 .10945 •99399 .12678 •99193 .14407 •98957 18 .09237 •99572 .10973 .99396 .12706 .99189 .14436 •98953 19 .09266 .99570 .11002 •99393 •12735 .99186 .14464 .98948 20 .09295 .99567 .11031 .99390 .12764 .99182 •14493 •98944 21 .09324 .99564 .11060 .99386 •12793 .99178 .14522 .98940 22 •09353 .99562 .11089 •99383 .12822 •99175 .14551 •98936 23 .09382 •99559 .11118 .99380 .12851 .99171 .14580 .98931 24 .09411 .99556 .11147 •99377 .12880 .99167 .14608 .98927 25 .09440 •99553 .11176 •99374 .12908 .99163 .14637 .98923 26 .09469 .99551 .11205 •99370 •12937 .99160 .14666 .98919 27 .09498 .99548 .11234 •99367 .12966 .99156 .14695 .98914 28 .09527 •99545 .11263 •99364 .12995 .99152 •14723 .98910 29 .09556 •99542 .11291 .99360 .13024 .99148 .14752 .98906 30 •09585 .99540 .11320 •99357 •13053 .99144 .14781 .98902 31 .09614 •99537 .11349 •99354 .13081 .99141 .14810 .98897 32 .09642 •99534 .11378 •99351 .13110 •99137 .14838 .98893 33 .09671 •99531 .11407 •99347 .13139 •99133 .14867 .98889 34 .09700 .99528 .11436 •99344 .13168 .99129 .14896 .98884 35 .09729 .99526 .11465 •99341 •13197 .99125 .14925 .98880 36 .09758 .99523 .11494 •99337 .13226 .99122 •14954 .98876 37 .09787 .99520 .11523 •99334 • 13254 .99118 .14982 .98871 38 .09816 •99517 .11552 •99331 • 13283 .99114 .15011 .98867 39 .09845 •99514 .11580 ■99327 .13312 .99110 .15040 .98863 40 .09874 •995H .11609 •99324 .13341 .99106 .15069 .98858 4i .09903 .99508 .11638 .99320 •13370 .99102 .15097 .98854 42 .09932 .99506 .11667 •99317 •13399 .99098 .15126 .98849 43 .09961 •99503 .11696 •993T4 • 13427 .99094 • 15155 .98845 44 .09990 .99500 .11725 .99310 •13456 .99091 .15184 .98841 45 .10019 •99497 .11754 •99307 •13485 .99087 .15212 .98836 46 .10048 •99494 .11783 •99303 •T35I4 .99083 .15241 .98832 47 .10077 .99491 .11812 .99300 • 13543 •99079 .15270 .98827 48 .10106 .99488 .11840 .99297 .13572 •99075 .15299 .98823 49 .10135 .99485 .11869 .99293 .13600 .99071 •15327 .98818 50 .10164 .99482 .11898 .99290 .13629 .99067 .15356 .98814 51 .10192 .99479 .11927 .99286 .13658 .99063 • 15385 .98809 52 .10221 .99476 .11956 •99283 .13687 .99059 .15414 .98805 53 .10250 •99473 .11985 .99279 .13716 •99055 .15442 .98800 54 .10279 .99470 .12014 .99276 .13744 .99051 •i547i .98796 55 .10308 .99467 .12043 .99272 • 13773 •99047 .15500 .98791 56 .10337 .99464 .12071 .99269 .13802 •99043 .15529 .98787 57 .10366 .99461 .12100 .99265 .13831 .99039 •15557 .98782 58 •10395 •99458 .12129 .99262 .13860 •99035 .15586 .98778 59 .10424 •99455 .T2158 .99258 .13889 .99031 .15615 •98773 60 •10453 •99452 .12187 •99255 • 13917 .99027 .15643 .98769 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine & t° 8: ]° & jo 81 NATURAL SINES AND COSINES 493 ( )° 1( 3° 11° 12° / Sine Cosine Sine Cosine Sine .19081 Cosine Sine Cosine t o •15643 .98769 .17365 .98481 .98163 .20791 •97815 60 I .15672 .98764 •17393 .98476 .19109 •98157 .20820 .97809 59 2 .15701 .98760 .17422 .98471 .19138 .98152 .20848 .97803 58 3 .15730 •98755 .i745i .98466 .19167 .98146 .20877 •97797 57 4 .15758 •98751 •17479 .98461 .19195 .98140 .20905 •97791 56 5 .15787 .98746 .17508 •98455 .19224 •98135 .20933 .97784 55 6 .15816 .98741 •17537 .98450 .19252 .98129 .20962 .97778 54 7 -15845 •98737 •17565 •98445 .19281 .98124 .20990 .97772 53 8 .15873 .98732 .17594 .98440 .10309 .98118 .21019 •97766 52 9 .15902 .98728 .17623 •98435 .19338 .98112 .21047 .97760 5i IO .15931 .98723 .17651 .98430 .19366 .98107 .21076 •97754 5o ii •15959 .98718 .17680 .98425 .19395 .98101 .21104 •97748 49 12 .15988 .98714 .17708 .98420 .19423 ' .98096 .21132 •97742 48 13 .16017 .98709 •17737 .98414 .19452 .98000 .21161 •97735 47 14 .16046 .98704 .17766 .98409 .19481 .98084 .21189 .97729 46 15 .16074 .98700 •17794 .98404 .19509 .98079 .21218 •97723 45 16 .16103 .98695 .17823 .98399 •19538 .98073 .21246 •97717 44 i7 .16132 .98690 .17852 .98394 .19566 .98067 .21275 .97711 43 18 .16160 .98689 .17880 .98389 •19595 .98061 .21303 .97705 42 19 .16189 .98681 .17909 •98383 .19623 .98056 .21331 .97698 4i 20 .16218 .98676 •17937 .98378 .19652 .98050 .21360 .97692 40 21 .16246 .98671 .17966 •98373 .19680 .98044 .21388 .97686 39 22 .16275 .98667 .•17995 .98368 .19709 .98039 .21417 .97680 38 23 .16304 .98662 .18023 .98362 •19737 •98033 .21445 •97673 37 24 •16333 ■98657 .18052 •98357 .19766 .98027 •21474 .97667 36 25 .16361 .98652 .18081 •98352 .19794 .98021 .21502 .97661 35 26 .16390 .98648 .18109 •98347 .19823 .98016 .21530 •97655 34 27 .16419 .98643 .18138 .98341 .19851 .98010 •21559 .97648 33 28 .16447 .98638 .18166 •98336 .19880 .98004 •21587 .97642 32 29 .16476 •98633 .18195 •98331 .19908 .97987 .21616 .97636 3i 30 •16505 .98629 .18224 •98325 •19937 .97992 .21644 .97630 30 31 •16533 .98624 .18252 •98320 .19965 •97987 .21672 •97623 29 32 .16562 .98619 .18281 •98315 .19994 .97981 .21701 •976i7 28 33 .16591 .98614 .18309 .98310 .20022 •97975 .21729 .97611 27 34 .16620 .98609 .18338 .98304 .20031 •97969 .21758 .97604 26 35 .16648 .98604 .18367 .98299 .20079 •97963 .21786 •97598 25 36 .16677 .98600 •18395 .98294 .20108 •97958 .21814 •97592 24 37 .16706 •98595 .18424 .98288 .20136 •97952 .21843 •97585 23 38 •16734 .98590 .18452 .98283 .20165 .97946 .21871 •97579 22 39 .16763 •98585 .18481 .98277 .20193 •97940 .21899 •97573 21 40 .16792 .98580 .18509 .98272 .20222 •97934 .21928 •97566 20 4i .16820 •98575 .18538 .98267 .20250 .97928 •21956 .97560 19 42 .16849 .98570 .18567 .08261 .20279 .97922 .21985 •97553 18 43 .16878 .98565 •18595 .98256 •20307 .97916 .22013 •97547 17 44 .16906 .98561 .18624 .98250 .20336 .97910 .22041 •97541 16 45 •16935 .98556 .18652 .98245 .20364 •97905 .22070 •97534 15 46 .16964 •98551 .18681 .98240 •20393 .97899 .22098 .97528 14 47 .16992 .98546 .18710 •98234 .20421 •97893 .22126 •97521 13 48 .17021 .98541 .18738 .98229 .20450 .97887 •22155 •97515 12 49 .17050 .98536 .18767 .98223 .20478 .97881 .22183 .97508 11 50 .17078 •98531 •18795 .98218 .20507 •97875 .22212 .97502 10 5i .17107 .98526 .18824 .98212 •20535 .97869 .22240 .97496 9 52 .17136 .98521 .18852 .98207 .20563 .97863 .22268 •97489 8 53 .17164 .98516 .18881 .98201 .20592 •97857 .22297 •97483 7 54 .17193 .98511 .18910 .98196 .20620 •97851 .22325 •97476 6 55 .17222 .98506 .18938 .98190 .20649 •97845 •22353 •97470 5 56 .17250 .98501 .18967 .98185 .20677 •97839 .22382 •97463 4 57 .17279 .98496 .18995 .98179 .20706 •97833 .22410 •97457 3 58 .17308 .98491 .19024 •98174 .20734 .97827 .22438 •97450 2 59 .17336 .98486 .19052 .98168 .20763 .97821 .22467 •97444 1 60 •17365 .98481 .19081 •9S163 .20791 •978i5 .22495 •97437 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine r 8( )° 7< )° 7* 1° Ti ro 494 NATURAL SINES AND COSINES 13° 14° 15° 16° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine o .22495 •97437 .24192 .97030 .25882 •96593 •27564 .96126 I .22523 •97430 .24220 .97023 .25910 .96585 .27592 .96118 2 .22552 •97424 .24249 .97015 •25938 •96578 .27620 .96110 3 .22580 •97417 •24277 .97008 .25966 .96570 .27648 .96102 4 .22608 .97411 •24305 .97001 •25994 .96562 .27676 .96094 5 .22637 .97404 •24333 .96994 .26022 •96555 •27704 .96086 6 .22665 •97398 .24362 .96987 .26050 •96547 .27731 .96078 7 .22693 •97391 .24390 .96980 .26079 •96540 •27759 .96070 8 .22722 •97384 .24418 .96973 .26107 •96532 .27787 .96062 9 .22750 •97378 .24446 .96966 •26135 .96524 .27815 .96054 IO .22778 •97371 •24474 •96959 .26163 •96517 •27843 .96046 ii .22807 •97365 .24503 •96952 .26191 .96509 .27871 .96037 12 .22835 •97358 •24531 .96945 .26219 .96502 .27899 .96029 *3 .22863 •97351 .24559 •96937 .26247 •96494 .27927 .96021 14 .22892 •97345 .24587 •96930 .26275 .96486 •27955 •96013 15 .22920 •97338 .24615 .96923 •26303 •96479 •27983 .96005 16 .22948 •97331 .24644 .96916 .26331 •96471 .28011 •95997 17 .22977 .97325 .24672 .96909 .26359 .96463 .28039 •95989 18 .23005 .97318 .24700 .96902 .26387 .96456 .28067 •95981 IQ •23033 •973H .24728 .96894 .26415 .96448 .28095 •95972 20 .23062 •97304 .24756 .96887 .26443 .96440 .28123 •95964 21 .23090 .97298 .24784 .96880 .26471 •96433 .28150 •95956 22 .23118 .97291 .24813 .96873 .26500 .96425 .28178 •95948 23 .23146 .97284 .24841 .96866 .26528 .96417 .28206 .95940 24 •23175 .97278 .24869 .96858 .26556 .96410 .28234 •9593i 25 .23203 .97271 .24897 .96851 .26584 .96402 .28262 •95923 26 .23231 .97264 .24925 .96844 .26612 •96394 .28290 •95915 27 .23260 .97257 •24954 .96837 .26640 .96386 .28318 •95907 28 .23288 •97251 .24982 .96829 .26668 .96379 .28346 .95898 29 .23316 •97244 .25010 .96822 .26696 •96371 •28374 .95890 30 •23345 •97237 .25038 .96815 .26724 •96363 .28402 .95882 31 •23373 •97230 .25066 .96807 .26752 •96355 .28429 •95874 32 .23401 .97223 •25094 .96800 .26780 ■96347 •28457 .95865 33 .23429 •97217 .25122 •96793 .26808 •96340 .28485 •95857 34 •23458 .97210 .25151 .96786 .26836 .96332 .28513 •95849 35 .23486 .97203 •25179 •96778 .26864 •96324 .28541 .95841 36 •23514 .97196 .25207 .96771 .26892 .96316 .28569 •95832 37 •23542 .97189 •25235 .96764 .26920 .96308 .28597 .95824 38 •23571 .97182 •25263 .96756 .26948 •96301 .28625 .95816 39 •23599 .97176 .25291 •96749 .26976 .96293 .28652 .95807 40 •23627 .97169 •25320 .96742 .27004 .96285 .28680 •95799 4i •23656 .97162 .25348 •96734 .27032 .96277 .28708 •9579i 42 .23684 •97155 .25376 •96727 .27060 .96269 .28736 .95782 43 .23712 .97148 .25404 .96719 .27088 .96261 .28764 •95774 44 .23740 .97141 •25432 .96712 .27116 •96253 .28792 •95766 45 •23769 •97134 •25460 .96705 .27144 .96246 .28820 •95757 46 •23797 .97127 .25488 .96697 .27172 .96238 .28847 •95749 47 •23825 .97120 .25516 .96690 .27200 .96230 .28875 •95740 48 •23853 •97H3 •25545 .96682 .27228 .96222 .28903 •95732 49 .23882 .97106 •25573 •96675 .27256 .96214 .28931 •95724 50 .23910 .97100 .25601 .96667 .27284 .96206 .28959 •95715 5i •23938 .97093 .25629 .96660 .27312 .96198 .28987 •95707 52 .23966 .97086 .25657 •96653 •27340 .96190 .29015 .95698 53 •23995 •97079 .25685 •96645 .27368 .96182 .29042 .95690 54 .24023 .97072 •25713 .96638 .27396 •96174 .20070 .95681 55 .24051 .97065 .25741 .96630 .27424 .96166 .29098 .95673 56 .24079 .97058 •25769 .96623 •27452 .96158 .29126 .95664 57 .24108 •97051 .25798 .96615 .27480 .96150 •29154 •95656 58 .24136 .97044 .25826 .96608 .27508 .96142 .29182 •95647 59 .24164 •97037 •25854 .96600 •27536 .96134 .29209 •95639 60 .24192 •97030 .25882 •96593 •27564 .96126 .29237 •95630 e Cosine Sine Cosine Sine Cosine Sine Cosine Sine 76 >° 11 >° 74 1° J 7£ s° NATURAL SINES AND COSINES 495 17° 18° 19° 20° Sine Cosine Sine Cosine Sine Cosine Sine Cosine f .29237 •95630 .30902 .95106 .32557 •94552 .34202 •93969 60 .29265 .95622 .30929 •95097 •32584 .94542 .34229 •93959 59 .29293 •95613 •30957 .95088 .32612 •94533 •34257 •93949 58 .29321 •95605 •30985 •95079 .32639 •94523 .34284 •93939 57 .29348 •95596 .31012 •95070 .32667 •94514 •343 11 •93929 56 .29376 •95588 .31040 .95061 .32694 •94504 •34339 •93919 55 .29404 •95579 .31068 •95052 .32722 •94495 •34366 •93909 54 .29432 •95571 •31095 •95043 •32749 •94485 •34393 •93899 53 .29460 •95562 .31123 •95033 •32777 •94476 •34421 .93889 52 .294S7 •95554 ■31151 •95024 •32804 .94466 .34448 •93879 51 .29515 •95545 .31178 •95015 .32832 •94457 •34475 .93869 50 .29543 •95536 .31206 .95006 •32859 •94447 •34503 •93859 49 •29571 .95528 .31233 •94997 .32887 .94438 •34530 .93849 48 •29599 •95519 .31261 .94988 .32914 .94428 •34557 •93839 47 .29626 •955H .31289 .94979 .32942 .94418 •34584 •93829 46 .29654 •95502 .31316 •94970 .32969 •94409 •34612 •93819 45 .29682 •95493 .31344 .94961 •32997 •94399 .34639 •93809 44 .29710 •95485 •31372 •94952 •33024 .94390 .34666 •93799 43 •29737 •95476 •31399 •94943 •33051 •9438o .34694 •93789 42 •29765 •95467 •31427 •94933 •33079 •94370 •34721 •93779 41 .29793 •95459 •31454 •94924 •33106 •9436i •34748 •93769 40 .29821 •95450 .31482 •94915 •33134 •94351 •34775 •93759 39 .29849 •95441 •31510 .94906 .33161 •94342 •34803 •93748 38 .29876 •95433 •31537 •94897 .33189 •94332 •34830 •93738 37 .29904 .95424 •31565 .94888 .33216 •94322 •34857 •93728 36 •29932 •95415 •31593 .94878 .33244 •94313 .34884 •937i8 35 .29960 •95407 .31620 .94869 •33271 •94303 .34912 •937o8 34 .29987 .95598 .31648 .94860 •33298 .94293 •34939 .93698 33 •30015 •95389 •31675 .94851 ■33326 .94284 .34966 .93688 32 •30043 •9538o •31703 .94842 ■33353 .94274 •34993 •93677 31 .30071 •95372 .31730 •94832 •3338i .94264 •3502I •93667 30 .30098 •95363 .31758 •94823 •334o8 .94254 •35048 •93657 29 .30126 •95354 .31786 .94814 •33436 •94245 •35075 •93647 28 .30154 •95345 •31813 .94805 •33463 •94235 •35I02 •93637 27 .30182 •95337 .31841 •947Q5 •33490 •94225 •35130 .93626 26 .30209 •95328 .31868 .94786 •335i8 •94215 •35157 •93616 25 .30237 •95319 .31896 •94777 •33545 .94206 •35184 .93606 24 .30265 •953io .31923 .94768 ■33573 .94196 •352H •93596 23 .30292 •95301 .31951 •94758 •33600 .94186 ■35239 •93585 22 •30320 •95293 •31979 •94749 •33627 .94176 •35266 •93575 21 •30348 .95284 .32006 •94740 •33655 .94167 •35293 •93565 20 •30376 •95275 •32034 •94730 .33682 •94157 •35320 •93555 19 •30403 .95266 .32061 •94721 •337IO .94147 •35347 •93544 18 •30431 •95257 .32089 .94712 •33737 •94137 •35375 •93534 17 •30459 .95248 .32116 .94702 •33764 .94127 •35402 •93524 16 .30486 .95240 .32144 •94693 •33792 .94118 •35429 •93514 15 •30514 •95231 .32171 .94684 •33819 .94108 •35456 •93503 14 •30542 .95222 .32199 •94674 •33846 .94098 •35484 •93493 13 •30570 •95213 •32227 .94665 .33874 .94088 •355H •93483 12 •30597 .95204 •32254 •94656 •33901 .94078 •35538 •93472 11 .30625 •95195 .32282 .94646 .33929 .94068 •35565 •93462 10 .30653 •95i86 •32309 •94637 •33956 .94058 •35592 •93452 9 .30680 •95177 •32337 .94627 •33983 •94049 •35619 •93441 8 •30708 .95168 •32364 .94618 .34011 •94039 •35647 •93431 7 .30736 •95159 .32392 .94609 •34038 .94029 •35674 .93420 6 •30763 •95i5o .32419 •94599 •34065 .94019 •35701 .93410 5 •30791 .95142 •32447 •94590 •34093 .94009 ■35728 •93400 4 .30819 '95133 •32474 .9458o .34120 •93999 •35755 •93389 3 .30846 .95124 .32502 •94571 •34147 '93989 •35782 ■93379 2 •30874 •95H5 •32529 •9456i •34175 •93979 •358io .93368 1 .30902 .95106 •32557 •94552 .34202 .93969 •35837 •93358 Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 7! 2° 7 1° 71 r 6< r 496 NATURAL SINES AND COSINES 2 L° 22° 23° 24° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine o .35837 •93358 •3746i .92718 •39073 .92050 .40674 .91355 I ^35864 •93348 .37488 .92707 .39100 •92039 .40700 •91343 2 .35891 •93337 •37515 .92697 •39127 .92028 •40727 •91331 3 .35918 •93327 •37542 .92686 •39153 .92016 •40753 •91319 4 •35945 •933i6 •37569 •92675 .39180 .92005 .40780 •91307 5 •35973 .93306 •37595 .92664 •39207 .91994 .40806 .91295 6 .36000 •93295 .37622 •92653 •39234 .91982 •40833 .91283 7 .36027 •93285 •37649 .92642 .39260 .91971 .40860 .91272 8 .36054 •93274 •37676 .92631 •39287 •91959 .40886 .91260 9 .36081 .93264 •37703 .92620 •39314 .91948 .40913 .91248 IO .36108 •93253 •37730 .92609 •39341 .91936 •40939 .91236 n •36135 •93243 •37757 .92598 •39367 .91925 .40966 .91224 12 .36162 .93232 •37784 •92587 •39394 .91914 .40992 .91212 13 .36190 .93222 .378ii .92576 .39421 .91902 .41019 .91200 14 .36217 .93211 .37838 •92565 •39448 .91891 .41045 .91188 15 .36244 .93201 .37865 •92554 •39474 .91879 .41072 .91176 16 .36271 .93190 •37892 •92543 •39501 .91868 .41098 .91164 17 .36298 .93180 •37919 •92532 •39528 .91856 .41125 .91152 18 •36325 .93169 •37946 .92521 •39555 .91845 .41151 .91140 19 •36352 •93159 •37973 .92510 •3958i •91833 .41178 .91128 20 •36379 .93148 •37999 .92499 .39608 .91822 .41204 .91116 21 .36406 •93137 .38026 .92488 •39635 .91810 . -41231 .91104 22 •36434 •93127 .38053 .92477 .39661 .91799 .41257 .91092 23 •36461 .93116 .38080 .92466 .39688 .91787 .41284 .91080 24 .36488 .93106 •38107 •92455 •39715 •91775 .41310 .91068 25 •36515 •93095 •38134 .92444 •39741 .91764 •41337 .91056 26 •36542 .93084 .38161 •92432 •39768 •91752 .41363 .91044 27 •36569 •93074 .38188 .92421 •39795 .91741 .41390 .91032 28 •36596 .93063 •38215 .92410 .39822 .91729 .41416 .91020 29 .36623 •93052 •38241 •92399 .39848 .91718 .41443 .91008 30 •36650 .93042 .38268 ,92388 •39875 .91706 .41469 .90996 31 •36677 •93031 •38295 •92377 .39902 .91694 .41496 .90984 32 .36704 .93020 .38322 .92366 .39928 .91683 .41522 .90972 33 .36731 .93010 •38349 .92355 •39955 .91671 •41549 .90960 34 •36758 .92999 •38376 .92343 .39982 .91660 •41575 .90948 35 .36785 .92988 •38403 •92332 .40008 .91648 > .41602 .90036 36 .36812 .92978 •38430 .92321 •40035 .91636 .41628 .90924 37 .36839 .92967 .38456 .92310 .40062 .91625 •41655 .90911 38 .36867 .92956 .38483 .92299 .40088 .91613 .41681 .90899 39 .36894 .92945 .38510 .92287 .40115 .91601 .41707 .90887 P .36921 •92935 •38537 .92276 .40141 .91590 .41734 •90875 *i .36948 .92924 •38564 .92265 .40168 .91578 .41760 .90863 *2 •36975 .92913 •38591 .92254 •40195 .91566 .41787 •90851 tf .37002 .92902 .38617 •92243 .40221 •91555 .41813 •90839 H .37029 .92892 •38644 .92231 .40248 •91543 .41840 .90826 15 •37056 .92881 .38671 .92220 •40275 •91531 .41866 .90814 tf •37083 .92870 .38698 .92209 .40301 •91519 .41892 .90802 17 .37110 .92859 •38725 .92198 .40328 .91508 .41919 .90790 *8 •37137 .92849 •38752 .92186 .40355 .91496 .41945 .90778 \9 .37164 .92838 .38778 •92175 .40381 .91484 .41972 .90766 ;o •37I9I .92827 .38805 .92164 .40408 .91472 .41998 •90753 ;i .37218 .92816 .38832 .92152 •40434 .91461 .42024 .90741 )2 •37245 .92805 .38859 .92141 .40461 .91449 .42051 .90729 ;3 .37272 .92794 .38886 .92130 .40488 •91437 •42077 .90717 '4 .37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 '5 .37326 •92773 .38939 .92107 .40541 .91414 .42130 .90692 6 •37353 .92762 .38966 .92096 •40567 .91402 .42156 .90680 7 •3738o •92751 •38993 .92085 .40594 .91390 .42183 .90668 8 •37407 .92740 .39020 .92073 .40621 •91378 .42209 •90655 9 •37434 .92729 .39046 .92062 .40647 .91366 .42235 •90643 KD •3746i .92718 •39073 .92050 .40674 •91355 .42262 .90631 / Cosine Sine Cosine 1 Sine Cosine Sine Cosine Sine 6* ;° 6^ TO 66 >° 61 ;° NATURAL SINES AND COSINES 497 25° 26° 27° 28° / Sine Cosine Sine Cosine Sine Cosine Sine Cosine / o .42262 •90631 •43837 .89879 •45399 .89101 .46947 .88295 60 I .42288 .90618 •43863 .89867 •45425 .89087 •46973 .88281 59 2 •42315 .90606 .43889 .89854 •45451 .89074 .46999 .88267 58 3 .42341 •90594 .43916 .89841 •45477 .89061 .47024 .88254 57 4 .42367 .90582 •43942 .89828 •45503 .89048 •47050 .88240 56 5 .42394 .90569 .43968 .89816 •45529 •89035 .47076 .88226 55 6 .42420 •90557 •43994 .89803 •45554 .89021 .47101 .88213 54 7 .42446 .90545 .44020 .89790 •4558o .89008 .47127 .88199 53 8 •42473 •90532 .44046 .89777 .45606 .88995 .47153 .88185 52 9 •42499 .90520 .44072 .89764 •45632 .88981 .47178 .88172 5i IO •42525 •90507 .44098 •89752 .45658 .88968 .47204 .88158 5o ii •42552 •90495 .44124 •89739 .45684 .88955 .47229 .88144 49 12 .42578 .90483 •44I5I .89726 •457IO .88942 .47255 .88130 48 13 .42604 .90470 •44177 •89713 •45736 .88928 .47281 .88117 47 14 .42631 .90458 •44203 .89700 •45762 .88915 .47306 .88103 46 15 •42657 .90446 .44229 .89687 •45787 .88902 •47332 .88089 45 16 .42683 •90433 •44255 .89674 •45813 .88888 •47358 .88075 44 17 .42709 .90421 .44281 .89662 •45839 .88875 •47383 .88062 43 18 .42736 .90408 •44307 .89649 .45865 .88862 •47409 .88048 42 19 .42762 •90396 •44333 .89636 .45891 .88848 •47434 .88034 41 20 .42788 •90383 •44359 .89623 •45917 .88835 .47460 .88020 40 21 .42815 •90371 .44385 .89610 •45942 .88822 .47486 .88006 39 22 .42841 •90358 .44411 •89597 .45968 .88808 ■475" •87993 38 23 .42867 •90346 •44437 .89584 •45994 •8S795 •47537 .87979 37 24 .42894 •90334 •44464 •89571 .46020 .88782 •47562 •87965 36 25 .42920 .90321 •44490 •89558 .46046 .88768 .47588 •87951 35 26 .42946 .90309 .44516 •89545 .46072 •88755 .47614 •87937 34 27 .42972 .90296 •44542 •89532 .46097 .88741 •47639 .87923 33 28 •42999 .90284 .44568 .89519 .46123 .88728 •47665 .87909 32 29 •43025 .90271 •44594 .89506 .46149 .88715 .47690 .87896 3i 30 •43051 •90259 .44620 89493 .46175 .88701 •477i6 .87882 30 31 •43077 .90246 .44646 .89480 .46201 .88688 •47741 .87868 29 32 .43104 •90233 .44672 .89467 .46226 .88674 •47767 .87854 28 33 •43130 .90221 .44698 •89454 .46252 .88661 •47793 .87840 27 34 •43156 .90208 •44724 .89441 .46278 .88647 .47818 .87826 26 35 .43182 .90196 •44750 .89428 .46304 .88634 •47844 .87812 25 36 •43209 •90183 •44776 .89415 •46330 .88620 .47869 .87798 24 37- •43235 .90171 .44802 .89402 •46355 .88607 •47895 .87784 23 38 .43261 .90158 .44828 .89389 .46381 .88593 .47920 .87770 22 39 •43287 .90146 •44854 .89376 .46407 .88580 .47946 .87756 21 40 •43313 •90133 .44880 .89363 •46433 .88566 •47971 •87743 20 4i •43340 .90120 .44906 •89350 .46458 .88553 •47997 •87729 19 42 •43366 .90108 •44932 •89337 .46484 .88539 .48022 •87715 18 43 •43392 •90095 •44958 .89324 .46510 .88526 .48048 .87701 17 44 .43418 .90082 .44984 .89311 •46536 .88512 .48073 .87687 16 45 •43445 .90070 .45010 .89298 .46561 .88499 .48099 .87673 15 46 •43471 •90057 •45036 .89285 .46587 .88485 .48124 .87659 14 47 •43497 .90045 .45062 .89272 •46613 .88472 .48150 •87645 13 48 •43523 .90032 .45088 .89259 •46639 .88458 •48i75 .87631 12 49 •43549 .90019 .45H4 .89245 .46664 .88445 .48201 .87617 11 50 •43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 10 5i .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 9 52 .43628 .89981 45192 .89206 .46742 .88404 •48277 .87575 8 53 •43654 .89968 •452i8 .89193 •46767 .88390 •48303 .87561 7 54 .43680 .89956 •45243 .89180 •46793 .88377 .48328 .87546 6 55 •437o6 •89943 •45269 .89167 .46819 .88363 •48354 .87532 5 56 •43733 •89930 •45295 •89153 .46844 .88349 •48379 .87518 4 57 •43759 .89918 •45321 .89140 .46870 .88336 .48405 •87504 3 58 •43785 .89905 •45347 .89127 .46896 .88322 .48430 .87490 2 59 .43811 .89892 •45373 .89114 .46921 .88308 .48456 .87476 1 60 •43837 .89879 •45399 .89101 •46947 .88295 .48481 .87462 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 6^ [° 6c 1° 65 1° 6] 49 8 NATURAL SINES AND COSINES 29° 30° 31° 32° / Sine Cosine Sine ( "OSINE Sine Cosine Sine Cosine f o .48481 .87462 .50000 86603 •51504 •85717 .52992 .84805 60 I .48506 .87448 .50025 86588 •51529 .85702 •53017 .84789 59 2 •48532 .87434 .50050 86573 •51554 .85687 •53041 •84774 58 3 .48557 .87420 .50076 86559 •51579 .85672 •53066 •84759 57 4 .48583 .87406 .50101 86544 .51604 .85657 •53091 .84743 56 5 .48608 .87391 .50126 86530 .51628 .85642 •531 15 .84728 55 6 .48634 .87377 •50151 86515 .51653 .85627 •53140 .84712 54 7 .48659 .87363 .50176 86501 .51678 .85612 •53164 .84697 53 8 .48684 .87349 .50201 86486 •51703 •85597 •53189 .84681 52 9 .48710 •87335 .50227 86471 .51728 .85582 •53214 .84666 51 IO •48735 .87321 .50252 86457 •51753 .85567 •53238 .84650 50 ii .48761 .87306 •50277 86442 •51778 .85551 •53263 •84635 49 12 .48786 .87292 •50302 86427 •51803 .85536 .53288 .84619 48 13 .48811 .87278 •50327 86413 .51828 .85521 .53312 .84604 47 14 •48837 .87264 •50352 86398 .51852 .85506 •53337 .84588 46 15 .48862 .87250 •50377 86384 .51877 .85491 •53361 •84573 45 16 .48888 •87235 •50403 86369 .51902 .85476 .53386 •84557 44 17 .48913 .87221 .50428 86354 •51927 .85461 .53411 .84542 43 18 .48938 .87207 •50453 86340 •51952 •85446 •53435 .84526 42 19 .48964 •87193 •50478 86325 •51977 •85431 .53460 .84511 41 20 .48989 .87178 •50503 86310 .52002 .85416 .53484 .84495 40 21 .49014 .87164 .50528 86295 .52026 .85401 •53509 .84480 39 22 .49040 •87150 •50553 86281 .52051 .85385 •53534 .84464 38 23 .49065 .87136 •50578 86266 .52076 .85370 .53558 .84448 37 24 .49090 .87121 .50603 86251 .52101 •85355 .53583 •84433 36 25 .49116 .87107 .50628 86237 .52126 •85340 •53607 .84417 35 26 .49141 •87093 •50654 86222 •52151 •85325 •53632 .84402 34 27 .49166 •87079 •50679 86207 .52175 •85310 •53656 .84386 33 28 .49192 .87064 .50704 86192 .52200 •85294 •5368i .84370 32 29 .49217 •87050 .50729 86178 •52225 •85279 •53705 •84355 3i 30 .49242 .87036 •50754 86163 ♦52250 .85264 •53730 •84339 30 31 .49268 .87021 .50779 86148 •52275 .85249 •53754 •84324 29 32 .49293 .87007 .50804 86133 .52299 •85234 •53779 .84308 28 33 .49318 .86993 .50829 86119 •52324 .85218 •53804 .84292 27 34 •49344 .86978 .50854 86104 •52349 •85203 .53828 .84277 26 35 .49369 .86964 •50879 86089 •52374 .85188 •53853 .84261 25 36 •49394 .86949 .50904 86074 •52399 .85173 •53877 .84245 24 37 .49419 .86935 .50929 86059 .52423 •85157 •53902 .84230 23 38 •49445 .86921 •50954 86045 .52448 .85142 •53926 .84214 22 39 .49470 .86906 •50979 86030 •52473 .85127 •53951 .84198 21 40 •49495 .86892 .51004 86015 .52498 .85112 •53975 .84182 20 41 •49521 .86878 .51029 86000 .52522 .85096 .54000 .84167 19 42 .49546 .86863 .51054 85985 •52547 .85081 .54024 .84151 18 43 •49571 .86849 .51079 85970 •52572 .85066 •54049 •84135 17 44 .49596 .86834 .51104 85956 •52597 •85051 •54073 .84120 16 45 .49622 .86820 .51129 85941 .52621 •85035 •54097 .84104 15 46 .49647 .86805 •5H54 85926 .52646 .85020 .54122 .84088 14 47 .49672 .86791 •5ii79 859TI .52671 .85005 .54146 .84072 13 48 .49697 .86777 .51204 85896 .52696 .84989 •54I7I .84057 12 49 •49723 .86762 .51229 8s88i .52720 •84974 •54195 .84041 11 50 •49748 .86748 .51254 85866 .52745 .84959 .54220 .84025 10 51 •49773 •86733 .51279 85851 .52770 .84943 •54244 .84009 9 52 .49798 .86719 •51304 85836 •52794 .84928 .54269 •83994 8 53 .49824 .86704 •51329 85821 .52819 .84913 •54293 •83978 7 54 .49849 .86690 •51354 85806 .52844 .84897 •54317 .83962 6 55 •49874 .86675 •51379 85792 .52869 .84882 •54342 .83946 5 56 .49899 .86661 .51404 85777 .52893 .84866 .54366 •83930 4 57 .49924 .86646 .51429 85762 .52918 .84851 •54391 .83915 3 58 .49950 .86632 •51454 85747 .52943 .84836 •54415 .83899 2 59 •49975 .86617 •51479 85732 .52967 .84820 •54440 .83883 1 60 .50000 .86603 •51504 85717 .52992 .84805 •54464 .83867 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine t 6( )° 59° 5£ *° 51 to NATURAL SINES AND COSINES 499 33° 34° 35° 36° / Sine Cosine Sine Cosine Sine Cosine Sine Cosnie / o .54464 .83867 .55919 .82904 .57358 .81915 •58779 .80902 60 I .54488 .83851 •55943 .82887 .57381 .81899 .58802 .80885 59 2 •54513 .83835 .55968 .82871 •57405 .81882 .58826 .80867 58 3 •54537 .83819 .55992 .82855 •57429 .81865 .58849 .80850 57 4 •5456i .83804 .56016 .82839 •57453 .81848 .58873 .80833 56 5 •54586 .83788 .56040 .82822 •57477 .81832 .58896 .80816 55 6 .54610 •83772 .56064 .82806 .57501 .81815 .58920 ..80799 54 7 •54635 •83756 .56088 .82790 .57524 .81798 •58943 .80782 53 8 •54659 .83740 .56112 .82773 •57548 .81782 .58967 .80765 52 9 •54683 •83724 .56136 .82757 •57572 .81765 .58990 .80748 5i IO .54708 .83708 .56160 .82741 .57596 .81748 .59014 .80730 5o ii •54732 83692 .56184 .82724 •57619 .81731 •59037 .80713 49 12 •54756 .83676 .56208 .82708 •57643 .81714 .59061 .80696 48 13 •5478i .83660 .56232 .82692 •57667 .81698 .59084 .80679 47 14 •54805 .83645 .56256 .82675 .57691 .81681 .59108 .80662 46 15 .54829 .83629 .56280 .82659 •57715 .81664 •59131 .80644 45 16 •54854 .83613 •56305 .82643 .57738 .81647 •59154 .80627 44 17 •54878 .83597 •56329 .82626 .57762 .81631 •59178 .80610 43 18 .54902 .83581 .56353 .82610 .57786 .81614 .59201 .80593 42 19 •54927 .83565 .56377 •82593 .57810 •81597 •59225 .80576 4i 20 •54951 .83549 .56401 .82577 .57833 .81580 .59248 .80558 40 21 •54975 .83533 •56425 .82561 •57857 .81563 .59272 .80541 39 22 •54999 .83517 •56449 .82544 .57881 .81546 •59295 .80524 38 23 •55024 •83501 •56473 .82528 .57904 •81530 •59318 •80507 37 24 •55048 .83485 •56497 .82511 .57928 .81513 •59342 .80489 36 25 •55072 •83469 .56521 .82495 •57952 .81496 •59365 .80472 35 26 •55097 .83453 .56545 .82478 •57976 .81479 •59389 •80455 34 27. •55121 •83437 •56569 .82462 •57999 .81462 .59412 .80438 33 28 •55145 .83421 •56593 .82446 .58023 •81445 •59436 .80420 32 29 •55169 •83405 .56617 .82429 •58047 .81428 •59459 .80403 3i 30 •55194 .83389 .56641 .82413 .58070 .81412 .59482 .80386 30 31 .55218 .83373 .56665 .82396 .58094 •81395 •595o6 .80368 29 32 •55242 .83356 .56689 .82340 .58118 .81378 .59529 .80351 28 33 .55266 •83340 .56713 .82363 .58141 .81361 •59552 •80334 27 34 •55291 •83324 .56736 .82347 .58165 .81344 .59576 .80316 26 35 •55315 .83308 .56760 •82330 .58189 .81327 •59599 .80299 25 36 •55339 •83292 .56784 .82314 .58212 .81310 .59622 .80282 24 37 •55363 .83276 .56808 .82297 .5S236 .81293 •59646 .80264 23 38 .55388 .83260 .56832 .82281 .58260 .Si 176 .59669 .80247 22 39 •55412 .83244 .568=16 .82264 •58283 .81259 •59693 .80230 21 40 •55436 .83228 .56880 .82248 .58307 .81242 •597i6 .80212 20 4i •5546o .83212 .56904 .82231 •58330 .81225 •59739 .80195 19 42 .55484 •83195 .56928 .82214 •58354 .81208 •59763 .80178 18 43 •55509 •83179 •56952 .82198 .58378 .81191 .59786 .80160 17 44 •55533 .83163 .56976 .82181 .58401 .81174 .59809 •80143 16 45 •55557 •83147 .57000 .82165 •58425 .81157 •59832 .80125 15 46 .55581 •83131 .57024 .82148 •58449 .81140 •59856 .80108 14 47 •55605 •83115 •57047 .82132 .58472 .81123 ! •59879 .80091 13 48 ■55630 .83098 j •57071 .82115 .58496 .81106 .59902 .80073 12 49 •55654 .83082 ! •57095 .82098 •58519 .81089 [ •59926 .80056 11 5o .55678 .83066 •57H9 .82082 •58543 .81072 •59949 .80038 10 5i •55702 .83050 •57143 .82065 .58567 .81055 •59972 .80021 9 52 •55726 •83034 •57167 .82048 .58590 .81038 •59995 .80003 8 53 •55750 .83017 •57191 .82032 .58614 .81021 .60019 .79986 7 54 •55775 .83001 •57215 .82015 .58637 .81004 .60042 .79968 6 55 •55799 .82985 •57238 .81999 .58661 .80987 ! .60065 •79951 5 56 •55823 .82969 •57262 .81982 .58684 .80970 .60089 •79934 4 57 •55847 •82953 .57286 .81965 .58708 •80953 .60112 .79916 3 58 •55871 .82936 .57310 .81949 •58731 .80936 •60135 •79899 2 59 •55895 .82920 •57334 .81932 .58755 .80919 .60158 .79881 1 60 •55919 .82904 •57358 .81915 .58779 .80902 .60182 .79864 / Cosine Sine Cosine Sine Corine Sine 1 Cosine Sine / 5( )° 51 )° 54 t° 1 5c ;° 5°° NATURAL SINES AND COSINES 37° 38° 39° 40° f Sine Cosine Sine Cosine Sine Cosine Sine Cosine / o .60182 .79864 .61566 .78801 .62932 .77715 .64279 .76604 60 I .60205 .79846 .61589 .78783 .62955 .77696 .64301 .76586 59 2 .60228 .79829 .61612 .78765 .62977 .77678 •64323 .76567 58 3 .60251 .79811 .61635 •78747 .63000 .77660 .64346 .76548 57 4 .60274 •79793 .61658 .78729 .63022 .77641 .64368 •76530 56 5 .60298 .79776 .61681 .78711 .63045 .77623 .64390 .76511 55 6 .6032 1« •79758 .61704 .78694 .63068 .77605 .64412 .76492 54 7 .60344 •79741 .61726 .78676 .63090 .77586 .64435 •76473 53 8 .60367 .79723 •61749 .78658 •63113 •77568 .64457 •76455 52 9 .60390 .79706 .61772 .78640 •63135 •77550 .64479 .76436 5i IO .60414 .79688 •61795 .78622 .63158 .77531 .64501 .76417 50 ii .60437 •79671 .61818 .78604 .63180 •77513 .64524 .76398 49 12 .60460 .79653 .61841 .78586 .63203 •77494 •64546 .76380 48 13 .60483 •79635 .61864 .78568 .63225 •77476 .64568 .76361 47 14 .60506 .79618 .61887 •78550 .63248 •77458 .64590 •76342 46 15 .60529 .79600 .61909 •78532 •63271 •77439 .64612 .76323 45 16 •60553 •79583 .61932 •78514 .63293 •77421 •64635 •76304 44 17 .60576 •79565 .61955 .78496 .63316 •77402 .64657 .76286 43 18 .60599 •79547 .61378 .78478 •63338 •77384 .64679 .76267 42 IQ .60622 •79530 .62001 .78460 .63361 .77366 .64701 .76248 4i 20 .60645 •79512 .62024 .78442 .63383 •77347 ■64723 .76229 40 21 .60668 •79494 .62046 .78424 .63406 •77329 .64746 .76210 39 22 .60691 •79477 .62069 •78405 .63428 •773io .64768 .76192 38 23 .60714 •79459 .62092 .78387 •63451 .77292 .64790 •76173 37 24 .60738 .79441 .62115 .78369 .63473 .77273 .64812 •76154 36 25 .60761 •79424 .62138 •78351 .63496 •77255 .64834 .76135 35 26 .60784 .79406 .62160 •78333 •63518 .77236 .64856 .76116 34 27 .60807 .79388 .62183 •78315 .63540 .77218 .64878 .76097 33 28 .60830 •79371 .62206 .78297 •63563 .77199 .64901 .76078 32 29 .60853 •79353 .62229 .78279 •63585 .77i8i .64923 .76059 31 30 .60876 •79335 .62251 .78261 .63608 .77162 .64945 .76041 30 31 .60899 .793x8 .62274 •78243 •63630 .77144 .64967 .76022 29 32 .60922 .79300 .62297 .78225 •63653 .77125 .64989 .76003 28 33 .60945 .79282 .62320 .78206 .63675 .77107 .65011 •75984 27 34 .60968 .79264 .62342 .78188 .63698 .77088 .65033 .75965. 26 35 .60991 .79247 •62365 .78170 .63720 .77070 •65055 •75946 25 36 .61015 .79229 .62388 .78152 •63742 .77051 .65077 •75927 24 37 .61038 .79211 .62411 .78134 •63765 •77033 .65100 •75908 23 38 .61061 •79193 •62433 ,78116 .63787 •77014 .65122 .75889 22 39 .61084 .79176 .62456 .78098 .63810 .76996 .65144 .75870 21 40 .61107 .79158 .62479 .78079 .63832 .76977 .65166 .75851 20 41 .61130 .79140 .62502 .78061 .63854 .76959 .65188 .75832 19 42 •61 153 .79122 .62524 .78043 .63877 .76940 .65210 .75813 18 43 .61176 •79105 •62547 .78025 .63899 .76921 .65232 •75794 17 44 .61199 •79087 •62570 .78007 .63922 •76903 •65254 •75775 16 45 .61222 .79069 .62592 .77988 •63944 .76884 •65276 •75756 15 46 .61245 •79051 .62615 •7797o .63966 .76866 .65298 .75738 14 47 .61268 •79033 .62638 .77952 .63989 .76847 •65320 .75719 13 48 .61291 '79016 .62660 •77934 .64011 .76828 •65342 .75700 12 49 .61314 .78998 .62683 •779i6 .64033 .76810 -65364 .75680 11 5o •61337 .78980 .62706 .77897 .64056 .76791 .65386 .75661 10 51 •6t^6o .78962 .62728 .77879 .64078 .76772 .65408 .75642 9 52 •61383 .78944 •62751 .77861 .64100 •76754 •65430 •75623 8 53 .61406 .78926 .62774 .77843 .64123 .76735 •65452 •75604 7 54 .61429 .78908 .62796 •77824 .64145 .76717 •65474 .75585 6 55 •6T451 .78891 .62819 .77806 .64167 .76698 .65496 •75566 5 56 •6i474 .78873 .62842 .77788 .64190 .76679 .65518 •75547 4 57 •6i497 .78855 .62864 •77769 .64212 .76661 .65540 •75528 3 58 .61520 .78837 .62887 •77751 .64234 .76642 .65562 •75509 2 59 •6i543 .78819 .62909 •77733 .64256 .76623 .65584 •75490 1 60 .61566 .78801 .62932 .77715 .64279 .76604 .65606 .75471 / Cosine Sine Cosine Sine Cosine Sine Cosine Sine / 52 1 51 50 49 NATURAL SINES AND COSINES SOI 4 1° 42° 43° 44° Sine Cosine Sine 1 Cosine Sine Cosine Sine Cosine f .65606 •75471 •66913 .74314 .68200 •73135 .69466 .71934 60 .65628 •75452 •66935 •74295 .68221 .73116 .69487 •71914 59 .65650 •75433 .66956 .74276 .68242 •73096 .69508 .71894 58 .65672 •75414 .66978 .74256 .68264 .73076 •69529 •71873 57 .65694 •75395 .66999 •74227 .68285 •73056 .69549 •71853 56 .65716 •75375 .67021 .74217 .68306 •73036 .69570 .71833 55 .65738 •75356 .67043 .74198 .68327 .73016 .69591 .71813 54 •65759 •75337 .67064 .74178 .68349 .72996 .69612 .71792 53 .65781 .75318 .67086 •7*159 .68370 .72:176 •69633 .71772 52 .65803 •75299 .67107 •74139 ■68391 •72957 .69654 •71752 5i .65825 .75280 .67129 '74120 .68412 .72937 •69675 •71732 50 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 .65869 .75241 .67172 .74080 .68455 •72897 .69717 •.71691 48 .65891 .75222 .67194 .74061 .68476 .72377 •69737 .71671 47 •65913 •75203 .67215 .74041 .68497 •72857 .69758 .71650 46 .65935 •75184 .67237 ./4022 .68518 •72837 •69779 .71630 45 •65956 •75i65 .67258 ./4002 .68539 .72817 .69800 .71610 44 .65978 .75146 .67280 .73983 .68561 .72797 .69821 •71590 43 .66000 .75126 •67301 •73963 .68582 .72777 .69842 •71569 42 .66022 •75io7 •67323 •73944 .68603 •72757 .69862 •71549 4i .66044 .75088 •67344 •73924 .68624 •72737 .69883 .71529 40 .66066 •75069 .67366 .73904 .68645 .72717 .69904 .71508 39 .66088 •75050 .67387 .73885 .68666 .72697 •69925 .71488 38 .66109 .75030 .67409 .73865 .68688 .72677 .69946 .71468 37 .66131 .75011 .67430 .73846 .68709 .72657 .69966 •71447 36 .66153 .74992 •67452 .73826 .68730 •72637 .69987 .71427 35 .66175 •74973 •67473 .73806 .68751 .72617 .70008 .71407 34 .66197 •74953 •67495 •73787 .68772 •72597 .70029 .71386 33 .66218 •74934 .67516 •73767 .68793 •72577 .70049 .71366 32 .66240 •74915 .67538 •73747 .68814 •72557 .70070 •71345 3i .66262 .74896 •67559 •73728 .68835 .72537 .70091 •71325 30 .66284 .74876 .67580 .737o8 .68857 .72517 .70112 .71305 29 .66306 .74857 .67602 .73688 .68878 •72497 •70132 .71284 28 .66327 .74838 .67623 .73669 j .68899 •72477 •70153 .71264 27 •66349 .74818 .67645 •73649 .68920 •72457 •70174 •71243 26 •66371 •74799 .67666 .73629 .68941 •72437 •70195 .71223 25 •66393 .74780 .67688 .73610 .68962 .72417 •70215 .71203 24 .66414 .7476o .67709 .73590 .68983 •72397 .70236 .71182 23 .66436 •74741 •67730 •73570 .69004 •72377 .70257 .71162 22 .66458 .74722 .67752 •73551 .69025 •72357 .70277 .71141 21 .66480 •74703 •67773 .73531 .69046 •72337 .70298 .71121 20 .66501 .74683 •67795 •735II .69067 •72317 •70319 .71100 19 .66523 .74664 .67816 •73491 .69088 .72297 •70339 .71080 18 •66545 .74644 •67837 •73472 .69109 .72277 .70360 •71059 17 .66566 •74625 .67859 •73452 .69130 .72257 .70381 • 71039 16 .66588 .74606 .67880 •73432 .69151 .72236 .70401 .71019 15 .66610 .74586 .67901 •73413 .69172 .72216 .70422 .70998 14 .66632 .74567 .67923 •73393 .69193 .72196 •70443 .70978 13 ■66653 .74548 •67944 •73373 .69214 .72176 .70463 •70957 12 •66675 •74528 •67965 •73353 •69235 .72156 .70484 •70937 11 .66697 •74509 .67987 •73333 .69256 .72136 •70505 .70916 10 .66718 •74489 .68008 •73314 .69277 .72116 •70525 .70896 9 .66740 •74470 .68029 •73294 .69298 •72095 .70546 •70875 8 .66762 •74451 .68051 •73274 .69319 .72075 .70567 •70855 7 .66783 •74431 .68072 •73254 •69340 •72055 .70587 .70834 6 .66805 .74412 .68093 •73234 .69361 •72035 .70608 .70813 5 .66827 •74392 .68115 •73215 .69382 .72015 .70628 .70793 4 .66848 •74373 .68136 •73195 .69403 •71995 .70649 .70772 3 .66870 •74353 .68157 •73175 .69424 •71974 .70670 .70752 2 .66891 •74334 .68179 •73155 •69445 •71954 .70690 •70731 1 .66913 •74314 .68200 •73135 .69466 •71034 .70711 .70711 Cosine Sine Cosine Sine Cosine Sink 1 Cosine Sine 1 4£ 1° 47 'O 46 1 45 502 NATURAL SECANTS AND CO-SECANTS 1° 2° 3° r Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. CO-SEC. r O Infinite. 1. 000 1 57-299 1 .0006 28.654 1. 0014 19.107 6c I 3437-7Q 1. 000 1 56.359 1 .0006 28.417 1. 0014 19.002 55 2 1718.90 1 .0002 55-450 1 .0006 28.184 1.0014 18.897 5* 3 1145.90 1 .0002 54-570 1.0006 27-955 1. 0014 18.794 57 4 859-44 1 .0002 53./18 1 .0006 27.730 1. 0014 18.692 56 5 687.55 1.0002 52.891 1.0007 27.508 1. 0014 18.591 55 6 572.96 1 .0002 52.090 1.0007 27.290 1. 0015 18.491 SA 7 491. 11 1.0002 51-313 1.0007 27.075 1.0015 18.393 53 8 429.72 1.0002 50.558 1.0007 26.864 1.0015 18.295 52 9 381.97 1.0002 49.826 1.0007 26.655 1.0015 18.198 5i IO 343-77 1 .0002 49.114 1.0007 26.450 1.0015 18.103 5c ii 312.52 1.0002 48.422 1.0007 26.249 1.0015 18.008 45 12 j 286.48 1.0002 47-750 1.0007 26.050 1.0016 17.914 48 13 264.44 1.0002 47.096 1.0007 25-854 1.0016 17.821 47 14 245-55 1.0002 46.460 1 .0008 25.661 1.0016 17.730 At 15 229.18 1.0002 45.840 1 .0008 25-471 1. 0016 17.639 45 16 214.86 1.0002 45.237 1 .0008 25.284 1. 0016 17-549 AA 17 202.22 1.0002 44.650 1 .0008 25.100 1. 0016 17.460 A3 18 190.99 1.0002 44-077 1 .0008 24.918 1. 001 7 17-372 42 19 180.73 1 .0003 43-520 1 .0008 24-739 1. 001 7 17.285 41 20 171.89 1 .0003 42.976 1.0008 24.562 1. 0017 17.198 4c 21 163.70 1 .0003 42.445 1 .0008 24-358 1. 001 7 17. 113 35 22 156.26 1 .0003 41.928 1.0008 24.216 1. 001 7 17.028 38 23 149.47 1.0003 41.423 1 0009 24.047 1. 001 7 16.944 37 24 1 143.24 1.0003 40.930 1.0009 23.880 1.0018 16.861 36 25 I37.5I 1 .0003 40.448 1.0009 23.716 1. 00 18 16.779 35 26 132.22 1 .0003 39-978 1 .0009 23-553 1.0018 16.698 34 27 127.32 1 .0003 39-5i8 1 .0009 23-393 1. 00 18 16.617 33 28 122.78 1 .0003 39.069 1 1.0009 23-235 1. 0018 16.538 32 29 118.54 1 .0003 38.631 1 .0009 23.079 1.0018 16.459 3i 30 H4-59 1.0003 38.201 1.0009 22.925 1.0019 16.380 30 31 110.90 1 .0003 37.782 1. 0010 22.774 1. 0019 16.303 29 32 IQ7-43 1 .0003 37-371 1. 00 10 22.624 1.0019 16.226 28 33 104.17 1 .0004 36.969 I.OOIO 22.476 1. 0019 16.150 27 34 IOI.II 1 .0004 36.576 1. 00 10 22.330 1.0019 16.075 26 35 98.223 1 .0004 36.191 I.OOIO 22.186 1.0019 16.000 25 36 95*495 1 .0004 35.8i4 I.OOIO 22.044 1.0020 15-926 24 37 92.914 1 .0004 35-445 I.OOIO 2 1 .904 1.0020 15.853 23 38 1. 000 1 92.469 1 .0004 35-o84 I.OOIO 21.765 1.0020 I5.780 22 39 1. 000 1 88.149 1 .0004 34-729 I. OOI I 21.629 1.0020 I5.708 21 40 1. 000 1 85.946 1 .0004 34-382 I. OOI I 21.494 1.0020 I5.637 20 4i 1. 000 1 83.849 1 .0004 34.042 I. OOI I 21.360 1. 002 1 15.566 19 42 1. 000 1 81.853 1 .0004 33-7o8 I. OOI I 21.228 1. 002 1 15.496 18 43 1. 000 1 79-950 1 .0004 33-38i I. OOI I 2 1 .098 1. 002 1 I5.427 17 44 1. 000 1 78.133 1 .0004 33.060 I. OOI I 20.970 1. 002 1 15.358 16 45 1. 000 1 76.396 1.0005 32.745 1. 00 1 1 20.843 1. 002 1 15.290 15 46 1. 000 1 74.736 1 .0005 32.437 1. 0012 20.717 1.0022 15.222 14 47 1. 000 1 73-146 1 .0005 32.134 I. OOI2 20.593 1.0022 15-155 13 48 1. 000 1 71.622 1 .0005 31-836 I.OOI2 20.471 1.0022 15-089 12 49 1. 000 1 71.160 1.0005 31-544 I. OO I 2 20.350 1.0022 I5-023 11 5o 1. 000 1 68.757 1 .0005 31-257 I. OOI2 20.230 1.0022 14.958 10 5i 1. 000 1 67.409 1 .0005 30.976 I. OOI2 20.112 1.0023 14.893 9 52 1. 000 1 66.113 1 .0005 30.699 I.OOI2 19-995 1.0023 14.829 8 53 1. 000 1 64.866 1 .0005 30.428 I.OOI3 19.880 1.0023 I4.765 7 54 1. 000 1 63.664 1 .0005 30.161 I.OOI3 19.766 1.0023 14.702 6 55 1. 000 1 62.507 1 .0005 29.899 I.OOI3 19-653 1.0023 14.640 5 56 1. 000 1 61.391 1.0006 29.641 I.OOI3 19-541 1.0024 14-578 4 57 1. 000 1 61.314 1.0006 29.388 1. 001 3 19-431 1.0024 I4.5I7 3 58 1. 000 1 59-274 1.0006 29.139 1.0013 19.322 1.0024 14.456 2 59 1. 000 1 58.270 1 .0006 28.894 1. 0013 19.214 1.0024 14-395 1 60 1. 000 1 57-299 1 .0006 28.654 1. 0014 19.107 1.0024 14-335 / Co-sec. 1 Sec. Co-sec- Sec. Co-sec. Sec. CO-SEC. Sec. 8< )° Si 1° 8' 7° 8( 3° 1 NATURAL SECANTS AND CO-SECANTS S03 40 5° 6 >° 7° / Sec. Co-sec. Sec. Co-sec. ! Sec.' Co-sec Sec. Co-sec. / 1.0024 14-335 1 .0038 11.474 1.0055 9.5668 1.0075 8.2055 60 I 1.0025 14.276 1.0038 11.436 1.0055 9.5404 1 1.0075 8.1861 59 2 1.0025 14.217 1.0039 n.398 1.0056 9-5141 1.0076 8.1668 58 3 1.0025 14-159 1.0039 11.360 1.0056 9.4880 1.0076 8.1476 57 4 1.0025 14.101 1.0039 n.323 1.0056 9.4620 1.0076 8.1285 56 5 1.0025 14.043 1.0039 11.286 1.0057 9.4362 1.0077 8.1094 55 6 1.0026 13-986 1 .0040 11.249 1.0057 9-4105 1.0077 8.0905 54 7 1.0026 I3-930 1 .0040 11-213 1.0057 9-3850 1.0078 8.0717 53 8 1.0026 13-874 1 .0040 11. 176 1.0057 9-3596 1.0078 8.0529 52 9 1.0026 13.818 1.0040 1 1. 140 1.0058 9-3343 1.0078 8.0342 5i 10 1.0026 13-763 1. 004 1 11. 104 j 1.0058 9.3092 1.0079 8.0156 5o 11 1.0027 13-708 1. 004 1 11.069 f 1.0058 9.2842 1.0079 7.9971 49 12 1.0027 I3.654 1. 0041 11-033 1 1.0059 9-2593 1.0079 7-9787 48 13 1.0027 13.600 1. 004 1 10.988 1.0059 9.2346 1 .0080 7.9604 47 14 1.0027 13-547 1 .0042 10.963 ! 1.0059 9.2100 1 .0080 7.9421 46 15 1.0027 13-494 1.0042 10.929 1 .0060 9.i855 1 .0080 7.9240 45 16 1.0028 13-441 1 .0042 10.894 1.0060 9.1612 1. 008 1 7-9059 44 17 1.0028 13-389 1.0043 10.860 1 .0060 9.I370 1. 008 1 7.8879 43 18 1.0028 13-337 1.0043 10.826 1. 006 1 9.1129 1.0082 7.8700 42 IQ 1.0028 13.286 1.0043 10.792 1. 006 1 9.0890 1 .0082 7-8522 4i 20 1.0029 13-235 1.0043 10.758 1. 006 1 9.0651 1.0082 7-8344 40 21 1.0029 13-184 1.0044 10.725 1 .0062 9.0414 1 .0083 7.8168 39 22 1.0029 13-134 1.0044 10.692 1 0062 9.0179 1 .0083 7.7992 38 23 1.0029 13-084 1 .0044 10.659 1 .0062 8.9944 1 .0084 7.7817 37 24 1.0029 13-034 1.0044 10.626 1 .0063 8.9711 1 .0084 7.7642 36 25 1.0030 12.985 1.0045 10.593 1 .0063 8.9479 1 .0084 7.7469 35 26 1.0030 12.937 1.0045 10.561 1 .0063 8.9248 1 .0085 7.7296 34 27 1.0030 12.888 1.0045 10.529 1.0064 8.9018 1.0085 7.7124 33 28 1 .0030 12.840 1 .0046 10.497 1 .0064 8.8790 1.0085 7-6953 32 29 1. 003 1 12.793 1 .0046 10.465 1 .0064 8.8563 1 .0086 7.6783 3i 30 1. 003 1 12.745 1 .0046 10.433 1.0065 8.8337 1 .0086 7.6613 30 3i 1. 003 1 12.698 1.0046 10.402 1 .0065 8.8112 1.0087 7.6444 29 32 1. 003 1 12.652 1.0047 10.371 1 .0065 8.7888 1.0087 7.6276 28 33 1.0032 12.606 1.0047 10.340 1 .0066 8.7665 1.0087 7.6108 27 34 1.0032 12.560 1.0047 10.309 1 .0066 8.7444 1.0088 7-5942 26 35 1.0032 12.514 1 .0048 10.278 1 .0066 8.7223 1.0088 7-5776 25 35 1.0032 12.469 1.0048 10.248 1.0067 8.7004 1 .0089 7.5611 24 37 1.0032 12.424 1.004S 10.217 1.0067 8.6786 1 .0089 7-5446 23 38 1-0033 12-379 1 .0048 10.187 1.0067 8.6S69 1 .0089 7.5282 22 39 1 -0033 12.335 1 .0049 10.157 1 .0068 B-6 3 53 1 .0090 7-5ii9 21 40 1.0033 12.291 1 .0049 10.127 1.0068 8.6138 1*0090 7-4957 20 41 1 -0033 12.248 1.0049 10.098 1.0068 8.5924 1 .0090 7-4795 19 42 1.0034 12.204 1.0050 10.068 1.0069 8.5711 1. 009 1 7-4634 18 43 1.0034 12. 161 1.0050 10.039 1 .0069 8.5499 1. 009 1 7-4474 17 44 1 .0034 12. 118 1.0050 10.010 ' 1.0069 8.5289 1.0092 7-4315 16 45 1.0034 12.076 1.0050 9.9812 1.0070 8.5079 1.0092 7-4I56 15 46 1 -0035 12.034 1. 00 5 1 99525 1.0070 8.4871 1.0092 7-3998 14 47 1 -0035 11.992 1 .005 1 9.9239 1.0070 8.4663 1 .0093 7-3840 !3 48 1 -0035 11.950 1. 005 1 9-8955 1.0071 8-4457 1 -0093 7-3683 12 49 1-0035 11.909 1.0052 9.8672 1. 007 1 8.4251 1 .0094 7-3527 11 5o 1 .0036 11.868 1.0052 9.8391 1.0071 8.4046 1 .0094 7-3372 10 51 1.0036 11.828 1.0052 9.8112 1.0072 8.3843 1.0094 7-3217 9 52 1 .0036 11.787 1 -0053 9-7834 1.0072 8.3640 1.0095 7-3063 8 53 1.0036 ci.747 1 -0053 9-7558 1.0073 8-3439 1 .0095 7.2909 7 54 1.0037 11.707 1 -0053 9.7283 1.0073 8.3238 1 .0096 7-2757 6 55 1.0037 11.668 1 -0053 9.7010 1.0073 8.3039 1 .0096 7.2604 5 56 1.0037 11.628 1.0054 9-6739 1.0074 8.2840 1.0097 7-2453 4 57 1.0037 n.589 1.0054 9.6469 1.0074 8.2642 1.0097 7.2302 3 58 1.0038 H-550 1.0054 9.6200 1.0074 8.2446 1.0097 7-2152 2 59 1.0038 11. 512 1 -0055 9-5933 1.0075 8.2250 1 .0098 7.2002 1 5o 1 .0038 11.474 1.0055 9.5668 1.0075 8.2055 1 .0098 7.1853 / Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. / 8. >° 8 1° 8c \° 8: 1° 5°4 NATURAL SECANTS AND CO-SECANTS 8° 9° 10° 11° r Sec. Co-sec. Sec. Co-sec. Sec. CO.SEC. Sec. Co-sec. / O 1.0098 7-1853 1.0125 6.3924 1.0154 5.7588 1.0187 5.2408 60 I 1 .0099 7.1704 1.0125 6.3807 1.0155 5-7493 1.0188 5-2330 59 2 1.0099 7-1557 1.0125 6.3690 1.0155 5-7398 1.0188 5.2252 58 3 1 .0099 7.1409 1.0126 6-3574 1.0156 5.7304 1.0189 5-2174 57 4 1. 0100 7.1263 1.0126 6.3458 1.0156 5-72IO 1. 0189 5.2097 56 5 I.OIOO 7.1117 1.0127 6-3343 1.0157 5-7II7 I.OIOO 5-2019 55 6 I.OIOI 7.0972 1.0127 6.3228 1.0157 5-7023 1.0191 5.1942 54 7 I.OIOI 7.0827 1.0128 6.3113 1.0158 5-6930 1.0191 5.1865 53 8 I.OI02 7.0683 1.0128 6.2999 1.0158 5.6838 1.0192 5.1788 52 9 I.OI02 7-Q539 1.0129 6.2885 1.0159 5-6745 1.0192 5.1712 5i IO I.OI02 7-0396 1.0129 6.2772 1.0159 5.6653 1.0193 5.1636 50 ii I.OIO3 7.0254 1.0130 6.2659 1.0160 5.6561 1.0193 5.1560 49 12 I.OIO3 7.01 12 1.0130 6.2546 1.0160 5.6470 1. 0194 5.1484 48 13 I.OIO4 6.9971 1.0131 6.2434 1.0161 5.6379 1.0195 5.1409 47 14 I.OIO4 6.9830 1.0131 6.2322 1.0162 5.6288 1.0195 5.1333 46 15 I. OIO4 6.9690 1.0132 6.2211 1.0162 5-6197 1,0196 5.1258 45 16 I.OI05 6.9550 1.0132 6.2100 1. 163 5-6107 1. 0196 5.1183 44 17 I.OI05 6.9411 1.0133 6.1990 1.0163 5-6017 1.0197 5.1109 43 18 I.OIOO 6.9273 1.0133 6.1880 1.0164 5.5928 1.0198 5.1034 42 19 I.OI06 6.9135 1.0134 6.1770 1.0164 5.5838 1. 0198 5.0960 4i 20 I.OI07 6.8998 1. 0134 6.1661 1.0165 5-5749 1.0199 5.0886 40 21 I.OI07 6.8861 I-OI35 6.1552 1. 0165 5.5660 1.0199 5.0812 39 22 I.OI07 6.8725 I-OI35 6.1443 1. 0166 5-5572 1.0200 5.0739 38 23 I.OIO8 6.8589 1.0136 6.1335 1.0166 5-5484 1. 0201 5.0666 37 24 I. OI08 6.8454 1.0136 6.1227 1.0167 5-5396 1. 020 1 5- 593 36 25 I.OIO9 6.8320 1.0136 6.1120 1.0167 5-53o8 1.0202 5.0520 35 26 I. OIO9 6.8185 1.0137 6.1013 1. 0168 5-5221 1.0202 5-0447 34 27 I. OIIO 6.8052 1.0137 6.0906 1.0169 5-5134 1.0203 50375 33 28 I.OIIO 6.7919 1.0138 6.0800 1. 0169 5-5047 1.0204 5.0302 32 29 I.OIII 6.7787 1.0138 6.0694 1. 01 70 5.4960 1.0204 5.0230 3i 30 I.OIII 6.7655 1.0139 6.0588 1. 01 70 5-4874 1.0205 5-0158 30 31 I.OIII 6.7523 1.0139 6.0483 1.0171 5.4788 1.0205 5-0087 29 32 I.OII2 6.7392 1.0140 6.0379 1.0171 5.4702 1.0206 5-0015 28 33 I.OII2 6.7262 1. 0140 6.0274 1.0172 5-4617 1.0207 4.9944 27 34 I.OII3 6.7132 1.0141 6.0170 1.0172 5-4532 1.0207 4-9873 26 35 I.OII3 6.7003 1.0141 6.0066 1.0173 5-4447 1.0208 4.9802 25 36 I.OII4 6.6874 1.0142 5-9963 1.0174 5-4362 1.0208 4-9732 24 37 I.OII4 6.6745 1.0142 5.9860 1.0174 5.4278 1.0209 4.9661 23 38 I.OII5 6.6617 1.0143 5-9758 1.0175 5-4194 1. 02 10 4-9591 22 39 I.OII5 6.6400 1.0143 5-9655 1.0175 5-4HO 1. 02 10 4.9521 21 40 I.OII5 6.6363 1.0144 5-9554 1.0176 5.4026 1. 02 1 1 4.9452 20 41 I.OIl6 6.6237 1.0144 5.9452 1. 01 76 5-3943 1. 02 1 1 4.9382 19 42 I.OIl6 6.6111 1.0145 5-9351 1.0177 5.3860 1.02 12 4.9313 18 43 I.OII7 6.5985 1.0145 5-9250 1.0177 5-3777 1.0213 4.9243 17 44 I.OII7 6.5860 1.0146 5-9150 1. 01 78 5-3695 1.0213 4-9175 16 45 I.OIl8 6.5736 1. 0146 5.9049 1.0179 5-36i2 1.0214 4.9106 15 46 I.OIl8 6.5612 1.0147 5.8050 1.0179 5-3530 1. 02 1 5 4-9037 14 47 I.OII9 6.5488 1.0147 5-8850 1.0180 5 3449 1 .02 1 5 4.8969 13 48 I.OII9 6.5365 1. 0148 5.875I 1. 0180 5-3367 1.0210 4.8901 12 49 I.OII9 6.5243 1.0148 5-8652 1.0181 5-3286 1. 02 16 4.8833 11 50 I.OI20 6.5121 1. 0149 5.8554 1.0181 5-3205 1. 02 1 7 4.8765 10 5i I.OI20 6.4999 1.0150 5-8456 1. 0182 5-3124 1.0218 4.8697 9 52 I.OI2I 6.4878 1.0150 5.8358 1.0182 5-3044 1.0218 4.8630 8 53 I.OI2I 6-4757 1.0151 5.8261 1.0183 5-2963 1.0219 4.8563 7 54 I.OI22 6.4637 I.OX5I 5.8163 1.0184 5.2883 1.0220 4.8496 6 55 I.OI22 6.4517 1.0152 5-8067 1.0184 5-2803 1.0220 4.8429 5 56 I.OI23 6.4398 1.0152 5.7970 1.0185 5-2724 1.0221 4.8362 4 57 I.OI23 6.4279 1.0153 5-7874 1.0185 5.2645 j.0221 4.8296 3 58 I.OI24 6.4160 I-OI53 5.7778 1. 0186 5.2566 T.0222 4.8229 2 59 I.OI24 6.4042 1.0154 5.7683 1. 0186 5-2487 1.0223 4.8163 1 60 I.OI25 6.3924 1.0154 5.7588 1.0187 5.2408 1.0223 4.8097 / CO-SEC. Sec. CO-SEC. Sec. Co-sec. Sec. Co-sec. Sec. / 81 8( )° 7t )° 7* 1° NATURAL SECANTS AND CO-SECANTS 5°5 12° 13° 14° / Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. o 1.0223 4.8097 1.0263 4-4454 1.0306 4-1336 I 1.0224 4.8032 1.0264 4.4398 1.0307 4.1287 2 1.0225 4.7966 1.0264 4.4342 1.0308 4.1239 3 1.0225 4.7901 1.0265 4.4287 1.0308 4.1191 4 1.0226 4-7835 1.0266 4.4231 1.0309 4.1144 5 1.0226 4.7770 1.0266 4.4176 1.0310 4.1096 6 1.0227 4.7706 1.0267 4.4121 1.0311 4.1048 7 1.0228 4.7641 1.0268 4.4065 1-0311 4.1001 8 1.0228 4-7576 1.0268 4.401 1 1. 03 1 2 4-0953 9 1.0229 4-7512 1.0269 43956 1.0313 4.0906 IO 1.0230 4.7448 1.0270 4.3910 1.0314 4.0859 ii 1.0230 4-7384 1.0271 4-3847 1.0314 4.0812 12 1.0231 4.7320 1.0271 4-3792 1-0315 4.0765 13 1.0232 4-7257 1.0272 4-3738 1.0316 4.0718 14 1.0232 4.7193 1.0273 4.3684 1-0317 4.0672 15 1.0233 4-7130 1.0273 4-3630 1-0317 4.0625 16 1.0234 4.7067 1.0274 4-3576 1.0318 4-0579 17 1.0234 4.7004 1.0275 4-3522 1.0319 4-0532 18 1.023s 4.6942 1.0276 4.3469 1.0320 4.0486 19 1.0235 4.6879 1.0276 4-3415 1.0320 4.0440 20 1.0236 4.6817 1.0277 4-3362 1.032 1 4.0394 21 1.0237 4-6754 1.0278 4.3309 1.0322 4.0348 22 1.0237 4.6692 1.0278 4-3256 1-0323 4.0302 23 1.0238 4.6631 1.0279 4-3203 1.0323 4.0256 24 1.0239 4.6569 1.0280 4.3I50 1.0324 4.0211 25 1.0239 4.6507 1.0280 4.3098 1.0325 4.0165 26 1.0240 4.6446 1.0281 4-3045 1.0326 4.0120 27 1.0241 4-6385 1.0282 4.2993 1.0327 4.0074 28 1.0241 4.6324 1.0283 4.2941 1.0327 4.0029 29 1.0242 4.6263 1.0283 4.2888 1.0328 3.9984 30 1.0243 4.6202 1.0284 4.2836 1.0329 3-9939 31 1.0243 4.6142 1.0285 4.2785 1.0330 3.9894 32 1.0244 4.6081 1.0285 4-2733 1.0330 3-9850 33 1.0245 4.6021 1.0286 4.2681 1-0331 3-9805 34 1.0245 4.5961 1.0287 4.2630 1.0332 3.976o 35 1.0246 4.5901 1.0288 4-2579 1 -0333 3.9716 36 1.0247 4.5841 1.0288 4.2527 1 -0334 3.9672 37 1.0247 4.5782 1.0289 4.2476 1.0334 3.9627 38 1.0248 4.5722 1.0200 4.2425 ; 1.0335 3-9583 39 1.0249 4-5663 1. 0291 4-2375 1.0336 3-9539 40 1.0249 4-5604 1.0291 4.2324 1.0337 3-9495 41 1.0250 4-5545 1.0292 4.2273 1.0338 3-9451 42 1-0251 4-5486 1.0293 4.2223 1-0338 3.9408 43 1.0251 4.5428 1.0293 4-2173 1.0339 3-9364 44 1.0252 4-5369 1.0294 4.2122 1.0340 3.9320 45 1.0253 4-53II 1.0295 4.2072 1. 0341 3-9277 46 1.0253 4-5253 1.0296 4.2022 1-0341 39234 47 1.0254 4-5195 1.0296 4.1972 , 1.0342 3-9199 48 1.0255 4-5137 1.0297 4.1923 1 -0343 3-9147 49 1.0255 4.5079 1.0298 4.1873 ; 1.0344 3.9104 50 1.0256 4.5021 1.0299 4.1824 1 -0345 3.9061 5i 1.0257 4.4964 1.0299 4.1774 1 -0345 3.9018 52 1.0257 4.4907 1.0300 4.I725 1.0346 3.8976 53 1.0258 4.4850 1. 0301 4.1676 1 -0347 3.8933 54 1.0259 4-4793 1.0302 4.1627 1.0348 3.8990 55 1.0260 4.4736 1.0302 4.I578 j 1.0349 3.8848 56 1.0260 4.4679 1-0303 4.1529 1.0349 3-8805 57 1.0261 4.4623 1.0304 4.1481 1-0350 3-8763 58 1.0262 4.4566 1-0305 4.1432 1.0351 3.8721 59 1.0262 4.4510 1-0305 4.1384 1.0352 3.8679 60 1.0263 4-4454 1 .0306 4.I336 1 -0353 3.8637 / CO-SEC. Sec. Co-sec. 1 Sec. Co-sec. Sec. 71 r° 7( 5° 71 >° 15° Sec. Co-sec. I-0353 I-0353 I.0354 I.0355 1.0356 I-0357 1-0358 1.0358 1.0359 1.0360 1.0361 1.0362 1.0362 1.0363 1.0364 1-0365 1.0366 1.0367 1.0367 1.0368 1.0369 1.0370 1-0371 1-0371 1.0372 1 -0373 1 -03 74 I-0375 1.0376 1.0376 1.0377 1.0378 I-0379 1.0380 1. 038 1 1.0382 1.0382 1.0383 1.0384 1.0385 1.0386 1.0387 1.0387 1.0388 1.0389 1.0390 1-0391 1.0392 I -0393 I-0393 1.0394 I.0395 1.0396 I.0397 1.0398 1.0399 1.0399 1 .0400 1. 040 1 1.0402 1.0403 CO-SEC. 3-8637 3-8595 3-8553 3.85I2 3.8470 56 3.8428 55 3.8387 54 3.8346 53 3.8304 52 3.8263 5i 3.8222 50 3.8181 49 3.8140 48 3.8100 47 3.8059 46 3.8018 45 3.7978 44 3-7937 43 3-7897 42 3.7857 4i 3.7816 40 3-7776 39 3.7736 38 3.7697 37 3.7657 36 3.7617 35 3-7577 34 3.7538 33 3.7498 32 3-7459 3i 3.7420 30 3.738o 29 3.7341 28 3.7302 27 3-7263 26 3.7224 25 3-7186 24 3 7147 23 3.7108 22 3.7070 21 3-7031 20 3.6993 19 3.6955 .18 3-6917 17 3-6878 t6 3.6840 15 3.6802 14 3-6765 13 3.6727 12 3.6689 11 3-6651 10 3.6614 9 3.6576 8 3-6539 7 3.6502 6 3.6464 5 3.6427 4 3-6390 3 3.6353 2 3.6316 1 3.6279 Sec. / 5°6 NATURAL SECANTS AND CO-SECANTS 1 S° 17° 1 8° 19° Sec. Co-sec. Sec. Co-sec. Sec. CO-SEC. Sec. CO-SEC. 1.0403 3.6279 1 -0457 3.4203 1-0515 3.2361 1.0576 3-07I5 1 .0404 3-6243 1.0458 3.4170 1.0516 3-2332 I-Q577 3.0690 1 .0405 3.6206 1.0459 3-4138 1-0517 3.2303 1.0578 3.0664 1 .0406 3.6169 1.0460 3.4106 1.0518 3.2274 I.0579 3.0638 1 .0406 3-6133 1. 046 1 3.4073 1.0519 3-2245 1.0580 3.0612 1.0407 5.6096 1. 0461 3.4041 1.0520 3.2216 1.0581 3-0586 1 .0408 3.6060 1.0462 3-4009 1.0521 3.2188 1.0582 3-056I 1 .0409 3.6024 1.0463 3-3977 1.0522 3.2159 1.0584 3-0535 1.0410 3-5987 1.0464 3-3945 1-0523 3.2131 1.0585 3.0509 1. 041 1 3-5951 1.0465 3-3913 1.0524 3.2102 1.0586 3.0484 1.0412 3-5915 1 .0466 3-388i 1.0525 3.2074 1.0587 3-0458 1-0413 3-5879 1.0467 3-3849 1.0526 3.2045 1.0588 3-0433 1-0413 3-5843 1.0468 3-3817 1.0527 3.2017 1.0589 3.0407 1. 0414 3-5807 1.0469 3-3785 1.0528 3.1989 1.0590 3.0382 1.0415 3-5772 1.0470 3-3754 1.0529 3.1960 1.0591 3-Q357 1. 0416 3-5736 1.0471 3.3722 1.0530 3.1932 1.0592 3-0331 1. 041 7 3-5700 1.0472 3.3690 1-0531 3.1904 1 -0593 3.0306 1.0418 3-5665 1 -0473 3-3659 1.0532 3-i8r 1.0594 3.0281 1.0419 3-5629 1.0474 3-3627 I-0533 3.1848 1 -Q595 3.0256 1.0420 3-5594 1 -0475 3-3596 I-0534 3.1820 1.0596 3.0231 1.0420 3-5559 1.0476 3-3565 I-0535 3.1792 1.0598 3.0206 1. 042 1 3-5523 1.0477 3-3<34 1.0536 3-I764 1.0599 3.0181 1.0422 3-5488 1.0478 3-35Q2 I-0537 3-I736 1 .0600 3-0156 1.0423 3-5453 1.0478 3-3471 1-0538 3.1708 1. 060 1 3-0131 1.0424 3-54i8 1.0479 3 -344Q I-0539 3.1681 1.0602 3.0106 1.0425 3S3^>3 1 .0480 3-3409 1.0540 3.1653 1 .0603 3.0081 1.0426 3-5348 1. 048 1 3.3378 1-0541 3-1625 1 .0604 3.0056 1.0427 3-5313 1.0482 3-3347 1.0542 3.I598 1.0605 3.0031 1.0428 3-5279 1.0483 3-33i6 1 -0543 3-I570 1 .0606 3.0007 1.0428 3-5244 1.0484 3.3286 1.0544 3-1543 1.0607 2.9982 1.0429 3-5209 1.0485 3-3255 1 -Q545 3-I5I5 1 .0608 2-9957 1.0430 3-5175 1.0486 3-3224 1.0546 3.1488 1 .0609 2-9933 1. 043 1 3-5140 1.0487 3-3194 1 -Q547 3-1461 1.0611 2 .9908 1.0432 3-5106 1.0488 3-3163 1.0548 3-1433 1.0612 2.9884 1 -0433 3-5072 1 .0489 3-3133 1.0549 3.1406 1.0613 2.9859 1.0434 3-5037 1.0490 3.3102 1-0550 3-1379 1.0614 2.9835 1-0435 3-5003 1. 049 1 3.3072 1-0551 3-1352 1.0615 2.9810 1.0436 3.4969 1.0492 3-3042 1.0552 3.I325 1.0616 2.9786 1.0437 3-4935 1 1.0493 3-30H 1 -0553 3.1298 1.0617 2.9762 1.0438 3.4901 j 1.0494 3.2981 i.o554 3-1271 1. 0618 2.9738 1.0438 3.4867 1.0495 3.2951 1 -0555 3.1244 1.0619 2-9713 1.0439 3-4833 1 .0496 3.2921 1-0556 3.1217 1.0620 2.9689 1 .0440 3-4799 1.0497 3.2891 1.0557 3-1190 1.0622 2.9665 1. 0441 3.4766 1 .0498 3.2861 1-0558 3-1163 1.0623 2. 9641 1.0442 3-4732 1.0499 3-2831 I-0559 3-H37 1.0624 2.9617 1.0443 3.4698 1.0500 3.2801 1.0560 3.1110 1.0625 2-9593 1 .0444 3.4665 1. 050 1 3.2772 1.0561 3-1083 1.0626 2.9569 1.0445 3-4632 1.0502 3.2742 1.0562 3-1057 1.0627 2-9545 1 .0446 3-4598 1-0503 3.2712 1-0563 3.1030 1.0628 2.9521 1.0447 3-4565 1.0504 3-2683 1.0565 3.1004 1.0629 2.9497 1 .0448 3-4532 1-0505 3-2653 1.0566 3-0977 1.0630 2.9474 1.0448 3.4498 1.0506 3.2624 1-0567 3-0951 1.0632 2.9450 1.0449 3-4465 1-0507 3-2594 1.0568 3-0925 1.0633 2.9426 1.0450 3-4432 1.0508 3-2565 1.0569 3.0898 1.0634 2.9402 1-0451 3-4399 1.0509 3.2535 1.0570 3.0872 1-0635 2.9379 1.0452 3-4366 1. 05 10 3.2506 1-0571 3.0846 1.0636 2-9355 1 -Q453 3-4334 1-0511 3-2477 1.0572 3.0820 1.0637 2.9332 1.0454 3-430I 1.0512 3.2448 I-0573 3-Q793 1.0638 2.9308 1 -0455 3.4268 1-0513 3.2419 1 -0574 3.0767 1.0639 2.9285 1.0456 3-4236 1-0514 3.2390 I-0575 3-074I 1. 0641 2.9261 1 -045 7 3-4203 1-0515 3.2361 1-0576 3-07I5 1.0642 2.9238 CO-SEC. Sec. Co-sec. Sec. CO-SEC. Sec. CO-SEC. Sec. Tc 1° 72 71 7C >° NATURAL SECANTS AND CO-SECANTS 5°7 2( )° 21° 22° 23° / Sec. Co-sec. Sec. CO-SEC. Sec. Co-sec. Sec. Co-sec. 1.0642 2.9238 1.0711 2.7904 1.0785 2.6695 1.0864 2.5593 I 1.0643 2.9215 1-0713 2.7883 1.0787 2.6675 1.0865 2-5575 2 1 .0644 2.9191 1.0714 2.7862 1.0788 2.6656 1.0866 2.5558 3 1.0645 2.9168 1.0715 2.7841 1.0789 2.6637 1.0868 2.5540 4 1.0646 2-9145 1.0716 2.7820 1.0790 2.6618 1.0869 2.5523 5 1.0647 2.9122 1-0717 2-7799 1.0792 2.6599 1.0870 2.5506 1.0648 2.9098 1.0719 2.7778 1.0793 2.6580 1.0872 2.5488 7 1.0650 2.9075 1.0720 2-7757 1.0794 2.6561 1.0873 2.5471 8 1. 065 1 2.9052 1.0721 2.7736 1 -0795 2.6542 1.0874 2.5453 9 1.0652 2.9029 1.0722 2.7715 1.0797 2.6523 1.0876 2.5436 1-0653 2.9006 1.0723 2.7694 1.0798 2.6504 1.0877 2.5419 1 1.0654 2.8983 1.0725 2.7674 1.0799 2.6485 1.0878 2.5402 2 1.0655 2.8960 1.0726 2.7653 1. 080 1 2.6466 1.0880 2.5384 3 1.0656 2.8937 1.0727 2.7632 1.0802 2.6447 1. 088 1 2-5367 4 1.0658 2.8915 1.0728 2.7611 1 .0803 2.6428 1.0882 2.5350 5 1.0659 2.8892 1.0729 2.7591 1.0804 2.6410 1.0884 2-5333 6 1 .0660 2.8869 1.0731 2.7570 1.0806 2.6391 1.0885 2.5316 7 1. 066 1 2.8846 1.0732 2.7550 | 1.0807 2.6372 1.0886 2.5299 8 1.0662 2.8824 I-Q733 2.7529 1.0808 2.6353 1.0888 2.5281 1.0663 2.8801 I-Q734 2.7509 1.0810 2-6335 1.0889 2.5264 1.0664 2.8778 1.0736 2.7488 1. 081 1 2.6316 1. 0891 2.5247 1 1.0666 2.8756 1-0737 2.7468 1.0812 2.6297 1.0892 2.5230 2 1.0667 2.8733 1.0738 2-7447 1.0813 2.6279 1.0893 2.5213 3 1.0668 2.8711 I-0739 2.7427 1.0815 2.6260 1.0895 2.5196 4 1.0669 2.8688 1.0740 2.7406 1.0816 2.6242 1.0896 2.5179 5 1.0670 2.8666 1.0742 2.7386 1. 081 7 2.6223 1.0897 2.5163 6 1. 067 1 2.8644 1 -0743 2.7366 1.0819 2.6205 1.0899 2.5146 7 1.0673 2.8621 1.0744 2.7346 1.0820 2.6186 1 .0900 2.5129 8 1.0674 2.8599 1 -0745 2.7325 1. 082 1 2.6168 1.0902 2.5112 9 1.0675 2.8577 1.0747 2.7305 1.0823 2.6150 1.0903 2.5095 1.0676 2.8554 1.0748 2.7285 1.0824 2.6131 1.0904 2.5078 1 1.0677 2.8532 1.0749 2.7265 1.0825 2.6113 1.0906 2.5062 2 1.0678 2.8510 1-0750 2.7245 1.0826 2.6095 1.0907 2.5045 3 1.0679 2.8488 1-0751 2.7225 1.0828 2.6076 1.0908 2.5028 4 1. 0681 2.8466 I-0753 2.7205 1.0829 2.6058 1.0910 2.5011 5 1.0682 2.8444 I-0754 2./I85 1.0830 2 .6040 1.0911 2.4995 6 1.0683 2.8422 10755 2.7165 1.0832 2.6022 1-0913 2.4978 7 1.0684 2 .8400 1.0756 2.7145 1.0833 2 .6003 1.0914 2.4961 8 1.0685 2.8378 1.0758 2.7125 1.0834 2.5985 1.0915 2-4945 9 1.0686 2.8356 I-Q759 2.7105 1.0836 2.5967 1. 091 7 2.4928 1.0688 2.8334 1.0760 2.7085 1.0837 2.5949 1.0918 2.4912 1 1.0689 2.8312 1.0761 2.7065 1.0838 2.5931 1.0920 2.4895 2 1.0600 2.8290 1.0763 2.7045 1 .0840 2.5913 1. 092 1 2.4879 3 1. 069 1 2.8269 1.0764 2.7026 1. 0841 2.5895 1.0922 2.4862 4 1.0692 2.8247 1.0765 2.7006 1.0842 2.5877 1.0924 2.4846 5 1 .0694 2.8225 1.0766 2.6986 1.0844 2.5859 1.0925 2.4829 6 1.0695 2.8204 1.0768 2.6967 1.0845 2.5841 1.0927 2.4813 7 1.0696 2.8182 1.0769 2.6947 1.0846 2.5823 1.0928 2-4797 3 1.0697 2.8160 1.0770 2.6927 1.0847 2.5805 1.0929 2.4780 P 1.0698 2.8139 1.0771 2.6908 1 .0849 2.5787 1-0931 2.4764 1.0699 2.8117 I-0773 2.6888 1.0850 2.5770 1.0932 2.4748 1 1.0701 2.8096 1.0774 2.6869 1. 085 1 2.5752 1.0934 2.4731 2 1.0702 2.8074 I-0775 2.6849 1.0853 2-5734 1 -0935 2.4715 3 1.0703 2.8053 1.0776 2.6830 1.0854 2.5716 1.0936 2.4699 4 1.0704 2.8032 1.0778 2.6810 1.0855 2.5699 1.0938 2.4683 5 1.0705 2.8010 1.0779 2.6791 1.0857 2.5681 1.0939 2.4666 6 1.0707 2.7989 1.0780 2.6772 1.0858 2.5663 1. 0941 2.4650 7 1.0708 2.7968 1.0781 2.6752 1.0859 2.5646 1.0942 2.4634 s 1.0709 2.7947 1.0783 2.6733 1. 086 1 2.5628 1 .0943 2.4618 9 1.0710 2.7925 1.0784 2.6714 1.0862 2.5610 1.0945 2.4602 1.0711 2.7904 1.0785 2.6695 1.0864 2-5593 1.0946 2.4586 Sec. / Co-sec. Sec. CO-SEC. Sec. Co-sec. Sec. Co-sec. 61 )° 61 3° 6' r° 64 3° 5 o8 NATURAL SECANTS AND CO-SECANTS 24° 2 5° 26° 27° Sec. Co-sec. Sec. Co-sec. Sec. 1.1126 Co-sec. Sec. Co-sec. i .0946 2.4586 1. 1034 2.3662 2.2812 1. 1223 2.2027 1.0948 2.4570 1.1035 2.3647 1.1127 2.2798 1. 1225 2.2014 1.0949 2-4554 1.1037 2.3632 1.1129 2.2784 1. 1226 2.2002 1-0951 2.4538 1. 1038 2.3618 1.1131 2.2771 1. 1228 2.1989 1.0952 2.4522 1. 1040 2.3603 1.1132 2.2757 1. 1230 2.1977 1 -0953 2.4506 1.1041 2.3588 1.1134 2.2744 1.1231 2.1964 1 -0955 2.4490 1.1043 2-3574 I.U35 2.2730 1.1233 2.1952 1.0956 2.4474 1. 1044 2.3559 1.1137 2.2717 1. 1235 2.1939 1.0958 2.4458 1. 1046 2-3544 1.1139 2.2703 1. 1237 2.1927 1.0959 2.4442 1. 1047 2.3530 1.1 140 2.2690 1. 1-38 2.1914 1.0961 2.4426 1.1049 2-3515 1.1142 2.2676 1. 1 240 2.1902 1.0962 2.4411 1. 1050 2.3501 1.1143 2.2663 1. 1242 2.1889 1.0963 2-4395 1. 1052 2.3486 1.1145 2.2650 1. 1243 2.1877 1.0965 2-4379 1. 1053 2.3472 1.1147 2.2636 1. 1245 2.1865 1.0966 2.4363 1. 1055 2-3457 1.1148 2 2623 1. 1247 2.1852 1.0968 2-4347 1. 1056 2-3443 1.1150 2.2610 1. 1248 2.1840 1.0969 2.4332 1. 1058 2.3428 1.1151 2.2596 1. 1250 2.1828 1. 097 1 2.4316 1. 1059 2.3414 I.H53 2.2583 1. 1252 2.1815 1.0972 2.4300 1.1061 2-3399 1.1*55 2.2570 1.1253 2.1803 1.0973 2.4285 1. 1062 2.3385 1-1156 2.2556 1. 1255 2.1791 1.0975 2.4269 1. 1064 2.3371 1.1158 2-2543 1. 1257 2.1778 1.0976 2.4254 1. 1065 2.3356 1.1159 2.2530 1. 1258 2.1766 1.0978 2.4238 1. 1067 2.3342 1.1161 2.2517 1. 1260 2.1754 1.0979 2.4222 1. 1068 2.3328 1.1163 2.2503 1. 1262 2.1742 1. 098 1 2.4207 1. 1070 2.3313 1.1164 2.2490 1. 1264 2.1730 1.0982 2.4191 1. 1072 2.3299 1.1166 2.2477 1. 1265 2.1717 1.0984 2.4176 1. 1073 2.3285 1.1167 2.2464 1. 1267 2.1705 1.0985 2.4160 1. 1075 2.3271 1.1169 2.2451 1. 1269 2.1693 1.0986 2.4145 1. 1076 2.3256 1.1171 2.2438 1. 1270 2.1681 1.0988 2.4130 1. 1078 2.3242 1.1172 2.2425 1. 1272 2.1669 1.0989 2.4114 1. 1079 2.3228 1.1174 2.2411 1. 1274 2.1657 1. 099 1 2.4099 1.1081 2.3214 1.1176 2.2398 1. 1275 2.1645 1.0992 2.4083 1. 1082 2.3200 1.1177 2.2385 1. 1277 2.1633 1.0994 2.4068 1. 1084 2.3186 1.1179 2.2372 1. 1279 2.1620 1.0995 2.4053 1. 1085 2.3172 1.1180 2.2359 1.1281 2.1608 1.0997 2.4037 1. 1087 2.3158 1.1182 2.2346 1. 1282 2.1596 1.0998 2.4022 1. 1088 2.3143 1.1184 2-2333 1. 1284 2.1584 1. 1000 2.4007 1. 1090 2.3129 1.1185 2.2320 1. 1286 2.1572 I.IOOI 2.3992 1. 1092 2-3II5 1.1187 2.2307 1. 1287 2.1560 1. 1003 2.3976 1. 1093 2.3101 1.1189 2.2294 1. 1289 2.1548 1. 1004 2.3961 1. 1095 2.3087 1.1190 2.2282 1.1291 2.1536 1. 1005 2.3946 1. 1096 2.3073 1.1192 2.2269 1. 1293 2.1525 1. 1007 2.3931 1. 1098 2.3059 1.1193 2.2256 1. 1294 2.1513 1. 1008 2.3916 1. 1099 2.3046 1.1195 2.2243 1. 1296 2.1501 I.IOIO 2.3901 I.IIOI 2.3032 1.1197 2.2230 1. 1298 2.1489 1. ton 2.3886 ; I.II02 2.3018 1.1 198 2.2217 1. 1299 2.1477 1.1013 2.3871 I.IIO4 2.3004 1. 1 200 2.2204 1.1301 2.1465 1.1014 2.3856 ! I.IIOO 2.2990 1. 1202 2.2192 1-1303 2.1453 1.1016 2.3841 I.II07 2.2976 1. 1203 2.2179 1. 1305 2.1441 1.1017 2.3826 I.IIO9 2.2962 1. 1205 2.2166 1. 1306 2.1430 1.1019 2.3811 i. mo 2.2949 1. 1207 2.2153 1. 1308 2.1418 1. 1020 2.3796 1. Ilia 2.2935 1. 1208 2.2141 1.1310 2.1406 1. 1022 2.3781 1.1113 2.2921 1.1210 2.2128 1.1312 2.1394 1. 1023 2.3766 1. HIS 2.2907 1.1212 2.2115 1-1313 2.1382 1. 1025 2.3751 1.1116 2.2894 1.1213 2.2103 1-1315 2.1371 1. 1026 2.3736 1.1118 2.2880 1.1215 2.2090 1.1317 2.1359 1. 1028 2.3721 1.1120 2.2866 1.1217 2.2077 1.1319 2.1347 1. 1029 2.3706 1.1121 2.2853 1.1218 2.2065 1. 1320 2-1335 1.1031 2.3691 1.1123 2.2839 1. 1220 2.2052 1. 1322 2.1324 1. 1032 2.3677 j 1.1 124 2.2825 1. 1222 2.2039 1-1324 2.1312 1. 1034 2.3662 1.1126 2.2812 1. 1223 2.2027 1. 1326 2.1300 Sec. CO-SEC. Sec. Co-sec. Sec. CO-SEC. Sec. CO-SEC. 1 6£ >° 1 64 1° 6S 1° 62 NATURAL SECANTS AND CO-SECANTS 509 28° 29° 30° 3] L° Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. 1. 1666 Co-sec. 1. 1326 2.1300 I.J433 2.0627 I.I547 2 .0000 1. 9416 1. 1327 2.1289 1.1435 2.0616 I-I549 1.9990 1. 1668 1.9407 1.1329 2.1277 I-I437 2.0605 I.I55I 1 .9980 1. 1670 1-9397 1-1331 2.1266 I- 1439 2.0594 1. 1553 1.9970 1. 1672 1.9388 1.1333 2.1254 1.1441 2.05S3 I-I555 1.9960 1. 1674 1.9378 I.I334 2.1242 I-I443 2.0573 I-I557 1.9950 1. 1676 1.9369 1. 1336 2.1231 I-I445 2.0562 1. 1559 1.9940 1. 1678 1.9360 1.1338 2.1219 1. 1446 2.0551 1.1561 1.9930 1.1681 I-9350 1. 1 340 2.1208 1. 1448 2.0540 1. 1562 1.9920 1.1683 I-934I 1.1341 2.1196 1. 1450 2.0530 1-1564 1. 9910 1.1685 1-9332 I-I343 2.1185 I.I452 2.0519 1. 1566 1 .9900 1. 1687 1.9322 1. 1345 2.1173 I.I454 2.0508 1.1568 1.9890 1. 1689 I.93I3 I.I347 2.1162 1. 1456 2 .0498 1. 1570 1.9880 1.1691 1.9304 I-I349 2.1150 I.I458 2.0487 I-I572 1.9870 1-1693 1-9295 I-I350 2.1139 I-I459 2.0476 I-I574 1.9860 1.1695 1.9285 I-I352 2.1127 1.1461 2.0466 I-I576 1.9850 1. 1697 1.9276 I-I354 2.1116 1. 1463 2.0455 1.1578 1 .9840 1. 1699 1.9267 1. 1356 2.1104 1. 1465 2.0444 1. 1580 1.9830 1.1701 1.9258 I-I357 2.1093 1. 1467 2.0434 1. 1582 1.9820 1-1703 1.9248 1 .1359 2.1082 1. 1469 2.0423 1.1584 1. 98 1 1 1-1705 1.9239 1-1361 2.1070 1.1471 2.0413 1.1586 1. 980 1 1. 1707 1.9230 1. 1363 2.1059 I-I473 2.0402 1.1588 1.9791 1. 1709 1.9221 1.1365 2.1048 1. 1474 2.0392 1. 1590 1.9781 1.1712 1. 92 1 2 1. 1366 2.1036 1. 1476 2.0381 1. 1592 1.9771 1.1714 1.9203 1. 1 368 2.1025 1.1478 2.0370 I-I594 1.9761 1.1716 I-9I93 i-i37o 2.1014 1. 1480 2.0360 1. 1596 1-9752 1.1718 1.9184 1. 1372 2.1002 1. 1482 2.0349 1.1598 1.9742 1. 1720 I-9I75 I-I373 2.0991 1. 1484 2-0339 1. 1600 1.9732 1. 1722 1.9166 1. 1375 2.0980 1. i486 2.0329 1. 1602 1.9722 1. 1724 I-9I57 I-I377 2.0969 1. 1488 2.0318 1. 1604 i-97i3 1. 1726 1.9148 I- 1379 2.0957 1. 1489 2.0308 1. 1606 1.9703 1. 1728 I-9I39 1,1381 2.0946 1.1491 2.0297 1. 1608 1.9693 1. 1730 1.9130 1. 1382 2-0935 J. 1493 2.0287 1.1610 1.9683 1. 1732 1.9121 1. 1384 2.0924 I-I495 2.0276 1.1612 1.9674 I-I734 1.9112 1.1386 2.0912 1. 1497 2.0266 1.1614 1.9664 I-I737 1.9102 1. 1 388 2.0901 1. 1499 2.0256 1.1616 1.9654 I-I739 1.9093 1. 1 390 2.0890 1.1501 2.0245 1.1618 1.9645 1.1741 1.9084 I-I3QI 2.0879 I-I503 2.0235 1. 1620 1-9635 I-I743 1.9075 I.I393 2.0868 1-1505 2.0224 1. 1622 1.9625 I-I745 1 .9066 1 -1395 2.0857 1-1507 2.0214 1. 1624 1.9616 I-I747 I.9057 I- 1397 2.0846 1. 1508 2.0204 1. 1626 1.9606 1. 1749 1 .9048 I-I399 2.0835 1-1510 2.0T94 1.1628 1.9596 I-I75I 1.9039 1.1401 2.0824 1.1512 2.0183 1. 1630 1.9587 I- 1753 1.9030 1. 1402 2.0812 I-I5I4 2.0173 1. 1632 1-9577 1. 1756 1. 902 1 I. 1404 2.0801 1.1516 2.0163 1. 1634 1.9568 1. 1758 1. 9013 1. 1406 2.0790 1.1518 2.0152 1. 1636 1.9558 1. 1760 1 .9004 1. 1408 2.0779 1. 1520 2.0142 .1.1638 1.9549 1. 1762 1.8995 1.1410 2.0768 1. 1522 2.0132 1. 1640 1-9539 1. 1764 1.8986 I.1411 2.0757 1. 1524 2.0122 1. 1642 1 -9530 1. 1766 1.8977 1.1413 2.0746 1. 1526 a.oni 1. 1644 1.9520 1. 1768 1.8968 1.1415 2.0735 1. 1528 2.0101 1. 1646 1.9S10 1. 1770 1.8959 I.1417 2.0725 I-I530 2.0091 1. 1648 1,9501 1. 1772 1.8950 1.1419 2.0714 I-I53I 2.0081 1. 1650 1. 9491 1. 1775 1. 8941 1.1421 2.0703 I-I533 2.0071 1. 1652 1.9482 1. 1777 1.8932 1. 1422 2.0692 1.1535 2.0061 1. 1654 1-9473 1. 1779 1.8924 1. 1424 2.0681 I-I537 2.0050 1.1656 1.9463 1.1781 1.8915 1. 1426 2.0670 I-I539 2.0040 1. 1658 1-9454 1.1783 1.8906 1. 1428 2.0659 1.1541 2.0030 1. 1660 1.9444 1.1785 1.8897 1.1430 2.0648 I-I543 2.0020 1. 1662 1-9435 1.1787 1.8888 1. 1432 2.0637 1 -1545 2.0010 1. 1664 1.9425 1. 1790 1.8879 1. 1433 2.0627 T-I547 2.0000 1. 1666 1. 9416 1. 1792 1.8871 CO-SEC. Sec. Co-SEC. Sec. Co-sec. Sec. CO-SEC. Sec. 6 L° 61 3° 5< )° 5! 3° 5i° NATURAL SECANTS AND CO-SECANTS ! 32° 33° 34° 35° •Sec. Co-sec. Sec. CO-SEC. Sec. Co-sec. Sec. C( 5-SEC. f i. 1792 1.8871 1. 1924 I.8361 1.2062 1.7883 1.2208 1 7434 60 1. 1794 - 1.8862 1. 1926 1.8352 1.2064 1.7875 1. 2210 1 7427 59 1. 1796 1.8853 1. 1928 I.8344 1.2067 1.7867 1. 2213 1 7420 58 1. 1798 1.8844 1.1930 I.8336 1.2069 1.7860 1. 2215 1 •7413 57 1. 1800 1.8836 I-I033 I.8328 1.2072 1.7852. 1. 2218 1 •7405 56 1. 1802 1.8827 I-I935 I.8320 1.2074 1.7844 1.2220 1 ■7398 55 1. 1805 1.8818 I.I937 I.83H 1.2076 1-7837 1.2223 1 739i 54 1.7807 1.8809 1 -1939 I.8303 1.2079 1.7829 1.2225 1 7384 53 1. 1809 1. 880 1 1. 1942 I.8295 1. 2081 1. 7821 1.2228 1 7377 52 1.1811 1.8792 1. 1944 I.8287 1.2083 1.7814 1.2230 1 7369 5i 1.1813 1.8783 1. 1946 I.8279 1.2086 1.7806 1.2233 1 7362 50 1.1815 1.8785 1. 1948 I.8271 1.2088 1.7798 1.2255 1 7355 49 1.1818 1.8766 I-T95I I.8263 1. 2091 1.7791 1.2238 1 7348 48 1. 1820 1.8757 I-I953 I.8255 1.2093 1.7783 1.2240 1 734i 47 1. 1822 1.8749 I-I955 I.8246 1.2095 1.7776 1.2243 1 7334 46 1. 1824 1.8740 1. 1958 I.8238 1.2098 1.7768 1.2245 1 7327 45 1. 1826 1-8731 1. i960 I.8230 1. 2100 1.7760 1.2248 1 7319 44 1. 1828 1.8723 1. 1962 I.8222 1. 2 103 1-7753 1.2250 1 7312 43 1.1831 1.8714 1. 1964 I.8214 1.2105 1-7745 1.2253 1 7305 42 1-1833 1.8706 1. 1967 I.8206 1.2107 1.7738 1.2255 1 7298 4i 1.183s 1.8697 1. 1969 I.8198 1.2110 I-773Q 1.2258 1 7291 40 1.1837 1.8688 1.1971 I.8190 1.2112 1-7723 1.2260 1 7284 39 1. 1839 1.8680 1. 1974 I.8182 1.2115 I-77I5 1.2263 1 7277 38 1.1841 1.8671 1. 1976 I.8174 1.2117 1.7708 1.2265 1 7270 37 1. 1844 1.8663 1. 1978 I.8166 1.2119 1.7700 1.2268 1 7263 36 1. 1846 1.8654 1. 1980 I.8158 1. 2122 1.7693 1.2270 1 7256 35 1.1848 1.8646 1. 1983 I.8150 1.2124 1.7685 1.2273 1 7249 34 1.1850 1.8637 1. 1985 I.8142 1. 2127 1.7678 1.2276 1 7242 33 1.1852 1.8629 1. 1987 I.8134 1. 2129 1.7670 1.2278 1 7234 32 1.185s 1.8620 1. 1990 I.8126 1.2132 1.7663 1. 2281 1 7227 3i 1.1857 1. 861 1 1. 1992 I.8118 1.2134 I.7655 1.2283 1 7220 30 1.1859 1.8603 1. 1994 I.8IIO 1.2136 1.7648 1.2286 1 7213 29 1.1861 1.859s 1. 1997 I.8102 1.2159 1.7640 1.2288 1 7206 28 1.1863 1.8586 1. 1999 I.8094 1.2141 1-7633 1. 2291 1 7199 27 1. 1866 1.8578 1. 200 1 I.8086 1.2144 1.7625 1.2293 1 7192 26 1. 1868 1.8569 1.2004 I.8078 1. 2146 1.7618 1.2296 1 7185 25 1. 1870 1.8561 1.2006 I.8070 1.2149 1. 7610 1.2298 1 7178 24 1. 1872 1.8552 1.2008 I.8062 1.2151 1.7603 1.2301 1 7171 23 1. 1874 1.8544 1.2010 I.8054 i.? 153 1-7596 1.2304 1 7164 22 1.1877 1.8535 1. 2013 I.8047 1.2156 1.7588 1.2306 1 7157 21 1. 1879 1.8527 1.2015 I.8039 1.2158 1.7581 1.2309 1 7i5i 20 1.1881 1.8519 1. 2017 I.8031 1.2161 1-7573 1.2311 1 7144 19 1.1883 1.8510 1.2020 I.8023 1.2163 1.7566 1.2314 1 7137 18 1. 1886 1.8502 1.2022 I.8015 1. 2166 1-7559 1.2316 1 7130 17 1. 1888 1.8493 1.2024 I.8007 1.2168 I.755I 1.2319 1 7123 16 1. 1890 1.8485 1.2027 I.7999 1.2171 1-7544 1.2322 1 7116 15 1. 1892 1.8477 1.2029 I.7992 1.2173 1-7537 1.2324 1 7109 14 1. 1894 1.8468 1. 2031 I.7984 I-2I75 1.7529 1.2327 1 7102 13 1. 1897 1 .8460 1.2034 1.7976 1. 2178 1.7522 1.2329 1 7095 12 1. 1899 1.8452 1.2036 I.7968 1. 2 180 I-75I4 1.2332 1 7088 11 1.1901 .1.8443 1.2039 I.7960 1.2183 I.7507 1.2335 1 7081 10 1. 1903 1-8435 1. 2041 1-7953 1.2185 1.7500 1.2337 1 7075 9 1. 1906 1.8427 1.2043 1-7945 1.2188 1-7493 1.2340 1 7068 8 1. 1908 1.8418 1.2046 1-7937 1. 2190 I-7485 1.2342 1 7061 7 1.1910 1. 8410 1.2048 1.7929 1.2193 1.7478 1-2345 1 7054 6 1.1912 1.8402 1.2050 1.7921 1. 2 1 95 1.7471 1.2348 1 7047 5 1.1915 1.8394 1.2053 1.7914 1. 2198 I-7463 1.2350 1 7040 4 1.1917 1.8385 1.2055 1.7906 1.2200 I.7456 1-2353 1 7033 3 1.1919 1-8377 1.2057 1.7898 1.2203 1.7449 1.2355 1 7027 2 1.1921 1.8369 1.2060 1.7891 1.2205 1.7442 1.2358 1 7020 1 1. 1.922 1.8361 1.2062 1.7883 1.2208 1-7434 1. 2361 1 7013 Co-sec. Sec. CO-SEC. Sec. CO-SEC. Sec. CO-SEC. Sec. f 5 7° 1 5 5° 5, 5° 54° NATURAL SECANTS AND CO-SECANTS 5" 36 37 ( 38 39 Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. 1.2361 1-7013 1. 2521 1.6616 1.2690 1.6243 1.2867 1.^890 1.2363 1.7006 1.2524 1. 6610 1.2693 1.6237 1. 2871 1.5884 1.2366 1.6999 1.2527 1.6603 1.2696 1.6231 1.2874 1-5879 1.2368 1.6993 1.2530 1-6597 1.2699 1.6224 1.2877 1-5873 1.2371 1.6986 1.2532 1.6591 1.2702 1.6218 1.2880 1.5867 1-2374 1.6979 1-2535 1.6584 1.2705 1.6212 1.2883 1.5862 1.2376 1.6972 1.2538 1.6578 1.2707 1.6206 1.2886 1.5856 1-2379 1.6965 1-2541 1.6572 1. 2710 1.6200 1.2889 1.5850 1.2382 1.6959 1-2543 1-6565 1-2713 1.6T94 1.2892 1-5845 1.2384 1.6952 1.2546 1-6559 1. 2716 1.6188 1.2895 1-5839 1.2387 1.6945 1.2549 1.6552 1.2719 1.6182 1.2898 1.5833 1.2389 1.6938 1.2552 1.6546 1.2722 1.6176 1. 2901 1.5828 1.2392 1.6932 1-2554 1.6540 1.2725 1. 61 70 1.2904 1.5822 1-2395 1.6925 1-2557 1-6533 1.2728 1.6164 1.2907 1.5816 1-2397 1.6918 1.2560 1.6527 1.2731 1.6159 1. 2910 1.5811 1.2400 1. 691 2 1.2563 1.6521 1-2734 i.6i53 1.2913 1.5805 1.2403 1.6905 1.2565 1.6514 1-2737 1.6147 1. 2916 1-5799 1.2405 1.6898 1.2568 1.6508 1-2739 1.6141 1. 2919 1-5794 1.2408 1. 6891 1-2571 1.6502 1.2742 i.6i35 1.2922 1.5788 1.2411 1.6885 1-2574 1 .6496 1-2745 1.6129 1.2926 1.5783 1.2413 1.6878 1.2577 1 .6489 1.2748 1.6123 1.2929 1-5777 1. 2416 1. 687 1 1.2579 1 .6483 1.2751 1.6117 1.2932 I-577I 1.2419 1.6865 1.2582 1.6477 1-2754 1.6111 1.2935 1.5766 1. 242 1 1.6858 1.2585 1.6470 1.2757 1.6105 1.2938 1.5760 1.2424 1. 685 1 1.2588 1 .6464 1.2760 1.6099 1. 2941 1-5755 1.2427 1.6845 1.2591 1.6458 1.2763 1.6093 1.2944 1-5749 1.2429 1.6838 1.2593 1-6452 1.2766 1.6087 1.2947 1-5743 1.2432 1. 683 1 1.2596 1 .6445 1.2769 1. 6081 1.2950 1-5738 1.2435 1.6825 1.2599 1.6439 1.2772 1.6077 1-2953 1.5732 1-2437 1. 6818 1.2602 1-6433 1.2775 1.6070 1.2956 1.5727 1.2440 1.6812 1.2605 1.6427 1.2778 1.6064 1.2960 1.5721 1-2443 1.6805 1.2607 1.6420 1. 2781 1.6058 1.2963 i-57i6 1.2445 1.6798 1. 2610 1. 6414 1.2784 1.6052 1.2966 1.5710 1.2448 1.6792 1.2613 1 .6408 1.2787 1 .6046 1.2969 1-5705 1.2451 1.6785 1. 2616 1.6402 1.2790 1 .6040 1.2972 1.5699 1.2453 1.6779 1. 2619 1.6396 1.2793 1.6034 1.2975 1.5694 1.2456 1.6772 1.2622 1.6389 1-2795 1.6029 1.2978 1.5688 1.2459 1.6766 1.2624 1.6383 1.2798 1.6023 1. 2981 1.5683 1. 2461 1-6759 1.2627 1.6377 1. 2801 1. 601 7 1.2985 1.5677 1.2464 1.6752 1.2630 1-6371 1.2804 1. 601 1 1.2988 1.5672 1.2467 1.6746 1.2633 1.6365 1.2807 1.6005 1. 2991 1.5666 1.2470 1.6739 1.2636 1.6359 1. 2810 1 .6000 1.2994 1.5661 1.2472 1.6733 1.2639 1.6352 1.2813 1-5994 1.2997 1.5655 1-2475 1.6726 1. 2641 1 .6346 1.2816 1.5988 1.3000 1.5650 1.2478 1.6720 1.2644 1.6340 1.2819 1.5982 1.3003 1.5644 1.2480 1-6713 1.2647 1-6334 1.2822 1-5976 1 .3006 1-5639 1.2483 1.6707 1.2650 1.6328 1.2825 I-597I 1,3010 1.5633 1.2486 1.6700 1-2653 1.6322 1.2828 I-5965 1.3013 1.5628 1.2488 1.6694 1.2656 1.6316 1.2831 1-5959 1. 3016 1.5622 1.2490 1.66S7 1.2659 1.6309 1.2834 1-5953 1.3019 1-5617 1.2494 1. 6681 1. 2661 1.6303 ; 1.2837 1-5947 1.3022 1.5611 1.2497 1.6674 1.2664 1.6297 1.2840 1.5942 1-3025 1.5606 1.2499 1.6668 1.2667 1.6291 1.2843 1-5936 1.3029 1.5600 1.2502 1. 6661 1.2670 1.6285 1.2846 I-593Q 1-3032 1-5595 1.2505 1.6655 1.2673 1.6279 1.2849 1.5924 1-3035 I-5590 1.2508 1.6648 1.2676 1.6273 1.2852 I-59I9 1.3038 1.5584 1.2510 i .6642 1.2679 1.6267 1.2855 I-59I3 1. 3041 1-5579 1-2513 1.6636 1.2681 1.6261 1.2858 I.5907 1 .3044 1-5573 1.2516 1.6629 1.2684 1.6255 1.2861 1.5901 1.3048 1.5568 1.2519 1.6623 1.2687 1.6249 1.2864 1.5896 1-3051 I-5563 1. 2521 1.6616 1.2690 1.6243 1.2867 1.5890 1-3054 1-5557 CO-SEC. Sec. Co-sec. Sec. Co-sec. Sec. Co-sec. Sec. 5 3° 5 2° 5 1° 5 0° 512 NATURAL SECANTS AND CO-SECANTS 4C 1° 4] L° 42° 43° Sec. Co-sec. Sec. CO-SEC. Sec. Co-sec. Sec. CO-SEC. 1.3054 1-5557 1.3250 1-5242 1.3456 1-4945 1.3673 1.4663 1-3057 1-5552 1-3253 1-5237 1.3460 1.4940 1-3677 1.4658 1.3060 L5546 1.3257 1-5232 1.3463 1-4935 1.3681 1.4654 1.3064 I-554I 1.3260 1.5227 1-3467 1.4930 1.3684 1.4649 1.3067 I.5536 1.3263 1.5222 1.3470 1.4925 1.3688 1.4644 1.3070 I.5530 1.3267 1-5217 1-3474 1. 492 1 1.3692 1 .4640 1-3073 1-5525 1.3270 1. 5212 1-3477 1.4916 1-3695 1.4635 1.3076 I-5520 1-3274 1.5207 1.3481 1. 49 1 1 1.3699 1.4631 1.3080 I-55I4 1.3277 1.5202 I-3485 1.4906 1-3703 1.4626 1.3083 1.5509 1.3280 1.5197 1.3488 1. 490 1 1-3707 1.4622 1.3086 I-5503 1.3284 1.5192 1.3492 1.4897 1.3710 1. 461 7 1.3089 1.5498 1.3287 1.5187 1-3495 1.4892 i-37i4 1.4613 1.3092 1-5493 1.3290 1.5182 1-3499 1.4887 i-37i8 1.4608 1.3096 I-5487 1.3294 1-5177 I-3502 1.4882 1.3722 1 .4604 1.3099 1.5482 1.3297 1.5171 1.3506 1.4877 1-3725 1-4599 1. 3102 1-5477 1.3301 1.5166 I-3509 1.4873 1-3729 1-4595 1-3105 I.S47I 1.3304 1.5161 I-35I3 1.4868 1-3733 1.4590 1.3109 1.5466 1.3307 1.5156 I-35I7 1.4863 1-3737 1.4586 1.3112 1.5461 I-33H 1.5151 1.3520 1.4858 1 -3740 1.4581 i.3"5 1-5456 I-33I4 1.5146 1-3524 1.4854 1-3744 1-4577 i.3"8 1-5450 I.33I8 1.5141 1-3527 1.4849 1.3748 1-4572 1.3121 1-5445 I-332I 1.5136 I-353I 1.4844 1-3752 1.4568 1.3125 1.5440 I.3324 1.5131 1-3534 1.4839 1-3756 1-4563 1.3128 1.5434 1.3328 1.5126 1-353^ 1.4835 1-3759 1-4559 1.3131 1.5429 I-333I 1.5121 1-3542 1.4830 1.3763 1-4554 1-3134 1.5424 1-3335 1.5116 1-3545 1.4825 1.3767 I-4550 1.3138 I-54I9 I.3338 1.5111 1-3549 1. 482 1 I.377I 1-4545 1.3141 I.54I3 1-3342 1.5106 1-3552 1.4816 1-3774 1 -4541 1-3144 1.5408 1-3345 1.5101 1.3556 1. 481 1 1.3778 1-4536 1.3148 1.5403 I-3348 1.5096 1.3560 1.4806 1.3782 1-4532 1.3151 I.5398 1-3352 1.5092 1.3563 1.4802 1.3786 1-4527 1.3154 1.5392 1-3355 1.5087 1.3567 1-4797 1.3790 1-4523 1.3157 1.5387 1-3359 1.5082 I-357I 1.4792 1-3794 1.4518 1.3161 1.5382 1.3362 1.5077 1-3574 1.4788 1-3797 1 4514 1.3164 1-5377 1.3366 1.5072 1-3578 1.4783 1. 3801 1. 4510 1.3167 I.537I 1.3369 1.5067 i.358i 1.4778 1.3805 I-4505 1.3170 1.5366 1.3372 1.5062 I-3585 1-4774 1.3809 1.4501 i-3i74 i.536i 1-3376 1-5057 1,3589 1.4769 1-3813 1.4496 1.3177 1-5356 1-3379 1.5052 1-3592 1.4764 1.3816 1.4492 1.3180 I-535I I-3383 1.5047 1-3596 1.4760 1.3820 1.4487 1.3184 1-5345 1.3386 1.5042 1.3600 1-4755 1.3824 1.4483 1.3187 I-5340 ■ I-3390 1-5037 1.3603 1-4750 1.3828 1.4479 1.3190 1-5335 1-3393 1.5032 1.3607 1.4746 1.3832 1.4474 1-3193 1 -533Q 1-3397 1.5027 1.3611 1. 4741 1.3836 1.4470 1-3197 1.5325 1 .3400 1*5022 1.3614 1.4736 1.3839 1.4465 1.3200 I-53I9 1.3404 1.5018 1. 36 1 8 1-4732 1.3843 1. 446 1 1.3203 I-53I4 I-3407 1-5013 1.3622 1.4727 1.3847 1-4457 1.3207 1-5309 1. 341 1 1.5008 1.3625 1-4723 1.3851 1.4452 1. 3210 I-5304 I-34I4 1-5003 1.3629 1.4718 1.3855 1.4448 1-3213 1.5299 1.3418 1.4998 1-3633 I-47I3 1.3859 1-4443 1-3217 1.5294 1. 342 1 1-4993 1.3636 1.4709 1.3863 1-4439 1.3220 I.5289 1.342 5 1.4988 1.3640 1.4704 1.3867 1-4435 1.3223 1.5283 1.3428 1.4983 1.3644 1 .4699 1.3870 1.4430 1.3227 1.5278 1-3432 1.4979 1-3647 1.4695 1.3874 1.4426 1.3230 1-5273 1-3435 1.4974 1-3651 1.4690 1.3878 1.4422 1.3233 1.5268 1-3439 1 .4969 1-3655 1.4686 1.3882 I-44I7 1.3237 1.5263 1.3442 1.4964 1.3658 1. 4681 1.3886 I-44I3 1.3240 1.5258 1.3446 1-4959 1.3662 1.4676 1.3890 1 .4408 1-3243 1-5253 1-3449 1-4954 1.3666 1.4672 1.3894 1.4404 1-3247 1.5248 1.3453 1.4949 1.3669 1.4667 1.3898 1.4400 1.3250 1.5242 I.3456 1-4945 1.3673 1.4663 1.3902 1-4395 Co-sec. Sec. CO-SEC. Sec. Co-sec. Sec. CO-SEC. Sec. 4< }° 4, B° 4' 7 q 4( 3° NATURAL SECANTS AND CO-SECANTS 513 440 440 440 i Sec. Co-sec. 60 f 21 Sec. Co-sec. 39 41 Sec 1 .4065 Co-sec. 1.3902 i-4395 1.3984 1-4305 1. 4221 I 1-3905 I-439I 59 22 1.3988 1. 430 1 38 42 1.4069 1.4217 2 1.3909 1-4387 58 23 1.3992 1.4297 37 43' 1-4073 1. 4212 3 i-39i3 1.4382 57 24 1.3996 1.4292 36 44 1 4077 1.4208 4 i-39i7 1-4378 56 25 1.4000 1.4288 35 45 1. 408 1 1.4204 5 1.3921 1-4374 55 26 1.4004 1.4284 34 46 1.4085 1.4200 6 1-3925 I.4370 54 27 1 .4008 1.4280 33 47 1 .4089 1.4196 7 1.3929 I-4365 53 28 1. 4012 1.4276 32 48 1.4093 1.4192 8 1-3933 I-436i 52 29 1. 4016 1.4271 31 49 1.4097 1.4188 9 1-3937 1-4357 51 30 1.4020 1.4267 30 50 1.4101 1-4183 10 1 -3941 1-4352 50 31 1.4024 1.4263 29 51 1.4105 1.4179 11 1-3945 1.4348 49 32 1.4028 1.4259 28 52 1. 4109 1-4175 12 1-3949 1-4344 48 33 1.4032 1.4254 27 53 1.4113 1-4171 13 1-3953 1-4339 47 34 1.4036 1.4250 26 54 1.4117 1.4167 14 1-3957 1-4335 46 35 1.4040 1.4246 25 55 1. 4122 1.4163 15 1.3960 I-433I 45 36 1.4044 T.4242 24 56 1.4126 1-4159 16 1.3964 1-4327 44 37 1.4048 1.4238 23 57 1-4130 1-4154 17 1.3968 1.4322 43 38 1.4052 1-4233 22 58 1-4134 1.4150 18 1.3972 1.4318 42 39 1.4056 1.4229 21 59 1,4138 1. 4146 19 1.3976 I-43I4 41 40 1 .4060 1.4225 20 60 1.4142 1. 4142 20 1 .3980 1.4310 40 / / i ' / CO-SEC. Sec. Co-sec. Sec. Co-sec. Sec 4, 5° 4 5" 4 5° to 18 n 16 15 13 T2 II IO 9 8 7 6 5 4 3 2 1 o Si4 LOGARITHMS OF NUMBERS* Table 6i IOO I 2 3 4 5 6 7 8 9 Diff. 432 000000 0434 0868 1301 1734 2166 2598 3029 346i 3891 IOI 4321 475i 5181 5609 6038 6466 6894 7321 7748 8174 428 I02 8600 9026 9451 9876 *0300 *0724 *ii47 *i 57 o *i993 *24i5 424 I03 012837 3259 3680 4100 452i 4940 536o 5779 6197 6616 420 IO4 7033 745i 7868 8284 8700 9116 9532 9947 *036i *o775 416 I05 021 189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 I06 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 I07 9384 9789 *oi95 *o6oo *ioo4 *i4o8 *l8l2 *22l6 *26i9 *302I 404 I08 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 400 IO9 7426 7S25 8223 8620 9017 '9414 9811 *0207 *o6o2 *0998 397 N. Diff. 434 I 2 3 4 5 6 7 8 9 Diff. 43 87 130 174 217 260 304 347 391 434 433 43 87 130 173 217 260 303 346 390 433 432 43 86 130 173 216 259 302 346 389 432 43i 43 86 129 172 216 259 302 345 388 43i 430 43 86 129 172 215 258 301 344 387 430 429 43 86 129 172 215 257 300 343 386 429 428 43 86 128 171 214 257 300 342 385 428 427 43 85 128 171 214 256 299 342 384 427 426 43 85 128 170 213 256 298 34i 383 426 425 43 85 128 170 213 255 298 340 383 425 424 42 85 127 170 212 254 297 339 382 424 423 42 85 127 169 212 254 296 338 381 423 422 42 84 127 169 211 253 295 338 380 422 421 42 84 126 168 211 253 295 337 379 421 420 42 84 126 168 210 252 294 336 378 420 W 419 42 84 126 168 210 251 293 335 377 4 ig H < 418 42 84 125 167 209 251 293 334 376 418 417 42 83 125 167 209 250 292 334 375 4i7 416 42 83 125 166 208 250 291 333 374 416 4i5 42 83 125 166 208 249 291 332 374 4i5 _) 414 4i 83 124 166 207 248 290 33i 373 414 < 413 4i 83 124 165 207 248 289 330 372 413 z 412 4i 82 124 165 206 247 288 330 37i 412 O 411 4i 82 123 164 206 247 288 329 37o 411 H 410 4i 82 123 164 205 246 287 328 369 410 ti 409 4i 82 123 164 205 245 286 327 368 409 O 408 4i 82 122 163 204 245 286 326 367 408 O 407 41 81 122 163 204 244 285 326 366 407 406 4i 81 122 162 203 244 284 325 365 406 405 4i 81 122 162 203 243 284 324 365 405 404 40 81 121 162 202 242 283 323 364 404 403 40 81 121 161 202 242 282 322 363 4°3 402 40 80 121 161 201 241 281 322 362 402 401 40 80 120 160 201 241 281 321 361 401 400 40 80 120 160 200 240 280 320 360 400 399 40 80 120 160 200 239 279 319 359 399 398 40 80 119 159 199 239 279 3i8 358 398 397 40 79 119 159 199 238 278 318 357 397 396 40 79 119 158 198 238 277 3i7 356 396 395 40 79 119 158 198 237 277 316 356 395 394 39 79 118 158 197 236 276 3i5 355 394 393 39 79 118 157 197 236 275 3i4 354 393 392 39 78 118 i57 196 235 274 314 353 392 39i 39 78 117 156 196 235 274 313 352 39i 390 39 78 117 156 195 234 273 312 35i 390 389 39 78 117 156 195 233 272 311 35o 389 388 39 78 116 155 194 233 272 310 349 388 L Diff. I 2 3 4 5 6 7 8 9 Diff. *From Allen's "Field and Office Tables." Copyright, 1903, 1914. by C. F. Allen. LOGARITHMS OF NUMBERS SIS N. I 2 3 4 5 6 7 8 9 Diff. no 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 in 5323 5714 6105 6495 6885 7275 ■7664 8053 8442 8830 390 112 9218 9606 9993 ^0380 *0766 *H53 *I538 *i924 ^'2309 *2694 386 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 383 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 *0320 379 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 117 8186 8557 8928 9298 9668 *oo38 *0407 ^0776 *ii45 *I5*4 37o 118 071882 2250 2617 2985 3352 37i8 4085 4451 4816 5182 366 ng 5547 59'i2 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 9543 9904 *o266 #0626 ^0987 *i 34 7 *i707 *2o67 #2426 360 121 082785 3*44 3503 3861 4219 4576 4934 5291 5647 6004 357 122 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 123 9905 *0258 *o6n ♦0963 *i3i5 *i667 *20l8 *237o *272I #3071 352 124 093422 3772 4122 447i 4820 5169 5518 5866 6215 6562 349 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 387 39 77 116 155 194 232 271 310 348 387 386 39 77 116 154 193 232 270 309 347 386 385 39 77 116 154 193 231 270 308 347 3 £ 5 384 38 77 115 154 192 230 269 307 346 384 383 38 77 115 153 192 230 268 306 345 383 382 38 76 115 153 191 229 267 306 344 382 381 38 76 114 152 191 229 267 305 343 381 380 38 76 114 152 190 228 266 304 342 380 379 38 76 114 152 190 227 265 303 34i 379 378 38 76 113 151 189 227 265 302 340 378 377 38 75 "3 151 189 226 264 302 339 377 376 38 75 113 150 188 226 263 301 338 376 375 38 75 113 150 188 225 263 300 338 375 CO < 0. 374 37 75 112 150 187 224 262 299 337 374 373 37 75 112 149 187 224 261 298 336 373 372 37 74 112 149 186 223 260 298 335 372 37i 37 74 in 148 186 223 260 297 334 37i 37o 37 74 in 148 185 222 259 296 333 37o < z 36g 37 74 in 148 185 221 258 295 332 369 368 37 74 no 147 184 221 258 294 33i 368 367 37 73 no 147 184 220 257 294 330 367 366 37 73 no 146 183 220 256 293 329 366 H 365 37 73 no 146 183 219 256 292 329 365 ft 364 36 73 109 146 182 218 255 291 328 364 O 0. a, 363 36 73 109 145 182 218 254 290 327 363 362 36 72 109 145 181 217 253 290 326 362 361 36 72 108 144 181 217 253 289 325 361 360 36 72 108 144 180 2l6 252 288 324 360 359 36 72 108 144 180 215 251 287 323 359 358 36 72 107 143 179 215 251 286 322 358 357 36 7i 107 143 179 214 250 286 321 357 356 3 5 7i 107 142 178 214 249 285 320 356 355 36 7i 107 142 178 213 249 284 320 355 354 35 7i 106 142 177 212 248 283 319 354 353 35 7i 106 141 177 212 247 282 3i8 353 352 35 70 106 141 176 211 246 282 317 352 35i 35 70 105 140 176 211 246 281 316 351 350 35 70 105 140 175 210 245 280 315 350 349 35 70 105 140 175 209 244 279 314 349 348 35 70 104 139 174 209 244 278 313 348 347 35 69 104 139 174 208 243 278 312 347 Diff. I 2 3 4 5 6 7 8 9 Diff. 5i6 LOGARITHMS OF NUMBERS N. I 2 3 4 5 6 7 8 9 Diff. 346 125 096910 7257 7604 7951 8298 8644 8990 9335 9681 *0026 126 10037 1 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 34i 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 ♦0253 338 129 1 1 0590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 1 13943 4277 461 1 4944 5278 561 1 5943 6276 6608 6940 333 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 *0245 330 *I32 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 325 134 7105 7429 7753 8076 8399 8722 9°45 9368 9690 *OOI2 323 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 3i8 137 6721 7037 7354 *o5o8 7671 7987 8303 8618 8934 9249 9564 316 138 9879 *oi94 *0822 *H36 *I450 *I763 ^2076 *2 3 8 9 *2702 314 139 1430 1 5 3327 3639 3951 4263 4574 4885 5196 5507 58l8 3ii N. Diff. I 2 3 4 5 6 7 8 9 Diff. 347 35 69 104 139 174 208 243 278 312 347 346 35 69 104 138 173 208 242 277 311 346 345 35 69 104 138 173 207 242 276 311 345 344 34 69 103 138 172 206 241 275 310 344 343 34 69 T03 137 172 206 240 274 309 343 342 34 68 103 137 171 205 239 274 308 342 34i 34 68 102 136 171 205 239 273 307 341 340 34 68 102 136 170 204 238 272 306 340 339 34 68 102 136 170 203 237 271 305 339 338 34 68 IOI 135 169 203 237 270 304 338 337 34 67 IOI 135 169 202 236 270 303 337 336 34 67 IOI 134 168 202 235 269 302 336 W 335 34 67 IOI 134 168 201 235 268 302 335 H 334 33 67 IOO 134 167 200 234 267 301 334 N 333 33 67 IOO 133 167 200 233 266 300 333 < 332 33 66 IOO 133 166 199 232 266 299 332 a< 33i 33 66 99 132 166 199 232 265 298 33i < 330 33 66 99 132 165 198 231 264 297 330 329 33 66 99 132 165 197 230 263 296 3 2 9 2 328 33 66 98 131 164 197 230 262 295 328 O 327 33 65 98 131 164 196 229 262 294 327 H O 326 33 65 98- 130 163 196 228 261 293 326 325 33 65 98 130 163 195 228 260 293 325 324 32 65 97 130 162 194 227 259 292 324 £ 323 32 65 . 97 129 162 194 226 258 291 323 322 32 64 97 129 161 193 225 258 290 322 ptf 321 32 64 96 128 161 193 225 257 289 321 & 320 32 64 96 128 160 192 224 256 288 320 319 32 64 96 128 160 191 223 255 287 3i9 318 32 64 95 127 159 191 223 254 286 3i8 3i7 32 63 95 127 159 190 222 254 285 3i7 316 32 63 95 126 158 190 221 253 284 316 3i5 32 63 95 126 158 189 221 252 284 315 3i4 3i 63 94 126 157 188 220 251 283 3M 3i3 3i 63 94 125 157 188 219 250 282 313 312 3i 62 94 125 156 187 218 250 281 312 311 3i 62 93 124 156 187 218 249 280 3" 310 3i 62 93 124 155 186 217 248 279 310 309 3i 62 93 124 155 185 216 247 278 309 308 3i 62 92 123 154 185 216 246 277 308 307 3i 61 92 123 154 184 215 246 276 307 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 517 N. I 2 3 4 5 6 7 8 9 Diff. 140 146128 6438 6748 7058 7367 7676 7985 8294 8603 891 1 309 141 9219 9527 9835 *0I43 *0449 *Q756 *io63 *i37o *i676 *i982 307 142 1522SS 2594 2900 3205 35io 3815 4120 4424 4728 5°32 305 M3 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 144 S362 8664 8965 9266 9567 9868 *oi68 ^0469 *o769 *io68 301 M5 161 368 1667 1967 2266 2564 2863 3161 3460 375S 4055 299 146 4353 4650 4947 5244 5541 5S38 6i34 6430 6726 7022 297 I4 £ 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 148 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 149 31S6 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 150 176091 6381 6670 6959 7248 7536 .7825 8113 8401 8689 289 151 8977 9264 9552 9839 *OI26 *04i3 ^0699 *0986 *1272 *I558 287 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 154 7521 7803 8084 8366 8647 8928 9209 9490 9771 *oo5i 281 155 190332 0612 0892 1171 145 1 1730 2010 22S9 2567 2846 279 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 157 5900 6176 6453 6729 7005 7281 7556 7S32 8107 .8382 276 158 8657 8932 9206 948i 9755 *0029 #0303 *Q577 *0850 *II24 274 159 201397 1670 1943 2216 24S8 2761 3033 3305 3577 3848 272 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 306 3i 61 92 122 153 184 214 245 275 306 305 3i 61 92 122 153 183 214 244 275 305 304 30 61 . 9i 122 152 182 213 243 274 304 303 30 61 91 121 152 182 212 242 273 303 302 30 60 91 121 151 181 211 242 272 302 301 30 60 90 120 151 181 211 241 271 301 300 30 60 90 120 150 180 210 240 270 300 299 30 60 90 120 150 179 209 239 269 299 298 30 60 89 119 149 179 209 238 268 298 297 30 59 89 119 149 178 208 238 267 297 296 30 59 89 11S 148 178 207 237 266 .296 CO 295 30 59 89 118 148 177 207 236 266 295 294 29 59 88 118 147 176 206 235 265 294 < 293 29 59 88 117 147 176 205 234 264 293 ft 292 29 58 88 117 146 175 204 234 263 292 < 291 29 58 87 116 146 175 204 233 262 291 290 29 58 87 116 145 i74 203 232 261 290 z 28g 29 58 87 116 145 173 202 231 260 28g 288 29 58 86 115 144 173 202 230 259 288 H 287 29 57 86 115 144 172 201 230 258 287 X 286 29 57 86 114 143 172 200 229 257 286 285 29 57 86 114 143 171 200 228 257 285 a* 284 28 57 85 114 142 170 199 227 256 284 283 28 57 85 113 142 170 198 226 255 283 tt & 282 28 56 85 113 141 169 197 226 254 282 281 28 56 84 112 141 169 197 225 253 281 280 28 56 84 112 140 168 196 224 252 280 279 278 28 56 84 112 140 167 195 223 251 279 28 56 83 in 139 167 195 222 250 278 277 28 55 83 in 139 166 194 222 249 277 276 28 55 83 no 138 166 193 221 248 276 275 28 55 83 no 138 165 193 220 248 275 274 27 55 82 no 137 164 192 219 247 274 273 27 55 82 109 137 164 191 218 246 273 272 27 54 82 109 136 163 190 218 245 272 271 27 54 81 108 136 163 190 217 244 271 Diff. Diff.. I 2 3 4 5 6 7 8 9 5i LOGARITHMS OF NUMBERS N. O I 2 3 4 5 6 7 8 9 Diff. 1 60 204120 439i 4663 4934 5204 5475 5746 6016 6286 6556 271 161 6S26 7096 7365 7634 7904 8i73 8441 8710 8979 9247 269 162 9515 9783 ^0051 *03 1 9 *0586 *o853 *II2I *I388 *i654 *I92I 267 163 212188 2454 2720 2986 3252 35i8 3783 4049 4314 4579 266 164 4844 5109 5373 5638 5902 6166 643O 6694 6957 7221 264 165 217484 7747 8010 8273 8536 8798 9060 9323 9585 9846 262 166 220108 0370 0631 0S92 "53 1414 1675 1936 2196 2456 261 167 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 168 5309 5568 5826 60S4 6342 6600 6858 7"5 7372 7630 258 169 7887 8144 8400 8657 8913 9170 9426 9682 9938 *oi 9 3 256 170 230449 0704 0960 1215 1470 1724 1979 2234 2488 2742 255 171 2996 3250 3504 3757 401 1 4264 4517 4770 5023 5276 253 172 5528 578i 6033 6285 6537 6789 704I 7292 7544 7795 252 173 8046 8297 8548 8799 9049 9299 9550 9800 *oo5o *03oo 250 174 240549 0799 1048 1297 1546 1795 2044 2293 2541 2790 249 175 243038 32S6 3534 3782 4030 4277 4525 4772 5019 5266 248 176 5513 5759 6006 6252 6499 6745 699I 7237 7482 7728 246 177 7973 8219 8464 8709 8954 919S 9443 9687 9932 *oi76 245 178 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 243 179 2853 3096 3338 358o 3822 4064 4306 4548 4790 5031 242 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 272 27 54 82 109 136 163 190 218 245 272 271 27 54 81 108 136 163 190 217 244 271 270 27 54 81 108 135 162 189 216 243 270 269 27 54 81 108 135 161 188 215 242 26g 268 27 54 80 107 134 161 188 214 241 268 267 27 53 80 107 134 160 187 214 240 267 266 - 27 53 80 106 133 160 186 213 239 266 265 27 53 80 106 133 159 186 212 239 265 264 26 53 79 106 132 158 185 211 238 264 ro 263 26 53 79 105 132 158 184 210 237 263 H 262 26 52 79 105 131 157 183 210 236 262 ti 261 26 52 78 104 131 157 183 209 235 261 < a, < Z O 260 26 52 78 104 130 156 182 208 234 260 259 26 52 78 104 130 155 181 207 233 259 258 26 52 77 103 129 155 181 206 232 258 257 26 5i 77 103 129 154 180 206 231 257 256 26 5i 77 102 128 154 179 205 230 256 255 26 5i 77 102 128 153 179 204 230 255 H 254 25 5i 76 102 127 152 178 203 229 254 & 253 25 5i 76 IOI 127 152 177 202 228 253 O (X O 252 25 50 76 IOI 126 151 176 202 227 252 251 25 50 75 100 126 151 176 201 226 251 tf 250 25 50 75 IOO 125 150 175 200 225 250 & 249 25 50 75 IOO 125 149 174 199 224 249 248 25 50 74 99 124 149 174 198 223 248 247 25 49 74 99 124 148 173 198 222 247 246 25 49 74 9? 123 148 172 197 221 246 245 25 49 74 9 ^ 123 147 172 196 221 245 244 24 49 73 98 122 146 171 i95 220 244 243 24 49 73 97 122 146 170 194 219 243 242 24 48 73 97 121 145 169 194 218 242 241 24 48 72 96 121 145 169 193 217 241 240 24 48 72 96 120 144 168 192 216 240 , Diff. I 2 3 4 5 6 7 8 9 Dill. LOGARITHMS OF NUMBERS 519 - I 2 3 4 5 6 7 8 9 Diff. i8o 255273 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 181 7679 7918 S158 8398 8637 SS77 9116 9355 9594 9833 239 182 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 183 2451 26S8 2925 3162 3399 3636 3873 4109 4346 4582 237 184 4818 5054 5290 5525 576i 5996 6232 6467 6702 6937 235 185 267172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 186 9513 9746 9980 *02I3 ^446 *o679 *09I2 *ii44 *i377 *i6o9 233 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 232 188 4158 4389 4620 4850 5081 53i 1 5542 5772 6002 6232 230 189 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 , 190 278754 S982 9211 9439 9667 9895 *OI23 *Q35i *o 57 8 *o8o6 228 191 281033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 ig2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 193 5557 5782 6007 6232 6456 66S1 6905 7130 7354 7578 225 194 7802 8026 8249 8473 8696 8920 9 T 43 9366 9589 9812 223 195 290035 0257 04S0 0702 0925 1147 1369 i59i 1813 2034 222 196 2256 2478 2699 2920 3Hi 3363 3584 3804 4025 4246 221 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 6665 6S84 7104 7323 7542 7761 7979 8198 8416 8635 219 199 8853 9071 9289 9507 9725 9943 *oi6i ^378 ^595 *o8i 3 218 200 301030 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 201 3196 34i2 3628 3844 4059 4275 4491 4706 4921 5136 216 202 535i 5566 578i 599 6 6211 6425 6639 6854 7068 7282 215 203 7496 7710 7924 8i37 8351 8564 8778 8991 9204 9417 213 204 9630 9843 *oo56 *0268 *048i *o693 *o9o6 *iii8 *i33° *I542 1 212 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 239 238 24 48 72 96 120 143 167 191 215 239 24 48 71 95 119 i43 167 190 214 238 237 24 47 71 95 119 142 166 190 213 237 236 24 47 71 94 118 142 165 189 212 236 235 24 47 71 94 118 141 165 188 212 235 234 23 47 70 94 117 140 164 187 211 234 233 23 47 70 93 117 140 163 186 210 233 (W 232 23 46 70 93 116 139 162 186 209 232 < 231 23 46 69 92 116 139 162 185 208 231 230 23 46 69 92 115 138 161 184 207 230 a< 229 23 46 69 92 115 i37 160 183 206 229 _j 228 23 46 68 9i 114 137 160 182 205 228 < 227 23 45 68 9i 114 136 159 182 204 227 z 226 23 45 68 90 113 136 158 181 203 226 225 23 45 68 90 113 135 158 180 203 225 *-* 224 22 45 67 90 112 134 157 179 202 224 223 22 45 67 89 112 i34 156 178 201 223 O 222 22 44 67 89 in 133 155 178 200 222 cu 221 22 44 66 88 in 133 155 177 199 221 0. 220 22 44 66 8S no 132 154 176 198 220 219 22 44 66 88 no 131 153 175 197 2ig 218 22 44 65 87- 109 131 153 174 196 218 217 22 43 65 87 109 130 152 174 i95 217 216 22 43 65 86 108 130 151 173 194 216 215 22 43 65 86 108 129 151 172 194 215 214 21 43 64 86 107 128 150 171 193 214 j 213 21 43 64 85 107 128 149 170 192 213 „ 212 21 42 64 85 106 127 148 170 191 212 Diff. I 2 3 4 5 6 1 7 8 9 | Diff - S20 LOGARITHMS OF NUMBERS N. I 2 3 4 5 6 7 8 9 Diff. 205 311754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 206 3S67 4078 4289 4499 4710 4920 5130 534o 555i 5760 210 207 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 208 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 2og 320146 0354 0562 0769 0977 1 184 1391 1598 1805 2012 207 210 322219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 212 6336 6541 6745 6950 7i55 7359 7563 7767 7972 8176 204 213 8380 8583 8787 8991 9194 9398 9601 9805 *ooo8 *02II 203 214 330414 0617 0819 1022 1225 1427 1630 1832 2034 2236 202 215 332438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 216 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 217 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 218 8456 8656 8855 9054 9253 9451 9650 9849 *oo47 *0246 199 219 340444 0642 0841 1039 1237 1435 1632 1830 2028 2225 198 220 342423 2620 2817 3014 3212 3409 3606 3802 3999 4196 197 221 4392 4589 4785 4981 5178 5374 5570 5766 5962 6i57 196 222' 6353 6549 6744 6939 7i35 7330 7525 7720 7915 8110 195 223 8305 8500 8694 8889 9083 9278 9472 9666 9860 *oo54 194 224 350248 0442 0636 0829 1023 1216 1410 1603 1796 1989 193 225 352183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 226 4108 43oi 4493 4685 4876 5068 5260 5452 5643 5834 192 227 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 228 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 229 9835 *0025 *02I5 *0404 ^593 *0783 *0972 *n6i *I35° **539 189 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 188 231 3612 3800 3988 4176 4363 455i 4739 4926 5ii3 5301 188 232 5488 5675 5862 6049 6236 6423 6610 6796 S 83 7169 187 233 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 234 9216 9401 9587 9772 9958 *oi43 *0328 *05i3 *o698 *o883 185 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 212 21 42 64 85 106 127 148 170 191 212 211 21 42 63 84 106 127 148 169 190 211 210 21 42 63 84 105 126 147 168 189 210 209 21 42 63 84 105 125 146 167 188 209 208 21 42 62 83 104 125 146 166 187 208 207 21 4i 62 83 104 124 145 166 186 207 tt 206 21 4i 62 82 103 124 144 165 185 206 < 205 21 4i 62 82 103 123 144 164 185 205 Pa 204 20 4i 61 82 102 122 143 163 184 204 j 203 20 4i 61 81 102 122 142 162 183 203 < 202 20 40 61 81 101 121 141 162 182 202 O 201 20 40 60 80 ior 121 141 161 181 201 200 20 40 60 80 100 120 140 160 180 200 1—1 H 199 20 40 60 80 100 119 139 159 179 199 P* 198 20 40 59 79 99 119 139 158 178 198 O 197 20 39 59 79 99 118 138 158 177 197 04 ig6 20 39 59 78 98 118 137 157 176 196 O 195 20 39 59 78 98 117 137 156 176 195 194 19 39 58 78 *97 116 136 155 i75 194 193 19 39 58 77 97 116 135 154 i74 193 ig2 19 38 58 77 96 115 134 154 i73 192 191 19 38 57 76 96 115 134 153 172 191 190 19 38 57 76 95 114 133 152 171 190 lB % 19 38 57 76 95 113 132 151 170 189 188 19 38 56 75 94 113 132 150 169 188 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 521 N. O I 2 3 4 5 6 7 8 9 Diff. 235 371068 1253 1437 1622 1806 1991 2175 2360 2544 2728 184 236 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 237 4748 4932 5"5 5298 5481 5664 5846 6029 6212 6394 183 238 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 1S2 239 8398 8580 8761 8943 9124 9306 9487 9668 9849 *0030 181 240 3802 1 1 0392 0573 0754 0934 i"5 1296 1476 1656 1837 181 241 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 180 242 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 243 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 244 739o 7568 7746 7923 8101 8279 8456 8634 881 1 8989 178 245 389166 9343 952o 9698 9875 *oo5i *0228 *04Q5 *0 5 S2 *0759 177 246 39°935 1112 1288 1464 1641 1817 1993 2169 2345 2521 176 247 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 397940 8114 8287 8461 8634 8808 8981 9154 * 932 ? 9501 173 251 9674 9847 *0020 *OI92 ^365 *o 53 8 *oyii *o883 *io56 *I228 173 252 401401 1573 1745 1917 20S9 2261 2433 2605 2777 2949 172 253 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 254 4834 5005 5176 5346 5517 5688 5858 6029 6199 637O 171 255 406540 6710 688l 7051 7221 7391 756i 7731 7901 8070 170 236 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 257 9933 *oio2 *027I *0440 *o6o9 *0777 *0946 *ni4 "•'1283 *i45i 169 258 41 1620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 259 33oo 3467 3635 3803 3970 4137 4305 4472 4039 4806 167 260 414973 5140 5307 5474 5641 5808 5974 6141 6308 6474 167 261 6641 6807 6973 7139 7306 7472 7638 7804 797o 8i35 166 262 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 263 9956 *OI2I *0286 *045i *o6i6 *o78i *0945 *IIIO *i275 *I439 165 264 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 187 19 37 56 75 94 112 131 150 168 187 186 19 37 56 74 93 112 130 149 167 186 x ! 5 19 37 56 74 93 in 130 148 167 185 184 18 37 55 74 92 no 129 147 166 184 183 18 37 55 73 92 no 128 146 165 183 CO 182 18 36 55 73 9i 109 127 146 164 182 H 181 18 36 54 72 9i 109 127 145 163 181 tt 180 18 36 54 72 90 108 126 144 162 . 180 < 179 18 36 54 72 90 107 125 143 161 179 178 18 36 53 7i 89 107 125 142 160 178 h3 177 18 35 53 7i 89 106 124 142 159 177 < 176 18 35 53 70 88 106 123 141 158 176 Z 175 18 35 53 70 88 105 123 140 158 175 O 174 17 35 52 70 87 104 122 139 157 174 H 173 17 35 52 69 87 104 121 138 156 173 X 172 17 34 52 69 86 103 120 138 155 172 O 171 17 34 5i 68 86 103 120 i37 154 171 O 170 17 34 5i 68 85 102 119 136 153 170 K i6g 17 34 5i 68 85 101 118 i35 152 i6g 0. 168 17 34 50 67 84 101 118 134 151 168 167 17 33 50 67 84 100 117 i34 150 167 166 17 33 50 66 83 100 116 133 149 166 165 17 33 50 66 83 99 116 132 149 165 164 Diff. 16 I 33 49 66 82 98 115 131 14S 164 2 3 4 5 6 7 8 9 Diff. 522 LOGARITHMS OF NUMBERS N. O I 2 3 4 5 6 7 8 9 — 1 Diflf. 265 423246 34io 3574 3737 3901 4065 4228- 4392 4555 4718 164 266 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 163 267 651 1 6674 6836 6999 7161 7324 7486 7648 781 1 7973 162 268 8135 8297 8459 8621 8783 8944 9106 9268 9429 ,959i 162 269 9752 9914 *oo75 *0236 *o 39 8 *0559 *0720 *o88i *I042 *I203 161 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 271 2969 3130 3290 3450 3610 377o 3930 4090 4249 4409 160 272 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159 273 6163 6322 6481 6640 6799 6957 7116 7275 7433 7592 159 274 775i 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 275 439333 9491 9648 9806 9964 *OI22 *o279 ^437 *0594 *o752 158 276 440909 1066 1224 1381 1538 1695 1852 2009 2166 2323 157 277 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 278 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 279 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 280 447158 7313 7468 7623 7778 7933 8088 8242 8397 8552 155 281 8706 8861 9015 9170 9324 9478 9633 9787 9941 *oo95 154 282 450249 0403 0557 0711 0865 1018 1 172 1326 1479 1633 154 283 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 284 33i8 347i 3624 3777 3930 4082 4235 4387 454o 4692 153 285 454845 4997 5150 5302 5454 5606 5758 59io 6062 6214 152 286 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 287 7882 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 288 9392 9543 9694 9845 9995 *oi46 ^0296 *0447 *c>597 *0748 151 289 460898 1048 1 198 1348 1499 1649 1799 1948 2098 2248 150 290 462398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 291 3893 4042 4191 434o 4490 4639 4788 4936 5085 5234 149 292 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 293 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 294 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 295 469822 9969 *on6 *o26>3 *04io *o557 *o704 *o85i ^0998 *ii45 147 296 471292 1438 1585 1732 1878 2025 2171 2318 2464 2610 146 297 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 298 4216 4362 4508 4653 4799 4944 5090 5235 538i 5526 146 299 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 N. Diflf. I 2 3 4 5 6 7 8 9 Diflf. 164 164 16 33 49 66 82 98 115 131 148 163 16 33 49 65 82 98 114 130 147 163 162 16 32 49 65 81 97 113 130 146 162 161 16 32 48 64 81 97 113 129 145 161 160 16 32 48 64 80 96 112 128 144 160 159 16 32 48 64 80 95 in 127 143 159 < 158 16 32 47 ^ 79 95 in 126 142 158 & 157 16 3i 47 63 79 94 no 126 141 157 J 156 16 3i 47 62 78 94 109 125 140 156 < 155 16 3i 47 62 78 93 109 124 140 155 z 154 15 3i 46 62 77 92 108 123 139 154 O 153 15 3i 46 61 77 92 107 122 138 153 1— 1 H 04 152 J 5 I 30 46 61 76 91 106 122 137 152 151 15 30 45 60 76 91 106 121 136 151 O 150 15 30 45 60 75 90 105 120 135 150 O P-i 149 15 30 45 60 75 § 9 104 119 134 149 148 15 30 44 59 74 89 104 118 133 148 147 15 29 44 59 74 88 103 118 132 147 146 15 29 44 5 § 73 88 102 117 131 146 145 15 29 44 58 73 87 I02 116 131 M5 144 14 29 43 58 72 86 IOI 115 130 144 143 14 29 43 57 72 86 IOO 114 129 143 Diflf. I 2 3 4 5 6 7 8 9 Diflf. LOGARITHMS OF NUMBERS 523 N. O I 2 3 4 5 6 7 8 9 Diff. 300 477121 7266 741 1 7555 7700 7844 7989 8133 8278 8422 145 301 8566 8711 8855 8999 9M3 9287 943i 9575 9719 9863 144 302 480007 0151 0294 0438 0582 0725 0869 1012 1 156 1299 144 303 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 304 2874 3016 3159 3302 3445 3587 3730 3872 4015 4i57 143 305 484300 4442 4585 4727 4869 501 1 5153 5295 5437 5579 142 306 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 142 307 7138 7280 7421 7503 7704 7845 7986 8127 8269 8410 141 308 8551 8692 8833 8974 ,9H4 9255 9396 9537 9677 9818 141 309 9958 *oo99 *0239 *038o *0520 *o66i *o8oi *094i *io8i *I222 140 310 491362 1502 1642 1782 1922 2062 2201 2341 2481 2621 140 3ii 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 312 4155 4294 4433 4572 4711 4850 4989 5128 5267 5406 139 313 5544 5683 5822 5960 6099 6238 6376 6515 6653 679I 139 3M 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 315 4983 1 1 8448 8586 8724 8862 8999 9137 9275 9412 .9550 138 316 9687 9824 9962 *oo99 *o236 *©374 *05 1 1 *o648 *0785 ^0922 137 317 50i 59 1 196 1333 1470 1607 1744 1880 2017 2154 229I 137 3i8 2427 2564 2700 2837 2973 3109 3246 3382 35i8 3655 136 319 3791 3927 4063 4199 4335 4471 4607 4743 4878 5014 136 320 505150 5286 5421 5557 5693 5828 5964 6099 6234 637O 136 321 6505 6640 6776 691 1 7046 7181 7316 745i 7586 7721 135 322 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 323 9203 9337 9471 9606 9740 9874 *ooo9 *oi43 *o277 *04ii 134 324 510545 0679 0813 0947 1081 1215 1349 1482 1616 I750 134 325 511883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 326 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 327 4548 4681 4813 4946 5079 5211 5344 5476 5609 5741 133 328 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 132 329 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 8646 8777 8909 9040 9171 9303 9434 9566 9697 131 331 9828 9959 *oo9o *022I *0353 *o484 *o6i5 *0745 *o876 *ioo7 131 332 5 2II 38 1269 1400 I530 1661 1792 1922 2053 2183 2314 131 333 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 334 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 335 525045 5174 5304 5434 5563 5693 5822 595i 6081 6210 129 336 6339 6469 6598 6727 6856 6985 7114 7243 7372 75oi 129 337 7630 7759 7888 8016 8i45 8274 8402 8531 8660 8788 129 338 8917 9045 9*74 9302 9430 9559 9687 9815 9943 *0072 128 339 530200 0328 0456 0584 0712 0840 0968 1096 1223 1351 128 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 142 14 28 43 57 71 85 99 114 128 142 tn 141 14 28 42 56 71 85 99 113 127 141 < 140 14 28 42 56 70 84 98 112 126 140 139 14 28 42 56 70 83 97 in 125 139 fc 138 14 28 4i 55 69 83 97 no 124 138 < O 137 14 27 4i 55 69 82 96 no 123 137 136 14 27 4i 54 68 82 95 109 122 136 135 14 27 4i 54 68 81 95 108 122 135 134 13 27 40 54 67 80 94 107 121 134 133 13 27 40 53 67 80 93 106 I20 133 O 0. 132 13 26 40 53 66 79 92 106 119 132 131 13 26 39 52 66 79 92 105 Il8 131 130 13 26 39 52 65 78 9i 104 117 130 129 13 26 39 52 65 77 9° 103 Il6 129 ft 1 * 128 13 26 38 5i 64 77 90 102 115 128 127 13 25 38 5i 64 76 89 102 114 127 Diff. I 2 3 4 5 6 7 8 9 Diff. 524 LOGARITHMS OF NUMBERS N. O I 2 3 4 5 6 7 8 9 Diff. 34° 531479 1607 1734 1862 1990 2117 2245 2372 2500 2627 128 341 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 127 342 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 127 343 5294 5421 5547 5674 5800 5927 6053 6i8-> 6306 6432 126 344 6558 6685 681 1 6937 7063 7189 7315 7441 7567 7693 126 345 537819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 346 9076 9202 9327 9452 9578 9703 9829 99J4 *oo79 *0204 125 347 540329 0455 0580 0705 0830 0955 1080 1205 1330 1454 125 348 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 349 2825 2950 3074 3199 3323 3447 357i 3696 3820 3944 124 350 544068 4192 43i6 4440 4564 4688 4812 4936 5060 5183 124 35i 5307 543i 5555 5678 5802 5925 6049 6172 6296 6419 124 352 6543 6666 6789 6913 7036 7159 7282 8512 7405 7529 7652 123 353 7775 7898 8021 8144 8267 8389 8635 8758 8881 123 354 9003 9126 9249 937i 9494 9616 9739 9861 9984 *oio6 123 355 550228 0351 0473 0595 0717 0840 0962 1084 1206 1328 122 356 145° 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 357 2668 2790 291 1 3033 3155 3276 3398 3519 3640 3762 121 358 3883 4004 4126 4247 4368 4489 4610 473i 4852 4973 121 359 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 360 556303 6423 6544 6664 6785 6905 7026 7146 7267 7387 I20 361 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 I20 362 8709 8829 8948 9068 9188 93o8 9428 9548 9667 9787 I20 363 9907 *0026 *oi46 *0265 *o 3 8 5 *0504 *o624 *o743 *o863 ♦0982 II 9 364 561 101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 365 562293 2412 2531 2650 2769 2887 3006 3125 3244 3362 II 9 366 348i 360O 37i8 3837 3955 4074 4192 43i 1 4429 4548 * n 2 367 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 368 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 n8 36g 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 n8 37o 568202 8319 8436 8554 8671 8788 8905 9023 9140 9257 117 37i 9374 949I 960S 9725 9842 9959 *oo76 *oi93 ^309 *0426 117 372 570543 0660 0776 0893 IOIO 1 126 1243 1359 1476 1592 117 373 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 n6 374 2872 2988 3 J o4 3220 3336 3452 3568 3684 3800 3915 116 375 574031 4H7 4263 4379 4494 4610 4726 4841 4957 5072 116 376 5188 5303 5419 5534 5 6 50 5765 5880 5990 6111 6226 115 377 6341 6457 6572 66S7 6802 6917 7032 7147 . 7262 7377 115 378 7492 7607 7722 7836 7951 8066 81S1 8295 8410 8525 115 379 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 128 13 26 38 51 64 77 90 102 115 128 127 13 25 38 51 64 76 89 102 114 127 00 126 13 25 38 50 63 76 88 IOI 113 126 125 13 25 38 50 63 75 88 100 113 125 < 124 12 25 37 50 62 74 87 99 112 124 Ah 123 12 25 37 49 62 74 86 98 in 123 122 12 24 37 49 61 73 85 98 no 122 (U 121 12 24 36 48 61 73 85 97 109 121 O 120 12 24 36 48 60 72 84 96 108 120 Oh iiq 12 24 36 48 60 7i 83 95 107 119 118 12 24 35 47 59 7i 83 94 106 118 117 12 23 35 47 59 70 82 94 105 117 116 12 23 35 46 58 7o 81 93 104 116 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 525 N. O I 2 3 4 5 6 7 8 9 Diff. ! 380 579784 9S9S *OOI2 *OI26 *024I *°355 *0469 *o 5 8 3 *o697 *o8n 114 381 580925 1039 1153 1267 1381 1495 1608 1722 1836 1950 114 382 2063 2177 229I 2404 2518 2631 2745 2858 2972 3085 114 383 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 384 4331 4444. 4557 4670 4783 4896 5009 5122 5235 5348 113 385 585461 5574 5686 5799 5912 6024 6i37 6250 6362 6475 113 386 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 387 771 1 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 388 8832 8944 9056 9167 9279 939i 9503 9615 9726 9838 112 389 9950 *oo6i *oi73 *o284 *0396 *0507 *o6i9 *0730 *o842 *0953 112 390 59i° 6 5 1176 1287 1399 1510 162 1 1732 1843 1955 2066 in 391 2177 2288 2399 2510 2621 2732 2843 2954 3064 3^75 in 392 3286 3397 35o8 3618 3729 3840 3950 4061 4171 4282 in 393 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 no 394 5496 5606 5717 5827 5937 6047 6i57 6267 6377 6487 no 395 596597 6707 6817 6927 7037 7146 7256 7366 7476 7586 no 396 7695 7805 79H 8024 8i34 8243 8353 8462 8572 8681 no 397 8791 8900 9009 9119 9228 9337 9446 . 955 £ 9665 9774 109 398 9883 9992 *OIOI *02IO ^319 ♦0428 *0537 *o646 *0755 *o864 109 399 600973 1082 1 191 I299 1408 1517 1625 1734 1843 I95i 109 400 602060 2169 2277 2386 2494 2603 2711 2819 2928 3036 108 401 3144 3253 336i 3469 3577 36S6 3794 3902 4010 4118 108 402 4226 4334 4442 4550 4658 4766 4874 49S2 5089 5197 108 403 5305 5413 5521 5628 5736 5844 5951 6059 6166 6274 108 404 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 107 405 607455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 406 8526 8633 8740 8847 8954 9061 9167 9274 938i 9488 107 407 9594 9701 9808 9914 *002I *OI28 *o234 *o 3 4i *0447 *0554 107 408 610660 0767 0873 0979 IO86 1192 1298 1405 1511 1617 106 409 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 2890 2996 3102 3207 3313 3419 3525 3630 3736 106 411 3842 3947 4053 4159 4264 4370 4475 4581 4686 4792 106 412 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 413 5950 6055 6160 6265 637O 6476 6581 6686 6790 6895 105 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 415 618048 8i53 8257 8362 8466 8571 8676 8780 8884 8989 105 416 9093 9198 9302 9406 9511 9615 9719 9824 9928 *0032 104 A1 l 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 104 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2IIO 104 4 ig 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 104 N. Diff. I 2 3 4 5 6 7 8 9 Diff. "5 12 23 35 46 58 69 81 92 104 "5 114 n 23 34 46 57 68 80 91 103 114 W "3 n 23 34 45 5 J 68 79 90 102 "3 H < 112 n 22 34 45 56 67 78 90 IOI 112 in n 22 33 44 56 67 78 89 IOO in no n 22 33 44 55 66 77 88 99 no O Oh log n 22 33 44 55 65 76 87 98 109 108 n 22 32 43 54 65 76 86 97 108 107 n 21 32 43 54 64 75 86 96 107 106 11 21 32 42 53 64 74 85 95 106 105 n 21 32 42 53 63 74 84 95 105 104 10 21 3i 42 52 62 73 83 94 104 103 10 21 3i 4i 52 62 72 82 93 103 Diff. I 2 3 4 5 6 7 8 9 Diff. 526 LOGARITHMS OF NUMBERS N. I 2 3 4 5 fc 7 8 9 Diff. 103 420 623249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 4282 4385 4488 4591 4695 4798 4901 5004 5i°7 5210 103 422 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 103 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 103 424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 102 4 2 5 628389 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 426 9410 9512 9613 9715 9817 9919 *002I *OI23 *6224 #0326 102 427 630428 0530 0631 0733 0835 0936 IO38 1139 1241 1342 102 428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 IOI 429 2457 2559 2660 2761 2862 2963 3064 3x65 3266 3367 IOI 430 633468 3569 3670 3771 3872 3973 4074 4175 4276 4376 IOI 43i 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 IOI 432 5484 5584 5685 5785 5^86 5986 6087 6187 6287 6388 IOO 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 739° IOO 434 7490 7590 7690 7790 7890 7990 809O 8190 8290 8389 IOO 435 638489 8589 8689 8789 8888 8988 9088 9188 9287 9387 IOO 436 9486 9586 9686 9785 9885 9984 *oo84 *oi83 *028 3 $0382 99 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 99 440 643453 3551 3650 3749 3847 3946 4044 4H3 4242 434o 98 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 5422 552i 5619 5717 5815 5913 6011 6110 6208 6306 98 443 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 7383 748i 7579 7676 7774 7872 7969 8067 8165 8262 98 445 648360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 446 9335 9432 9530 9627 9724 9821 9919 *ooi6 *oii3 *02I0 97 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 Il8l 97 448 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 97 449 2246 2343 2440 2536 2633 2730 2826 2923 3019 3Il6 97 450 653213 3309 3405 3502 3598 3695 379i 3888 3984 4080 96 45i 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 96 452 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 453 6098 6194 6290 6386 64S2 6577 6673 6769 6864 6960 96 454 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 6580 1 1 8107 8202 8298 8393 8488 8584 8679 8774 887O 95 456 8965 9060 * 9I5 £ 9250 9346 9441 9536 9631 9726 982I 95 457 9916 *OOII *oio6 *020I *0296 ^391 #0486 *058i *o676 *077i 95 458 660865 0960 1055 II5O 1245 1339 1434 1529 1623 1718 95 459 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 95 460 662758 2852 2947 3041 3135 3230 3324 34i8 3512 3607 94 461 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 462 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 463 558i 5675 5769 5862 5956 6050 6143 6237 6331 6424 94 464 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 94 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 104 10 21 31 42 52 62 73 83 94 104 CO H 103 10 21 31 41 52 62 72 82 93 103 102 10 20 31 41 51 61 7i 82 92 102 < 101 10 20 30 40 51 61 7i 81 9i IOI & 100 10 20 30 40 50 60 70 80 90 IOO cu 99 10 20 30 40 50 59 69 79 89 gg q8 10 20 29 39 49 59 69 78 88 98 tf 97 10 19 29 39 49 58 68 78 87 97 04 96 10 19 29 38 48 58 67 77 86 96 95 10 19 29 38 48 57 67 76 86 95 Diff. X 2 3 4 5 6 7 8 9 Diff. 1 LOGARITHMS OF NUMBERS 527 In. ! 465 I 2 3 4 5 6 7 8 9 Diff. 667453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 466 S3S6 8479 8572 8665 8759 8852 8945 9038 9131 9224 93 467 9317 9410 9503 9596 9689 9782 9875 9967 *oo6o *oi53 93 468 670246 0339 0431 0524 0617 0710 0802 0895 0988 10S0 93 469 1 173 1265 1358 145 1 1543 1636 1728 182 1 1913 2005 93 470 672098 2190 2283 2375 2467 2560 2652 2744 2836 2929 92 471 3021 3ii3 3205 3297 3390 3482 3574 3666 3758 3850 92 472 3942 4°34 4126 4218 4310 4402 4494 4586 4677 4769 92 473 4861 4953 5045 5137 5228 5320 54i2 5503 5595 5687 92 n74 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 475 676694 6785 6876 6968 7059 7151 7242 7333 7424 75i6 91 476 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 9i 477 8518 8609 8700 8791 8882 8973 9064 9155 9246 9337 9i 478 9428 9519 9610 9700 9791 9882 9973 *oo63 *oi54 *0245 9i 479 680336 0426 0517 0607 0698 0789 0879 0970 1060 1151 9i 480 681241 1332 1422 1513 1603 1693 1784 1874 1964 2055 90 481 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 90 482 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 1 483 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 484 4845 4935 5025 5ii4 5204 5294 5383 5473 5563 5652 90 485 685742 5831 5921 6010 6100 6189 6279 6368 6458 6547 89 486 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 487 7529 7618 7707 7796 7886 7975 8064 8i53 8242 8331 89 488 8420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 489 9309 9398 9486 9575 9664 9753 9841 9930 *ooi9 *oio7 89 490 690196 0285 0373 0462 0550 0639 0728 0816 0905 0993 89 491 1081 1170 1258 1347 1435 1524 1612 1700 1789 1877 88 492 1965 2053 2142 2230 2318 2406 2494 2583 2671 2759 88 493 2847 2935 3023 3111 3i99 3287 3375 3463 355i 3639 88 494 c 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 88 495 694605 4693 47Sl 4868 4956 5044 5131 5219 5307 5394 88 496 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 87 497 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 87 498 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 87 499 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883 87 500 698970 9057 9144 9231 93i7 9404 9491 9578 9664 9751 87 501 9838 9924 *OOII *oo98 *oi84 *027I *o 35 8 *0444 *o53i *o6i7 ll 502 700704 0790 0877 0963 1050 1136 1222 1309 1395 1482 86 503 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 86 504 2431 2517 2603 2689 ' 2775 2861 2947 3033 3ii9 3205 86 505 703291 3377 3463 3549 3635 3721 3807 3893 3979 4065 86 506 415 1 4236 4322 4408 4494 4579 4665 4751 4837 4922 86 507 5008 5094 5179 5265 5350 5436 5522 5607 5693 5778 86 508 5864 5949 6035 6120 6206 6291 6376 6462 6547 6632 85 509 6718 6803 6888 6974 7059 7144. 7229 7315 7400 7485 85 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 94 9 19 28 38 47 5 $ 66 75 85 94 CO 93 9 19 28 37 47 56 65 74 84 93 tf 92 9 18 28 37 46 55 64 74 83 92 < 9i 9 18 27 36 46 55 64 73 82 9i ft 90 9 18 27 36 45 54 63 72 81 90 cC 89 9 18 27 36 45 53 62 7i 80 89 88 9 18 26 35 44 53 62 70 79 88 tt 87 9 17 26 35 44 52 61 70 78 87 d. 86 ( 9 17 26 34 43 52 60 69 77 86 85 9 17 26 34 43 5i 60 68 77 85 Diff. I 2 3 4 5 6 7 8 9 Diff. 528 LOGARITHMS OF NUMBERS ' N. I 2 3 4 5 6 7 8 9 Diff. 5io 707570 7655 7740 7826 791 1 7996 8081 8166 8251 8336 85 5" 8421 8506 8591 8676 8761 8846 8931 9 OI 5 9100 9185 85 512 9270 9355 9440 9524 9609 9694 9779 9863 9948 *oo33 85 513 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 85 514 0963 1048 1132 1217 1301 1385 1470 1554 1639 1723 84 5i5 71 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 84 5i6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 84 517 349 1 3575 3659 3742 3826 3910 3994 4078 4162 4246 84 5i8 4330 4414 4497 4581 4665 4749 4833 4916 5000 5084 84 519 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 84 520 716003 6087 6170 6254 6337 6421 6504 6588 6671 6754 83 521 6838 6921 7004 70S8 .7171 7254 7338 7421 7504 7587 83 522 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 83 523 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 83 524 9331 9414 9497 9580 9663 9745 9828 991 1 9994 *oo77 83 525 720159 0242 0325 0407 0490 0573 0655 0738 0821 0903 83 526 0986 1068 1151 1233 1316 1398 148 1 1563 1646 1728 82 527 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 528 2634 2716 2798 28SI 2963 3045 3127 3209 3291 3374 82 529 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 53o 724276 4358 4440 4522 4604 4685 4767 4849 493i 5013 82 53i 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 532 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 533 6727 6809 6890 6972 7053 7134 7216 7297 7379 746o 81 534 7541 7623 7704 7785 ^866 7948 8029 8110 8191 8273 81 535 728354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 536 9165 9246 9327 9408 94-89 9570 9651 9732 9813 9893 81 537 9974 *oo55 *oi36 *02IJ ^0298 *o 37 8 *0459 *0540 *062I *070 2 81 53C 730782 0863 0944 1024 1 105 1 186 1266 1347 1428 1508 8i 539 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 54o 732394 2474 2555 2635 2715 2796 2876 2956 3037 3ii7 80 54i 3197 3278 3358 3438 35i8 3598 3679 3759 3839 3919 80 542 3999 4079 4160 4240 4320 4400 4480 456o 4640 4720 80 543 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 80 544 5599 5679 5759 5838 59i8 5998 6078 6i57 6237 6317 80 545 736397 6476 6556 6635 6715 6795 6874 6954 7034 7ii3 80 546 7193 7272 7352 7431 75" 759o 7670 7749 7829 7908 79 547 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 79 1 548 8781 8860 8939 9018 9097 9177 9256 . 933 5 9414 9493 79 549 9572 9651 9731 9810 9889 9968 *oo47 *OI26 *0205 *0284 79 55c- 740363 0442 0521 0600 0678 0757 0836 0915 0994 1073 79 55i 1152 1230 1309 1388 1467 1546 1624 1703 1782 i860 79 552 1939 2018 2096 2175 2254 2332 241 1 2489 2568 2647 79 553 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 554 3510 3588 3667 3745 3^23 3902 3980 4058 4136 4215 78 N. Diff. I 2 3 4 5 6 7 8 9 Diff. w [h 86 9 17 26 . 34 43 52 60 69 77 86 X 85 9 17 26 34 43 51 60 68 77 85 < 84 8 17 25 34 42 50 59 67 76 84 fc 83 8 17 25 33 42 50 58 66 75 83 . 82 8 16 25 33 4i 49 57 66 74 82 cu 81 8 16 24 32 4i 49 57 65 73 81 04 80 8 16 24 32 40 48 56 64 72 80 cw 79 8 16 24 32 40 47 55 63 7i 79 1 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 529 N. 555 O I 2 3 4 5 6 7 8 9 Diff. 744293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 556 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 557 5S55 5933 601 1 60S9 6167 6245 6323 6401 6479 6556 78 558 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 559 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 8266 8343 8421 8498 8576 8653 873i 8808 8885 77 561 8963 9040 9118 9 J 95 9272 9350 9427 9504 9582 9659 77 562 9736 9S14 9891 9968 *oo45 *OI23 *0200 *0277 *0354 *043i 77 5 6 3 750508 0586 0663 0740 0817 0894 O97I 1048 1125 1202 77 564 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 565 752048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 566 2S16 2893 2970 3047 3123 3200 3277 3353 3430 35o6 77 567 35S3 3660 3736 3S13 3889 3966 4042 4119 4195 4272 77 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 569 5112 5i89 5265 5341 5417 5494 5570 5646 5722 5799 76 57o 755875 5951 6027 6103 6180 6256 6332 6408 6484 6560 76 57i 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 573 8i55 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 574 8912 89S8 9063 9*39 9214 9290 9366 9441 9517 9592 76 575 759668 9743 9819 9894 9970 *oo45 *OI2I *oi96 *0272 *Q347 75 576 760422 0498 0573 0649 0724 0799 0875 0950 IO25 IIOI 75 577 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 75 578 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 579 2679 2754 2S29 2904 2978 3053 3128 3203 3278 3353 75 580 763428- 3503 3578 3653 3727 3802 3877 3952 4027 4101 75 581 4176 4251 4326 4400 4475 455o 4624 4699 4774 4848 75 582 4923 4998 5072 5H7 5221 5296 5370 5445 552o 5594 75 583 5669 5743 5818 5892 5966 6041 6lI5 6190 6264 6338 74 584 6413 6487 6562 6636 6710 6785 6859 6933 7007 70S2 74 585 767156 7230 73°4 7379 7453 7527 760I 7675 7749 7823 74 586 7898 7972 8046 8120 8i94 8268 8342 8416 8490 8564 74 587 8638 8712 87S6 8860 8934 9008 9082 9^6 9230 9303 74 588 9377 945i 9525 9599 9673 9746 9820 9894 996S *0O42 74 589 7701 15 0189 0263 0336 0410 0484 0557 0631 0705 0778 74 590 770852 0926 0999 1073 1 146 1220 I293 1367 1440 1514 74 59i 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 73 592 2322 2395 2468 2542 2615 2688 2762 2S35 2908 2981 73 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 594 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 774517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 596 5246 5319 5392 5465 5538 5610 56S3 5756 5829 5902 73 597 5974 6047 6120 6193 6265 6338 641 1 6483 6556 6629 73 598 6701 6774 6S46 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7S62 7934 8006 8079 72 N. Biff. 1 1 2 3 4 5 6 7 8 9 Diff. CO 78 1 8 16 23 31 39 47 55 62 70 78 tt 77 8 15 23 3i 39 46 54 62 69 77 < 76 8 15 23 30 38 46 53 61 68 76 cu 75 8 15 23 30 38 45 53 60 68 75 74 7 15 22 | 30 37 44 52 59 67 74 73 7 15 22 j 29 37 44 5i 58 66 73 72 7 14 22 I 29 36 43 50 58 65 72 Diff, Diff. I 2 3 4 5 6 7 8 9 53o LOGARITHMS OF NUMBERS N. 600 O I 2 3 4 5 6 7 8 9 Diff. 778i5i 8224 8296 8368 8441 8513 8585 8658 8730 8802 72 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 959 6 9669 9741 9813 9885 9957 *0029 *OIOI *oi73 *o245 72 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 604 1037 1 109 Il8l 1253 1324 1396 1468 1540 1612 1684 72 605 781755 1827 1899 197 1 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2688 2759 2831 2902 2974 3046 3ii7 72 607 3189 3260 3332 3403 3475 3546 3618 3689 376i 3832 71 608 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 609 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 785330 54oi 5472 5543 5615 5686 5757 5828 5899 597o 71 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 613 7460 753i 7602 7673 7744 78i5 7885 7956 8027 8098 71 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 6*5 788875 8946 9016 9087 9157 9228 9299 9369 9440 95io 71 616 958i 9651 9722 9792 9S63 9933 *ooo4 *oo74 *oi44 *02I5 70 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1129 1 199 1269 1340 1410 1480 I550 1620 70 6ig 169 1 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3162 3231 33oi 3371 344i 35i 1 358i 3651 3721 70 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 70 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 581 1 70 625 79588o 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 626 6574 6644 6713 67S2 6852 6921 6990 7060 7129 7198 69 627 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890 69 628 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 629 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 799341 9409 9478 9547 9616 9685 9754 9823 9892 9961 69 631 800029 0098 0167 0236 0305 0373 0442 051 1 0580 0648 69 632 0717 0786 0854 0923 0992 106 1 1129 1 198 1266 1335 69 633 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 634 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 68 635 802774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 636 3457 3525 3594 3662 3730 3798 3867 3935 4003 407 i 68 637 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 638 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 639 5501 5569 5637 5705 5773 5S41 5908 5976 6044 6112 68 640 806180 6248 6316 6384 6451 6519 6587 6655 6723 6790 68 641 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 68 642 7535 7603 7670 7738 7806 7873 7941 8008 8076 8i43 68 643 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 644 8886 8953 9021 9088 9 J 56 9223 9290 9358 9425 9492 67 645 809560 9627 9694 9762 9829 9896 9964 *oo3i ^0098 *oi65 67 646 810233 0300 0367 0434 0501 0569 0636 0703 0770 0837 67 647 0904 0971 1039 1106 "73 1240 1307 1374 144 1 1508 67 648 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 649 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 N. Diff. I 2 3 4 5 6 7 8 9 Diff. w 73 7 15 22 29 37 44 51 58 66 73 H 72 7 14 22 29 36 43 50 58 6s 72 0< 7i 7 14 21 28 36 43 50 57 64 7i . 70 7 14 21 28 35 42 49 56 63 70 O 6q 7 14 21 2S 35 4i 48 55 62 69 & 68 7 14 20 27 34 4i 48 54 61 68 Diff. 1 Diff. I 2 3 4 5 * 7 8 9 LOGARITHMS OF NUMBERS S3 1 N, O 1 2 3 4 5 6 7 8 9 Diff. 67 650 812913 2980 3047 3114 3181 3247 33H 3381 3448 35i4 651 35Si 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 652 4248 43i4 43Si 4447 4514 4581 4647 4714 4780 4847 67 653 4913 4980 5046 5113 5i79 5246 5312 5378 5445 55i 1 66 654 5578 5525 64 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1 166 64 678 1230 1294 1358 1422 i486 1550 1614 1678 1742 1806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 2573 2637 2700 2764 2828 2892 2956 3020 3083 64 681 3147 3211 3275 3338 3402 3466 3530 3593 3657 372i 64 682 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 683 4421 4484 4548 461 1 4675 4739 4802 4866 4929 4993 64 684 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 685 835691 5754 5817 5881 5944 6007 6071 6i34 6197 6261 63 686 6324 6387 6451 65H 6577 6641 6704 6767 6830 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838849 8912 8975 9038 9101 9164 9227 9289 9352 9415 63 6gi 9478 954i 9604 9667 9729 9792 9855 9918 9981 *oo43 63 692 840106 0169 0232 0294 0357 0420 0482 0545 0608 0671 63 693 0733 0796 0859 0921 0984 1046 1 109 1172 1234 1297 63 694 1359 1422 1485 1547 1610 1672 1735 1797 i860 1922 63 695 841985 2047 2110 2172 2235 2297 2360 2422 2484 2547 62 6g6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 698 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 62 699 4477 4539 4601 4664 4726 4788 4850 4912 4974 5 36 62 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 67 7 13 20 27 34 40 47 54 60 67 h 66 7 13 20 26 33 40 46 53 59 66 65 7 13 20 26 33 39 46 52 59 65 6 l A 6 13 19 26 32 38 45 5i 58 64 04 63 6 13 19 25 32 .38 44 50 57 63 62 6 12 19 25 3i 37 43 50 56 62 Diff. I 2 3 4 5 6 7 8 9 Diff. 532 LOGARITHMS OF NUMBERS N. I 2 3 4 5 6 7 8 9 Diff. 62 700 845098 5160 5222 5284 5346 5408 5470 5532 5594 5656 701 5718 578o 5842 5904 5966 6028 6090 6151 6213 6275 62 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 703 6955 7017 7079 714I 7202 7264 7326 7388 7449 75i 1 62 704 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 848189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 707 9419 9481 9542 9604 9665 9726 9788 9849 991 1 9972 61 708 850033 0095 0156 0217 0279 0340 0401 0462 0524 0585 61 7og 0646 0707 0769 0830 0891 0952 1014 1075 1 136 1 197 61 710 851258 1320 1381 1442 1503 1564 1625 1686 1747 1809 61 711 1870 193 1 1992 2053 2114 2175 2236 2297 2358 2419 61 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 714 3698 3759 3820 3881 394i 4002 4063 4124 41S5 4245 61 715 854306 4367 4428 4488 4549 4610 4670 473i 4792 4852 6r 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 717 5519 558o 5640 570I 576i 5822 5882 5943 6003 6064 61 718 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 719 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 7393 7453 7513 7574 7634 7694 7755 78i5 7875 60 721 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 722 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 723 9138 9198 9258 93i8 9379 9439 .,9499 .9559 9619 9679 60 724 9739 9799 9859 9918 9978 *oo38 *oo98 *oi58 *02l8 *0278 60 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 0937 0996 1056 1116 1176 1236 1295 1355 14 15 1475 60 727 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 728 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 729 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 3382 3442 3501 356i 3620 3680 3739 3799 3858 59 731 39i7 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 45i 1 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 733 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 59 734 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 735 866287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 736 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 59 738 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 59 739 8644 8703 8762 8821 8879 8938 8997 9056 9114 9 J 73 59 740 869232 9290 9349 9408 9466 9525 9584 9642 9701 9760 59 74i 9818 9877 9935 9994 *oo53 *OIII *oi70 *0228 ^0287 *0345 59 742 870404 0462 0521 0579 0638 0696 0755 0813 0872 0930 58 743 0989 1047 1 106 1 164 1223 1281 1339 1398 1456 1515 58 744 1573 1631 1690 1748 1806 1865 1923 I98 1 2040 2098 58 745 872156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 58 747 3321 3379 3437 3495 3553 361 1 3669 3727 3785 3844 58 748 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 749 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 58 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 62 6 12 19 25 31 37 43 50 56 62 ft, 61 6 12 18 24 31 37 43 49 55 61 60 6 12 18 24 30 36 42 48 54 60 O 59 6 12 18 24 30 35 4i 47 53 59 58 6 12 17 23 29 35 4i 46 52 58 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 533 N. O I 2 3 4 5 6 7 8 9 Diff. 75° 875061 5ii9 5177 5235 5293 535i 5409 5466 5524 5582 58 75i 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 752 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 753 6795 6853 6910 6968 7026 7083 7141 7199 7256 73i4 58 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 755 877947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 756 8522 8579 8637 8694 8752 8809 8866 8924 8981 9°39 57 757 9096 9*53 9211 9268 9325 9383 9440 9497 9555 9612 57 758 9669 9726 97S4 9841 9898 9956 *ooi3 *oo7o *OI27 *oi85 57 759 880242 0299 0356 0413 0471 0528 0585 0642 0699 0756 57 760 880814 0871 0928 0985 1042 1099 1156 1213 1271 1328 57 761 1385 1442 1499 1556 1613 1670 1727 1784 184 1 1898 57 762 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 763 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 57 764 3093 3150 3207 3264 3321 3377 3434 349i 3548 3605 57 765 883661 37i8 1 3775 3832 3888 3945 4002 4059 4115 4172 57 766 4229 4285 ■ 4342 4399 4455 4512 4569 4625 4682 4739 57 767 4795 4852 1 4909 4965 5022 5078 5135 5192 5248 5305 57 768 536i 54i8 5474 5531 5587 5644 5700 5757 5813 5870 57 769 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 56 770 886491 6547 6604 6660 6716 6773 6829 6885 6942 6998 56 771 7054 7111 7167 7223 7280 7336 7392 7449 7505 756i 56 772 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 56 773 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 56 774 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 56 775 889302 9358 9414 9470 9526 9582 9638 9694 9750 9806 56 776 9862 9918 9974 *oo30 *oo86 *oi4i *oi97 *0253 #0309 $0365 56 777 890421 0477 0533 0589 0645 0700 0756 0812 0868 0924 56 778 0980 1035 1091 1 147 1203 1259 1314 1370 1426 1482 56 779 1537 1593 1649 I705 1760 1816 1872 1928 1983 2039 56 780 892095 2150 2206 2262 2317 2373 2429 2484 2540 2595 56 781 2651 2707 2762 2818 2873 2929 2985 3040 3096 3i5i 56 782 3207 3262 33i8 3373 3429 3484 354o 3595 3651 3706 56 783 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 55 784 43^6 437i 4427 4482 4538 4593 4648 4704 4759 4814 55 7 l 5 894870 4925 4980 5036 5091 5146 5201 5257 5312 5367 55 786 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 787 5975 6030 6085 6140 6i95 6251 6306 6361 6416 6471 55 788 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 789 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 897627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 791 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 55 792 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 793 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 794 9821 9875 9930 9985 *oo39 *oo94 *oi49 *0203 *0258 *03I2 55 795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859 55 796 0913 0968 1022 1077 1131 1 186 1240 1295 1349 1404 55 797 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 798 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 799 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 N. OQ H Diff. I 2 3 4 5 6 7 8 9 Diff. 57 6 II 17 23 29 34 40 46 51 57 CU 56 6 11 17 22 28 34 39 45 50 56 • 55 6 II 17 22 28 33 39 44 50 55 54 5 II 16 22 27 32 38 43 49 54 [ Diff. I 2 3 4 5 6 7 8 9 Diff. .1 534 LOGARITHMS OF NUMBERS N. O I 2 3 4 5 6 1 , 7 8 l 9 Diff. 800 903090 3144 3199 3253 3307 3361 34i6 3470 3524 3578 54 801 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 802 4174 4229 4283 4337 439i 4445 4499 4553 4607 4661 54 803 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 804 5256 53io 5364 54i8 5472 5526 558o 5634 5688 5742 54 805 905796 5850 5904 5958 6012 6066 6119 6173 6227 6281 54 806 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 54 807 6874 6927 6981 7°35 7089 7143 71Q6 7250 7304 7358 54 808 741 1 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 809 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 54 811 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 54 812 9556 9610 9663 9716 9770 9823 9877 9930 9984 *oo37 53 813 910091 0144 0197 0251 0304 0358 041 1 0464 0518 0571 53 814 0624 0678 0731 0784 0838 0891 0944 0998 1051 1 104 53 815 911158 1211 1264 1317 1371 1424 1477 1530 1584 1637 53 816 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 53 817 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 53 818 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 53 819 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 820 913814 3867 3920 3973 4026 4079 4132 4184 4237 4290 53 821 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 822 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 53 823 5400 5453 5505 5558 561 1 5664 57i6 5769 5822 5875 53 824 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 53 ? 2 2 916454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 826 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 53 827 7506 7558 761 1 7663 7716 7768 7820 7873 7925 7978 52 828 8030 8083 8i35 8188 8240 8293 8345 8397 8450 8502 52 829 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 919078 9130 9183 9235 9287 934o 9392 9444 9496 9549 52 831 9601 9653 9706 9758 9810 9862 9914 9967 *ooi9 *oo7i 52 832 920123 0176 0228 0280 0332 0384 0436 0489 0541 0593 52 833 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 834 1 166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 835 921686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 836 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 837 2725 2777 2829 2881 2933 2985 3°37 3089 3 x 4o 3192 52 838 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 52 839 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 4331 4383 4434 4486 4538 4589 4641 4693 4744 52 841 4796 4848 4899 4951 5°°3 5°54 5106 5157 5209 5261 52 842 53i2 5364 5415 5467 55i8 557o 5621 5673 5725 5776 52 843 5828 5879 593i 5982 6034 6085 6137 6188 6240 6291 5i 844 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 5i 2 4 l 926857 6908 6959 701 1 7062 7114 7165 7216 7268 7319 5i 846 737o 7422 7473 7524 7576 . 7627 7678 7730 778i 7832 5i 847 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 5i 848 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 5i 849 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 5 1 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 55 H 55 6 11 17 22 28 33 39 44 50 Ph 54 5 11 16 22 27 32 38 43 49 54 • 53 5 11 16 21 27 32 37 42 48 53 52 5 10 16 21 26 31 36 42 47 52 Diff. I 2 3 4 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 535 N. I 2 3 4 5 6 7 8 9 1 \ Diff. 51 850 929419 9470 9521 9572 9623 9674 9725 9776 9827 9879 851 9930 9981 *0032 ♦0083 *oi34 *oi85 *0236 *0287 *o 33 8 *0389 51 852 930440 0491 0542 0592 0643 0694 0745 0796 0847 0898 5i 853 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 51 854 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 855 931906 2017 2068 2118 2169 2220 2271 2322 2372 2423 5 1 856 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 51 857 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 51 858 3487 3538 3589 3639 3690 3740 379i 3841 3892 3943 5i 859 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 5i 860 934498 4549 4599 4650 4700 4751 480 r 4852 4902 4953 50 861 5003 5054 5104 5154 5205 5255 53o6 5356 54o6 5457 50 862 5507 5558 5608 5658 5709 5759 5809 5860 59io 596o 50 863 601 1 6061 6111 6162 6212 6262 6313 6363 6413 6463 50 864 65H 6564 6614 6665 6715 6765 6815 6S65 6916 6966 50 865 937016 7066 7117 7167 7217 7267 7317 7367 7418 7468 50 866 75i8 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 867 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 5° 868 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 50 869 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 939519 9569 9619 9669 9719 9769 9819 9869 9918 9968 50 871 940018 0068 0118 0168 0218 0267 0317 0367 0417 0467 50 872 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 50 873 1014 1064 1114 1 163 1213 1263 1313 1362 1412 1462 50 874 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 50 * 75 942008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50 876 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 50 877 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 49 878 3495 3544 3593 3643 3692 3742 379i 3841 3890 3939 49 879 3989 4038 4088 4137 41S6 4236 4285 4335 4384 4433 49 880 944483 4532 458i 4631 4680 4729 4779 4828 4877 4927 49 881 4976 5025 5074 5 12 4 5173 5222 5272 5321 537o 5419 49 882 5469 55iS 5567 5616 5665 5715 5764 5813 5862 5912 49 883 596i 6010 6059 6108 6157 ' 6207 6256 6305 6354 6403 49 884 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 49 S 5 946943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 886 7434 7483 7532 758i 7630 7679 7728 7777 7826 7875 49 887 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 888 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 889 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 949390 9439 9488 9536 9585 9634 9683 9731 978o 9829 49 891 9878 9926 9975 *0024 *oo73 *OI2I *oi70 *02I9 *o267 *03i6 49 892 950365 0414 0462 051 1 0560 060S 0657 0706 0754 0803 49 893 0851 0900 0949 0997 1046 I095 1 143 1192 1240 1289 49 894 1338 1386 1435 1483 1532 I580 1629 1677 1726 1775 49 895 951823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48. 896 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 48 897 2792 2841 2S89 2938 2986 3034 3083 3131 3180 3228 48 898 3276 3325 3373 3421 347o 3518 3566 3615 3663 37i 1 48 899 3760 3808 3856 3905 3953 4OOI 4049 409S 4146 4194 48 N. Diff. I 2 3 4 5 6 7 8 9 Diff. CO H 5i 5 TO 15 20 26 31 36 41 46 5i 50 1 5 | 10 15 20 25 30 35 40 45 50 49 5 to 15 20 25 29 34 39 44 49 48 5 1 10 14 19 24 29 34 38 43 48 Diff. 1 2 3 4 5 6 7 8 9 Diff. 536 LOGARITHMS OF NUMBERS N. I 2 3 4 5 6 7 8 9 Diff. 900 954243 4291 4339 4387 4435 4484 4532 4580 4628 4677 48 901 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 48 902 5207 5255 5303 535i 5399 5447 5495 5543 5592 5640 48 903 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 904 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 905 956649 6697 6745 6793 6840 6888 6936 6984 7032 7080 48 906 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 907 7607 7655 7703 775i 7799 7847 7894 7942 7990 8038 48 908 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 909 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 9089 9137 9185 9232 9280 9328 9375 9423 947i 48 911 9518 9566 9614 9661 9709 9757 9804 9852 9900 9947 48 912 9995 *0042 ^0090 ^0138 *oi85 *0233 *028o #0328 #0376 ^423 48 913 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 48 914 0946 0994 1041 1089 1 136 1 184 1231 1279 1326 1374 48 915 961421 1469 1516 1563 1611 1658 1706 1753 1801 1848 47 916 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 47 917 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 918 2843 2890 2937 2985 3°3 2 3079 3126 3174 3221 3268 47 919 3316 3363 34io 3457 3504 3552 3599 3646 3693 374i 47 920, 963788 3835 3882 3929 3977 4024 4071 4118 4165 4212 47 921 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 47 922 4731 4778 4825 4872 4919 4966 5 OI 3 5061 5108 5155 47 923 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 47 924 5672 5719 5766 5813 5860 59°7 5954 6001 6048 6095 47 925 966142 6189 6236 6283 6329 6376 6423 6470 6517 6564 47 926 661 1 6658 6705 6752 6799 6845 6892 6939 6986 7033 47 927 7080 7127 7173 7220 7267 73H 7361 7408 7454 75oi 47 928 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 47 929 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 968483 8530 8576 8623 8670 8716 8763 8810 8856 8903 47 931 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 47 932 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 47 933 9882 9928 9975 *002I *oo68 *oii4 *oi6i *0207 *o254 *0300 47 934 970347 0393 0440 O486 0533 0579 0626 0672 0719 0765 46 935 970812 0858 0904 095I 0997 1044 1090 1137 1183 1229 46 936 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 937 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 938 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 939 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 940 973128 3174 3220 3266 3313 3359 3405 3451 3497 3543 46 941 3590 3636 3682 3728 3774 3820 3866 39*3 3959 4005 46 942 4051 4097 4H3 4189 4235 4281 4327 4374 4420 4466 46 943 4512 4558 4604 465O 4696 4742 4788 4834 4880 4926 46 944 4972 5018 5064 5IIO 5156 5202 5248 5294 534o 5386 46 945 975432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 947 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 949 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 N. Diff. I 2 3 4 5 6 7 8 9 Diff. 49 c/j H 49 5 10 15 20 25 29 34 39 44 b 48 5 10 14 19 24 29 34 38 43 48 47 5 9 14 19 24 28 33 38 42 47 46 5 9 14 18 23 28 32 37 4i 46 Diff. I 2 3 4 | 5 6 7 8 9 Diff. LOGARITHMS OF NUMBERS 537 N. 950 • O I 2 3 4 5 6 7 8 9 Diff. 977724 7769 7815 7861 7906 7952 7998 8043 8089 8135 46 951 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 953 9°93 9138 9184 9230 275 9321 9366 9412 9457 9503 46 954 9548 9594 9639 9685 973o' 9776 9821 9867 9912 9958 46 955 980003 0049 0094 0140 0185 0231 0276 0322 0367 0412 45 956 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 45 957 0912 0957 1003 1048 1093 1139 1 184 1229 1275 1320 45 958 1366 1411 1456 1 501 1547 1592 1637 1683 1728 1773 45 959 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 g6o 982271 2316 2362 2407 2452 2497 2543 2588 2633 2678 45 96 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 962 3175 3220 3265 3310 3356 3401 3446 3491 3536 358i 45 963 -3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 45 964 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 965 984527 4572 46i7 4662 4707 4752 4797 4842 4887 4932 45 966 4977 5022 5067 5112 5i57 5202 5247 5292 5337 5382 45 967 5426 5471 55i6 5561 5606 5651 5696 574i 5786 5830 45 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 969 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 986772 6817 6861 6906 6951 6996 7040 7085 7130 7i75 45 97i 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 973 8113 8i57 8202 8247 8291 8336 8381 8425 8470 8514 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 989005 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 976 945o 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 9895 9939 9983 *0028 ^0072 *oii7 *oi6i *0206 *0250 *o294 44 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0738 44 979 0783 0827 0871 0916 0960 1004 1049 1093 1137 1 182 44 980 991226 1270 1315 1359 1403 1448 1492 1536 1580 1625 44 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 9 & 993436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 9 ll 4317 436i 4405 4449 4493 4537 4581 4625 4669 4713 44 988 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 989 5196 5240 5284 5328 5372 54i6 546o 5504 5547 5591 44 990 995635 5679 5723 5767 5811 5854 5898 5942 5986 6030 44 991 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 992 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 7386 7430 7474 7517 756i 7605 7648 7692 7736 7779 44 995 997823 7867 7910 7954 7998 8041 8085 8129 8172 8216 44 996 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 44 997 8695 8739 8782 8826 8869 8913 8956 9000 9°43 9087 44 998 9*3* 9174 9218 9261 9305 9348 9392 9435 9479 9522 44 999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 N. Diff. I 2 3 4 5 ' 7 8 9 Diff. 46 1 5 9 14 18 23 28 32 37 41 46 Oh 45 I 5 j 9 i 14 18 23 27 32 36 41 45 1 PR. 44 1 4 1 9 I 13 18 22 26 3i 35 40 44 43 4 9 1 13 17 22 26 30 34 39 43 Diff. Diff. I 1 2 3 4 5 6 7 8 9 538 LOGARITHMS OF NUMBERS N. I 2 3 4 5 6 7 8 9 Diff. IOOO 000000 0043 0087 0130 0174 0217 0260 0304 0347 0391 43 IOOI 0434 0477 0521 0564 0608 0651 0694 0738 0781 0824 43 I002 0868 0911 0954 0998 1041 1084 1128 1171 1214 1258 43 1003 1301 1344 1388 1431 1474 1517 1561 1604 1647 1690 43 1004 1734. 1777 1820 1863 1907 1950 1993 2036 2080 2123 43 1005 002166 2209 2252 2296 2339 2382 2425 2468 2512 2555 43 1006 2598 2641 2684 2727 2771 2814 2857 2900 2943 2986 43 1007 3029 3073 3116 3159 3202 3245 3288 333i 3374 34i7 43 1008 3461 3504 3547 3590 3633 3676 3719 3762 3805 3848 43 ioog 3891 3934 3977 4020 4063 4106 4149 4192 4235 4278 43 IOIO 004321 4364 4407 4450 4493 4536 4579 4622 4665 4708 43 IOII 4751 4794 4837 4880 4923 4966 5009 5°52 5095 5138 43 1012 5181 5223 5266 5309 5352 5395 5438 548i 5524 5567 43 1013 5609 5652 5695 5738 578i 5824 5867 5909 5952 5995 43 1014 6038 6081 6124 6166 6209 6252 6295 6338 6380 6423 43 1015 006466 6509 6552 6594 6637 6680 6723 6765 6808 6851 43 1016 6894 6936 6979 7022 7065 7107 7150 7i93 7236 7278 43 1017 7321 7364 7406 7449 7492 7534 7577 7620 7662 7705 43 1018 7748 7790 7833 7876 7918 7961 8004 8046 8089 8132 43 1019 8174 8217 8259 8302 8345 8387 8430 8472 8515 8558 43 1020 008600 8643 8685 8728 8770 8813 8856 8898 8941 8983 43 102 1 9026 9068 9111 9153 9196 9238 9281 9323 9366 9408 42 1022 9451 9493 9536 9578 9621 9663 9706 9748 9791 9833 42 1023 9876 9918 9961 *ooo3 *oo45 *oo88 *oi30 *oi73 *02I5 *02 5 8 42 1024 010300 0342 0385 0427 0470 0512 0554 0597 0639 0681 42 1025 010724 0766 0809 0851 0893 0936 0978 1020 1063 1 105 42 1026 1 147 1190 1232 1274 1317 1359 1401 1444 i486 1528 42 1027 1570 1613 1655 1697 1740 1782 1824 1866 1909 1951 42 1028 1993 2035 2078 2120 2162 2204 2247 2289 2331 2373 42 1029 2415 2458 2500 2542 2584 2626 2669 2711 2753 2795 42 1030 012837 2879 2922 2964 3006 3048 3090 3132 3174 3217 42 103 1 3259 33oi 3343 3385 3427 3469 35ii 3553 3596 3638 42 1032 3680 3722 3764 3806 3848 3890 3932 3974 4016 4058 42 1033 4100 4142 4184 4226 4268 43io 4353 4395 4437 4479 42 1034 4521 4563 4605 4647 4689 4730 4772 4814 4856 4898 42 1035 014940 4982 5024 5066 5108 5150 5192 5234 5276 53i8 42 1036 5360 5402 5444 5485 5527 5569 561 1 5653 5695 5737 42 1037 5779 5821 5863 5904 5946 5988 6030 6072 6114 6156 42 1038 6197 6239 6281 6323 6365 6407 6448 6490 6532 6574 42 1039 6616 6657 6699 6741 6783 6824 6866 6908 6950 6992 42 1040 017033 7075 7117 7159 7200 7242 7284 7326 7367 7409 42 1041 7451 7492 7534 7576 7618 7659 7701 7743 7784 7826 42 1042 7868 7909 795i 7993 8034 8076 8118 8i59 8201 8243 42 1043 8284 8326 8368 8409 8451 8492 8534 8576 8617 8659 42 1044 8700 8742 8784 8825 8867 8908 8950 8992 9033 9075 42 1045 019116 9158 9199 9241 9282 9324 9366 9407 9449 9490 42 1046 9532 9573 9615 9656 9698 9739 9781 9822 9864 9905 4i 1047 9947 9988 *oo30 *oo7i *oii3 *oi54 *oi95 *0237 *0278 *0320 4i 1048 020361 0403 0444 0486 0527 0568 0610 0651 0693 0734 41 1049 0775 0817 0858 0900 0941 0982 1024 1065 1 107 1 148 41 1050 02 1 189 1231 1272 1313 1355 1396 1437 1479 1520 1561 4i N. Diff. I 2 9 3 4 5 6 7 8 9 Diff. 44 CO H 0. 44 4 13 18 22 26 31 35 40 43 4 9 13 17 22 26 30 34 39 43 a* 42 4 8 13 17 21 25 29 34 38 42 4i 4 8 12 16 21 25 29 33 37 4i 1 Diff. I 2 3 4 5 6 7 8 9 Diff. 0° LOGARITHMIC SIN., COS., TAN. AND COT * i^ , Table 62 79 M. o 1 2 3 4 5 6 7 8 9 10 11 12 13 M 15 Sin. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 6.463726 .764756 .940847 7.065786 7.162696 .241877 .308824 .366816 .417968 7.463726 .505118 .542906 .577668 •609853 7.639816 •667845 .694173 .718997 .742478 7764754 •785943 .806146 .825451 •843934 7.861662 .878695 .895085 .910879 .926119 7.940842 .955082 .968870 .982233 .995198 8.007787 .020021 .031919 .043501 .054781 8.065776 .076500 .086965 .097-83 .107167 8.116926 .126471 .135810 •144953 •153907 8.162681 .171280 .179713 .187985 .196102 8.204070 .211895 .219581 .227134 •234557 8.241855 D, i' Cos. 5017.17 2 934.85 2082.32 1615.17 1319-68 1115-78 966.53 852.53 762.63 689.87 629.80 579-37 536.42 499.38 467.15 438.80 413.73 39L35 371.27 353.15 336.72 321.75 308.05 295.47 283.88 273.17 263.23 254-00 245.38 237.33 229.80 222.72 216. c8 209.82 203.90 198.30 193.03 188.00 183.25 178.73 174-42 170.30 166.40 162.65 159-08 155.65 152.38 149-23 146.23 143.32 *40.55 137.87 135.28 132.80 130.42 128.10 125.88 123.72 121.63 Cos. 10.000000 .000000 .000000 .000000 .000000 10.000000 9-999999 •999999 .999999 •999999 9.999998 •999998 .999997 .999997 .999996 9.999996 •999995 •999995 •999994 •999993 9-999993 .999992 .999991 •999990 •999989 9.999989 .999988 •999987 .999986 •999985 9.999983 .999982 .999981 .999980 .999979 9.999977 •999976 •999975 •999973 •999972 9.999971 .999969 .999968 .999966 .999964 9.999963 .999961 •999959 •999958 .999956 9-999954 •999952 .999950 •999948 •999946 9.999944 .999942 •999940 . 99993 S •999936 9-999934 D. 1 Sin. D. 1' .00 .00 .00 .00 .02 .00 .00 .00 .02 .00 .02 .00 .02 .00 .02 .00 .02 .02 .00 .02 .02 .02 .02 .00 .02 .02 .02 .02 .03 .02 .02 .02 .02 •03 .02 .02 •03 .02 .02 •03 .02 •03 •03 .02 •03 •03 .02 •03 •03 •03 •03 •03 .03 •03 •03 .03 .03 • •03 .03 D. i> Tan. D. 1". 6.463726 .764756 •940847 7.065786 7.162696 .241878 .308825 .366817 •417970 7.463727 .505120 .542909 •577672 .609857 7.639820 .667849 .694179 .719003 .742484 7-764761 •785951 .806155 .825460 .843944 7.861674 .878708 •895099 .910894 •926134 7.940858 .955100 .968889 •982253 .995219 8.007809 .020044 •031945 •043527 .054809 8.065806 .076531 .086997 .097217 .107203 8. 1 16963 .126510 •135851 .144996 •153952 8.162727 .171328 .179763 .188036 .196156 8.204126 •2H953 219641 .227195 .234621 8.241921 15017.17 ! 2934.85 2082.32 1615.17 I3I9.70 III5.78 966.53 852.55 762.62 689.88 629.82 579-38 536.42 499-38 467.15 438.83 413.73 39L35 37L28 353.17 336.73 321.75 308.07 295.50 283.90 273.I8 263.25 254-00 245.40 237.37 229.82 222.73 2I6.IO 209.83 203.92 198.35 I93.03 188.03 183.28 178.75 174.43 170.33 166.43 162.67 159.12 155-68 152.42 149.27 146.25 143.35 140.58 137.88 135.33 132.83 130.45 128.13 125.90 123.77 121.67 Cot. 3.536274 •235244 •059153 2.934214 2.837304 .758122 .691175 •633183 2.536273 .494880 .457091 .422328 •390143 2.360180 •332151 .305821 .280997 •257516 2.235239 .214049 • I 93845 .174540 .156056 2.138326 .121292 .104901 .089106 .073866 2.059142 .044900 .031 11 1 .017747 .004781 I.992191 •979956 .968055 •956473 .945191 1.934194 •923469 .913003 •902783 •892797 1.883037 .873490 .864149 .855004 .846048 1.837273 .828672 .820237 .811964 .803844 I-795874 .788047 •780359 •772805 •765379 1.758079 .582030 51 Cot. D. Tan, 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 II 27 26 25 24 23 22 21 20 J! 17 16 I IS) M 13 j 12 11 10 M *From Allen's 1903, 1914. by C. "Field and Office Tables.' F. Allen. Copyright, 89 LOGARITHMIC SINES 178 M. Sin. o 1 2 3 4 5 6 I 9 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 8. 241855 • 249°33 . 256094 . 263042 .269881 8.276614 . 283243 . 289773 . 296207 . 302546 8.308794 .3H954 .321027 .327016 • 332924 8.338753 • 344504 .350181 .355783 .361315 8. 366777 .372171 • 377499 . 382762 . 387962 8.393101 .398179 .403199 . 408161 . 413068 8.417919 .422717 . 427462 .432156 . 436800 8.441394 • 445941 .450440 .454893 • 459301 8.463665 • 467985 .472263 • 47649S . 480693 8. 484848 .488963 .493040 . 497078 . 501080 8. 505045 • 508974 .512867 .516726 .520551 8. 524343 .528102 .531828 • 535523 .539186 8.542819 Cos. D. i' 119.63 117.68 115.80 113.98 112.22 110.48 108. 83 107. 23 105. 65 104. 13 102.67 101. 22 99.82 98.47 97.15 95.85 94.62 93.37 92. 20 91.03 89.90 88.80 87.72 86.67 85.65 84.63 83.67 82.70 81.78 80.85 79-97 79.o8 78.23 77.40 76.57 75.78 74.98 74.22 73-47 72.73 72. 00 71.30 70.58 69.92 69.25 68.58 67-95 67.30 66.70 66.08 65.48 64.88 64.32 63.75 63.20 62.65 62. 10 61.58 61.05 60.55 D. 1". Cos. D. 1". Tan. D. 1 999934 999932 999929 999927 999925 999922 999920 999918 999915 999913 999910 999907 999905 999902 999899 999897 999894 999891 999885 999879 999876 999873 999870 999867 999864 999861 999858 999854 999851 999848 999844 999841 999838 999834 999831 999827 999824 999820 999816 999813 999809 999805 999801 999797 999794 999790 999786 999782 999778 999774 999769 999765 99976i 999757 999753 999748 999744 999740 999735 Sin. ,„ •°3 .05 .03 • 03 .05 .03 •03 .05 .03 .05 .05 .03 .05 .05 .03 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .05 .07 .05 .05 .07 ■ .05 .05 .07 .05 .07 .05 .07 .07 .05 .07 .07 .07 .07 .05 .07 .07 .07 .07 .07 .08 •07 .07 .07 .07 .08 .07 .07 .08 D. 1'.'. ,241921 ,249102 ,256165 ,263115 . 269956 , 276691 ■ 283323 ,289856 , 296292 . 302634 , 308884 315046 321 122 ,3271 14 333025 338856 3446io ,350289 355895 361430 366895 372292 377622 . 388092 8. 393234 .398315 .403338 . 408304 .413213 8. 418068 . 422869 .427618 .432315 . 436962 8. 441560 .446110 .450613 • 455070 • 459481 8. 463849 .468172 . 472454 . 476693 . 480892 8. 485050 .489170 • 493250 • 497293 .501298 8. 505267 . 509200 .513098 .516961 . 520790 8.524586 • 528349 . 532080 •535779 • 539447 8. 543084 Cot. Cot. 119.68 117.72 115.83 114.02 112.25 110.53 108. 88 107. 27 105. 70 104. 17 102. 70 101. 27 99.87 98.52 97.18 95.90 94.65 93-43 92.25 91.08 89.95 88.83 87.78 86.72 85.70 84.68 83.72 82.77 81.82 80.92 80.02 79.15 78.28 77-45 76.63 75.83 75.05 74.28 73.52 72.80 72.05 71.37 70.65 69.98 69. 30 68.67 68.00 67.38 66.75 66.15 65-55 64.97 64.38 63.82 63.27 62.72 62.18 61.65 61.13 60.62 D. I". 1.758079 60 • 750898 59 .743835 .736885 • 730044 1.723309 .716677 .710144 . 703708 .697366 1.691116 . 684954 . 678878 . 672886 . 666975 1. 661 144 • 655390 .649711 .644105 .638570 1. 633105 . 627708 .622378 .617111 .611908 1 . 606766 .601685 . 596662 .591696 .586787 1. 581932 .577131 .572382 .567685 .563038 1.558440 . 553890 • 549387 . 54493° .540519 I.536I5I .531828 • 527546 • 523307 .519108 1. 5H950 • 510830 • 506750 . 502707 . 498702 1-494733 . 490800 . 486902 • 483039 .479210 I-4754H .471651 . 467920 .464221 .460553 1. 456916 Tan. 91 88° COSINES, TANGENTS, AND COTANGENTS 177' M. Sin. 8.542819 I . 546422 2 • 549995 3 • 553539 4 • 557054 5 8. 560540 6 • 563999 7 .567431 8 . 570836 9 .574214 10 8.577566 11 . 580892 12 .584193 13 • 587469 *4 .590721 15 8. 593948 16 .597152 17 .600332 18 . 603489 19 . 606623 20 8. 609734 21 .612823 22 .615891 23 .618937 24 .621962 25 8. 624965 26 . 627948 27 .630911 28 .633854 29 . 636776 30 8. 639680 3i . 642563 32 . 645428 33 . 648274 34 .651102 35 8. 65391 1 36 . 656702 37 .659475 38 . 662230 39 .664968 40 8. 667689 4i .670393 42 . 673080 43 .675751 44 . 678405 45 8.681043 46 .683665 47 . 686272 48 .688863 49 . 691438 50 8. 693998 5i • 696543 52 .699073 53 • 701589 54 . 704090 55 8. 706577 5b •709049 5 l .711507 5« .713952 59 .716383 60 8. 718800 Cos. D. i' 60.05 59-55 59.07 58.58 58.10 57.65 57.20 56.75 56.30 55.87 55-43 55- 02 54.60 54.20 53.78 53.40 53.oo 52.62 52.23 51-85 51.48 5i.i3 5o.77 50.42 50.05 49.72 49.38 49.05 48.70 48.40 48.05 47-75 47-43 47.13 46.82 46.52 46.22 45.92 45- 63 45-35 45.07 44.78 44.52 44.23 43-97 43.70 43-45 43.18 42.92 42.67 42.42 42.17 41.93 41.68 41.45 41.20 40.97 40.75 40. 52 40.28 D. 1". Cos. D. i' Tan. D. 999735 999731 999726 999722 999717 999713 999708 999704 999699 999694 999689 999685 999680 999675 999670 999665 999660 999655 999650 999645 999640 999635 999629 999624 999619 999614 999608 999603 999597 999592 999586 999581 999575 999570 999564 999558 999553 999547 999541 999535 999529 999524 999518 99951,2 9995o6 9995oo 999493 999487 999481 999475 999469 999463 999456 999450 999443 999437 999431 999424 999418 99941 1 999404 Sin. .07 ,08 07 08 07 ,08 07 08 08 08 07 08 08 08 08 08 08 08 08 08 08 10 08 08 08 10 08 10 08 10 08 10 08 10 10 08 10 10 10 10 08 10 10 10 10 12 10 10 10 10 10 12 10 12 10 10 12 10 12 12 D. 8. 543084 . 546691 . 550268 • 553817 • 557336 8. 560828 .564291 .567727 .57H37 .574520 8.577877 .581208 .584514 .587795 .591051 8. 594283 • 597492 .600677 • 603839 . 606978 8.610094 .613189 .616262 .619313 • 622343 8.625352 . 628340 .631308 . 634256 .637184 8. 640093 . 642982 .645853 . 648704 •651537 8. 654352 .657149 . 659928 . 662689 • 665433 8.668160 . 670870 • 673563 . 676239 . 678900 8.681544 . 684172 . 686784 . 689381 • 691963 8. 694529 .697081 . 699617 . 702139 . 704646 8. 707140 .709618 .712083 • 7H534 .716972 8. 719396 Cot. 60. 12 59.62 59.15 58.65 58.20 57.72 57-27 56.83 56.38 55-95 55.52 55.io 54.68 54.27 53.87 53.48 53.o8 52.70 52.32 51-93 51.58 51.22 50.85 50.50 50.15 49.80 49-47 49.13 48.80 48.48 48.15 47.85 47.52 47.22 46.92 46.62 46.32 46.02 45-73 45.45 45.17 44.88 44.60 44-35 44.07 43.8o 43-53 43.28 43.03 42.77 42.53 42.27 42.03 41.78 41.57 41.30 41.08 40.85 40.63 40.40 D. i". Cot. 1. 456916 • 453309 • 449732 .446183 . 442664 1. 439172 • 435709 .432273 . 428863 .425480 1. 422123 .418792 . 415486 .412205 • 408949 1. 405717 . 402508 • 399323 .396161 . 393022 1.389906 .386811 .383738 .380687 • 377657 1.374648 .371660 . 368692 . 365744 .362816 I.359907 .357oi8 . 354H7 .351296 • 348463 1.345648 . 342851 . 340072 -33731 1 .334567 1. 331840 • 329130 .326437 . 323761 .321100 1. 318456 .315828 .313216 .310619 . 308037 I-30547I • 302919 . 300383 .297861 • 295354 1.292860 . 290382 .287917 . 285466 . 283028 1 . 280604 Tan. 92 87 LOGARITHMIC SINES 176* M. o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 2 Z 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 93' Sin. 8.718800 . 721204 . 723595 • 725972 • 728337 8. 730688 . 733027 • 735354 • 737667 . 739969 8. 742259 .744536 . 746802 • 749055 .751297 8.753528 • 755747 • 757955 .760151 • 762337 8. 76451 1 . 766675 .768828 . 770970 .773101 8. 775223 • 777333 • 779434 .781524 .783605 8.785675 • 787736 .789787 . 791828 .793859 8.795881 .797894 • 799897 . 801892 . 803876 8.805852 .807819 .809777 .811726 .813667 8.815599 .817522 .819436 .821343 . 823240 8.825130 .827011 .828884 . 830749 .832607 8.834456 . 836297 .838130 .839956 .841774 8.8 43585 Cos. D. 1' 40.07 39.85 39.62 39.42 39.18 38.98 38.78 38.55 38.37 38.17 37-95 37.77 37-55 37-37 37.18 36.98 36.80 36.60 36.43 36.23 36.07 35-88 35.70 35.52 35-37 35.17 35-02 34.83 34.68 34.50 34-35 34.18 34-02 33.85 33-70 33-55 33.38 33-25 33.07 32.93 32.78 32.63 32.48 32.35 32. 20 32.05 31.90 31.78 31.62 31.50 31.35 31.22 31.08 30.97 30.82 30.68 30.55 30.43 30.30 30.18 D. 1". Cos. 999404 999398 999391 999384 999378 999371 999364 999357 999350 999343 999336 999329 999322 999315 999308 999301 999294 999287 999279 999272 999265 999257 999250 999242 999235 999227 999220 999212 999205 999197 999189 999181 999174 999166 999158 999150 999142 999134 999126 9991 18 9991 10 999102 999094 999086 999077 999o69 999061 999053 999044 999036 999027 999019 999010 999002 998993 998984 998976 998967 998958 998950 998941 Sin. D. 1". Tan. D. 1". 8. 719396 .721806 . 724204 . 726588 • 728959 8.73I3I7 • 733663 • 735996 .738317 . 740626 8. 742922 • 745207 • 747479 • 749740 ■ 751989 8. 754227 . 756453 . 758668 . 760872 . 763065 8. 765246 • 767417 .769578 .771727 . 773866 8. 775995 .778114 . 780222 . 782320 . 784408 8. 786486 .788554 .790613 . 792662 • 794701 8. 796731 • 798752 . 800763 . 802765 .804758 8. 806742 .808717 .810683 .812641 .814589 8.816529 .818461 . 820384 . 822298 . 824205 8. 826103 . 827992 . 829874 .831748 .833613 8.835471 .837321 .839163 . 840998 . 842825 8. 844644 Cot. D. i' 40.17 39-97 39-73 39.52 39.30 39.10 38.88 38.68 38.48 38.27 38.08 37.87 37-68 37.48 37.30 37- 10 36.92 36.73 36.55 36.35 36.18 36.02 35.82 35.65 35.48 35.32 35.13 34-97 34.80 34.63 34-47 34.32 34.15 33.98 33-83 33-68 33o2 33-37 33-22 33-07 32.92 32.77 32.63 32.47 32.33 32. 20 32.05 31.90 31.78 31.63 31.48 31-37 31.23 31.08 30.97 30.83 30.70 30.58 30.45 30.32 D. 1". Cot. 1.280604 60 .278194 59 . 275796 58 .273412 57 .271041 56 1.268683 55 . 266337 54 . 264004 53 .261683 52 . 259374 5i 1.257078 50 . 254793 49 .252521 48 . 250260 47 .248011 46 1.245773 45 . 243547 44 .241332 43 .239128 42 . 236935 4i 1.234754 40 . 232583 39 . 230422 38 . 228273 37 .226134 36 1.224005 35 .221886 34 .219778 33 .217680 32 •215592 3i 1.213514 30 .211446 29 . 209387 28 . 207338 27 . 205299 26 I . 203269 25 . 201248 24 . 199237 23 . 197235 22 . 195242 21 I. 193258 20 . i9 I2 83 19 . 189317 18 • 187359 17 .185411 16 1.183471 15 • 181539 14 . 1 79616 13 . 177702 12 • 175795 11 1. 173897 10 . 172008 % . 1 70126 . 168252 7 . 166387 6 1. 164529 5 . 162679 4 . 160837 3 . 159002 2 .157175 1 1. 155356 Tan. 86' COSINES, TANGENTS, AND COTANGENTS I75 c M. Sin. D. i". Cos. D. i". Tan. D. i". Cot. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 8. 843585 .8453S7 .847183 .848971 .850751 .854291 . 856049 .857S01 • 859546 8.S61283 . 863014 .864738 .866455 .868165 8. 869868 .871565 • 873255 .874938 .876615 8. 878285 . 879949 .881607 • 883258 .884903 8. 886542 .888174 S. 89142 1 893035 894643 896246 897842 899432 901017 902596 904169 9 5736 907297 90S853 910404 .911949 .913488 .915022 •916550 . 918073 ■9I959I .921103 .922610 ,924112 . 925609 .927100 .928587 . 930068 .931544 .933015 .934481 .935942 • 937398 ■ 938850 . 940296 Cos. 30-03 29. 93 29. So 29.67 29-57 29.43 29.30 29. 20 29.08 28.95 28.85 28.73 28.62 28.50 28.38 28.28 28.17 28.05 1 27.95 27.83 27-73 27.63 27.52 27.42 27.32 27.20 27. 12 27.00 26. 90 26.80 26.72 26. 60 26.50 26.42 26.32 26.22 26. 12 26. 02 25.93 25.85 25.75 25.65 25-57 25.47 I 25.38 25.30 25.20 25.12 25.03 24.95 24.85 24.78 24.68 24. 60 24.52 24.43 24.35 24.27 24. 20 24. 10 D. 1 996941 998932 998923 998914 998905 998896 998887 998869 99S860 998851 99S841 998832 998823 998813 998804 998795 998785 998776 99S766 998757 998747 99S738 998728 998718 998708 998699 9986S9 998679 998669 998659 998649 998639 998629 998619 998609 998599 998589 998578 998568 998558 998548 998537 998527 998516 99S506 998495 998485 998474 998464 998453 998442 998431 998421 998410 998399 998388 998377 998366 998355 99S344 Sin. 15 15 15 15 15 15 15 15 15 15 17 15 15 17 15 15 17 15 17 *5 17 15 17 17 17 15 17 17 17 17 17 17 17 17 17 17 17 iS ^7 17 17 iS 17 iS 17 iS 17 iS 17 iS iS 18 17 iS 18 18 ^8 18 18 18 D. 1' 8. 844644 • 846455 . 848260 . 850057 .851846 8. 853628 .855403 .857171 .858932 . 860686 8.862433 .864173 .865906 .867632 • 869351 8.871064 .872770 . 874469 . 876162 .877849 8. 879529 .881202 .882S69 .884530 . 886185 8.887833 . 889476 .891112 . 892742 • 894366 8.895984 .897596 . 899203 .900803 . 902398 8. 903987 • 905570 .907147 .908719 . 910285 8. 91 1846 .913401 .9H95i • 916495 .918034 8.919568 .921096 .922619 .924136 • 925649 8.927156 . 928658 .930155 .931647 .933134 8. 934616 .936093 .937565 • 939032 . 940494 8.941952 Cot. '30. iS 30.08 29.95 29.82 29.70 29.58 29.47 29.35 29.23 29. 12 29.00 28.88 28.77 28.65 28.55 28.43 28.32 28.22 28.12 28.00 27.88 27.78 27.68 27.58 27.47 27.38 27.27 27.17 27.07 26.97 26.87 26.78 26.67 26.58 26.48 26.38 26.28 26. 20 26. 10 26.02 25.92 25.83 25.73 25-65 25.57 25.47 25. 38 25.28 25.22 25.12 25.03 24.95 24.87 24.78 24.70 24.62 24.53 24.45 24.37 24.30 D. 1". i. 155356 • x 53545 . 151740 • 149943 . H8154 1. 146372 • 144597 . 142829 . 141068 • I393H 1. 137567 • 135827 . 134094 . 132368 • 130649 1. 128936 . 127230 .125531 . 123838 . 122151 1. 120471 .118798 .117131 .115470 .113815 1. 112167 .110524 . 10S888 . 107258 . 105634 1. 104016 . 102404 . 100797 .099197 .097602 1. 096013 . 094430 .092853 .091281 .089715 1. 088154 . 086599 .085049 .083505 .081966 1 . 080432 . 078904 .077381 • 075864 .074351 1 . 072844 .071342 .069845 .068353 .066866 1.065384 . 063907 .062435 .060968 • 059506 1.058048 Tan. 94' 85 5° LOGARITHMIC SINES 174^ M. Sin. D. r 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 8. 940296 .941738 •943174 .944606 . 946034 8. 947456 • 948874 .950287 .951696 .953100 8. 954499 • 955894 . 957284 . 958670 .960052 8. 961429 .962801 .964170 • 965534 .966893 8.968249 . 969600 . 970947 . 972289 . 973628 8.974962 . 976293 .977619 .978941 . 980259 8.981573 . 982883 .984189 .985491 . 986789 8. 988083 .989374 . 990660 .991943 • 993 222 8. 994497 .995768 .997036 .998299 .999560 9.000816 . 002069 .003318 .004563 . 005805 9. 007044 .008278 .009510 .010737 .011962 9.013182 .014400 .015613 .016824 .018031 9.019235 Cos. 24.03 23.93 23.87 23.80 23.70 23.63 23.55 23.48 23.40 23.32 23-25 23.17 23.10 23.03 22.95 22.87 22.82 22.73 22.65 22.60 22.52 22.45 22.37 22.32 22.23 22.18 22. 10 22.03 21.97 21.90 21.83 21.77 21.70 21.63 21-57 21.52 21.43 21.38 21.32 21.25 21.18 21.13 21.05 21.02 20.93 20.88 20.82 20.75 20. 70 20.65 20.57 20.53 20.45 20.42 20.33 20.30 20. 22 20.18 20. 12 20.07 D. 1". Cos. 998344 998333 998322 9983 1 1 998300 998289 998277 998266 998255 998243 998232 998220 998209 998i97 998186 998174 998163 998151 998139 998128 9981 16 998104 998092 998080 998068 998056 998044 998032 998020 998008 997996 997984 997972 997959 997947 997935 997922 997910 997897 997885 997872 997860 997847 997835 997822 997809 997797 997784 997771 997758 997745 997732 997719 9977o6 997693 997680 997667 997654 997641 997628 997614 Sin. D. 1". .18 .18 .18 .18 .18 . 20 .18 .18 . 20 .18 .20 .18 .20 .18 .20 .18 .20 . 20 .18 .20 . 20 .20 . 20 . 20 .20 .20 .20 .20 .20 .20 .20 .20 .22 .20 .20 .22 .20 .22 .20 .22 .20 .22 .20 •22 .22 .20 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .22 .23 Tan, D. i' 8.941952 . 943404 • 944852 • 946295 • 947734 8. 949168 • 950597 .952021 .953441 • 954S56 8. 956267 • 957674 . 959075 • 960473 .961866 8.963255 • 964639 .966019 • 967394 . 968766 8. 970133 .971496 .972855 • 974209 • 97556o 8. 976906 .978248 • 979586 . 98092 1 .982251 8.983577 • 984899 .986217 • 987532 . 988842 8.990149 .991451 .992750 • 994045 • 995337 8. 996624 .997908 . 999188 9.000465 .001738 9. 003007 . 004272 .005534 . 006792 . 008047 9. 009298 .010546 .011790 .013031 .014268 9.015502 .016732 .017959 .019183 . 020403 9.021620 Cot. D. 1". 24.20 24.13 24.05 23.98 23.90 23.82 23.73 23.67 23.58 23.52 23- 45 23.35 23.30 23.22 23.15 23.07 23.00 22.92 22.87 22.78 22.72 22.65 22. 57 22.52 22.43 22.37 22.30 22.25 22. 17 22. 10 22.03 21.97 21.92 21.83 21.78 21.70 21.65 21.58 21.53 21.45 21.40 21.33 21.28 21.22 21.15 21.08 21.03 20.97 20.92 20.85 20.80 20.73 20.68 20.62 20.57 20.50 20.45 20.40 20.33 20.28 D. 1". Cot. 1.058048 .056596 .055148 • 053705 . 052266 1.050832 .049403 .047979 • 046559 .045144 I.043733 .042326 . 040925 • 039527 .038134 1.036745 . 035361 .033981 . 032606 .031234 1.029867 . 028504 .027145 .025791 . 024440 1.023094 .021752 .020414 .019079 .017749 1. 016423 .015101 .013783 .012468 .011158 1. 009851 . 008549 . 007250 . 005955 .004663 1.003376 .002092 .000812 o. 999535 . 998262 o. 996993 . 995728 • 994466 . 993 2 o8 .991953 o. 990702 . 989454 .988210 .986969 .985732 o. 984498 . 983268 .982041 .980817 .979597 o. 978380 Tan. 95' 84' COSINES, TANGENTS, AND COTANGENTS jjj M. Sin. D. i". Cos. D. i". Tan. D. i". Cot o i 2 3 4 5 6 I 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 9.019235 . 020435 .021632 .022825 .024016 9. 025203 .026386 .027567 . 028744 .029918 9.031089 .032257 .033421 .034582 ■035741 9. 036896 . 03S04S .039197 .040342 .041485 9. 042625 .043762 .044895 . 046026 .047154 9. 048279 . 049400 .050519 .051635 .052749 9- 053859 . 054966 .056071 .057172 .058271 9.059367 .060460 .061551 . 062639 .063724 9.064806 .065885 . 066962 .068036 .069107 9.070176 .071242 . 072306 .073366 .074424 9. 075480 .076533 .077583 .078631 • 079676 9.080719 .081759 .082797 .083832 . 084864 9. 085894 Cos. D. i' 9.997614 997601 997588 997574 99756i 997547 997534 997520 997507 997493 99748o 997466 997452 997439 997425 9974 1 1 997397 997383 997369 997355 997341 997327 997313 997299 997285 997271 997257 997242 997228 997214 997199 997i85 997170 997156 997141 997127 9971 12 997098 997083 997068 997053 997039 997024 997009 996994 996979 996964 996949 996934 996919 996904 996874 996843 996828 996812 996797 996782 996766 996751 Sin. .22 .22 .23 .22 .23 .22 .23 .22 .23 .22 .23 •23 .22 .23 .23 .23 .23 .23 .23 .23 .23 .23 •23 •23 .23 .23 .25 .23 •23 .25 •23 .25 .23 .25 •23 .25 .23 .25 .25 .25 .23 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 .27 .25 .25 .27 .25 .25 .27 •25 D. 1". 9.021620 . 022834 . 024044 .025251 .026455 9- 027655 . 028852 . 030046 .031237 .032425 9. 033609 .034791 .035969 .037144 .038316 9- 039485 .040651 .041813 .042973 .044130 9. 045284 .046434 .047582 .048727 . 049869 9.051008 .052144 .053277 • 054407 • 055535 9. 056659 . 057781 .058900 .060016 .061130 9. 062240 . 063348 • 064453 .065556 .066655 9. 067752 .068846 . 069938 .071027 .072113 9.073197 .074278 .075356 .076432 .077505 9. 078576 .079644 .080710 .081773 .082833 9. 083891 .084947 .086000 . 087050 . 088098 9.089144 Cot. 20.23 20. 17 20. 12 20.07 20.00 9-95 9.90 9.85 9.80 9-73 9.70 9.63 9.58 9-53 9.48 9-43 9-37 9-33 9.28 9-23 9.17 9.13 9.08 9-03 8.98 8-93 8.88 8.83 8.80 8.73 8.70 8.65 8.60 8.57 8.50 8.47 8.42 8.38 8.32 8.28 8.23 8.20 8.15 8.10 8.07 8.02 7-97 7-93 7.88 7.85 7.80 7-77 7.72 7.67 7.63 7.60 7-55 7-50 7-47 7-43 D. i". o. 978380 .977166 .975956 •974749 • 973545 o. 972345 .971148 .969954 .968763 .967575 o. 966391 . 965209 .964031 . 962856 . 9616S4 0.960515 •959349 .958187 .957027 .955870 0.954716 .953566 .952418 .951273 .950131 o. 948992 .947856 .946723 • 945593 .944465 o. 943341 .942219 .941100 •939984 .938870 0.937760 . 936652 •935547 •934444 .933345 o. 932248 .931154 .930062 .928973 . 927887 o. 926803 .925722 .924644 . 923568 .922495 0.921424 . 920356 .919290 .918227 .917167 o. 916109 .9!5053 .914000 .912950 .911902 0.910856 Tan. 60 59 58 57 56 55 54 53 ^52 5i 96' 83' LOGARITHMIC SINES I72< M. 10 ii 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 Sin, 9. 085894 . 086922 .087947 . 088970 . 089990 9.091008 . 092024 .093037 . 094047 . 095056 9. 096062 . 097065 . 098066 .099065 . 100062 9. 101056 . 102048 • 103037 . 104025 . 105010 9. 105992 . 106973 . 107951 . 108927 . 109901 9. 110873 . 1 1 1842 .112809 •1 13774 •"4737 h 1 15698 .116656 .117613 .118567 .119519 ). 120469 .121417 . 122362 . 123306 . 124248 9. 125187 .126125 . 127060 . 127993 . 128925 9- 129854 . 130781 . 131706 . 132630 • I3355I 9- I34470 • 135387 • 136303 .137216 .138128 9- 139037 • 139944 . 140850 • I4I754 • 142655 9- 143555 D. 1". Cos. 17-13 17.08 17.05 17. 00 16.97 16.93 16.88 16.83 16.82 16.77 16.72 16.68 16.65 16.62 16.57 16.53 16.48 16.47 16.42 16.37 16.35 16.30 16.27 16.23 16. 20 16.15 16. 12 16.08 16.05 16.02 15-97 15-95 15-9° 15.87 15-83 15.80 15-75 15.73 15-70 15.65 15-63 15.58 15.55 15-53 15.48 15-45 15.42 15.40 15.35 15.32 15.28 15.27 15.22 15.20 15.15 15.12 15.10 15-07 15.02 15.00 Cos. 9.996751 • 996735 . 996720 . 996704 . 996688 9. 996673 • 996657 . 996641 • 996625 .996610 9.996594 . 996578 . 996562 . 996546 • 996530 9.996514 •996498 . 996482 . 996465 .996449 9- 996433 .996417 . 996400 .996384 .996368 9-99635I • 996335 .996318 . 996302 .996285 9. 996269 .996252 • 996235 .996219 . 996202 9.996185 .996168 .996151 .996134 .996117 9.996100 .996083 . 996066 . 996049 • 996032 9. 996015 .995998 . 995980 • 995963 . 995946 9.995928 •9959i 1 .995894 .995876 .995859 9. 995841 • 995823 . 995806 .995788 •995771 9- 995753 D. 1" Sin. D. 1' .27 .25 .27 .27 •25 .27 .27 .27 .25 .27 .27 .27 .27 .27 .27 .27 .27 .28 .27 .27 .27 .28 .27 .27 .28 .27 .28 .27 .28 .27 .28 .28 .27 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 ,28 ,28 30 28 28 30 28 .30 .28 .30 .30 ,28 30 28 30 Tan. 9.089144 . 090187 .091228 . 092266 . 093302 9. 094336 • 095367 .096395 . 097422 .098446 9. 099468 . 100487 . 10 1 504 . 1025 19 . 103532 9. 104542 • 105550 . 106556 . io7559 . 108560 9- 10 9559 .110556 .111551 .112543 •II3533 9. 114521 .115507 . 1 1649 1 .117472 .118452 9. 1 19429 . 120404 .121377 . 122348 .123317 9. 124284 • 125249 . 1262 1 1 . 127172 . 128130 9. 129087 . 130041 . 130994 • I3I944 • 132893 9- 133839 • 134784 • 135726 . 136667 . 137605 9- 138542 • 139476 . 140409 . 141340 . 142269 9- 143196 . 144121 • 145044 . 145966 . 146885 9. 147803 D. 1' 17.38 17.35 17.30 17.27 17.23 17.18 17.13 17. 12 17.07 17.03 16.98 16.95 16.92 16.88 16.83 16.80 16.77 16.72 16.68 16.65 16.62 16.58 16.53 16.50 16.47 16.43 16.40 16.35 16.33 16.28 16.25 16. 22 16.18 16.15 16.12 16.08 16.03 16.02 15.97 15.95 15.90 15.88 15.83 15.82 15.77 15.75 15.70 15.68 15.63 15.62 15.57 15.55 15.52 15.48 15.45 15.42 15.38 15.37 15.32 15.30 D. 1". Cot. o. 910856 60 .909813 .908772 • 907734 .906698 o. 905664 . 904633 • 903605 . 902578 .901554 o. 900532 50 .899513 . 898496 .897481 . 896468 0.895458 • 894450 . 893444 .892441 . 891440 o. 890441 . 889444 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 .887457 . 886467 0.885479 • 884493 .883509 . 882528 .881548 0.880571 30 .879596 i 29 .878623 28 .877652 . 876683 0.875716 .874751 .873789 . 872828 .871870 o. 870913 • 869959 . 869006 . 868056 . 867107 0.866161 .865216 .864274 .863333 . 862395 0.861458 . 860524 .859591 . 858660 .857731 o. 856804 .855879 . 854956 . 854034 .853115 0.852197 Cot. D. 1". Tan. 27 26 25 24 23 22 21 20 19 18 17 16 15 M 13 12 11 M. 97 82 COSINES, TANGENTS, AND COTANGENTS 171' M. Sin. o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 98° 9. 143555 144453 H5349 146243 147 1 36 14S026 148915 149802 150686 151569 152451 153330 154208 155083 155957 156830 157700 158569 159435 1 6030 1 161164 162025 162885 163743 164600 165454 166307 167159 168008 168856 169702 170547 171389 172230 173070 173908 174744 175573 176411 177242 178072 178900 179726 1 8055 1 181374 182196 183016 183834 18465 1 185466 186280 187092 187903 188712 189519 190325 191130 191933 192734 193534 194332 Cos. D. 4-97 4-93 4.90 4.88 4.83 4.82 4.78 4-73 4.72 4.70 4.65 63 58 57 55 50 45 45 38 4-35 4-33 4.30 4.28 4.23 4.22 4.20 4.15 4.13 4. 10 4.08 4.03 4.02 4.00 3-97 3-93 3-90 3-88 3-85 3-83 3.80 3-77 3-75 3.72 3.70 3.67 3.63 3.62 3.58 3-57 3-53 3.52 3.48 3-45 3-43 3-42 3.38 3-35 3-33 3.30 D. i". Cos. 9- 995753 •995735 .995717 .995699 . 995681 9. 995664 . 995646 . 995628 .995610 .995591 9- 995573 • 995555 • 995537 •995519 • 995501 9. 995482 • 995464 • 995446 • 995427 • 995409 9- 995390 • 995372 • 995353 . 995334 . 9953i6 9- 995297 . 995278 . 995260 • 995241 . 995222 9- 995203 .995184 .995165 .995146 .995127 9.995108 . 995089 • 995070 .995051 • 995032 9-995013 • 994993 • 994974 • 994955 • 994935 9. 9949i6 . 994896 • 994877 .994857 • 994838 9.994818 . 994798 . 994779 • 994759 • 994739 9. 994720 . 994700 . 994680 . 994660 • 994640 9.994620 Sin. D. 1" -30 .30 .30 .30 .28 .30 .30 .30 .32 .30 .30 .30 .30 .30 .32 .30 .30 .32 .30 .32 •30 .32 • 32 .30 .32 • 32 •30 .32 • 32 .32 .32 • 32 .32 .32 .32 • 32 .32 .32 .32 • 32 .33 .32 .32 • 33 .32 • 33 • 32 •33 .32 •33 •33 .32 •33 •33 •32 •33 •33 •33 •33 •33 D. 1" Tan. 147803 148718 149632 150544 I5H54 152363 153269 I54I74 155077 155978 156877 157775 15867 1 159565 160457 161347 162236 163123 164008 164S92 165774 166654 167532 168409 169284 170157 171029 171899 172767 173634 174499 175362 176224 177084 177942 178799 179655 180508 181360 182211 183059 183907 184752 185597 186439 187280 188120 188958 189794 190629 191462 192294 193124 193953 1947S0 195606 196430 197253 198074 198894 199713 Cot. D. 1 Cot. 5.25 5-23 5.20 5.17 5.15 5-io 5.08 5.05 5.02 4.98 4-97 4-93 4.90 4.87 4.83 4.82 4.78 4-75 4-73 4.70 4.67 4-63 4.62 4.58 4-55 4-53 4-50 4-47 4-45 4.42 4.38 4-37 4-33 4.30 4.28 4.27 4.22 4.20 4.18 4.13 4.13 4.08 4.08 4.03 02 3.87 3.83 3.82 3.78 3-77 3- 73 3-72 3-68 3.67 3.65 D. 1". 0.852197 . 851282 .850368 • 849456 .848546 o. 847637 .846731 . 845826 .844923 . 844022 0.843123 . 842225 .841329 . 840435 • 839543 0.838653 .837764 c 836877 .835992 .835108 o. 834226 . 833346 . 832468 .831591 .830716 o. 829843 .828971 .828101 .827233 . 826366 o. 825501 . 824638 .823776 .822916 . 822058 0.821201 . 820345 .819492 . 818640 .817789 0.8 1694 1 .816093 . 815248 . 814403 .813561 0.812720 .811880 .811042 .810206 .809371 0.808538 . 807706 . 806S76 .806047 . 805220 o. 804394 . 803570 . 802747 .801926 .801106 o. S00287 Tan. 8i° LOGARITHMIC SINES i7o ( M. 10 ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 Sin. 9- 194332 . 195129 • 195925 . 196719 .197511 9. 198302 . 199091 • 199879 . 200666 . 201451 9. 202234 . 203017 . 203797 • 204577 • 205354 9.206131 . 206906 . 207679 . 208452 . 209222 9. 209992 .210760 .211526 .212291 .213055 9.213818 .214579 .215338 .216097 .216854 9.217609 .218363 .219116 .219868 .220618 9.221367 .222115 .222861 . 223606 .224349 9. 225092 .225833 .226573 .227311 . 228048 9. 228784 .229518 . 230252 . 230984 .231715 9.232444 .233172 .233899 . 234625 • 235349 9. 236073 .236795 .237515 . 238235 . 238953 9. 239670 Cos. D. 1". 13.28 13.27 13.23 13.20 13.18 13.15 13.13 13.12 13.08 13-05 13.05 13.00 13.00 12.95 12.95 12.92 12.88 12.88 12.83 12.83 12.80 12.77 12,75 12.73 12.72 12.68 12.65 12.65 12.62 12.58 12.57 12.55 12.53 12.50 12.48 12.47 12.43 12.42 12.38 12.38 12.35 12.33 12.30 12.28 12.27 12.23 12.23 12.20 12.18 12.15 12.13 12. 12 12. 10 12.07 12.07 12.03 12.00 12.00 11.97 11-95 D. 1". Cos. 9. 994620 . 994600 . 994580 . 994560 • 994540 9.994519 •994499 •994479 • 994459 .994438 9.994418 • 994398 . 994377 . 994357 . 994336 9.994316 . 994295 • 994274 . 994254 . 994233 9.994212 .994191 .994171 .994150 .994129 9. 994108 • 994087 .994066 . 994045 . 994024 9. 994003 .993982 • 993960 •993939 • 9939i8 9.993897 .993875 • 993854 .993832 .993811 9- 993789 .993768 .993746 . 993725 • 993703 9.993681 . 993660 •993638 . 993616 •993594 9- 993572 .993550 .993528 • 993506 .993484 9.993462 •993440 .993418 • 993396 • 993374 9-993351 Sin. D. 1' • 33 • 33 • 33 •33 • 35 • 33 • 33 •33 • 35 • 33 •33 • 35 •33 • 35 •33 •35 • 35 • 33 • 35 • 35 • 35 • 33 ■ 35 • 35 • 35 • 35 • 35 • 35 • 35 • 35 • 35 • 37 • 35 • 35 •35 •37 •35 •37 >35 •37 -35 • 37 ■ 35 37 ■37 ■35 ■37 ■37 37 37 37 37 37 37 37 37 37 37 37 38 D. r Tan. 9. 199713 . 200529 .201345 .202159 .202971 9. 203782 . 204592 . 205400 . 206207 . 207013 9. 207817 .208619 . 209420 .210220 .211018 9.211815 .212611 .213405 .214198 .214989 9.215780 .216568 .217356 .218142 .218926 9. 219710 . 220492 .221272 . 222052 . 222830 9. 223607 . 224382 .225156 . 225929 . 226700 9.227471 . 228239 .229007 . 229773 • 230539 9. 231302 . 232065 . 232826 . 233586 .234345 9- 235103 • 235859 .236614 .237368 .238120 9. 238872 . 239622 .240371 .241118 . 241865 9. 242610 • 243354 . 244097 . 244839 .245579 9.246319 Cot. D. 1". 13.60 13.60 13.57 13.53 13.52 13.50 13.47 13.45 13.43 13.40 13.37 13.35 13.33 13.30 13.28 13.27 13.23 13.22 13.18 13.18 13.13 13.13 13.10 13.07 13.07 13-03 13.00 13.00 12.97 12.95 12.92 12.90 12.88 12.85 12.85 12.80 12.80 12.77 12.77 12.72 12.72 12.68 12.67 12.65 12.63 12.60 12.58 12.57 12.53 12.53 12.50 12.48 12.45 12.45 12.42 12.40 12.38 12.37 12.33 12.33 D. 1". Cot. o. 800287 .799471 .798655 .797841 • 797029 0.796218 • 795408 .794600 •793793 .792987 0.792183 .791381 . 790580 . 789780 . 788982 o. 788185 ,787389 .786595 .785802 .785011 o. 784220 . 783432 . 782644 .781858 . 781074 o. 780290 .779508 . 778728 .777948 .777170 o. 776393 .775618 . 774844 .774071 .773300 o. 772529 .771761 . 770993 .770227 . 769461 o. 768698 .767935 .767174 .766414 . 765655 o. 764897 .764141 . 763386 . 762632 . 761880 0.761 128 . 760378 .759629 . 758882 . 758135 o. 757390 . 756646 . 755903 . 755161 .754421 o. 753681 Tan. 99 l 80' io v COSINES, TANGENTS, AND COTANGENTS 169' M. Sin. 10 11 12 13 14 15 16 i7 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 9. 239670 . 240386 .241101 .241814 . 242526 9- 2 43237 • 243947 . 244656 • 245363 . 246069 9. 246775 . 247478 .2481S1 . 248883 . 249583 9. 250282 . 250980 .251677 •252373 . 253067 9. 253761 . 254453 .255144 • 255834 • 256523 9.257211 .257898 .258583 . 259268 .259951 9. 260633 .261314 . 261994 . 262673 .263351 9.264027 . 264703 .265377 .266051 . 266723 9- 267395 . 268065 . 268734 . 269402 .270069 9- 270735 .271400 . 272064 . 272726 .273388 9- 274049 . 274708 • 275367 . 276025 . 276681 9- 277337 . 277991 .278645 . 279297 . 279948 9. 280599 Cos. D. i' •93 92 ,88 ,87 85 83 82 78 77 77 72 72 70 67 65 63 62 60 57 57 53 52 50 48 47 45 42 42 38 37 35 33 32 30 27 27 23 23 ,20 ,20 17 15 13 12 , 10 ,08 07 03 03 .02 0.98 0.98 o.97 o.93 o.93 0.90 o. 90 0.87 0.85 0.85 D. 1". Cos. 9-993351 • 993329 • 993307 • 993284 . 993262 9. 993240 .993217 .993195 .993172 .993149 9.993127 .993104 .993081 • 993059 . 993036 9- 993oi3 .992990 • 992967 • 99 2 944 .992921 9.992898 .992875 . 992852 . 992829 . 992806 9- 992783 .992759 . 992736 .992713 . 992690 9. 992666 . 992643 .992619 . 992596 . 992572 9- 992549 . 992525 .992501 . 992478 . 992454 9-992430 . 992406 . 992382 . 992359 . 992335 9. 99231 1 . 992287 . 992263 .992239 .992214 9.992190 .992166 .992142 .992118 • 992093 9. 992069 • 992044 . 992020 .991996 .991971 9.991947 Sin. D. 1". • 37 • 37 .38 • 37 • 37 .38 • 37 .38 .38 •37 .38 .38 • 37 .38 .38 .38 .38 .38 .38 .38 .38 .38 .38 .38 .38 .40 .38 .38 .38 .40 .38 .40 .38 .40 .38 .40 .40 .38 .40 .40 .40 .40 .38 .40 .40 .40 .40 .40 .42 .40 .40 .40 .40 .42 .40 .42 .40 .40 .42 .40 D. 1" Tan. 9.246319 • 247057 • 247794 • 248530 . 249264 9- 249998 . 250730 .251461 .252191 . 252920 9- 253648 • 254374 .255100 . 255824 . 256547 9. 257269 • 257990 .258710 . 259429 . 260146 9. 260863 .261578 . 262292 . 263005 .263717 9. 264428 .265138 . 265847 . 266555 .267261 9. 267967 .268671 . 269375 . 270077 . 270779 9.271479 .272178 . 272876 • 273573 . 274269 9- 274964 .275658 .276351 • 277043 . 277734 9. 278424 .279113 . 279801 . 280488 .281174 9.281858 . 282542 . 283225 . 283907 . 284588 9. 285268 . 285947 . 286624 . 287301 - 287977 9. 288652 Cot. D. 1". 2.30 2.28 2.27 2.23 2.23 2. 20 2.18 2.17 2.15 2.13 2. 10 2. 10 2.07 2.05 2.03 2.02 2.00 .98 -95 •95 .92 .90 .88 .87 .85 .83 .82 .80 • 77 •11 • 73 • 73 •70 .70 .67 .65 .63 .62 .60 .58 • 57 • 55 • 53 .52 .50 .48 • 47 .45 -43 .40 .40 .38 • 37 •35 •33 .32 .28 .28 •27 •25 D. 1". Cot. o. 753681 752943 752206 75M7o 750736 750002 749270 748539 747809 747080 746352 745626 744900 744176 743453 742731 742010 741290 740571 739854 739137 738422 737708 736995 736283 735572 734862 734153 733445 732739 732033 731329 730625 729923 729221 728521 727822 727124 726427 725731 725036 724342 723649 722957 722266 721576 720887 720199 719512 718826 718142 717458 716775 716093 7I54I2 714732 7M053 713376 712699 712023 711348 Tan. M. IOO .79' ir LOGARITHMIC SINES 168 M Sin< o i 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 l7 o 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 H 38 39 40 4i 42 43 44 45 46 *l 48 49 50 5i 52 53 54 55 56 57 58 59 60 IOI 9. 280599 .281248 .281897 .282544 .283190 9.283836 . 284480 .285124 . 285766 . 286408 9. 287048 . 288326 .288964 . 289600 9. 290236 . 290870 .291504 .292137 . 292768 9- 293399 . 294029 . 294658 . 295286 .295913 9. 296539 .297164 .297788 .298412 . 299034 9- 299655 . 300276 . 300895 .301514 .302132 9. 302748 .303364 .303979 .304593 . 305207 9- 305819 . 306430 .307041 • 307650 • 308259 9.308867 . 309474 . 310080 • 310685 .311289 9- 3 1 1893 .312495 .313097 .313698 .314297 9-3I4897 .315495 . 316092 .3166S9 .317284 9.317879 D. 1' 0.82 0.82 0.78 0.77 0.77 o.73 o.73 0.70 0.70 0.67 67 03 o o, 0.63 0.60 0.60 0.57 0.57 0.55 0.52 0.52 0.50 0.48 0.47 0.45 0.43 0.42 0.40 o. 40 0.37 0.35 0.35 0.32 0.32 0.30 o. 27 0.27 0.25 0.23 0.23 o. 20 0.18 0.18 0.15 0.15 o. 13 O. 12 O. IO O.08 o. 07 0.07 0.03 0.03 0.02 9.98 IO.OO 9-97 9-95 9-95 9.92 9.92 Cos. j D. i' Cos. 991947 991922 991897 991873 991848 991823 991799 991774 991749 991724 991699 991674 991649 991624 991599 991574 991549 991524 991498 991473 991448 991422 991397 991372 991346 991321 991295 991270 991244 991218 991 193 991 167 991 141 99III5 991090 991064 991038 991012 990986 990960 990934 990908 990882 990855 990829 990803 990777 990750 990724 99o697 990671 990645 990618 990591 990565 990538 9905 1 1 990485 990458 990431 990404 Sin. D. if. Tan. 288652 289326 289999 290671 291342 292013 292682 293350 294017 294684 295349 296013 296677 297339 298001 298662 299322 299980 300638 301295 301951 302607 303261 3039H 304567 305218 305869 306519 307168 307816 308463 309109 309754 310399 31 1042 311685 312327 312968 313608 314247 314885 315523 316159 316795 317430 318064 318697 319330 319961 320592 321222 321851 322479 323106 323733 324358 324983 325607 326231 326853 327475 D. 1". I J Cot. D. 1.23 1. 22 1. 20 1. 18 1. 18 i- 15 1. 13 1. 12 1. 12 1.08 1.07 1.07 1.03 1.03 1.02 1. 00 0-97 0.97 o.95 o.93 o.93 0.90 0.88 0.88 0.85 0.85 0.83 0.82 0.80 0.78 0.77 o.75 o.75 0.72 0.72 0.70 0.68 0.67 0.65 0.63 0.63 0.60 0.60 0.58 0-57 o.55 o.55 0.52 0.52 0.50 0.48 0.47 o.45 o.45 0.42 0.42 0.40 0.40 o.37 0-37 D. 1' Cot. 0.71 1348 .710674 .710001 • 709329 . 708658 o. 707987 .707318 . 706650 • 705983 .705316 o. 704651 . 703987 . 703323 . 702661 . 701999 o. 701338 . 700678 . 700020 . 699362 . 698705 o. 698049 .■697393 . 696739 . 696086 . 695433 o. 69^782 .694131 .693481 . 692832 .692184 0.691537 .690891 . 690246 .689601 . 688958 0.688315 . 687673 . 687032 .686392 .685753 0.6851 15 . 684477 .683841 . 683205 . 682570 0.681936 .681303 . 680670 . 680039 . 679408 o. 678778 .678149 .677521 . 676894 . 676267 0.675642 .675017 .674393 . 673769 .673147 o. 672525 Tan,. 78' 12° COSINES, TANGENTS, AND COTANGENTS zQf M. 10 ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 Sin. 9.317879 .318473 .319066 .319658 . 320249 9.320840 .321430 .322019 . 322607 .323194 9. 323780 .324366 . 324950 • 325534 .326117 9. 326700 .327281 . 327862 . 328442 . 329021 9- 329599 .330176 • 330753 .331329 •331903 9- 332478 •333051 • 333 6 24 •334195 • 334767 9- 335337 • 3359o6 • 336475 • 337043 .337610 9-338176 • 338742 • 339307 •339871 • 340434 9. 340996 .341558 .342119 • 342679 • 343239 9- 343797 • 344355 • 3449 12 • 345469 . 346024 9-346579 •347134 . 347687 . 348240 • 348792 9- 349343 • 349893 • 350443 • 350992 •35i54o 9. 352088 Cos. D. i' 9.90 9.88 9.87 9.85 9.85 9.83 9.82 9.80 9.78 9-77 9-77 9-73 9-73 9.72 9.72 9.68 9.68 9.67 9.65 9.63 9.62 9.62 9.60 9-57 9-58- 9-55 9-55 9.52 9-53 9.50 9.48 9.48 9-47 9-45 9-43 9-43 9.42 9.40 9.38 9-37 9-37 9-35 9-33 33 30 30 28 28 25 9- 9- 9- 9- 9- 9- 9.25 9.25 9. 22 9. 22 9. 20 9-13 D. 1 Cos. 990404 99037S 990351 990324 990297 990270 990243 990215 990188 990161 99oi34 990107 990079 990052 990025 989997 989970 9S9942 989915 989887 D. 1' o 989832 989804 989777 989749 989721 989693 989665 989637 989610 989582 989553 989525 989497 989469 989441 989413 989385 989356 989328 989300 989271 989243 989214 989186 989157 989128 989100 989071 989042 989014 988985 988956 988927 988840 98881 1 988782 988753 988724 Sin. 43 45 45 45 45 45 47 45 45 45 45 47 45 45 47 45 47 45 47 45 47 47 45 47 47 47 47 47 45 47 48 47 47 47 47 47 47 48 47 47 48 47 48 47 48 48 47 48 48 47 Tan. D. i' D. i' 9- 327475 • 328095 •328715 • 329334 • 329953 9- 330570 .331187 .331803 .332418 • 333033 9- 333646 • 334259 .334871 • 335482 • 336093 9. 336702 •3373II .337919 . 338527 •339!33 9- 339739 . 340344 . 34094S • 341552 .342155 9.342757 • 343358 • 343958 • 344558 • 345157 9. 345755 • 346353 • 346949 • 347545 .348141 9- 348735 • 349329 • 349922 .350514 .351106 9- 351697 .352287 .352876 • 353465 • 354053 9- 354640 • 355227 .355813 • 356398 . 356982 9.357566 .358149 • 358731 • 359313 • 359893 9. 360474 .361053 .361632 .362210 .362787 9- 363364 Cot. 10.33 10.33 10.32 10.32 10.28 10.28 10. 27 10.25 10.25 10. 22 10. 22 10. 20 10.18 10.18 10.15 10.15 10.13 10.13 10. 10 10. 10 10.08 10.07 10.07 10.05 10.03 10. 02 10.00 10.00 9.98 9-97 9-97 9-93 9-93 9-93 9.90 9.90 9.88 9.87 9.87 9.85 9.83 9.82 9.82 9.80 9.78 9.78 9-77 9.75 9-73 9-73 9.72 9.70 9.70 9.67 9.68 9.65 9-65 9.63 9.62 9.62 Cot. D. 1 o. 672525 .671905 .671285 . 670666 . 670047 o. 669430 .668813 .668197 .667582 . 666967 o. 666354 .665741 .665129 .664518 . 663907 o. 663298 . 662689 .662081 .661473 . 660867 0.660261 .659656 • 659052 . 658448 . 657845 o. 657243 .656642 . 656042 • 655442 . 654843 o. 654245 . 653647 •653051 •652455 • 651859 0.651265 .650671 . 650078 . 649486 . 648894 o. 648303 .647713 .647124 • 646535 • 645947 o. 645360 •644773 .644187 . 643602 .643018 o. 642434 .641851 .641269 . 640687 .640107 o. 639526 . 638947 .638368 • 637790 .637213 o. 636636 Tan. 60 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 2 i 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 I 7 6 5 4 3 2 1 o M. 77' 13* LOGARITHMIC SINES i66° M. Sin. o i 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 9. 352088 • 352635 .353i8i .353726 .354271 9-3548I5 • 355358 .355901 .356443 • 356984 9- 357524 • 358064 . 358603 • 359HI • 359678 9.360215 . 360752 .361287 .361822 . 362356 9.362889 • 363422 • 363954 . 364485 • 365016 9- 365546 .366075 . 366604 .367131 • 367659 9.368185 .368711 . 369236 .369761 .370285 9. 370808 .371330 .371852 .372373 .372894 9-3734H • 373933 • 374452 • 374970 . 375487 9- 376003 .376519 . 377035 •377549 . 378063 9- 378577 • 379089 . 379601 .380113 . 380624 9-38ii34 .381643 .382152 . 382661 .383168 9- 383675 Cos. D. i' D. i' Cos. 9.988724 988695 988666 988636 988607 988578 988519 988489 988460 988430 988401 988371 988342 988312 988282 988252 988223 988193 988163 988133 988103 988073 988043 988013 987983 987953 987922 987892 987862 987832 987801 987771 987740 987710 987679 987649 987618 987588 987557 987526 987496 987465 987434 987403 987372 987341 987310 987279 987248 987217 987186 987155 987124 987092 987061 987030 986998 986967 986936 986904 Sin. D. 1". .48 .48 .50 .48 .48 .50 .48 .50 .48 •50 .48 .50 .48 .50 .50 .50 .48 .50 •50 •50 ■ -50 .50 .50 •50 .50 •50 .52 .50 .50 .50 .52 .50 .52 .50 .52 .50 .52 .50 .52 .52 .50 .52 .52 •52 .52 .52 .52 .52 .52 .52 .52 .52 • 52 • 53 .52 .52 • 53 .52 • 52 • 53 D. 1" Tan. 9. 363364 . 363940 .364515 . 365090 . 365664 9- 366237 .366810 .367382 • 367953 • 368524 9. 369094 . 369663 . 370232 • 370799 .371367 9-37I933 • 372499 • 373064 • 373629 .374193 9- 374756 .375319 .375881 • 376442 - 377003 9- 377563 .378122 .378681 • 379239 • 379797 9- 380354 .380910 .381466 . 382020 .382575 9.383129 .383682 • 384234 .384786 • 385337 9.385888 .386438 .386987 • 387536 . 388084 9.388631 .389178 . 389724 . 390270 . 390815 9. 391360 •39I903 • 392447 • 392989 •393531 9- 394073 .394614 .395154 • 395694 • 396233 9. 396771 Cot. D. i' D. 1". Cot. o. 636636 . 636060 .635485 .634910 • 634336 o. 633763 .633190 .632618 . 632047 .631476 o. 630906 • 630337 . 629768 .629201 .628633 o. 628067 .627501 . 626936 .626371 . 625807 o. 625244 .624681 .624119 .623558 .622997 0.622437 .621878 .621319 .620761 . 620203 0.619646 .619090 .618534 .617980 .617425 0.616871 .616318 .615766 .615214 . 614663 0.614112 .613562 .613013 .612464 .611916 0.61 1369 .610822 .610276 . 609730 .609185 o. 608640 . 608097 . 607553 .607011 . 606469 o. 605927 . 605386 . 604846 . 604306 . 603767 o. 603229 Tan. 103 7 6« COSINES, TANGENTS, AND COTANGENTS 165' M. Sin, D. 1". Cos. o 1 2 3 4 5 6 7 8 9 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 I 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 9-383675 .384182 .384687 .385192 • 385697 9. 386201 . 386704 . 337207 • 387709 .388210 9. 38871 1 .389211 .389711 .390210 . 390708 9.391206 .391703 .392199 . 392695 .393191 9- 3936S5 .394179 • 394673 .395166 • 395653 9. 396150 .396641 .397132 .397621 .398111 9. 398600 .399088 • 399575 . 400062 .400549 9.401035 .401520 . 402005 . 402489 .402972 9- 403455 • 403938 . 404420 .404901 • 405382 9- 405^62 .406341 . 406820 • 407299 .407777 9.408254 .408731 . 409207 . 409682 .410157 9.410632 .411106 .411579 .412052 .412524 9.412996 Cos. 8.45 8.42 8.42 8.42 8.40 8.38 8.38 8.37 8-35 8.35 8-33 8-33 8.32 8.30 8.30 8.28 8.27 8.27 8.27 8.23 8.23 8.23 8.22 8.20 8.20 8.18 8. 18 8.15 8.17 8.15 8.13 8.12 8.12 8.12 8. 10 8.08 8.08 8.07 8.05 8.05 8.05 8.03 8.02 8.02 8.00 7.98 7.98 7.98 7-97 7-95 7-95 7-93 ! 7.92 1 7.92 7.92 7.90 7.88 7.88 7.87 7.87 D. 1". 9. 986904 .986873 .986841 . 986S09 .986778 9.9S6746 .986714 . 986683 .986651 .986619 9.986587 • 986555 . 9S6523 .986491 • 986459 9. 986427 • 986395 . 986363 .986331 . 986299 9. 986266 • 986234 . 986202 .986169 .986137 9.986104 . 986072 . 986039 . 986007 • 985974 9. 985942 • 985909 .985876 .985843 .985811 9- 985778 .985745 .985712 . 985679 . 985646 9-985613 .9855S0 .985547 .985514 • 985480 9- 985447 .985414 .985381 .985347 .985314 9. 985280 .985247 •985213 .985180 .985146 9-985113 .985079 • 9S5045 .985011 • 984978 9. 984944 Sin. D. 1' Tan. D. 1 '. D. 1 9.396771 397309 397846 3983S3 398919 399455 399990 400524 401058 401 59 1 402124 402656 403 1 S7 403718 404249 40477.8 40530S 405836 406364 406892 407419 407945 408471 408996 409521 410045 410569 4 1 1092 411615 412137 41265S 4I3I79 413699 414219 4H738 415257 415775 416293 416810 417326 417842 418358 418873 419387 4 1990 1 420415 420927 421440 421952 422463 422974 423484 423993 424503 42501 1 425519 426027 426534 427041 427547 428052 Cot. D. Cot. T " 1 ' 0.603229 . 60269I .602154 .60l6l7 .6OIO8I o. 600545 .60COIO . 599476 . 598942 .598409 o. 597876 50 .597344 .596813 . 596282 •595751 o. 595222 • 594692 .594164 • 593636 .593108 o. 592581 • 592055 •591529 .591004 . 590479 °. 589955 .589431 . 588908 .588385 .587863 o. 587342 .586821 . 586301 .585781 . 585262 o. 584743 . 584225 .583707 .583190 . 582674 0.582158 .581642 .581127 .580613 . 580099 o. 579585 . 579073 .578560 .578048 . 577537 o. 577026 .576516 . 576007 • 575497 • 574989 0.574481 •573973 • 573466 .572959 • 572453 0.571948 Tan. 75' i5 ( LOGARITHMIC SINES i6 4 < M. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 Sin. 9.412996 .413467 .413938 .414408 .414878 9-415347 .415815 .416283 .416751 .417217 9.417684 .418150 .418615 .419079 .419544 9. 420007 . 420470 • 420933 .421395 .421857 9.422318 .422778 .423238 . 423697 .424156 9.424615 .425073 • 425530 . 425987 • 426443 9. 426899 .427354 . 427809 .428263 .428717 9.429170 • 429623 .430075 . 430527 .430978 9. 431429 .431879 • 432329 .432778 . 433226 9- 433675 .434122 .434569 .435016 • 435462 9- 435908 . 436353 .436798 • 437242 .437686 9.438129 . 438572 • 439014 • 439456 • 439897 9- 440338 D. 1' Cos. 85 85 83 S3 82 80 80 So 77 78 77 75 73 75 72 72 72 70 70 68 67 67 65 65 65 63 62 62 60 60 58 58 57 57 55 55 53 53 52 52 50 50 48 47 48 45 45 45 43 43 42 42 40 40 38 38 37 37 35 35 D. 1". Cos. ). 984944 .984910 . 984876 .984842 9.984774 • 984740 . 984706 . 984672 .984638 9. 984603 .984569 .984535 .984500 . 984466 9. 984432 .984397 .984363 .984328 .984294 9. 984259 . 984224 .934190 • 984155 .984120 9. 984085 . 984050 • 984015 .983981 • 983946 9-9839" .983875 . 983840 .983805 . 983770 9-983735 . 983700 . 983664 .983629 • 983594 9- 983558 .983523 .983487 .983452 .983416 9.983381 • 983345 . 983309 • 983273 .983238 9. 983202 .983166 .983130 . 983094 .983058 9. 983022 . 982986 . 982950 .982914 . 982878 9. 982842 D. 1' Sin. • 57 • 57 • 57 • 57 -57 • 57 • 57 -57 -57 .58 • 57 • 57 .58 •57 • 57 .58 -57 .58 • 57 .58 .58 -57 .58 .58 .58 .58 .58 • 57 .58 .58 .60 .58 .58 .58 ■58 .58 .60 .58 .58 .60 .58 .60 .58 .60 .58 ,60 .60 .60 58 .60 ,60 60 60 60 60 60 60 60 60 60 Tan. D. 1' Cot. 9- 9- 9- D. 1". 9- 428052 428558 429062 429566 , 430070 430573 ■431075 431577 432079 432580 433080 43358o 434080 434579 435078 435576 436073 436570 437067 437563 438059 438554 439048 439543 440036 440529 .441022 .441514 . 442006 ■ 442497 . 442988 • 443479 . 443968 . 444458 ■444947 ■ 445435 ■ 445923 ,446411 , 446898 ■ 447384 447870 448356 448841 449326 449810 450294 450777 451260 451743 452225 452706 453187 453668 454148 454628 455107 455586 456064 456542 457019 457496 Cot. 8.43 8.40 8.40 8.40 8.38 8.37 8-37 8-37 8.35 8-33 8.33 8.33 8.32 8.32 8.30 8.28 8.28 8.28 8.27 8.27 8.25 8.23 8.25 8.22 8.22 8.22 8.20 8.20 8.18 8.18 8.18 8.15 8.17 8.15 8.13 8.13 8.13 8.12 8.10 8.10 8.10 8.08 8.08 8.07 8.07 8.05 8.05 8.05 8.03 8.02 8.02 8.02 8.00 8.00 7.98 7.98 7-97 7-97 7-95 7-95 D. 1". 0.571948 .57M42 • 570938 • 570434 .569930 0.569427 . 568925 .568423 .567921 .567420 o. 566920 .566420 • 565920 .565421 . 564922 o. 564424 . 563927 . 563430 . 562933 .562437 0.561941 . 561446 .560952 . 560457 . 559964 0.559471 .558978 .558486 .557994 . 557503 0.557012 .556521 .556032 . 555542 . 555053 o. 554565 . 554077 . 553589 .553102 .552616 o. 552130 .551644 .551159 . 550674 • 550 1 90 o. 549706 . 549223 . 548740 . 548257 .547775 o. 547294 .546813 .546332 .545852 . 545372 o. 544893 .544414 . 543936 . 543458 . 542981 o. 542504 105' Tan. M. i6< COSINES, TANGENTS, AND COTANGENTS 163* M. o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 50 5i 52 53 54 56 5 l 58 59 60 Sin. 9. 440338 . 44077S .441218 .441658 . 442096 9- 442535 • 442973 .443410 .443847 .444284 9. 444720 • 445155 . 445590 . 446025 • 446459 9. 446893 . 447326 • 447759 .448191 . 448623 9. 449054 . 449485 .449915 • 450345 • 450775 9. 451204 .451632 . 452060 .452438 .452915 9. 453342 • 453768 •454194 .454619 . 455044 35 9.455469 36 . 455893 37 .456316 • 456739 .457162 40 9. 457584 41 I .458006 • 458427 .458848 . 459268 9.459688 .46010S • 460527 . 460946 . 461364 9.461782 .462199 . 462616 . 463032 • 463448 9. 463S64 • 464279 . 464694 .46510S .465522 9- 465935 Cos. D. 1". 7-33 7.33 7-33 7.30 7-32 7-3o 7.28 7.28 7.28 7.27 7.25 7.25 7.25 7.23 7.23 7.22 7. 22 7. 20 7. 20 7.18 Cos. D. 1' 7.13 7.13 7-i3 7. 12 7. 12 7. 10 7. 10 7.08 7.0S 7.08 7.07 7-05 7-05 7.05 7-03 7.03 7.02 7.02 00 00 00 6-95 6-95 6-93 6-93 6.93 6.92 6.92 6.90 6.90 I 6.88 D. i". 9. 982842 982805 982769 982733 982696 982660 982624 982587 982551 982514 982477 982441 982404 982367 982331 982294 982257 982220 982183 982146 982109 982072 982035 981998 981961 981924 981886 981849 981812 981774 981737 981700 9S1662 981625 981587 981549 981512 981474 981436 981399 981361 981323 981285 981247 981209 981171 981 133 981095 981057 981019 980981 980942 980904 980866 980827 980789 980750 980712 980673 980635 980596 Sin. .62 .60 .60 .62 .60 .60 .62 .60 .62 .62 .60 .62 .62 .60 .62 .62 .62 .62 .62 .62 .62 .62 .62 .62 .62 .63 .62 .62 .63 .62 .62 .63 .62 .63 .63 .62 .63 .63 .62 .63 .63 .63 .63 .63 .63 .63 .63 .63 .63 .63 .65 .63 .63 .65 .63 .65 .63 .65 .63 .65 Tan. D. 1' 9- 457496 • 457973 • 458449 • 458925 • 459400 9- 459875 . 460349 . 460823 .461297 .461770 9.462242 .462715 .463186 •463658 .464128 9- 464599 . 465069 • 465539 . 466008 • 466477 9. 466945 .467413 . 467880 • 468347 . 46SS14 9. 469280 • 469746 .470211 . 470676 .471141 9.471605 . 472069 • 472532 • 472995 • 473457 9-4739I9 .474381 • 474842 • 475303 .475763 9. 476223 .476683 .477142 . 477601 • 478059 9.478517 •478975 • 479432 . 479889 • 480345 9. 480801 .481257 .481712 .482167 .482621 9- 483075 • 483529 • 4839S2 • 484435 . 484887 9- 485339 Cot. D. 1' Cot. 7-95 7-93 7-93 7.92 7.92 7.90 7.90 7.90 7. 88 7.87 7.88 7.85 7.87 7.83 7.85 7.83 7.83 7.82 7.82 7.80 7.80 7.78 7.78 7.78 7-77 7-77 7-75 7-75 7-75 7-73 7-73 7.72 7.72 7.70 7.70 7.70 7.68 7.68 7.67 7.67 7.67 7.65 7.65 7.63 7.63 7.63 7.62 7.62 7.60 7.60 7.60 7.58 7.58 7.57 57 57 55 55 53 55 D. o. 542504 • 542027 .541551 041075 . 540600 0.540125 .539651 •539177 .538703 . 538230 o. 537758 • 537285 .536814 . 536342 • 535872 0.535401 .534931 . 534461 . 533992 . 533523 o. 533055 . 532587 .532120 .531653 .531186 o. 530720 . 530254 . 529789 • 529324 . 528859 o. 528395 .527931 . 527468 .527005 • 526543 o. 526081 . 525619 .525158 . 524697 . 524237 o. 523777 .523317 .522858 . 522399 .521941 0.521483 .521025 . 520568 .520111 .519655 o. 519199 .518743 .518288 .517833 .517379 0.516925 .516471 .516018 •515565 .515113 o. 514661 Tan. 60 59 58 57 56 55 54 53 52 5i 50 49 4 8 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 32 3i 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 M 13 12 11 7 6 5 4 3 2 1 o M. >6° 73' 17° LOGARITHMIC SINES i6,2 £ M. o i 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 4 l 48 49 50 5i 52 53 54 55 56 57 58 59 60 IO7 Sin. 9- 465935 . 466348 . 466761 .467173 .467585 9. 467996 . 468407 .468817 . 469227 .469637 9. 470046 . 470455 . 470863 .47T271 .47T679 9. 472086 .472402 . 47-1898 .473304 .473710 9.474II5 •4745*9 .474923 .475327 .475730 9-476 T 33 . 476536 .476938 .477340 .477741 9. 478142 • 478542 . 478042 . 470342 •479 7 4i 9.480140 • 480539 • 480937 .481334 .481731 9.482128 .482525 .482921 .483316 .483712 9. 484107 . 484501 . 484895 .485289 .485682 9. 486075 . 486467 . 486860 . 487251 . 487643 9. 488034 . 488424 .488814 . 489204 . 489593 9. 489982 Cos. D. i' Cos. D. i' 6.88 6.88 6.87 6.87 6.85 6.85 6.83 6.83 6.83 6.82 6.82 6.80 6.80 6.80 6.78 6.77 6.77 6.77 6.77 6-75 6.73 6-73 6.73 6.72 6.72 6.72 6.70 6.70 6.68 6.68 6.67 6.67 6.67 6.65 6.65 6.65 6.63 6.62 6.62 6.62 6.62 6.60 6.58 6.60 6.58 6.57 6-57 6.57 6-55 6-55 6-53 6.55 6.52 6.53 6.52 6.50 6.50 6.50 6.48 6.48 D. I". 9. 980596 .980558 .980519 . 980480 . 980442 9. 980403 . 980364 . 980325 . 980286 .980247 9. 980208 .980169 .980130 . 980091 . 980052 9.980012 . 9799^3 • 979934 . 979895 .979855 9.979816 . 979776 •979737 .979697 . 979658 9.979618 • 979579 • 979539 • 979499 . 979459 9.979420 .97938o • 979340 . 9793oo . 979260 9. 979220 .979180 .979140 .979100 . 979059 9.979019 • 978979 .978939 .978858 9.978817 .978777 .978737 . 978696 .978655 9.978615 .97S574 . 978533 .978493 . 978452 9. 97841 1 . 978370 . 978329 . 978288 • 978247 9. 978206 Sin. .63 .65 .65 .63 .65 .65 .65 .65 .65 .65 .65 .65 .65 .65 .67 .65 .65 .65 .67 .65 .67 .65 .67 .65 .67 .65 .67 .67 .67 .65 .67 .67 .67 .67 .67 .67 .67 .67 ,68 .67 .67 .67 ,68 .67 ,68 .67 ,67 ,68 ,68 ,67 ,68 68 67 ,68 68 68 68 68 68 68 Tan. 9- 485339 .485791 . 486242 .486693 .487143 9- 487593 .488043 . 488492 . 488041 • 489390 9.480838 . 490286 • 490^33 .491180 .491627 9. 492073 .492519 .492965 .493410 .493854 9. 494299 •494743 .495186 . 495630 . 496073 9- 490515 . 496957 • 497399 .497841 .498282 9. 498722 .499163 • 499603 . 500042 .500481 9. 500920 .501359 .501797 . 502235 . 502672 9.503109 • 503546 . 503982 .504418 . 504854 9. 505289 • 505724 .506159 .506593 . 507027 9.507460 .507893 . 508326 . 508759 . 509191 9. 509622 .510054 .510485 .510916 .511346 9. 51 1776 Cot. D. 1". Cot, 7-53 52 52 50 50 50 48 48 48 47 7-47 7-45 7-45 7-45 7-43 7-43 7-43 7.42 7.40 7.42 7.40 7.38 7.40 7.38 7-37 7-37 7-37 7-37 7-35 7.33 7-35 7.28 7.28 7.28 7.27 7.27 7. 22 7. 22 7. 22 7. 22 7. 20 7.18 7.20 7.18 7.i8 7.17 7.17 D. o. 514661 .514209 .513758 • 513307 .512857 0.512407 .511957 .511508 .511059 .510610 o. 510162 .509714 .509267 . 508820 .508373 o. 507927 .507481 . 507035 . 506590 . 506146 0.505701 .505257 .504814 . 504370 . 503927 o. 503485 . 503043 .502601 . 502159 .501718 0.501278 . 500837 . 500397 .499958 .499519 o. 499080 .498641 . 498203 .497765 .497328 0.496891 . 496454 .496018 .495582 . 495146 0.49471 I .494276 .493841 .493407 .492973 0.492540 .492107 .491674 .491241 . 490809 o. 490378 .489946 .489515 . 489084 .488654 o. 488224 Tan. 72° I l8° COSINES, TANGENTS, AND COTANGENTS xfo* M. Sin. D. i". Cos. D. i". Tan. D. i". Cot. 9. 489982 •49037I • 49°759 .491147 .491535 9.491922 . 492308 . 49 26 95 .493081 .493466 10 9. 493851 .494236 .494621 . 495005 .495388 9- 495772 .496154 . 496537 .496919 • 497301 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 ! 36 I 37 1 38| 39 I 9- 40 4i 42 43 44 45 46 47 1 48 49 5o; 5i I 52 53 54 55 56 57 58 59 60 497682 498064 498444 .498825 . 499204 9- 499584 •499963 . 500342 .500721 . 501099 9. 501476 .501854 .502231 . 502607 . 502984 9- 503360 . 503735 .504110 • 504485 . 504860 9- 505234 • 505608 .505981 • 506354 • 506727 9. 507099 .507471 • 507843 .508214 .508585 9. 508956 . 509326 .509696 .510065 .510434 9. 510803 .511172 .511540 .511907 .512275 9.512642 6.48 6.47 6.47 6.47 6.45 6.43 6-45 6.43 6.42 6.42 6.42 6.42 6.40 6.38 6.40 6-37 6.38 6.37 6.37 6-35 6.37 6-33 6-35 6.32 6-33 6.32 6.32 6.32 6.30 6.28 6.30 6.28 6.27 6.28 6.27 6.25 6.25 6.25 6.25 6.23 6.23 6. 22 6. 22 6. 22 Cos. 6.18 6.17 6.17 6.15 6.15 6.15 6.15 6.13 6. 12 6.13 6.12 D. 1". 9. 97S206 .978165 .978124 .978083 . 978042 9.978001 • 977959 . 9779 J 8 .977877 •977835 9- 977794 •977752 .977711 . 977669 . 977628 9.977586 • 977544 • 977503 .977461 .977419 9- 977377 • 977335 . 977293 .977251 . 977209 9.977167 .977125 .977083 . 977041 . 976999 9.976957 .976914 . 976872 . 976830 .976787 9- 976745 . 976702 . 976660 .976617 • 976574 9- 976532 . 976489 . 976446 • 976404 .976361 9.976318 • 976275 . 976232 .976189 . 976146 9. 976103 .976060 .976017 • 975974 • 975930 9- 975887 • 975844 . 975800 •975757 .975714 9. 975670 Sin. .68 .68 .68 .68 .68 •7o .68 .68 .70 .68 •70 .68 .70 .68 .70 .70 .68 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .70 .72 .70 .70 .72 .70 .72 .70 .72 .72 .70 .72 .72 .70 .72 .72 .72 .72 .72 .72 .72 • 72 .72 .72 • 73 .72 .72 • 73 .72 .72 •73 D. 1" 9.511776 .512206 •512635 .513064 .513493 9-5I392I .5H349 .514777 . 515204 .515631 9- 5 l6 o57 • 516484 .516910 .517335 .517761 9.518186 .518610 .519034 .519458 .519882 9. 520305 . 520728 .521151 .521573 • 521995 9.522417 .522838 • 523259 . 523680 . 524100 9.524520 • 524940 • 525359 • 525778 .526197 9. 526615 . 527033 • 527451 .527868 .528285 9. 528702 .529119 • 529535 .529951 • 530366 9- 530781 .531196 .531611 • 532025 • 532439 9- 532853 • 533266 • 533679 • 534092 • 534504 9- 5349i6 • 535328 • 535739 • 536150 • 536561 9- 536972 Cot. 7.13 7. 12 7. 12 7. 10 7.12 7. 10 7.08 7. 10 7.08 7.07 7.07 7.07 7.07 7- °5 7.05 7.05 7-03 7-03 7- °3 7.02 7.02 7.02 7.00 7.00 7.00 6.98 6.98 6.98 6.97 6-97 6.97 6-95 6-95 6-95 6-95 6-93 6.93 6.92 6.92 6.92 6. 92 6.90 6.90 6.90 6.88 6.88 6.88 6.87 6.87 6.87 6.85 6.85 6.85 6.85 D. 1' o. 488224 • 487794 • 487365 . 486936 . 486507 o. 486079 .485651 . 485223 .484796 .484369 o. 483943 .483516 . 483090 . 482665 . 482239 o. 481814 . 481390 . 480966 . 480542 .480118 o. 479695 .479272 .478849 .478427 . 478005 o. 477583 .477162 .476741 .476320 .475900 o. 47548o . 475060 .474641 .474222 . 473803 o. 473385 . 472967 . 472549 .472132 .471715 0.471298 . 470881 .470465 . 470049 . 469634 0.469219 . 468804 .468389 . 467975 .467561 o. 467147 . 466734 .466321 . 465908 . 465496 o. 465084 . 464672 .464261 . 463S50 . 463439 o. 463028 Tan. 108 M. 71' itf LOGARITHMIC SINES i6o c M. Sin. io ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 Go IOg 512642 513009 513375 5I374I 514107 514472 514837 515202 515566 515930 516294 516657 517020 5173S2 517745 518107 51846S 518829 519190 5I955I 5I99II 520271 520631 520990 521349 521707 522066 522424 522781 523138 523495 523852 52420S 524564 524920 525275 525630 525984 526339 526693 527046 527400 527753 528105 528458 528810 529161 529513 529864 530215 530565 530915 53*265 53*614 531963 532312 532661 533009 533357 533704 534052 Cos. D. 1' 6. 12 6. 10 6. 10 6. 10 6.08 6.08 6.08 6.07 6.07 6.07 6.05 6.05 6.00 6. 00 5.98 5.98 5-97 5.98 5-97 5-95 5-95 5-95 5-95 5-93 5-93 5-93 5.92 5.92 5-90 5-92 5.90 5-88 5-9o 5. 88 5-87 5-88 5.87 5- 85 •5.87 5.85 5.85 5-8 3 5.83 5.83 5.82 5.82 5.82 5.82 5- 80 5.8o 5.78 5.80 D. I". Cos. 975670 975627 975583 975539 975496 975452 975408 975365 975321 975277 975233 975189 975145 975101 975057 975013 974969 974925 974880 974836 974792 974748 974703 974659 974614 974570 974525 974481 974436 974391 974347 974302 974257 974212 974167 974122 974077 974032 9739S7 973942 973897 973852 973807 973761 973716 973671 973625 97358o 973535 973489 973444 973398 973352 973307 97326i 973215 973169 973124 973078 973032 9729S6 Sin. D. r D. 1 Tan. D. 1 Cot. • 536972 •537382 .537792 . 538202 .538611 • 539020 .539429 . 539837 .540245 .540653 .541061 .541468 .541875 . 542281 . 542688 • 543094 .543499 . 543905 .5443io .544715 .545119 . 545524 .545928 .546331 .546735 ■547138 .547540 •547943 .548345 .548747 .549149 . 549550 .549951 . 550352 . 550752 .551153 .551552 .55I95 2 .552351 .552750 .553M9 .553548 . 553946 .554344 .554741 . 555139 . 555536 . 555933 .556329 .556725 .557121 .557517 .557913 .558308 . 558703 . 559097 .559491 .5598S5 ■ 560279 . 560673 .561066 Cot. 6.83 6.83 6.83 6.82 6.82 6.82 6.80 6.80 6.80 6.80 6.78 6. 78 6.77 6.78 6.77 6-75 6.77 6.75 6.75 6.73 6.75 6.73 6. 72 6-73 6. 72 6. 70 6. 72 6. 70 6.70 6.70 6.68 6.68 6.6S 6.67 6.68 6.65 6.67 6.65 6.65 6.65 6.65 6.63 6.63 6.62 6.63 6.62 6.62 6.60 6.60 6.60 6.60 6.60 6.58 6.58 6-57 6.57 6-57 6-57 6-57 6 -55 D. 1' o. 463028 .462618 . 462208 .461798 . 461389 o. 460980 .460571 . 460163 .459755 .459347 o. 458939 .458532 .458125 .457719 .457312 o. 456906 .456501 • 456095 . 455690 .455285 o. 454881 .454476 . 454072 . 453669 .453265 o. 452862 . 452460 . 452057 . 451655 .451253 o. 450851 . 450450 . 450049 .449648 . 449248 o. 448847 . 448448 . 448048 . 447649 .447250 o. 446851 . 446452 . 446054 .445656 .445259 o. 444861 .444464 .444067 .443671 .443275 0.442879 . 442483 .442087 .441692 .441297 o. 440903 .440509 .440115 .439721 .439327 °. 43S934 Tan. M. 70' 20 ° COSINES, TANGENTS, AND COTANGENTS 159 M. Sin. D. 1". Cos. D. 1". Tan. D. 1". Cot 9- 534052 • 554399 • 534745 . 535092 • 535438 9- 535783 .536129 • 536474 .536818 • 537163 9- 537507 .537851 .538194 .538538 .538S80 9- 539223 • 539565 • 539907 . 540249 • 54O590 9- 540931 .541272 .541613 .541953 .542293 9. 542632 .542971 .5433io • 543649 . 543987 9- 544325 • 544663 • 5450oo .545338 • 545674 9. 54601 1 .546347 • 546683 .547019 • 547354 9- 547689 . 548024 • 548359 . 54S693 • 549027 9. 549360 • 549693 • 550026 • 550359 • 550692 9.551024 •551356 .551687 .552018 • 552349 9. 552680 •5530io •553341 • 553670 . 554000 9- 554329 5-78 5-77 5.78 5-77 5-75 5-77 5-75 5-73 5-75 5-73 5. 5- 5- 5- 5- 5- 5-/ Cos. 5-7o 5-68 5-68 5-68 5-68 5.67 5.67 5.65 5.65 5.65 5.65 5.63 5.63 5-6 3 5.62 5.63 5.60 5.62 5.60 5.60 5.60 5.58 5.58 5.58 5-58 5-57 5-57 5-55 5-55 5-55 5-55 5-55 5-53 5-53 5.52 5.52 5.52 5.52 5-50 5-52 5.48 5.5o 5.48 D. 1". 972986 972940 972S94 972S48 972802 972755 972709 972663 972617 972570 972524 972478 972431 972385 972338 972291 972245 972198 972151 972105 972058 97201 1 971964 971917 971870 97^23 971776 971729 971682 971635 971588 97i54o 97M93 971446 971398 97I35I 971303 971256 971208 971161 97III3 971066 971018 970970 970922 970874 970827 970779 970731 970683 970635 9705S6 970538 970490 970442 970394 970345 970297 970249 970200 970152 Sin. D. i' 9. 561066 .561459 .561851 . 562244 • 562636 9. 563028 .563419 .563811 . 564202 • 564593 9- 564983 • 565373 • 565763 .566153 . 566542 9. 566932 . 567320 . 567709 . 568098 . 568486 9-568873 . 569261 . 569648 . 570035 . 570422 9. 570809 .57H95 .571581 .571967 •572352 9- 572738 .573123 • 573507 • 573892 . 574276 9. 574660 • 575044 • 575427 .5758io • 576193 9- 576576 • 576959 •577341 • 577723 • 578104 9. 578486 .578867 • 579248 • 579629 . 580009 9. 580389 . 580769 .581149 .581528 • 5S1907 9. 5S2286 .582665 • 583044 . 5S3422 .583800 9-584I77 Cot. 6.55 6.53 6.55 6.53 6.53 6.52 6.53 6.52 6.52 6.50 6.50 6.50 6.50 6.48 6.50 6.47 6.48 6.48 6.47 6.45 6.47 6.45 6-45 6.45 6-45 6.43 6-43 6.43 6.42 6.43 6.42 6.40 6.42 6.40 6.40 6.40 6.38 6.38 6.38 6.38 6.38 6.37 6.37 6.35 6.37 6.35 6.35 6.35 6.33 6.33 6.33 6-33 6.32 6.32 6.32 6.32 6.32 6.30 6.30 6.28 D. o- 438934 .438541 .438149 • 437756 • 437364 o. 436972 .436581 .436189 • 435798 .435407 0.435017 • 434627 • 434237 • 433847 • 433458 o. 433068 . 432680 . 432291 .431902 .431514 0.431127 . 430739 . 430352 . 429965 . 429578 o. 429191 . 42S805 . 428419 . 428033 . 427648 0.427262 . 426877 . 426493 .426108 . 425724 o. 425340 . 424956 . 424573 .424190 .423807 o. 423424 . 423041 .422659 .422277 .421896 0.421514 .421133 .420752 .420371 .419991 o. 419611 .419231 .418851 .418472 < 418093 c.417714 .417335 .416956 .416578 .416200 0.415823 Tan. 110 69' 21 LOGARITHMIC SINES I58 c M. Sin. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 III 9- 554329 • 554658 .554987 .555315 • 555643 9- 555971 • 556299 . 556626 • 556953 .557280 9.557606 • 557932 .558258 .558583 .558909 9- 559234 • 559558 .559883 . 560207 • 560531 9- 560855 .561178 . 561501 .561824 . 562146 9. 562468 . 562790 .563112 • 563433 .563755 9-564075 .564396 .564716 . 565036 • 565356 9. 565676 • 565995 .566314 • 566632 .566951 9. 567269 .567587 • 567904 . 568222 • 568539 9.568856 .569172 .569488 . 569804 .570120 9- 570435 .570751 .571066 .571380 .571695 9. 572009 • 572323 .572636 .572950 • 573263 9- 573575 Cos. D. i' Cos. D. i' D. 1' 9.970152 . 970103 • 970055 . 970006 • 969957 9. 969909 . 969860 .969811 . 969762 .969714 9. 969665 .969616 • 969567 .969518 . 969469 9. 969420 . 969370 . 969321 .969272 . 969223 9.969173 .969124 . 969075 . 969025 . 968976 9. 968926 . 968877 . 968827 .968777 . 968728 9. 968678 . 968628 .968578 . 968528 .968479 9. 968429 .968379 .968329 . 968278 . 968228 9. 968178 .968128 . 968078 . 968027 . 967977 9. 967927 . 967876 .967826 • 967775 • 967725 9.967674 . 967624 .967573 .967522 .967471 9.967421 • 967370 .967319 . 967268 .967217 9.967166 Sin. .82 .80 .82 .82 .80 .82 .82 .82 .80 .82 .82 .82 .82 .82 .82 .83 .82 .82 .82 .83 .82 .82 .83 .82 .83 .82 .83 .83 .82 .83 .83 .83 .83 .82 .83 .83 .83 .85 .83 .83 .83 •83 .85 .83 .83 .85 .83 .85 .83 .85 .83 .85 .85 .85 .83 .85 .85 .85 .85 .85 D. 1". Tan. 9.584177 .584555 • 584932 • 585309 • 585686 9. 586062 • 586439 .586815 .587190 .587566 9. 587941 .588316 . 588691 . 589066 . 589440 9.589814 .590188 • 590562 •590935 .591308 9.591681 • 592054 • 592426 • 592799 .593i7i 9- 593542 .593914 .594285 • 594656 • 595027 9.595398 .595768 .596138 . 596508 .596878 9- 597247 .597616 • 597985 • 598354 .598722 9. 599091 • 599459 . 599827 .600194 . 600562 9. 600929 .601296 .601663 . 602029 • 602395 9.602761 .603127 . 603493 .603858 . 604223 9. 604588 • 604953 .605317 . 605682 . 606046 9.606410 Cot. D. 1' 6.30 6.28 6.28 6.28 6. 27 6.28 6.27 6.25 6.27 6.25 6.25 6.25 6.23 6.23 6.23 6.23 6. 22 6. 22 6. 22 6. 22 6. 20 6. 22 6.20 6.18 6.20 6.18 6.18 6.18 6.18 6. 17 6. 17 6. 17 6.17 6.15 6.15 6.15 6.13 6.15 6.13 6.13 6. 12 6. 12 6. 12 6. 12 6. 10 6. 10 6. 10 6. 10 6. 10 6.08 6.08 6.08 6.08 6.07 6.08 6.07 6.07 D. 1". Cot. 0.415823 .415445 .415068 .414691 .414314 0.413938 .413561 .413185 .412810 .412434 0.412059 .411684 .411309 .410934 .410560 o. 410186 .409S12 . 409438 . 409065 . 408692 0.408319 . 407946 • 407574 .407201 . 406829 o. 406458 . 406086 .405715 .405344 .404973 o. 404602 .404232 .403862 .403492 .403122 o. 402753 . 402384 .402015 .401646 .401278 o. 400909 .400541 .400173 . 399806 • 399438 0.399071 . 398704 . 398337 .397971 .397605 o. 397239 .396873 . 396507 .396142 • 395777 o.3954i2 • 395047 • 394683 , .394318 • 393954 o. 393590 Tan. 68° 22 COSINES, TANGENTS, AND COTANGENTS I57 t M. o i 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 Sin. 9- 573575 .573888 . 574200 .574512 • 574824 9- 575136 • 575447 .575758 . 576069 • 576379 9.576689 • 576999 • 577309 .577618 .577927 9. 578236 .578545 .578853 .579162 • 579470 9- 579777 .580085 . 580392 . 580699 .581005 9-581312 .581618 .581924 . 582229 • 582535 9. 582840 .583145 . 583449 .583754 .584058 9- 584361 . 584665 .584968 . 585272 .585574 9.585877 .586179 . 586482 .586783 587085 587386 587688 .587989 . 588 2S9 .588590 . 588890 .589190 . 589489 .589789 . 590088 • 590387 . 590686 . 590984 .591282 .591580 .591878 Cos. 9- D. 1' 5.22 5.20 5.20 5.20 5.20 5-12 5- 10 5.10 5.08 5.10 5.08 5.08 5.07 5.08 5-07 5-05 5-07 5-05 5-07 5.03 5-05 5-03 5.05 5.02 5.03 5.02 5-03 5.02 5.00 5.02 5.00 5.00 4.98 5.00 4.98 4.98 4.98 4-97 4-97 4-97 4-97 D. 1". Cos. 9.967166 .967115 . 967064 .967013 .966961 9.966910 . 966859 . 966808 • 966756 . 966705 9. 966653 . 966602 . 966550 . 966499 • 966447 9- 966395 • 966344 . 966292 . 966240 .966188 9. 966136 . 966085 . 966033 .965981 . 965929 9. 965876 . 965824 • 965772 . 965720 . 965668 9.965615 • 965563 .965511 • 965458 . 965406 9. 965353 .965301 .965248 .965195 .965H3 9. 965090 .965037 .964984 .964931 .964879 9. 964826 .964773 .964720 . 964666 .964613 9. 964560 • 964507 • 964454 . 964400 • 964347 9- 964294 . 964240 .964187 .964133 . 964080 9. 964026 Sin. D. i' Tan. D. 1 9.606410 . 606773 .607137 . 607500 . 607863 9. 608225 . 608588 . 608950 .609312 . 609674 9. 610036 .610397 .610759 .611120 .611480 9.611841 .612201 .612561 .612921 .613281 9-613641 . 614000 .614359 .614718 .615077 9- 6i5435 .615793 .616151 .616509 .616867 9.617224 .617582 .617939 .618295 .618652 9. 619008 .619364 . 619720 . 620076 . 620432 9. 620787 .621142 .621497 .621852 .622207 9.622561 .622915 . 623269 . 623623 .623976 9. 624330 . 624683 .625036 .625388 .625741 9. 626093 . 626445 . 626797 .627149 .627501 9. 627852 Cot. D. 1' 6.05 6.07 6.05 6.05 6.03 6.05 6.03 6.03 6.03 6.03 6.02 6.03 6.02 6.00 6.02 6.00 6.00 6.00 6.00 6.00 5.98 5.98 5.98 5.98 5.97 5.97 5-97 5-97 5-97 5-95 5.92 5.92 92 92 90 90 90 90 5.88 5.90 5.88 5-88 5.87 5-88 5.87 5.87 5.87 5.87 5.87 5.85 D. 1". Cot. o. 393590 .393227 .392863 • 392500 .392137 o.39i775 .391412 .391050 . 390688 . 390326 o. 389964 . 389603 . 389241 . 388880 .388520 o. 388159 .387799 .387439 . 387079 .386719 0.386359 . 386000 . 385641 .385282 . 384923 o. 384565 . 384207 .383849 .383491 .383133 o. 382776 .382418 . 382061 .381705 .381348 o. 380992 .380636 . 380280 .379924 .379568 0.379213 .378858 . 378503 .378148 .377793 0.377439 . 377085 .376731 .376377 . 376024 o. 375670 .375317 .374964 .374612 . 374259 o. 3739°7 •373555 .373203 .372851 .372499 0.372148 Tan. 112' 6 7 " n" LOGARITHMIC SINES i56 ( M. Sin. D. i" 10 il 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 9. 591878 .592176 .592473 . 592770 . 59306/ 9- 593363 • 593659 • 593955 .594251 . 594547 9. 594842 .595137 . 595432 • 595727 . 596021 9-5963I5 • 596609 • 596903 .597196 . 597490 9.597783 .598075 .598368 . 598660 • 598952 9- 599244 . 599536 • 599827 .600118 .600409 9. 600700 . 600990 .601280 .601570 .601860 9.602150 . 602439 .602728 .603017 .603305 9. 603594 . 603882 . 604170 • 604457 . 604745 9. 605032 .605319 . 605606 . 605892 .606179 9. 606465 . 606751 . 607036 . 607322 . 607607 9. 607892 .608177 .608461 . 608745 . 609029 9.609313 Cos. 4-97 4-95 4-95 4.95 4-93 4-93 4-93 4-93 4-93 4.92 4.92 4.92 4.92 4.90 4.90 4.90 4.90 4.88 4.90 4.88 4.87 4.88 4.87 4.87 4.87 4.87 4.85 4.85 4.85 4.85 4.83 4.83 4.83 83 S3 82 82 S2 So 4.82 4.80 4.80 4.78 4.80 4.78 4.78 4.78 4-77 4.78 4-77 4-77 4-75 4.77 4-73 D. i' Cos. 9. 964026 • 963972 .963919 .963865 .963811 9- 963757 .963704 . 963650 .963596 • 963542 9. 963488 • 963434 .963379 .963325 .963271 9.963217 .963163 .963108 . 963054 .962999 9. 962945 . 962890 . 962S36 .962781 .962727 9. 962672 .962617 . 962562 . 962508 .962453 9. 962398 • 962343 . 962288 . 962233 .962178 9.962123 . 962067 .962012 .961957 .961902 9.961846 .961791 .961735 .961680 .961624 9.961569 .961513 .961458 .961402 .961346 9.961290 .961235 .961179 .961123 .961067 9. 961011 • 960955 . 960899 . 960843 . 960786 9. 960730 Sin. D. i ; Tan. D. 1 9. 627852 . 628203 .628554 .628905 . 629255 9. 629606 • 629956 . 630306 .630656 .631005 9- 6 3i355 .631704 .632053 .632402 .632750 9- 633099 .633447 .633795 .634H3 .634490 9.634838 .635185 .635532 .635879 . 636226 9. 636572 .636919 .637265 .637611 • 637956 9. 638302 .638647 • 638992 • 639337 . 6396S2 9. 640027 .640371 .640716 . 641060 .641404 9.641747 . 642091 .642434 .642777 .643120 9. 643463 . 643806 .644148 .644490 . 644832 9.645174 .645516 .645857 .646199 . 646540 9. 646881 . 647222 . 647562 .647903 . 648243 9-648583 Cot. D. i' D. 1 Cot. 0.372148 .371797 .371446 .371095 • 370745 o. 370394 • 370044 . 369694 • 369344 .368995 o. 368645 . 368296 . 367947 .367598 . 367250 0.366901 . 366553 . 366205 . 365857 . 365510 0.365162 .364815 . 364468 .364121 .363774 o. 363428 .363081 .362735 . 362389 . 362044 0.361698 .361353 .361008 . 360663 .360318 o. 359973 • 359629 • 359284 . 358940 • 358596 o. 358253 • 357909 .357566 • 357223 .356880 o. 356537 .356194 . 355852 .3555io .355168 o. 354826 .354484 .354143 .3538oi • 35346o o.353i 19 . 352778 . 352438 .352097 .351757 0.351417 Tan. "3' 66' 24^ COSINES, TANGENTS, AND COTANGENTS 155' M. Sin. o 9. 6 °93i3 I . 609597 2 . 609880 3 .610164 4 ,610447 | 9.610729 6 .611012 7 .611294 8 .611576 9 .611S5S 10 9.612140 ii ,612421 12 .612702 13 .6129S3 M .613264 15 9- 61 3545 16 .613825 17 .614105 18 .614385 19 .614665 20 9.614944 21 .615223 22 .615502 23 .6157S1 24 .616060 25 9.616338 26 .616616 27 .616894 28 .617172 29 .617450 30 9.617727 31 .61S004 32 .618281 33 .618558 34 .61S834 ai 9.619110 36 ,619386 37 .619662 38 .619938 39 .620213 40 9. 620488 4X . 620763 42 .621038 43 .621313 44 .621587 4S 9.621861 46 .622135 47 . 622409 48 . 622682 49 . 622956 50 9. 623229 5i . 623502 52 .623774 53 . 624047 54 .624319 55 9.624591 56 . 624863 57 .625135 58 . 625406 59 . 625677 60 9. 625948 Cos. D. 1 4-73 4.72 4-73 4.72 4.70 4.72 4.70 4.70 4.70 4.70 4.68 4.68 4.68 4.68 4.68 4.67 4.67 4.67 4.67 4.65 4-65 4-65 4-65 4-^5 4-63 4-63 4.63 4.63 4-63 4.62 4.62 4.62 4.62 4.60 4.60 4.60 4.60 4.60 4.58 4.58 4-55 4-57 4-55 4-55 4-53 4-55 4-53 4-53 4-53 4-53 4.52 4o2 4.52 D. 1". Cos. 9. 960730 .960674 .960618 .960561 . 960505 9. 960448 . 960392 . 960335 . 960279 .960222 9.960165 .960109 .960052 • 959995 .959938 9. 959882 • 959825 .959768 .959711 .959654 9- 959596 •959539 .959482 .959425 • 95936S 9- 9593io .959253 .959195 .959138 . 9590S0 9- 959023 . 958965 . 958908 .958850 • 958792 9- 958734 • 958677 .958619 .958561 .958503 9- 958445 .9583S7 • 958329 .958271 .95S213 9-958I54 . 958096 .958038 •957979 .957921 9- 957863 . 957S04 • 957746 .957687 . 957628 9- 957570 •9575II .957452 •957393 • 957335 9.957276 Sin. D. 1' D. 1". Tan. D. 1". 9.648583 .648923 . 649263 .649602 • 649942 9.650281 . 650620 .650959 .651297 .651636 9.651974 .652312 .652650 . 652988 .653326 9. 653663 .654000 .654337 . 654674 .655011 9- 655348 .6556S4 . 656020 . 656356 . 656692 9. 657028 .657364 .657699 .658034 .658369 9.658704 • 659039 .659373 .659708 .660042 9. 660376 .660710 .661043 .661377 .661710 9. 662043 .662376 . 662709 . 663042 • 663375 9. 663707 . 664039 . 664371 . 664703 • 665035 ! 9.665366 1 . 665698 . 666029 . 666360 . 666691 9. 667021 . 667352 . 6676S2 .66S013 . 66S343 9.668673 Cot. D. 1' Cot. 0.351417 .351077 • 350737 .350398 . 350058 0.349719 • 3493So .349041 . 34S703 . 34S364 o. 348026 .347688 • 347350 .347012 . 346674 o. 346337 . 346000 .345663 . 345326 . 3449S9 o. 344652 .344316 . 343980 . 343644 .343308 o. 342972 . 342636 . 342301 .341966 .341631 0.341296 .340961 . 340627 .340292 . 339958 o. 339624 . 339290 .338957 . 338623 . 33S290 o. 337957 • 337624 .337291 • 336958 .336625 o. 336293 .335961 • 335629 • 335297 • 334965 o. 334634 • 334302 •333971 • 333640 • 333309 o. 332979 . 33264S .332318 .331987 .331657 0.331327 Tan. "4* 65' 25 c LOGARITHMIC SINES 154' M. Sin. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 9. 625948 .626219 .626490 . 626760 . 627030 9. 627300 .627570 .627840 .628109 .628378 9. 628647 .628916 .629185 .629453 .629721 9. 629989 .630257 .630524 • 630792 .631059 9.631326 .631593 .631859 .632125 .632392 9.632658 • 632923 .633189 .633454 .633719 9-633984 • 634249 .6345H .634778 . 635042 9- 635306 • 635570 • 635834 . 636097 . 636360 9. 636623 . 636886 .637148 .637411 .637673 9- 637935 .638197 .638458 .638720 .638981 9. 639242 .639503 . 639764 . 640024 . 640284 9. 640544 . 640804 .641064 .641324 .641583 9.641842 D. 1". Cos. Cos. 4.52 4.52 4.5o 4.48 4.48 4.48 4.48 4-47 4-47 4-47 4-47 4-45 4-47 4-45 4-45 4.45 4-43 4-43 4-45 4-43 4.42 4-43 4.42 4.42 4.42 4.42 4.42 4.40 4.40 4.40 4.40 4.40 4.38 4.38 4.38 4.38 4-37 4.38 4-37 4-37 4-37 4-35 4-37 4-35 4.35 4-35 4-35 4-33 4-33 4-33 4-33 4-33 4.33 4-32 4.32 D. 1". D. 1". 9.957276 .957217 .957158 • 957099 .957040 9.956981 .956921 . 956862 . 956803 .956744 9. 956684 .956625 .956566 • 956506 .956447 9- 956387 • 956327 .956268 . 956208 .956148 9. 956089 . 956029 .955969 • 955909 • 955849 9- 955789 .955729 .955669 .955609 .955548 9-955488 • 955428 .955368 .955307 .955247 9-955186 .955126 .955065 . 955005 . 954944 9- 954883 • 954823 . 954762 .954701 . 954640 9- 954579 .954518 . 954457 .954396 •954335 9- 954274 .954213 .954152 .954090 . 954029 9. 953968 • 9539° 6 . 953845 . 953783 .953722 9. 953660 Sin. .98 .98 1. 00 .98 .98 .98 1. 00 1. 00 1. 00 1. 00 1. 00 1. 00 00 00 00 00 00 00 00 00 02 00 Tan. 1. 00 1. 00 1.02 1. 00 1.02 1. 00 1.02 1. 00 1.02 1.02 1. 00 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.03 1.02 1.02 1.03 1.02 1.03 1.02 1.03 D. 1" 9.668673 . 669002 .669332 .669661 . 669991 9. 670320 . 670649 .670977 .671306 .671635 9.671963 .672291 .672619 • 672947 • 673274 9. 673602 .673929 .674257 .674584 .674911 9.675237 .675564 . 675890 .676217 . 676543 9. 676869 .677194 • 677520 . 677846 .678171 9. 678496 .678821 . 679146 .679471 • 679795 9.680120 . 680444 . 680768 .681092 .681416 9.681740 .682063 .682387 .682710 . 683033 9- 683356 .683679 .684001 . 684324 . 684646 9. 684968 . 685290 .685612 .685934 .686255 9. 686577 D. 1". Cot. .687219 . 687540 .687861 9. 688182 Cot. 5.48 5.50 5-48 50 48 48 47 48 48 47 5-47 5-47 5-47 5-45 5-47 5-45 5-47 5-45 5-45 5-43 5-45 5-43 5-45 5-43 5-43 5-42 5-42 5.42 5.42 5-40 5.42 5-40 5-40 5.40 5.40 5-40 5-38 5.40 5.38 5-38 5.38 5.38 5-37 5-38 5-37 5-37 5-37 5-37 5-37 5-35 5-37 5-35 5-35 5-35 5-35 5-35 D. 1". 0.331327 . 330998 . 330668 . 330339 . 330009 o. 329680 .329351 . 329023 . 328694 .328365 o. 328037 • 327709 .327381 .327053 . 326726 o. 326398 .326071 . 325743 .325416 . 325089 o. 324763 .324436 .324110 .323783 . 323457 0.323131 . 322806 . 322480 .322154 .321829 o. 321504 .321179 .320854 . 320529 . 320205 o. 319880 .319556 .319232 .318908 .318584 0.318260 .317937 .317613 .317290 .316967 0.316644 .316321 .315999 .315676 .315354 0.315032 .314710 .314388 .314066 .313745 0.313423 .313102 .312781 .312460 .312139 0.311818 Tan. 115' 64' 2 6 c COSINES, TANGENTS, AND COTANGENTS I53 c M. Sin. 10 ii 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 9- ,641842 .642101 , 642360 ,642618 . 642877 643135 ■ 643393 , 643650 ,643908 ,644165 ,644423 , 644680 644936 645193 645450 645706 645962 646218 646474 646729 646984 647240 . 647494 • 647749 . 648004 9.648258 .648512 . 648766 . 649020 . 649274 9. 649527 .649781 . 650034 .650287 • 650539 9- 650792 .651044 .651297 •651549 .651800 9. 652052 . 652304 .652555 . 652806 • 653057 9. 653308 .653558 • 653808 .654059 .654309 9- 654558 . 654808 .655058 . 655307 .655556 9*655805 . 656054 . 656302 .656551 • 656799 9. 657047 D. i' Cos. D. 1' 4.32 4.32 4.30 4.32 4.3o 4.3o 4.28 4.30 4.28 4.30 4.28 4.27 4.28 4.28 4.27 4.25 4-25 4.27 4-23 4.25 4.25 4-23 4.23 4.23 4-23 4-23 4.22 23 22 22 20 22 20 4- 4- 4- 4- 4- 4- 4. 22 4.20 4.18 4. 20 4. 20 4.18 4.18 4.18 4.18 4.17 4.17 4.18 4.17 4.15 4.17 4.17 4.15 4.15 4.15 4.15 4.13 4.15 4.13 4.13 Cos. D. 9. 95366o • 953599 •953537 . 953475 .953413 9- 953352 . 953290 • 953228 .953166 .953104 9. 953042 . 952980 .952918 • 952855 . 952793 9-95273I . 952669 . 952606 . 952544 .952481 9. 952419 • 952356 • 952294 .952231 .952168 9. 952106 .952043 .951980 .951917 .951854 9- 95i79i .951728 .951665 .951602 .951539 9-95I476 .951412 .951349 .951286 .951222 9-95II59 .951096 .951032 . 950968 • 950905 9- 950841 •950778 .950714 . 950650 .950586 9.950522 . 950458 .950394 .950330 .950- 6 9. 950202 .950138 .950074 .950010 • 949945 9.949S81 Sin. .02 .03 .03 .03 .02 •03 •03 .03 .03 •03 .03 • 03 •05 ■ 03 .03 •03 .05 .03 .05 .03 .05 .03 •05 •05 .03 .05 .05 .05 .05 .05 •05 .05 •05 • 05 .05 .07 •05 .05 .07 .05 .05 .07 .07 .05 .07 .05 .07 .07 .07 .07 .07 .07 .07 .07 • 07 .07 .07 .07 .08 .07 D. i' Tan. 9. 688182 . 688502 . 688823 .689143 . 689463 9.689783 .690103 . 690423 . 690742 . 691062 9-691381 .691700 .692019 • 692338 . 692656 9- 692975 • 693293 .693612 • 693930 . 694248 9- 694566 .694883 .695201 •695518 . 695836 9.696153 . 696470 . 696787 .697103 . 697420 9- 697736 . 698053 . 698369 . 698685 . 699001 9.699316 .699632 • 699947 . 700263 . 700578 9- 700893 .701208 .701523 .701837 .702152 9. 702466 . 702781 • 703095 • 703409 • 703722 9. 704036 • 704350 . 704663 • 704976 . 705290 9. 705603 . 705916 . 706228 . 706541 • 706854 9. 707166 Cot. D. 5-33 5-35 5-33 5-33 5-32 5-32 5-32 5-30 5-32 5-30 5-32 5-30 5.30 5.30 5-28 5-30 5-28 5.30 5-28 5-28 5-28 5-27 5-28 5-27 5-28 5-27 5-27 5-27 5-25 5-27 5-25 5-27 5.25 5-25 5-25 5-25 5-23 5-25 5-23 5-25 5-23 5-23 5-22 5-23 5.23 5-22 22 23 22 22 20 22 22 20 D. 1". Cot. 0.311818 .311498 .311177 .310857 .310537 o. 310217 . 309897 • 309577 . 309258 .308938 0.308619 . 308300 .307981 . 307662 • 307344 o. 307025 . 306707 .306388 . 306070 . 305752 o. 305434 .305117 . 304799 . 304482 . 304164 o. 303847 . 303530 .303213 . 302897 . 302580 o. 302264 . 301947 .301631 .301315 . 300999 o. 300684 . 300368 . 300053 . 299737 . 299422 o. 299107 . 298792 .298477 .298163 . 297848 o. 297534 .297219 . 296905 . 296591 . 296278 o. 295964 . 295650 . 295337 . 295024 . 294710 o. 294397 . 294084 . 293772 . 293459 .293146 o. 292834 Tan. 116 63* *7< LOGARITHMIC SINES 152' M. O I 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 117 9- 657047 . 657295 .657542 . 657790 .65S037 9.65S284 .658531 .658778 . 659 02 5 .659271 D. 1" 9- 659517 659763 . 660009 . 660255 . 660501 9. 660746 . 660991 .661236 .661481 .661726 9.661970 .662214 . 662459 . 662703 . 662946 9.663190 • 663433 • 663677 . 663920 .664163 9. 664406 . 664648 .664S91 .665133 • 665375 9.665617 .665859 .666100 . 666342 .666583 9. 666824 . 667065 . 667305 . 667546 . 667786 9. 668027 . 668267 . 668506 . 668746 . 668986 9. 669225 . 669464 • 669703 .669942 .670181 9.670419 . 670658 . 670896 .671134 .671372 9. 671609 Cos. 4.13 4.12 4.13 4.12 4.12 4.12 4.12 4.12 4. 10 4. 10 4. 10 4. 10 4. 10 4. 10 4.08 4.08 4.08 4.08 4.08 4.07 4.07 4.08 4.07 4.05 4.07 4-05 4.07 4-05 4-05 4-05 4.03 4-05 4.03 4-03 4-03 4-03 4.02 4.03 4.02- 4.02 4.02 4.00 4.02 4.00 4.02 4.00 3.93 4. 00 4.00 3.98 3.98 3.98 3.98 3.98 3-97 3.98 3-97 3-97 3-97 3.95 D. 1". Cos. 9. 949881 . 949816 • 949752 . 949688 • 949623 9- 949558 . 949494 • 949429 • 949364 . 949300 9- 949235 .949170 . 949105 . 949040 • 948975 9. 948910 . 948845 . 948780 .948715 . 948650 9. 948584 .948519 • 948454 .948388 • 948323 9- 948257 .948192 .948126 . 948060 •947995 9. 947929 . 947863 • 947797 .947731 . 947665 9. 947600 . 947533 . 947467 • 947401 . 947335 9. 947269 . 947203 .947136 • 947070 • 947004 9.946937 .946871 . 946804 .946738 . 946671 9. 946604 .946538 .946471 . 946404 . 946337 9. 946270 . 946203 .946136 . 946069 . 946002 9. 945935 Sin. D. 1". 1.08 1.07 1. 07 1.08 1.08 1.07 1.08 1.08 1.07 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1. 10 1.08 1.08 1. 10 1.08 1. 10 1.08 1. 10 1. 10 1.08 1. 10 1. 10 1. 10 1. 10 1. 10 1.08 1. 12 1. 10 1. 10 1. 10 1. 10 1. 10 1. 12 1. 10 1. 10 1. 12 1. 10 1. 12 1. 10 1. 12 1. 12 1. 10 1. 12 1. 12 1. 12 1. 12 1. 12 1. 12 1. 12 1. 12 1. 12 Tan. 9. 707166 . 707478 .707790 .708102 . 708414 9. 708726 • 709037 • 709349 . 709660 • 709971 9.710282 . 710593 . 710904 .711215 .711525 9-711836 . 712146 .712456 .712766 . 713076 9.713386 .713696 . 714005 .714314 .714624 9-7I4933 .715242 .715551 .715860 . 716168 9- 7i6477 .716785 .717093 .717401 • 717709 9.718017 .718325 .718633 . 718940 .719248 9- 719555 . 719862 . 720169 . 720476 . 720783 9.721089 .721396 .721702 . 722009 .722315 9. 722621 .722927 • 723232 . 723538 • 723844 9.724M9 • 724454 . 724760 . 725065 - 725370 9. 725674 Cot. D. 5.20 5.20 5.20 5.20 5.18 5-i8 5-i8 5-17 5.i8 5.17 5-i7 5-i7 5-i7 5.17 5 17 5 15 5 15 5 *7 5 15 5 15 5 15 5 15 5 13 5 15 5 13 5 13 5 13 5 13 5 13 5 13 5- 13 5- 12 5- 13 5- 12 5- 12 5- 12 5- 12 5- 12 5- IO 5- 12 5- IO 5- 12 5- IO 5- IO 5- 5- IO oS 5- IO 5- 5- 5- IO oS oS 5. 5- 5- IO oS 08 5.07 Cot. D. 1". 0. 292834 . 292522 . 292210 . 291898 . 291586 o. 291274 . 290963 .290651 . 290340 -. 290029 o. 289718 , 289407 . 289096 . 288785 .288475 o. 288164 .287854 . 287544 > 287234 . 286924 0.286614 . 286304 . 285995 . 285686 . 285376 o. 285067 . 284758 . 284449 . 284140 . 283832 o. 283523 .283215 . 282907 . 282599 . 282291 o. 281983 . 281675 . 281367 . 281060 . 280752 o. 280445 . 280138 .279831 . 279524 . 279217 o. 278911 . 278604 . 278298 . 277991 . 277685 o. 277379 . 277073 . 276768 . 276462 .276156 o. 275851 . 275546 . 275240 . 274935 . 274630 o. 274326 Tan. 62' 28° COSINES, TANGENTS, AND COTANGENTS 151* M. Sin. D. 1". Cos. D, 1". Tan. D. i". Cot M. Sin. 9.671609 I .671847 2 . 672084 3 .672321 4 .672558 5 9.672795 6 . 673032 7 .673268 8 • 673505 9 . 673741 10 9. 673977 11 .674213 12 . 674448 13 . 674684 M .674919 15 9-675I55 16 • 675390 17 . 675624 18 • 675859 19 .676094 20 9. 676328 21 . 676562 22 . 676796 23 . 677030 24 . 677264 25 9. 677498 26 .677731 27 . 677964 28 .678197 29 . 678430 3o 9.678663 3i .678895 32 .679 1 2S 33 . 679360 •34 • 679592 35 9. 679824 36 . 680056 37 . 680288 38 .680519 39 .680750 40 9. 6809S2 4i .681213 42 .681443 43 . 68 1 674 44 .681905 45 9. 6S2135 46 .682365 47 • 682595 48 . 6S2825 49 . 6S3055 50 9.683284 5i .683514 52 .683743 53 . 683972 54 .6S4201 55 9. 684430 56 . 684658 57 . 6S48S7 58 .685115 59 .685343 60 9.685571 Cos. 3-97 3-95 3-95 95 95 95 93 95 93 3-93 3.90 3-90 3. 88 3.87 3-88 3.87 3.87 3.87 3.87 3.87 3.85 3. 85 3-87 3.85 3.83 3.85 3.85 3.83 3.83 3.83 3.83 3.83 3.82 3.83 3.82 3.82 3.82 3.82 3.80 3.82 3.8o 3.8o 3.80 D. 1' 9- 945935 . 945868 . 945800 • 945733 . 945660 9- 945598 •945531 . 945464 • 945396 • 945328 9. 945261 .945193 .945125 • 945058 • 944990 9. 944922 . 944854 . 944786 .944718 • 944650 9. 944582 .9445M . 944446 • 944377 • 944309 9.944241 . 944172 .944104 . 944036 • 943967 9- 943899 • 94383° .943761 • 943693 • 943624 9- 943555 • 943486 .943417 • 943348 • 943279 9. 943210 •943 Mi . 943072 • 943003 .942934 9. 942864 • 942795 . 942726 .942656 . 942587 9-9425I7 • 942448 .942378 . 942308 • 942239 9.942169 .942099 .942029 •941959 .941889 9-94i8i9 Sin. 1. 12 1. 13 !•• 13 1. 12 1. 13 1. 13 1. 12 i- 13 i-i3 1. 13 i- 13 1. 13 1. 13 1. 13 1. 13 1. 13 1. 15 1. 13 1. 13 1. 15 i- 13 1. 13 1. 15 1. 13 1. 15 1. 15 1. 13 1. 15 1. 15 1. 15 1. 15 1. 15 1. 15 1. 15 i- 15 1. 15 1. 15 1. 15 1. 17 1. 15 i. 15 1. 17 i- 15 1. 17 15 ^7 17 15 17 17 17 17 1. 17 1. 17 D. I' 9- 725674 • 725979 . 726284 .726588 . 726892 9. 727197 • 727501 . 727805 . 728109 .728412 9. 728716 . 729020 .729323 . 729626 • 729929 9- 730233 • 730535 . 730838 .731141 .73M44 9.731746 • 732048 .732351 . 732653 .732955 9- 733257 • 733558 . 733860 • 734i62 • 734463 9- 734764 • 735o66 • 735367 .735668 • 735969 9- 736269 • 736570 . 736870 .737171 • 737471 9- 737771 .738071 • 738371 .738671 • 738971 9- 739271 • 739570 . 739870 . 740169 . 740468 9. 740767 . 741066 .741365 .741664 .741962 9. 742261 • 742559 .742858 .743156 • 743454 9- 743752 Cot. 5.08 5.08 5.07 5-07 5.08 5.07 5.07 5.07 5-05 5-07 5.o7 5-05 5-05 5-05 5- °7 5- °3 5-05 5-05 5.05 5-03 5.03 5.05 5- °3 5- °3 5.o3 5.02 5-03 5-03 5.02 5.02 5.03 5.02 5.02 5.02 5.00 5.02 5.00 5. 02 5.00 5.00 5.00 5.00 5- 00 5- 00 5.00 4.98 5.00 4.98 4.98 4.98 4.98 4- 98 4.9S 4-97 4.98 4-97 4.98 4-97 4-97 4-97 D. 1". o. 274326 .274021 .273716 .273412 j 57 .273108 I 56 0.272803 j 55 .272499 I 54 • 272195 .271891 . 271588 0.271284 50 . 270960 4g .270677 | 48 . 270374 . 270071 o. 269767 . 269465 . 269162 . 268859 .268556 o. 268254 . 267952 . 267649 . 267347 . 267045 o. 266743 . 266442 . 266140 . 265838 . 265537 o. 265236 30 . 264934 at .264633 a8 • 264332 37 . 264031 36 0.263731 35 .263430 I 34 .263130 . 262829 . 262529 o. 262229 20 .261929 j 19 . 261629 . 261329 .261029 o. 260729 . 260430 . 260130 .25983I j 12 .259532 " o. 259233 • 258934 . 258635 .258336 . 258038 o. 257739 . 257441 .257142 . 256844 . 256546 o. 256248 Tan. n8 ( 6i° 29' LOGARITHMIC SINES I50 l M. 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 Sin. 9- 6 S557i .685799 . 686027 . 686254 .686482 9. 686709 . 686936 .687163 .687389 .687616 9.687843 . 688069 .688295 .688521 . 688747 9. 688972 .689198 . 689423 . 689648 .689873 9. 690098 .690323 . 690548 .690772 . 690996 9.691220 .691444 .691668 .691892 .692115 9. 692339 .692562 .6927S5 . 69300S .693231 9- 693453 . 693676 .693898 .694120 . 694342 9 694564 . 694786 . 695007 . 695229 •695450 9.695671 . 695892 .696113 .696334 • 696554 9.696775 • 696995 .697215 • 697435 • 697654 9. 697874 . 698094 .698313 .698532 .698751 9. 698970 Cos. D. 1". Cos. 9418 19 941749 941679 941609 941539 941469 941398 941328 941258 941 187 941 1 17 941046 940975 940905 940834 940763 940693 940622 940551 940480 940409 940338 940267 940196 940125 940054 939982 9399 1 1 939840 939768 939697 939625 939554 939482 939410 939339 939267 939195 939123 939052 938980 938908 938836 938763 938691 938619 938547 938475 938402 938330 938258 938185 9381 13 938040 937967 937895 937822 937749 937676 937604 937531 D. 1". |j Sin. D. 1" D. 1". Tan. 9- 743752 . 744050 • 744348 • 744645 • 744943 9. 745240 . 745538 • 745835 .746132 • 746429 9. 746726 • 747023 .747319 .747616 .747913 9. 748209 . 748505 .748801 • 749097 • 749393 9. 749689 • 749985 .750281 .750576 . 750872 9-75ii67 .751462 .751757 • 752052 • 752347 9- 752642 - 752937 .753231 . 753526 • 753820 9-754II5 • 754409 . 754703 • 754997 •755291 9- 755585 .755878 • 756172 . 756465 • 756759 9- 757052 • 757345 • 757638 .757931 .758224 9.758517 . 758810 .759102 • 759395 • 759687 9- 759979 . 760272 • 760564 . 760856 .761148 9.76i439 Cot. D. 1". Cot. o. 256248 • 255950 • 255652 • 255355 • 255057 o. 254760 . 254462 . 254165 . 253868 .253571 o. 253274 . 252977 . 252681 . 252384 . 252087 o. 251791 . 251495 .251199 . 250903 . 250607 0.25031 1 . 250015 .249719 . 249424 .249128 o. 248833 . 248538 . 248243 . 247948 . 247653 o. 247358 . 247063 . 246769 . 246474 . 246180 o. 245885 . 245591 . 245297 . 245003 . 244709 o. 244415 . 244122 . 243828 . 243535 .243241 o. 242943 . 242655 . 242362 . 242069 . 241776 II o. 241483 10 .241190 . 240898 . 240605 .240313 o. 240021 . 239728 . 239436 .239144 . 238852 0.238561 D. 1 Tan. M. 119' 6o< 30 v COSINES, TANGENTS, AND COTANGENTS 149' M. Sin. 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 9. 698970 .699189 . 699407 . 699626 . 699S44 9. 700062 . 700280 . 7004 9S .700716 . 7°°933 9.701151 . 7 OI 3 68 . 701585 . 701802 . 702019 9. 702236 . 702452 . 702669 .702885 . 703 101 9- 703317 . 703533 • 703749 • 7039 6 4 .704179 25 ! 9-704395 o 1 2 3 4 5 6 I 9 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 . 704610 . 704825 . 705040 . 705254 9. 705469 • 705683 .705898 .706112 . 706326 9- 706539 • 706753 . 706967 .7071S0 . 707393 40 9. 41 . 42 . 43 • 44 • 45 9- 46 . 47 • 48 J • 49 • I 50 9- 5i 52 53 54 55 56 57 58 59 60 J20 9- 707606 707819 708032 708245 708458 708670 708882 709094 709306 709518 709730 709941 710153 710364 710575 710786 .710997 .711208 .711419 .711629 .711839 Cos. D. i> Cos. D. 1". Tan. D. i". 65 ^3 65 63 63 ^3 ^3 63 h 2 63 62 62 h 2 62 62 60 62 60 60 60 60 hO 58 58 hO 58 58 58 57 58 57 58 57 57 55 57 57 55 55 55 55 55 55 55 53 53 53 53 53 53 52 53 52 52 52 52 52 52 50 50 D. 1 9-937531 • 937458 . 937385 •937312 • 937238 9-937i65 . 937092 .937019 . 936946 . 936872 9. 936799 • 936725 . 936652 • 936578 • 936505 9-93643I .936357 . 936284 .936210 .936136 9. 936062 .935988 .9359H • 935840 • 935766 9- 935692 • 9356i8 • 935543 • 935469 . 935395 9- 935320 • 935246 .935171 . 935097 . 935022 9- 934948 . 934873 • 934798 . 934723 • 934649 9- 934574 .934^99 • 934424 • 934349 . 934-274 9- 934199 .934123 • 934048 • 933973 .933898 9. 933822 • 933747 .933671 - 933596 • 933520 •9-933445 • 933369 .933293 .933217 •933MI 9- 933o66 Sin. 22 22 22 23 22 22 22 22 23 22 23 22 23 22 23 23 22 23 23 23 23 23 23 23 23 23 25 23 23 25 23 25 23 25 23 25 25 25 23 25 25 25 25 25 25 27 25 25 25 ,27 25 27 25 ,27 25 .27 ,27 .27 .27 .25 D. 1 9.761439 .761731 . 762023 .762314 . 762606 9- 762897 .763188 • 763479 .763770 . 764061 9- 764352 . 764643 • 764933 • 765224 • 7655H 9. 765805 • 766095 . 766385 . 766675 . 766965 9- 767255 . 767545 • 767834 .768124 .768414 9- 768703 . 768992 . 769281 .769571 . 769860 9. 770148 . 770437 . 770726 .771015 .771303 9. 771592 .771880 .772168 • 772457 . 772745 9- 773033 .773321 . 773608 . 773896 . 774184 9.774471 • 774759 • 775046 . 775333 .775621 9- 775908 .776195 . 776482 . 776768 - 777055 9- 777342 - 77762S .777915 .778201 . 778488 9- 778774 Cot. 4.87 4.87 ^5 ^7 85 ^5 85 4 4 4 4 4 4.85 4.85 4.85 4.85 4-83 4.85 4.83 4.85 4.83 4.83 4.83 4-83 4.83 4.83 4.82 4.83 4-83 4.82 4.82 4.82 4.83 4.82 4.80 4. 82 4.82 4.82 4. So 4.82 4.80 4.80 4. 82 4.80 4.80 80 78 So 80 7^ So 78 7^ so 78 78 78 77 7^ 78 77 78 77 78 4-77 Cot. D. 1 o. 238561 . 238269 • 237977 . 237686 . 237394 0.237103 . 236812 • 236521 • 236230 • 235939 o. 235648 . 235357 . 235067 . 234776 . 234486 o. 234195 . 233905 . 233615 . 233325 . 233035 o. 232745 . 232455 . 232166 .231876' .231586 0.231297 . 231008 .230719 . 230429 . 230140, o. 229852 . 229563 . 229274 . 2289S5 . 228697 o. 228408 . 228120 . 227832 . 227543 . 227255 o. 226967 . 226679 . 226392 . 226104 . 225816 o. 225529 . 225241 . 224954 . 224667 . 224379 o. 224092 . 223805 .223518 .223232 . 222945 o. 222658 .222372 . 2220S5 . 221799 . 221512 o. 221226 Tan. M. 59' 31' LOGARITHMIC SINES I48 c M. Sin. 10 ii 12 13 M 15 16 : 7 a 19 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 46 47 ! 48 49 I 5o 5i 52 53 54 56 57 58 59 60 7 1 1839 712050 712260 7 1 2469 712679 712889 713098 71330S 7I35I7 713726 713935 714144 714352 714561 714769 714978 715186 715394 715602 715809 716017 716224 716432 716639 716846 717053 717259 717466 717673 717879 718085 718291 718497 718703 718909 719114 719320 719525 719730 719935 720140 720345 720549 720754 720958 721162 721366 721570 721774 721978 722181 722385 722588 722791 722994 723197 723400 723603 723805 724007 724210 Cos. D. 1". D. 1 Cos. 9. 933066 • 932990 .932914 .932838 • 932762 9. 9326S5 . 932609 • 932533 . 932457 . 932380 9. 932304 .932228 .932151 • 932075 .931998 9-93I92I .931845 .931768 .931691 .931614 9- 931537 .931460 •931383 .931306 .931229 9-93II52 .93io75 • 930998 .930921 . 930843 9- 930766 . 930688 .930611 • 930533 - 930456 9- 930378 . 930300 • 930223 .930145 . 930067 9- 929989 .929911 • 929833 .929755 • 929677 9- 929599 . 929521 . 929442 . 929364 . 929286 9. 929207 .929129 . 929050 .928972 .928893 9.928815 . 92S736 .928657 .928578 . 928499 9. 928420 Sin. D. i' 1.27 1.27 1.27 1.27 1.28 1.27 1.27 27 1.28 1.27 1, 1. 1. 1. 1. 1. 1. 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.28 1.30 1.28 30 28 1. 1. 1.30 1.28 1.30 1.30 1.28 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 3.30 1.32 1.30 1.30 1.32 1.30 1.32 1.30 1.32 1.30 1.32 1.32 1.32 1.32 1.32 D. 1". Tan. 9- 77S774 . 779060 • 779346 . 779632 .779918 9. 780203 . 780489 • 780775 .781060 .781346 9-781631 .781916 . 782201 . 782486 .782771 9. 783056 .783341 .783626 .783910 .784195 9- 784479 . 784764 . 785048 • 785332 . 785616 9- 785900 . 786184 D. 1' . 786752 . 787036 9. 787319 . 787603 . 787886 .788170 .788453 9-788736 . 7S9019 . 789302 .7895.85 . 789868 9.790151 . 790434 . 790716 • 790999 .791281 9-79I563 . 791846 .792128 . 792410 . 792692 9. 792974 . 793256 • 793538 .793819 .794101 9- 794383 . 794664 - 794946 . 795227 - 7955o8 9- 795789 Cot. 4-77 4-77 4-77 4-77 4-75 4-77 4-77 4-75 4-77 4-75 4-75 4-75 4-75 4-75 4-73 4-75 4-73 4-73 4-73 4-73 4-73 4-73 4-73 4-73 4.72 4-73 4-72 4-73 4.72 4.72 4.72 4-72 4.72 4-72 4-72 4.72 4.7o 4-72 4-70 4.70 4.72 4.70 4.70 4.70 4-70 4.70 4.7o 4.68 4.70 4.7o 4.68 4.70 4.68 4.68 4.68 D. 1". Cot. 0.221226 . 220940 . 220654 . 220368 . 220082 o. 219797 .219511 . 219225 .218940 .218654 o. 218369 .218084 .217799 .217514 .217229 o. 216944 .216659 .216374 . 216090 . 215805 o. 215521 .215236 .214952 . 214668 . 214384 o. 214100 .2j 3 Si6 .213532 . 213248 . 212964 o. 2126S1 .212397 . 212114 . 21 1830 .211547 0.21 1 264 . 2IO981 . 210698 .210415 . 2I0I32 o. 209849 . 209566 . 209284 . 20900 I . 208719 o. 208437 . 208154 . 207872 . 207590 . 207308 o. 207026 . 206744 . 206462 . 206181 . 205899 o. 205617 . 205336 . 205054 . 204773 . 204492 o. 20421 1 Tan. INI. 5 8° 3 2° COSINES, TANGENTS, AND COTANGENTS 147* M. Sin. D. 1". I Cos. D. 1". Tan. D. 1". Cot. 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 122 9,724210 .724412 .724614 .724816 . 725017 9.725219 . 725420 .725622 . 725823 . 726024 9. 726225 . 726426 . 726626 , 726827 . 727027 9. 727228 . 727428 . 727628 .727828 .728027 9, 728227 . 728427 , 728626 .728825 . 729024 9. 729223 . 729422 .729621 . 729820 . 730018 9. 730217 . 730415 . 730613 .730811 .731009 9. 731206 . 731404 .731602 .731799 . 731996 9- 732193 . 732390 . 732587 . 732784 • 732980 9.733177 • 733373 • 733569 • 733765 • 733961 9- 734157 . 734353 • 734549 . 734744 • 734939 9- 735135 . 735330 • 735525 .735719 • 7359H 9. 736109 Cos. 3-37 3-37 3-37 3-35 3-37 3-35 3-37 3-35 3-35 3-35 3-35 3-33 3-35 3-33 3-35 3-33 3-33 3-33 3-32 3-33 3-33 3-32 3.32 3-32 3.32 3-32 3-32 3-32 3-30 3-32 30 30 30 30 28 30 30 3.28 3-28 3.28 28 28 28 27 2S 27 27 27 27 27 3-27 3-27 3-25 3-25 3-27 3-25 D. 1". 9. 928420 . 928342 . 928263 .928183 .928104 9. 928025 . 927946 . 927867 .927787 . 927708 9. 927629 • 927549 .927470 . 927390 .927310 9.927231 .927151 .927071 . 926991 .926911 9. 926831 .926751 .926671 . 926591 .926511 9.9264.31 .926351 . 926270 .926190 .926110 9. 926029 • 925949 . 925868 .925788 • 925707 9. 925626 • 925545 • 925465 • 925384 • 925303 9.925222 .925141 . 925060 • 924979 . 924897 9.924816 • 924735 • 924654 .924572 • 92449 1 9. 924409 . 924328 . 924246 , 924164 . 924083 9. 924001 .923919 • 923837 • 923755 • 923673 9- 923591 Sin. D. i" 9- 795789 . 796070 .796351 . 796632 .796913 9. 797194 • 797474 • 797755 . 798036 .798316 9- 798596 . 798877 • 799157 . 799437 .799717 9- 799997 . 800277 . 800557 . 800836 .801116 9. 801396 . 801675 . 801955 . S02234 .802513 9. 802792 . 803072 •803351 . 803630 . 803909 9.804187 . 804466 . 804745 . 805023 • 805302 9. 805580 .805859 .806137 . 806415 . S06693 9. 806971 . 807249 • 807527 . 807805 . 808083 9. 808361 . 80S638 . 808916 . 809193 .809471 9. 809748 .810025 .810302 . 810580 .810857 9.811134 .811410 .Si 1687 .811964 .812241 9-812517 Cot. 4.68 4.68 4.68 4.68 4.68 4.67 4.68 4.68 4.67 4.67 4.68 4.67 4.67 4.67 4.67 4.67 4.67 4.65 4.67 4.67 4.65 4.67 4.65 4.65 4.65 4.67 4.65 4.65 4.65 4.63 4.65 4.65 4.63 4.65 4.63 4.65 4.63 4.63 4-63 4-63 4.63 4.63 4-63 4.63 4-63 4.62 4-63 ,4.62 4-63 4.62 4.62 4.62 4-63 4.62 4.62 4.60 4.62 4.62 4.62 4.60 D. i". 0.20421 1 60 . 203930 59 . 203649 5« - 203368 57 . 203087 56 0. 202806 55 .,202526 54 . 202245 53 . 201964 52 . 201684 51 0. 201404 50 . 201123 49 . 200843 48 . 200563 47 . 200283 40 0. 200003 45 . 199723 44 . 199443 43 . 199164 42 . I9S884 41 0. 198604 40 • 198325 39 . 198045 3« . 197766 37 . 197487 3b 0. 197208 35 . 196928 34 . 196649 33 . 196370 32 . 19609 I 3i 0. 195813 30 . 195534 29 . 195255 28 . 194977 27 . 194698 2b 0. 194420 25 . 194141 24 . 193863 23 . 193585 22 . 193307 21 0. 193029 20 . 192751 \l . 192473 . 192195 17 .191917 ib 0. 191639 15 . 191362 14 . 191084 13 . 190807 12 . 190529 11 0. 190252 10 . IS9975 1 . 189698 . 189420 7 . 189143 b 0. 1 88866 5 . 188590 4 .188313 3 . 188036 2 . 187759 1 0. 1S74S3 Tan. M. 57' 33 c LOGARITHMIC SINES 146° M, o 1 2 3 4 5 6 I 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 Sin. 9. 736109 . 736303 .736498 . 736692 . 736886 9. 737080 • 737274 • 737467 . 73766i • 737855 9. 738048 • 738241 • 738434 . 738627 . 738820 9- 739oi3 • 739206 • 739398 • 739590 • 739783 9- 739975 . 740167 • 740359 • 740550 . 740742 9- 740934 .741125 .741316 .741508 . 741699 9. 741889 . 742080 .742271 . 742462 . 742652 9.742842 • 743033 • 743223 • 743413 • 743602 9- 743792 • 743982 .744171 . 74436i • 744550 9- 744739 . 744928 .745117 • 7453o6 • 745494 9- 745683 .745871 . 746060 . 746248 • 746436 9- 746624 .746812 • 746999 • 747187 • 747374 9- 747562 D. 1 3-23 3-25 3.23 3-23 3.23 3-23 3.22 3.23 3.23 3-22 3.22 3.22 3-22 3.22 3-22 3.22 3- 20 3-20 3.22 3.20 3.20 3.20 3.18 3- 20 3.20 ! 3.i8 J 3.18 ; 3.20 j 3.18 3.17 3.18 3.18 3.18 3-17 3-17 3.18 3-i7 3.17 3-i5 3.17 3.17 3.15 3.17 3.15 3.15 3.15 Cos. Cos. 3.13 3.15 3.13 3.13 3.13 3.13 3.12 3.13 3.12 3.13 D. 1". 9- 923591 • 923509 . 923427 . 923345 . 923263 9-923181 . 923098 .923016 . 922933 . 922851 9. 922768 . 922686 . 922603 .922520 • 922438 9- 922355 . 922272 .922189 .922106 . 922023 9. 921940 .921857 .921774 .921691 .921607 9.921524 .921441 .921357 .921274 .921190 9. 921107 .921023 . 920939 . 920856 . 920772 9. 9206S8 . 920604 . 920520 • 920436 . 920352 9. 920268 . 920184 . 920099 .920015 .919931 9.919846 .919762 .919677 • 919593 . 919508 9. 919424 • 919339 .919254 .919169 • 919085 9.919000 .918915 . 918830 .918745 • 9*8659 9- 918574 D. 1". Tan. I Sin. 1-37 1-37 1-37 1-37 1-37 1.38 1-37 1.38. i-37 1.38 i-37 1.38 1.38 1-37 1.38 1.38 1.38 1.38 1.38 1.38 1.38 38 38 40 38 38 40 38 40 38 1.40 1.40 1.38 1.40 1.40 1.40 1. 40 1.40 1.40 1.40 1.40 1.42 1.40 1. 40 1.42 1. 40 1.42 1. 40 1.42 1.40 1.42 1.42 1.42 1.40 1.42 1.42 1.42 1.42 1.43 1.42 D. 1". 9.812517 .812794 .813070 •813347 .813623 9.813899 .814176 .814452 .814728 .815004 9.815280 • 815555 .815831 .816107 .816382 9.816658 • 816933 .817209 . 817484 •817759 9.818035 .818310 .818585 . 818860 .819135 9..819410 .819684 . 819959 . 820234 . 820508 9. 820783 .821057 .821332 .821606 .821880 9.822154 . 822429 . 822703 . 822977 • 823251 9. 823524 .823798 . 824072 • 824345 .824619 9. 824893 . 825166 • 825439 .825713 . 825986 9. 826259 .826S32 . 826805 . 827078 .827351 9. 827624 .827897 .828170 .828442 .828715 9. 828987 Cot. D. 1' Cot. 4.62 4.60 4.62 4.60 4.60 4.62 4.60 4.60 4.60 4.60 4.58 4.60 4.60 4.58 4.60 4.58 4. 60 4.58 4.58 4.60 4.58 4.58 4.58 4.58 4.58 4-57 4.58 4.58 4-57 4.58 4.57 4.58 4-57 4-57 4-57 58 57 57 57 55 4-57 4-57 4-55 4-57 4-57 4-55 4- 55 4-57 4-55 4-55 4-55 4- 55 4-55 4-55 4-55 4-55 4-55 4-53 4-55 4-53 o. 187483 j 60 . 187206 . 186930 . 186653 • 186377 o. 186101 . 185S24 . 185548 . 185272 . 184996 o. 184720 . 184445 . 184169 . 183893 . 183618 o. 183342 . 183067 . 182791 . 182516 . 182241 o. 181965 . 181690 .181415 . 181 140 . 180865 o. 180590 . 1803 16 . 1 8004 1 . 179766 • 179492 o. 179217 .178943 . 178668 . 178394 . 1 781 20 o. 177846 • 177571 . 177297 . 177023 . 176749 o. 176476 . 176202 . 175928 • 175655 • 175381 o. 175107 . 174834 . I 74561 . 174287 59 58 57 56 55 54 53 52 5i 50 49 48 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 M 13 174014 j 11 D. 1 o. 173741 • 173468 .173195 . 172922 . 172649 o. 172376 . 172103 . 171830 .171558 . 17 1 285 o. 171013 Tan. 10 9 8 7 6 5 4 3 2 1 123 56 c 34° COSINES, TANGENTS, AND COTANGENTS 145' M. Sin. D. 1". I Cos. D. 1". Tan. D. 1". Cot. 9. 747562 I . 747749 2 • 747936 3 .748123 4 . 748310 5 9. 748497 6 .748683 7 . 74S870 8 . 749 56 9 • 749243 10 9. 749429 11 • 7496i5 12 . 749S01 13 . 749987 14 .750172 15 9- 750358 ifa . 750543 17 . 750729 18 ■ 750914 19 . 751099 20 9. 751284 21 .75H69 22 . 75i654 23 .751839 24 • 752023 25 9- 752208 2b • 752392 27 • 752576 28 . 752760 29 . 752944 30 9.753128 3i .753312 32 • 753495 33 • 753679 34 .753862 35 9. 754046 3b • 754229 37 .754412 3» • 754595 39 .754778 40 9- 75496o 4i .755143 42 .755326 43 •755508 44 • 755690 45 9- 755872 46 • 756054 47 . 756236 48 .756418 49 . 756600 50 9. 756782 5i • 756963 52 •757144 53 .757326 54 . 757507 55 9. 757688 5b • 757869 57 • 758050 5« . 758230 59 .758411 60 9- 758591 Cos. 3.12 3.12 3 3 3 3 3 3 3 3 3.10 3.10 3-o8 3- 10 3.08 3-io 3- 08 3.08 3.08 3.08 3.08 3.08 3.07 3.08 3.07 3.07 3.07 3-07 3.07 3.07 3.05 3- 07 05 °7 05 3.05 3-05 3.05 3.03 3.05 3.05 3.03 3.03 3.03 3.03 3.03 3.03 3.03 3.03 02 02 03 02 02 3. 3. 3- 3- 3- 3.02 3- 02 3.00 3.02 3.00 D. i". 8574 8489 8404 8318 8233 S147 8062 7976 7891 7805 7719 7634 7548 7462 7376 7290 7204 7118 7032 6946 6859 6773 6687 6600 6514 6427 6341 6254 6167 6081 5994 5907 5820 5733 5646 5559 5472 5385 5297 5210 5123 5035 4948 4860 4773 4685 4598 45 10 4422 4334 4246 4158 4070 3982 3S94 3806 37i8 3630 354i 3453 3365 Sin. D. 1". 9. 828987 . 829260 • 829532 . 829805 . 830077 9- 830349 .830621 .830893 .831165 .83M37 9.831709 .831981 . 832253 . 832525 • 832796 9.833068 • 833339 .833611 .833882 • 834154 9- 834425 • 834696 • 834967 .835238 • 835509 9- 835780 .836051 • 836322 . 836593 . 836864 9.837134 . 837405 .837675 • 837946 .838216 9.838487 .838757 . 839027 • 839297 .839568 9.839838 .840108 .840378 . 840648 .840917 9. 841187 .841457 .841727 .841996 .842266 9- 842535 . 842805 • 843074 • 843343 .843612 9.843882 .844151 .844420 . 844689 • 844958 9- 845227 Cot. 4-55 4-53 4-55 4-53 4-53 4-53 4-53 4-53 4-53 4-53 4-53 4-53 4-53 4.52 4-53 4.52 4-53 4.52 4-53 4.52 4.52 4.52 4.52 4.52 4.52 4.52 4.52 4.52 4.52 4.50 4.52 4-5° 4-52 4-5° 4.52 4.50 4-5° 4.50 4o2 4-50 4.50 4.50 4-50 4.48 4- 50 4.50 4-5o 4.48 4.50 4.48 4.50 4.48 4.48 4.48 4.50 4.48 4.48 4.48 4.48 4.48 D. i". o. 171013 . 170740 . 170468 . 170195 • 169923 o. 16965 I . 169379 . 169107 . 168835 . 168563 o. 168291 . 168019 . 167747 . 167475 . 167204 o. 166932 . 166661 . 166389 .166118 . 165846 o. 165575 . 165304 . 165033 . 164762 . 164491 o. 164220 . 163949 . 163678 . 163407 . 163136 o. 162866 . 162595 . 162325 . 162054 . 161784 0.161513 . 161243 . 160973 . 160703 . 160432 o. 160162 . 159892 . 159622 . 159352 . 159083 o. 158813 . 158543 . 158273 . 158004 • 157734 o. 157465 .157195 . 156926 . 156657 . I563S8 o. 156118 . 155849 . 155580 .155311 . 155042 o. 154773 Tan. 124' 55' 35 v LOGARITHMIC SINES 144 M. Sin. o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 51 52 I 53 54 55 56 57 58 59 60 "5 9- 758591 .758772 •758952 .759132 •759312 9- 759492 • 759672 • 759852 .760031 .760211 9. 760390 • 760569 . 760748 . 760927 .761106 9.761285 . 761464 .761642 .761821 .761999 9.762177 . 762356 • 762534 .762712 . 762889 9. 763067 .763245 . 763422 . 763600 . 763777 9- 763954 .764131 • 764308 . 764485 . 764662 9.764838 • 765015 .765191 • 765367 • 765544 9- 765720 .765896 . 766072 . 766247 . 766423 9- 766598 • 766774 • 766949 .767124 .767300 9- 767475 • 767649 . 767824 • 767999 .768173 9.768348 . 768522 . 768697 .768871 • 769045 9.769219 Cos. D. 1' 2.98 2.98 2.98 2.98 2.98 2.98 2.97 2.98 2.97 2-97 2.98 2.97 2.97 2-95 2.97 2.97 2-95 2.97 2.95 2-95 2.95 2-95 2-95 2.95 2-93 2.95 2.93 2-93 2.95 2.93 2.93 2.93 2.92 2-93 2.92 2.93 2.92 2.92 2-93 2.92 2.90 2.92 2.92 2.90 2.92 2.90 2.92 2.90 2.90 2.90 D. 1 Cos. 9- 9^365 913276 913187 913099 913010 912922 912833 912744 912655 912566 912477 912388 912299 912210 912121 912031 911942 91 1853 911763 91 1674 911584 9^495 91 1405 911315 911226 91 1 136 91 1046 910956 910866 910776 910686 910596 910506 910415 910325 910235 910144 910054 909963 909873 909782 909691 909601 909510 909419 909328 909237 909146 909055 908964 908873 908781 908690 908599 908507 908416 908324 908233 908141 908049 907958 Sin. D. 1' Tan. D. 1". Cot. 48 48 47 48 47 48 48 43 48 48 48 43 43 48 50 48 48 50 48 50 48 50 50 43 50 50 50 50 50 50 50 50 52 50 50 52 50 52 50 52 52 50 52 52 52 52 52 52 52 52 53 52 52 53 52 53 52 53 53 52 D. 1". 9. 845227 * . 845496 .845764 • 846033 . 846302 9. 846570 .846839 .847108 .847376 • 847644 9- 847913 .848181 . 848449 .848717 . 848986 9. 849254 • 849522 • 849790 • 850057 • 850325 9- 850593 . 850861 .851129 ,851396 .851664 9-851931 .852199 . 852466 • 852733 . 853001 9.853268 • 853535 .853802 • 854069 • 854336 9. 854603 .854870 .855137 • 855404 .855671 9- 855938 . 856204 .856471 .856737 . 857004 9. 857270 .857537 .857803 . 858069 .858336 9. 858602 . 858868 .859134 . 859400 . 859666 9- 859932 .860198 . 860464 . 860730 .860995 9. 861 261 Cot. 4.48 4-47 4.48 4.48 4-47 4.48 4.48 4-47 4-47 4.48 4-47 4-47 4-47 4.48 4-47 4.47 4.47 4-45 4-47 4-47 4-47 4-47 4-45 4-47 4-45 4-47 4-45 4-45 4-47 4-45 4-45 4-45 4-45 4-45 4-45 4-45 4-45 4-45 4-45 4-45 4-43 4-45 4-43 4-45 4-43 4-45 4-43 4-43 4-45 4-43 4-43 4-43 4-43 4-43 4-43 4-43 4-43 4-43 4.42 4-43 D. 1". o. 154773 154504 154236 153967 153698 153430 153161 152892 152624 152356 152087 151S19 !5i55i 151283 151014 150746 150478 150210 149943 149675 149407 I49I39 14887 1 148604 148336 148069 147801 147534 147267 146999 146732 146465 146 1 98 14593 1 145664 145397 I45I30 144863 144596 144329 144062 143796 143529 143263 142996 142730 142463 142197 I4I93I 141664 141398 141132 140866 140600 140334 140068 139802 139536 139270 139005 138739 Tan. INI 54' 36 c COSINES, TANGENTS, AND COTANGENTS 143* M. 10 11 12 13 14 x 5 Sin. D. 1". 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 126' 9.7(39219 769393 769566 769740 769913 770087 770260 770433 770606 770779 770952 771125 771298 771470 771643 771815 771987 772159 772331 772503 772675 772S47 77301S 773190 77336 1 773533 773704 773875 774046 774217 774388 774558 774729 774899 775070 775240 7754IO 77558o 775750 775920 776090 776259 776429 776598 776768 776937 777106 777275 777444 777613 777781 777950 77S119 77S2S7 778455 77S624 778792 778960 779128 779295 779463 ! Cos. 2.88 2.88 2.87 2.S8 2.S7 2.S7 2.87 2.87 2.S7 2. 87 2.87 2.85 2. S7 2.85 2.87 2.85 2.85 2.85 2.85 2.85 2.83 2.S5 2.83 2.85 2.83 2.83 2. S3 2. S3 2.83 2. S3 2.82 2.83 2.82 2.83 2.82 2. 82 2.S2 2.82 2.S2 2. So 2.82 2. 82 2.80 2.80 2.82 2. SO 2.80 2. SO 2.78 2.8o Cos. D. Tan. D. 1". Cot. D. 1' 9. 90795S . 907S66 . 907774 . 907682 .907590 9. 907498 . 907406 ,907314 ,907222 .907129 907037 906945 906S52 '906760 906667 906575 9064S2 9063S9 906296 906204 9061 1 1 906018 905925 905832 905739 905645 905552 905459 905366 905272 905179 9 5oS5 904992 904S98 904804 9047 1 1 904617 904523 904429 904335 904241 904147 904053 903959 903S64 903770 903676 903581 903487 903392 903298 903203 903108 903014 902919 902824 902729 902634 902539 902444 902349 Sin. 53 53 53 53 53 53 53 53 55 53 53" 55 53 55 53 55 55 55 53 55 55 55 55 55 57 55 55 55 57 55 57 55 57 57 55 57 57 57 57 57 57 57 57 5^ 57 57 58 57 58 57 58 5S 57 58 58 58 58 58 58 5^ D. 1". 9.S61261 .861527 .861792 . 86205S .862323 9.S62589 . 862854 .863119 .863385 .863650 9- 863915 .864180 . S64445 .864710 • 864975 9. 865240 . 865505 . S65770 . 866035 . 866300 9. 866564 . 866829 . 867094 .867358 . 867623 9. 867887 .868152 .868416 . 8686S0 .868945 9. 869209 . 869473 .869737 .870001 . 870265 9. 870529 • 870793 .S71057 .871321 .S71585 9- 871849 .872112 . 872376 . 872640 .87 2903 9.873167 • 873430 .873694 .873957 . 874220 9. 874484 • 874747 . 875010 .S75273 • S75537 9. 875800 . 876063 .876326 .876589 .876852 9.S77114 Cot. 4.43 4.42 4.43 4.42 43 42 42 43 42 4.42 4.42 4.42 4.42 4.40 4.42 4.40 4.38 4.40 4.4o 4- 38 4.40 4.38 4.3S 4.38 4.40 4.38 4. 38 4- 38 4.38 4.38 4-37 D. i". o. 138739 • 138473 . 138208 . 137942 • 137677 o. 137411 • 137146 . I 3688 I . 136615 . 136350 o. 136085 50 • 135820 49 • 1 35555 48 • 135290 47 .135025 J 46 o. 134760 • 134495 . 134230 • 133965 . 133700 o. 133436 .133171 . 132906 . 132642 • 132377 o. 132113 . 131848 • 13 1 584 .131320 • 131055 o. 130791 . 130527 . 130263 • 129999 .129735 o. 1 2947 I . 129207 . 128943 . 128679 . 1 284 15 o. 128151 . I278S8 . 127624 . 1 27360 . 127097 o. 126833 .126570 . 126306 .126043 . 125780 o. 125516 .125253 . 1 24990 .124727 .124463 o. 124200 .123937 .123674 .123411 .123148 o. 1228S6 Tan. M. 53' 37^ LOGARITHMIC SINES 142 M. Sin. D.I"., I I Cos. D. i' o 1 2 3 4 5 6 7 8 9 10 11 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 9- 779463 .779631 •77979$ . 779966 .780133 9. 780300 . 780467 . 780634 .780801 . 780968 9.78ii34 .781301 .781468 .781634 .781800 9.781966 .782132 . 782298 . 782464 . 782630 9. 782796 . 782961 .783127 . 783292 • 783458 9. 783623 .783788 • 783953 .784118 . 784282 9. 784447 .784612 . 784776 . 784941 .785105 9. 785269 • 785433 • 785597 .785761 .785925 9. 786089 . 786252 .786416 .786579 .786742 9. 786906 . 787069 .787232 .787395 .787557 9. 787720 .787883 .788045 . 788208 .788370 9- 788532 . 788694 . 788856 .789018 .789180 9- 789342 Cos. 2.80 2.78 2.80 2.78 2.78 2.78 2.78 2.78 2.78 2.77 2.78 2.78 2.77 2.77 2.77 2/77 2.77 2.77 2.77 2.77 2.75 2.77 2.75 2.77 2.75 2.75 2.75 2-75 2-73 2.75 2-75 2.73 2-75 2.73 2.73 2-73 2.73 2-73 2-73 2.73 2.72 2-73 2.72 2.72 2-73 2.72 2.72 2.72 2.70 2.72 2.72 2.70 2.72 2.70 2.70 2.70 2.70 2.70 2.70 2.70 D. 1". 902349 902253 902158 902063 901967 901872 901776 90 168 1 901585 901490 901394 901298 901202 901 106 901010 900914 900818 900722 900626 900529 900433 9O0337 900240 900144 900047 899951 899854 899757 899660 899564 899467 899370 S99 2 73 899176 899078 898981 898787 898592 898494 898397 898299 898202 898104 898006 897908 897810 897712 897614 897516 897418 897320 897222 897123 897025 896926 896828 896729 896631 896532 Sin. .60 .58 .58 ,60 58 .60 58 60 58 60 60 60 60 .60 60 60 60 60 62 60 60 62 60 62 60 62 62 62 60 62 62 62 62 63 62 62 62 63 62 63 62 63 62 63 63 63 63 63 63 63 63 63 63 65 63 65 63 65 63 65 Tan. D. 1". Cot. 9. 8771 14 .877377 . 877640 . 877903 .878165 9. 878428 .878691 .878953 .879216 .879478 9. 879741 . 880003 . 880265 . 880528 . 880790 9.881052 .881314 .881577 .881839 .882101 9.882363 . 882625 . 882887 .883148 .883410 9.883672 .883934 .884196 .884457 . 884719 9. 884980 .885242 .885504 . 885765 . 886026 9. 886288 .886549 .886811 . 887072 .887333 9. 887594 .887855 .888116 . 888378 . 888639 9. 888900 . 889421 D. 1 • 889943 9. 890204 . 890465 • 890725 . 890986 .891247 9.891507 .891768 . 892028 . 892289 . 892549 9.892810 Cot. D. 1". 22886 22623 22360 22097 21835 21572 21309 21047 20784 20522 20259 19997 19735 19472 19210 18948 18686 18423 18161 17899 17637 17375 17113 16852 16590 16328 16066 15804 15543 15281 15020 14758 14496 14235 13974 13712 1 345 1 13189 12928 12667 12406 12145 1 1 884 11622 11361 1 1 100 10839 10579 10318 10057 09796 09535 09275 09014 08753 08493 08232 07972 0771 1 o745i 07190 Tan. 127 5* 38 c COSINES, TANGENTS, AND COTANGENTS 141' M. Sin. D. 1' Cos. 9. 789342 • 789504 . 7S9665 . 7S9827 . 7S9988 9. 790149 • 79°3 10 . 790471 . 790632 • 790793 9- 790954 .791115 •791275 . 791436 • 79I59 6 9- 791757 .791917 . 792077 . 792237 • 792397 9- 792557 .792716 . 792876 • 793035 • 793195 9- 793354 .7935H • 793673 • 793S32 • 793991 9- 794i5o . 7943oS • 794467 . 794626 • 794784 9. 794942 .795101 • 795259 •795417 • 795575 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 9. 795733 • 795891 • 796049 . 796206 • 796364 9. 796521 • 796679 .796836 • 796993 .797150 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 I28 ( 9- 797307 • 797464 . 797621 •797777 ■ 797934 9.798091 • 798247 • 798403 • 798560 .798716 9- 798872 Cos. 2. 70 2.68 2. 70 2.68 2.68 2.68 2.68 2.68 2.68 2.68 2.68 2.67 2.68 2.67 2.68 2.67 2.67 2.67 2.67 2.67 2.65 2.67 2.65 2.67 2.65 2.67 2.65 2.65 2.65 2.65 2.63 ^5 65 ^3 63 05 63 ^3 63 63 2.63 2.63 2.62 2.63 2.62 2.63 2.62 2.62 2.62 2.62 2.62 2.62 2.60 2.62 2.62 2.60 2.60 2.62 2.60 2.60 D. 1". 9. 896532 • 896433 • 896335 . 896236 .896137 9.896038 • 895939 . 895840 .S95741 .895641 9- s 95542 • s 95443 • 895343 • 895244 • 895145 9. 895045 • 894945 D. 1". • 894746 . 894646 9. 894546 .894446 • 894346 . 894246 .894146 9. 894046 • 893946 .893846 • 893745 •893645 9- 893544 • 893444 • 893343 • 893243 .893142 9.893041 . 892940 .892839 • 892739 . 892638 9.892536 •892435 .892334 . 892233 .892132 9. 892030 .891929 .891827 .891726 .891624 9. 891523 .891421 .891319 .891217 .891115 9-891013 .890911 . 890809 . 890707 . 890605 9- 890503 Sin. 1-65 1.63 1.65 T.65 1.65 1.65 1.65 1.65 1.67 1.65 1.65 1.67 1.65 1.65 1.67 1.67 1.65 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.68 1.67 1.68 1.67 1.68 1.67 1.68 1.68 1.68 1.68 1.67 1.68 1.70 1.68 1.68 1.68 1.68 1.70 1.68 1.70 1.68 1.70 1.68 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1. 70 1.70 D. 1". Tan. 9.892810 . 893070 .893331 . 893591 .893851 9. S94111 • 894372 . 894632 . S94892 .S95152 9.895412 • 895672 •805932 .896192 . 896452 9.896712 . 896971 .897231 . 897491 .897751 9. 898010 . 898270 .898530 . 898789 • 899049 9. 899308 .899568 .899827 . 900087 . 900346 9. 900605 . 900864 .901124 .901383 . 901642 9. 901901 . 902160 . 902420 . 902679 • 902938 9. 903197 • 9 3456 .903714 • 903973 . 904232 9. 904491 • 904750 . 905008 . 905267 .905526 9- 905785 . 906043 . 906302 . 906560 . 906819 9. 907077 • 907336 • 907594 • 907853 .908111 9. 908369 Cot. D. 1' Cot. 4.33 4-35 4-33 4-33 4-33 4-35 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4-33 4.32 4-33 4-33 4-33 4-32 4-33 4-33 4.32 4-33 4.32 4-33 4.32 4-33 4.32 4.32 4.32 4-33 4.32 4.32 4.32 4.32 4-33 4.32 4.32 4.32 4.32 4-3o 4-32 32 32 32 30 32 32 4 4 4 4 4 4 4.32 4.30 4.32 4.30 4.32 4.30 4.32 4.3o 4.32 4-3o 4.30 D. 1". o. 107190 . 106930 . 106669 . 106409 . 106 149 o. 105889 . 105628 . 105368 . 105108 . 104848 o. 104588 . 104328 . 104068 . 103808 . 103548 o. 103288 . 103029 . 102769 . 102509 . 102249 o. 101990 . 101730 . 101470 . IOI2II . I OO95 I o. 100692 . 100432 . 100173 .099913 . 099654 o. 099395 .099136 . 098876 . 098617 .098358 o. 098099 . 097840 . 097580 .097321 . 097062 o. 096803 . 096544 . 096286 . 096027 . 095768 o. 095509 . 095250 . 094992 . 094733 . 094474 o. 094215 . 093957 . 093698 .093440 .093181 o. 092923 . 092664 . 092406 .092147 .091889 o. 091631 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 32 3i 30 3 27 26 25 24 23 22 21 20 19 18 17 16 15 M 13 12 11 10 9 8 7 6 5 4 3 2 1 o M 51 39 < LOGARITHMIC SINES 140 M. o 1 2 3 4 5 6 7 8 9 10 ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 2 2g 30 31 32 33 34 35 36 3 l 38 39 40 41 42 43 44 45 46 A l 48 49 50 5i 52 53 54 55 56 5 l 58 Sin. D. i' Cos. 9. 798872 . 799028 • 799i84 • 799339 • 799495 9. 799651 . 799806 . 799962 .800117 . 800272 9. 800427 . 800582 . 800737 . 800892 .801047 9. 801 201 .801356 .80151 1 .801665 .801819 9.801973 .802128 .802282 . 802436 . 802589 9. 802743 . 802897 . 803050 . 803204 .803357 9. 80351 1 . 803664 .803817 . 803970 .804123 9.804276 .804428 .804581 • 804734 805039 805191 805343 805495 805647 805799 805951 806103 , 806254 . 806406 ,806557 , 806709 , 806860 ,807011 .807163 .807314 . 807465 .807615 . 807766 .807917 . 808067 Cos. 2.60 2.60 2.5S 2.60 2.60 2.58 2.60 2.58 2.58 2.58 2.58 2.58 2.58 2. 58 2.57 2.58 2.58 2-57 2-57 2.57 2.58 2-57 2.57 2-55 2.57 2-57 2-55 2-57 2-55 2-57 2-55 2-55 2-55 2-55 2-55 2.53 2-55 2.55 2-53 2-55 2-53 2.53 2.53 2.53 2-53 2-53 2.53 2.52 2-53 2.52 2-53 2.52 2.52 2-53 2.52 2.52 2.50 2.52 2.52 2.50 D. 1' 9. 890503 890400 890298 890195 890093 889990 889785 889682 8S9579 889477 889374 889271 889168 888961 888755 888651 888548 888444 888341 888237 888134 888030 887926 887822 887718 887614 887510 887406 887302 887198 887093 886989 886885 886780 886676 886571 886466 886362 886257 886152 886047 885942 885837 885732 885627 885522 885416 8853 1 1 885205 8S5100 884994 884783 884677 884572 884466 884360 884254 Sin. D. 1' 72 70 72 70 72 70 72 72 72 jo 72 72 72 73 72 72 72 73 72 73 72 73 72 73 73 73 73 73 73 73 73 73 75 73 73 75 73 75 75 73 75 75 75 75 75 75 75 75 77 75 77 75 77 75 77 77 75 77 77 77 D. i' Tan. D. 1". I Cot. 9. 908369 908628 909144 909402 909660 909918 910177 9 I0 435 910693 910951 91 1209 911467 911725 91 1982 912240 912498 912756 913014 913271 9 T 3529 913787 914044 914302 914560 914817 915075 9*5332 915590 915847 916104 916362 916619 916877 9 T 7i34 917391 917648 917906 918163 918420 918677 918934 919191 919448 919705 919962 920219 920476 920733 920990 921247 921503 921760 922017 922274 922530 922787 923044 923300 923557 923814 Cot. 4.32 4-3o 4-3o 4-3o 4-3o 4.30 4.32 4-3o 4-3o 4-30 4-3o 4.30 4.30 4.28 4-30 4-30 4-30 4-30 4.28 4-30 4- 3o 4.28 4.30 4-30 4.28 4-30 4.28 4.30 4.28 4.28 4.30 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.2.8 4.27 4.28 4.28 4.28 4.27 4.28 4.28 4.27 4.28 4.28 D. 1". 0.091631 .091372 .091114 . 090856 . 090598 o. 090340 . 090082 . 089823 . 089565 . 089307 o. 089049 . 08S79I .088533 .088275 . 088018 o. 087760 . 087502 . 087244 . 086986 . 086729 0.086471 .086213 . 085956 . 085698 .085440 o. 085183 . 084925 .084668 .084410 .084153 o. 083896 .083638 .083381 .083123 . 082866 o. 0S2609 .082352 .082094 .081837 .081580 0.081323 .081066 . 080809 . 080552 . 080295 o. 080038 .079781 .079524 . 079267 .079010 0.078753 .078497 . 078240 - 077983 .077726 0.077470 .077213 . 076956 . 076700 . 076443 0.076186 Tan. 129' 50' COSINES, TANGENTS, AND COTANGENTS 139' M. Sin. D. i". Cos. 10 11 12 13 14 15 16 r 7 18 19 20 21 22 23 24 25 26 2 2 28 29 30 31 32 33 34 36 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 808067 808218 808368 808519 808669 808819 808969 8091 19 809269 809419 809569 809718 0017 0167 0316 0465 0614 0763 0912 1061 1210 1358 1507 1655 1804 1952 2100 2248 2396 2544 2692 2840 2988 3135 3283 3430 3578 3725 3872 4019 4166 4313 4460 4607 4753 4900 5046 5193 5339 5485 5632 5778 5924 6069 6215 6361 6507 6652 6798 6943 Cos. 2.52 2.50 2.52 2.50 2.50 2.50 2.50 2.50 2.50 2.50 2.48 2.50 2.48 2.50 2.48 2.48 2.48 2.48 2.48 2.48 2.48 2.47 2.48 2-47 2.48 2.47 2.47 2.47 2.47 2.47 2.47 2.47 2-47 2.45 2.47 2-45 2.47 2-45 2.45 2.45 2.45 2.45 2-45 2-45 2-43 2.45 2.43 2.45 2-43 2.43 2.45 2.43 2.43 2.42 2.43 2.43 2-43 2.42 2.43 2.42 D. 1' 9.884254 .884148 . 884042 .883936 . 883829 9.883723 .883617 .883510 . 883404 .883297 9.883191 . 883084 . 882977 .882871 . 882764 9- 882657 .882550 .882443 .S82336 .882229 9.882121 . 882014 .881907 .881799 .881692 9.881584 .881477 .881369 .881261 .881153 9.881046 . 880938 .880722 .880613 9. 880505 .880397 . 880180 .880072 9. 879963 .879855 .879746 . 879637 . 879529 9.879420 .879311 . 879202 • 879093 . 878984 9.878875 . 878766 . 878656 .878547 .878438 9.878328 .878219 .878109 .877999 .877890 9.877780 Sin. D. T ' 77 77 77 78 77 77 78 77 78 77 78 78 77 78 78 78 78 78 78 80 78 78 80 78 80 78 80 80 80 78 80 80 80 82 80 80 80 82 80 82 80 82 82 80 82 82 82 82 82 82 82 83 82 82 83 82 83 83 82 83 Tan. D. 1' I 9-923814 . 924070 .924327 • 924583 . 924840 9. 925096 .925352 . 925609 .925865 .926122 9. 926378 . 926634 . 926890 .927147 . 927403 9. 927659 .927915 .928171 .928427 9. 928940 .929196 .929452 . 929708 • 929964 9. 930220 • 930475 .930731 • 930987 .931243 9- 93H99 .931755 .932010 .932266 .932522 9.932778 • 933033 • 933289 • 933545 . 933800 9. 934056 •9343.li • 934567 • 934822 • 935078 9- 935333 • 935589 • 935844 .936100 • 936355 9-9366ii . 936866 .937121 •937377 . 937632 9- 937887 .938142 .938398 • 938653 . 938908 9.939163 Cot. D. t". 4.'27 4- 4. 4.28 4.27 4.27 4.27 4.28 4-27 4.27 4.27 4.27 4.27 4.27 4.27 4.25 4.27 4.27 4.27 4.27 4.25 4.27 4.25 4.27 4.25 4.27 4.25 4.27 4.25 4.27 4.25 4.25 4.27 4.25 4.25 4.25 4.27 4.25 4.25 4.25 D. 1". Cot. 0.076186 . 075930 • 075^73 .075417 .075160 0.074904 . 074648 .074391 ■074135 .073878 0.073622 . 073366 .073110 .072853 • 072597 0.072341 .072085 .071829 .071573 .071316 0.071060 . 070804 • 070548 .070292 . 070036 o. 069780 .069525 . 069269 .069013 • 068757 0.068501 . 06S245 .067990 I 28 . 067734 . 067478 0.067222 . 066967 . 0667 1 1 . 066455 .066200 0.065944 . 065689 . 065433 .065178 . 064922 o. 064667 .064411 .064156 . 063900 . 063645 o. 0633S9 •063134 . 062879 . 062623 . 062368 0.062113 .061858 .061602 . 06*347 .061092 0.060837 Tan. I30 ( 49' 41 LOGARITHMIC SINES I38 c M. Sin. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 131 9-816943 .817088 .817233 .817379 .817524 9.817668 .817813 .817958 .818103 .818247 9.818392 .818536 .818681 .818825 .818969 9.819113 • S19257 .819401 .819545 .819689 9.819832 .819976 .820120 . 820263 . 820406 9. 820550 . 820693 .820836 .820979 .821122 9.821265 .821407 .821550 .821693 .821835 9-821977 .822120 .822262 .822404 .822546 9. 822688 .822830 .822972 .823114 .823255 9. 823397 .823539 . 823680 ,823821 . 823963 9. 824104 • 824245 . 824386 . 824527 . 824668 9. 824808 • 824949 . 825090 . 825230 .825371 9. 82551 1 Cos. D. 1 2.42 2.42 2.43 2.42 2.40 2.42 2.42 2.42 2.40 2.42 2.40 2.42 2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.38 2.40 2.40 2.38 2.38 2.40 2.38 2.38 2.38 2.38 2.38 2.37 2.38 2.38 2.37 2.37 2.38 2-37 2-37 2-37 2-37 2.37 2.37 2-37 2-35 2-37 2-37 2.35 2-35 2-37 2-35 2.35 2.35 2.35 2-35 2-33 2.35 2.35 2-33 2-35 2-33 D. I". Cos. 9- 87778o .877670 . 877560 .877450 .877340 9- 877230 .877120 .877010 . 876899 . 876789 9. 876678 . 876568 .876457 .876347 .876236 9.876125 . 876014 . 8759 4 .875793 , 875682 9.875571 • 875459 .875348 .875237 .875126 9- 875014 • 874903 .874791 . 874680 .874568 9.874456 .874344 . 874232 .874121 . 874009 9.873896 .873784 .873672 . 873560 .873448 9- 873335 .873223 .873110 . 872998 . 872885 9.872772 .872659 .872547 .872434 .872321 9. 872208 . 872095 .871981 .871868 .871755 9. 87 164 1 .871528 .8714M .871301 .871187 9.871073 Sin. D. 1". 1.83 1.83 1.83 1.83 1.83 1.83 1.83 1.85 1.83 1.85 1.83 1.85 1.83 85 85 85 83 85 85 1.85 1.87 1.85 1.85 1.85 1.87 1.85 1.87 1.85 1.8.7 1.87 1.87 1.87 1.85 1.87 1.88 1.87 1.87 1.87 1.87 1.88 1.87 1.88 1.87 1.88 1.88 1.88 1.87 1.88 1.88 1.88 1.88 1.90 1.88 1.88 1.90 1.88 1.90 1,88 1.90 1.90 D. 1" Tan. 9-939i63 . 939418 . 939673 • 939928 .940183 9. 940439 • 940694 . 940949 .941204 .941459 9.94I7I3 .941968 .942223 . 942478 • 942733 9. 942988 .943243 • 943498 • 943752 . 944007 9. 944262 .944517 .944771 . 945026 .945281 9- 945535 • 94579° . 946045 • 946299 - 946554 9. 946808 . 947063 • 947318 • 947572 .947827 9.948081 • 948335 . 94859 .948844 .949099 9- 949353 . 949608 . 949862 .950116 .95°37i 9. 950625 . 950879 .951133 .951388 . 951642 9. 951896 .95215° • 9524°5 • 952659 .952913 9.953167 .953421 • 953675 . 953929 .954183 9- 954437 Cot. D, 1' 4.25 4.25 4.25 4.25 4.27 4.25 4-25 4.25 4.25 4.23 4.25 4.25 4.25 4.25 4.25 4.25 4.25 4.23 4.25 4.25 4.25 4.23 4.25 4.25 4.23 4.25 4.25 4.23 4.25 4.23 4.25 4.25 4.23 4.25 4.23 4.25 4.23 4.23 4.25 4.23 4.23 4.23 4.25 4-23 4-23 4-23 4.25 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4-23 D. 1 Cot. o. 060837 .060582 . 060327 . 060072 .059817 o. 059561 . 059306 .059051 . 058796 .058541 o. 058287 .058032 .057777 .057522 . 057267 0.057012 . 056757 . 056502 . 056248 .055993 o. 055738 . 055483 . 055229 . 054974 .054719 o. 054465 . 054210 . 053955 .053701 .053446 0.053192 . 052937 .052682 . 052428 .052173 0.051919 .051665 .051410 .051156 . 050901 o. 050647 . 050392 .050138 . 049884 . 049629 o. 049375 .049121 . 048867 .048612 .048358 o. 048104 . 047850 . 047595 .047341 . 047087 o. 046833 . 046579 . 046325 .046071 .045817 o. 045563 Tan. 48' 4 2° COSINES, TANGENTS, AND COTANGENTS i 37 c M. Sin. D. i". Cos. D. r Tan. D. i> Cot. o i 2 3 4 5 6 7 8 9 io ii 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 5i 52 53 54 55 56 57 58 59 60 I32 ( .S25511 .825651 .825791 .825931 .826071 .826211 .826351 .826491 .826631 . 826770 . 826910 .827049 .827189 .827328 . 827467 . 827606 . 827745 . 827884 . 82S023 .828162 . 828301 .828439 .828578 .828716 .828855 .828993 .829131 . 829269 . 829407 ■ 829545 . 829683 .829821 ■ 829959 ■ 830097 . 830234 . 830372 . 830509 . 830646 .830784 . 830921 .831058 ■83 1 195 .831332 .831469 .831606 .831742 .831879 .832015 .832152 . 832288 . 832425 .832561 . 832697 .832833 . 832969 • 833105 .833241 • 833377 .833512 .833648 •833783 Cos. 2-33 2-33 2-33 2-33 2.33 2.33 2-33 2-33 2.32 2.33 2.32 2.33 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.32 2.30 2.32 2.30 2.32 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.28 2.30 2.28 2.28 2.30 2.28 2.28 2.28 2.28 2.28 2.28 2.27 2.28 2.27 2.28 2.27 2.28 2.27 2.27 2.27 2.27 2. 27 2. 2. 2. 2. 2. D. 1". 9.871073 .870960 . 870846 . 870732 . 870618 9. 870504 . 870390 . 870276 .870161 . 870047 9- 869933 .869S18 . 869704 .869589 • S69474 9. 869360 . 869245 . 869130 . 869015 . 868900 9.868785 . 868670 .868555 . 868440 . 868324 9. 868209 . 868093 . 867978 . 867862 .867747 9. 867631 .867515 . 867399 .867283 . 867167 9.867051 .866935 .866819 . 866703 . 866586 9. 866470 .866353 .866237 .866120 . 866004 9.865887 . 865770 .865653 .865536 . 865419 9. 865302 .865185 . 865068 • 864950 .864833 9.864716 .864598 .864481 .864363 . 864245 9.864127 Sin. 88 9- 90 90 90 90 90 9- 90 92 90 90 92 9- 90 92 92 90 92 9- 92 92 92 92 92 9- 92 92 93 92 93 9- 92 93 92 93 93 9- 93 93 93 93 93 9- 93 93 95 93 95 9- 93 95 93 95 95 9- 95 95 95 95 95 9- 95 97 95 95 97 9- 95 97 97 97 9- D. 1' 954437 , 954691 954946 955200 955454 9557o8 955961 956215 956469 956723 956977 957231 957485 957739 957993 958247 958500 958754 959008 959262 959516 959769 960023 960277 960530 960784 961038 961292 96i545 961799 962052 962306 962560 962813 963067 963320 963574 963828 964081 964335 964588 964842 965095 965349 965602 965855 , 966109 966362 966616 966869 967123 967376 967629 967883 968136 968389 968643 968896 969149 969403 969656 Cot. 4-23 4.23 4.23 4.23 4.23 4-23 4.23 4.23 4. 22 4-23 4-23 4.23 4.23 4.22 4-23 4-23 4. 22 4.23 4-23 4.23 4. 22 4 23 4.22 4.23 4.23 4.22 4.23 4.22 4.23 4.23 4.22 4.23 4.22 4.23 4.22 4.23 4.22 4.22 4-23 4.22 4.23 4.22 4.23 4.22 4.22 4.23 4. 22 4. 22 4-23 4. 22 4. 22 4-23 4.22 D. i> o. 045563 • 045309 • 045054 . 044800 • 044546 o. 044292 . 044039 .043785 •043531 . 043277 o. 043023 . 042769 •042515 .042261 . 042007 0.041753 .041500 .041246 . 040992 . 040738 o. 040484 .040231 • 039977 .039723 • 039470 0.039216 . 038962 . 038708 .038455 . 038201 o. 037948 . 037694 .037440 .037187 . 036933 o. 036680 . 0364 26 .036172 ' 035919 . 035665 o, 035412 •035158 .034905 .034651 . 034398 0.034145 . 033891 . 033638 . 033384 •033131 o. 032877 . 032624 .032371 .032117 .031864 0.031611 .031357 .031104 .030851 . 030597 o. 030344 Tan. 47 43 v LOGARITHMIC SINES 136' M. Sin. 10 11 12 13 14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 45 4 Z 48 49 50 5i 52 53 54 55 56 5 l 58 59 60 9.833783 ►833919 , 834054 .834189 .834325 9. 834460 .834595 . 834730 .834865 • 834999 9.835134 . 835269 • 835403 .835538 .835672 9- 835807 .835941 .836075 . 836209 . 836343 9. 836477 .836611 .836745 . 836878 .837012 9.837146 .837279 .S37412 • 837546 .837679 9.837812 .837945 . 838078 .838211 .838344 9- 838477 .838610 .838742 .838875 . 839007 9- 839140 839272 839404 . 839536 . 839668 9. 839800 .839932 . 840064 .840196 . 840328 9. 840459 .840591 . 840722 . 840854 . 840985 9.841116 .841247 .841378 .841509 .841640 9-841771 Cos. D. 1". | 2.27 2.25 2.25 2.27 2.25 2.25 2.25 2.25 2.23 2.25 2.25 2.23 2.25 2.23 2.25 2.23 2.23 2.23 2.23 2.23 2.23 2.23 2. 22 2.23 2.23 2.22 2. 22 2.23 2. 22 2. 22 2. 22 2. 22 2. 22 2. 22 2. 22 2. 22 2. 20 2. 22 2. 20 2. 22 2. 20 2. 20 2. 20 2. 20 2. 20 2. 20 2. 20 2. 20 2. 20 2.18 2. 20 2.18 2. 20 2.18 2.18 2.18 2.18 2.18 2.18 2.18 D. 1". Cos, 9.864127 .864010 . 863892 .863774 .863656 9-863538 . 863419 .863301 .863183 . 863064 9. 862946 . 862827 . 862709 . 862590 .862471 9- 862353 . 862234 .862115 .861996 .S61877 9.861758 .861638 .861519 .86 1 400 .861280 9. 861161 .861041 . 860922 .860802 . 860682 9. 860562 . 860442 . 860322 . 860202 . 860082 9. 859962 .859842 .859721 .859601 . 859480 9. 859360 • 859239 .859119 .858998 .858877 9.858756 .858635 .858514 .858393 .858272 9-858151 . 858029 .857908 .857786 .857665 9.857543 . 857422 • 857300 .857178 . 857056 9- 856934 Sin. D. i' •95 •97 •97 •97 •97 .98 •97 -97 .98 •97 • 97 .98 .98 • 97 .98 .98 .98 .98 .98 2. 00 .00 .98 2.00 1.98 2.00 2.00 2.00 2.00 2. 00 2.00 2.00 2.00 2.00 2.02 2.00 2.02 2.00 2.02 2.00 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.03 2.02 2.03 2.02 2.03 2.02 2.03 2.03 2.03 2.03 D. 1" Tan. 9- 969656 . 969909 .970162 .970416 . 970669 9. 970922 .971175 .971429 .971682 .971935 9. 972188 .972441 • 972695 • 972948 .973201 9- 973454 . 973707 • 9739 6 o .974213 . 974466 9. 974720 • 974973 . 975226 • 975479 • 975732 9- 975985 • 976238 .976491 • 976744 . 976997 9. 977250 • 977503 . 977756 . 978009 .978262 9.9785I5 . 978768 .979021 .979274 • 979527 9. 97978o . 980033 . 980286 .980538 . 980791 9. 981044 .981297 .981550 .981803 . 982056 9. 982309 . 982562 .982814 . 983067 .983320 9- 983573 . 983826 • 984079 .984332 .984584 9. 9S4837 Cot. D. 1". 4.22 4. 22 4-23 4. 22 22 22 22 22 22 22 22 4. 22 4.22 4.22 4. 22 4.22 4.22 4.22 4. 4- 4- 4- 4- 4. 22 4. 22 4. 22 4. 20 4. 22 22 22 22 22 22 22 D. 1 Cot. o. 030344 .030091 . 029838 • 029584 .029331 o. 029078 . 028825 .028571 .028318 , 028065 0.027812 .027559 .027305 . 027052 .026799 o. 026546 .026293 . 026040 .025787 . 025534 o. 025280 .025027 .024774 .024521 . 024268 0.024015 .023762 . 023509 .023256 . 023003 o. 022750 . 022497 .022244 .021991 .021738 0.021485 .021232 . 020979 .020726 . 020473 o. 020220 .019967 .019714 .019462 .019209 0.018956 .018703 .018450 .018197 ,017944 0.017691 .017438 .017186 .016933 .016680 0.016427 ,016174 .015921 .015668 .015416 0.015163 Tan. 133 4 6° 44 COSINES, TANGENTS, AND COTANGENTS 135 M. o i 2 3 4 5 6 7 8 9 io ii 12 13 M 15 16 17 18 19 20 21 22 23 2 4 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 4 Z 48 49 50 5i 52 53 54 55 56 57 58 59 60 Sin. 9.841771 .841902 .842033 .842163 . 842294 9. 842424 ■ 842555 . 8426S5 .842815 . 842946 9. 843076 . 843206 • 843336 . 843466 • 843595 9- S43725 . 843855 • 843984 .844114 .844243 9. 844372 • 844502 .844631 . 844760 . 844889 9.845018 .845147 . 845276 .845405 • 845533 9. 845662 . 845790 .845919 . 846047 .846175 9. 846304 .846432 846560 . 846688 .846816 9. 846944 .847071 .847199 . 847327 • 847454 9- 847582 .847709 .847836 . 847964 9. 848218 •848345 . 84S472 .848599 . 848726 9. 848852 . 848979 .849106 • 849232 • 849359 9. S49485 Cos. D. 1 2. iS 2.18 2.17 2.18 2.17 2.18 2.17 2.17 2.18 2. 17 2.17 2.15 2.17 2.15 2.15 2.17 2.15 15 2.13 2.15 2.13 2.13 2.15 2.13 2.13 2.13 2.13 2.13 2. 12 2. 12 2. 2. 2. 2. 2. 2. 2. 2. U. 1' Qos. 9- 856934 .856812 . 856690 . 856568 . 856446 9-856323 . 856201 . 856078 • 855956 .855833 9.8557II .8555S8 • 855465 •855342 .855219 9.855096 • 854973 .854850 .854727 . 854603 9. 854480 • 854356 • 854233 .854109 .853986 9.853862 .853738 .853614 • 853490 .853366 9- 853242 .853118 . 852994 . 852869 • 852745 9. 852620 . 852496 •852371 .852247 .852122 9-85I997 .851872 •851747 .851622 •851497 9.851372 .851246 .851121 • 850996 . 850870 9- 850745 .850619 • 850493 .850368 . 850242 9.850116 • 849990 . 849864 .849738 . 8496 1 1 9. S49485 Sin. D. 1". 2.03 2.03 2.03 2.03 2.05 2.03 2.05 2.03 2.05 2.03 2.05 2.05 2.05 2.05 2.05 2.05 2.05 2.05 2.07 2.05 2.07 2.05 2.07 2.05 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.07 08 07 08 07 08 07 08 oS 2, 2. 2. 2. 2. 2. 2. 2. 2.08 2.08 2.08 2.08 2.08 2. 10 2.08 2.08 2. 10 2.0S 2. 10 D. V Tan. 9. 984837 . 985090 • 985343 • 985596 .985848 9. 986101 • 986354 . 986607 . 986860 .987112 9- 987365 .987618 .987871 .988123 . 988376 9. 988629 . 98SS82 • 989134 .989387 . 989640 9- 989893 .990145 • 990398 .990651 • 990903 9. 991 156 • 991409 .991662 .991914 .992167 9. 992420 . 992672 • 992925 .993178 .993431 9- 993683 • 993936 .994 1 89 .994441 • 994694 9- 994947 .995199 • 995452 • 995705 • 995957 9.996210 • 996463 .996715 . 996968 .997221 9- 997473 - 997726 • 997979 .998231 • 998484 9- 998737 • 998989 • 999242 • 999495 • 999747 o. 000000 Cot. D. 1' 4. 22 4. 22 4. 22 4. 20 4. 22 4. 22 4. 22 4. 22 4. 20 4.22 4. 22 4. 22 4.20 4. 22 4. 22 4. 22 4. 20 4. 22 4. 22 4. 22 4. 20 4. 22 4.22 4. 20 4. 22 4.22 4. 22 4. 20 4. 22 4. 22 4. 20 4. 22 4. 22 4. 22 4. 20 4. 22 4. 22 4. 20 4. 22 4. 22 4. 20 4. 22 4. 22 4. 20 4. 22 4. 22 4.20 4. 22 4. 22 4. 20 4.22 4,22 4. 20 4.22 4. 22 4. 20 4. 22 4.22 4. 20 4.22 D. 1". Cot. 0.015163 .014910 .014657 .014404 .014152 0.013899 . 013646 •OI3393 .013140 .012888 0.012635 .012382 .012129 .011877 .011624 0.011371 .011118 .010866 .010613 . 010360 o. 010107 . 009855 . 009602 • 009349 .009097 o. 008844 .008591 .008338 . 008086 . 007833 o. 007580 . 007328 . 007075 . 006822 . 006569 0.006317 . 006064 .005811 .005559 . 005306 o. 005053 .004801 . 004548 . 004295 . 004043 o. 003790 . 003537 .003285 . 003032 .002779 o. 002527 .002274 .002021 .001769 .001516 0.001263 .001011 . 000758 . 000505 . 000253 0.000000 Tan. 134' 45 5^4 GENERAL REFERENCE TABLES Table 63. Giving the Weights of Different Materials per Cubic Foot 1 Material Ash timber Brick (pressed) " (common building) Cement (Portland) (Natural Concrete 1:2:4 Mixture (Trap rock) (Gravel) " (Limestone) " (Sandstone) (Cinder) 1:3:6 Mixture (about 5 lbs. less) Earth (common loam, loose and dry) " (common loam, moist and rammed) . . " (sand or gravel loose and dry) " (sand or gravel rammed) " (sand or gravel wet) Hemlock timber Hickory " Iron (cast) " (wrought) Maple timber Oak " (white) (black) Masonry (dressed granite or limestone) (mortar rubble) (dry " ) Pine (white) " (northern yellow) " (southern yellow) Steel Water Weight per Cu. Ft. 75 5o 40 150 125 to 90 to 56 155 152 150 145 no 70 IOO IOO 120 120 25 50 450 480 50 48 40 165 155 125 25 34 40 490 62.5 lbs. Miscellaneous Weights 1 bbl. Portland cement 376 lbs. 1 " natural " 235 1 gal. water " 8.345 " 1 For weight of road rocks, see Tables 21 and 22, page 99. MODULI OF ELASTICITY 585 Table 64. Giving Moduli of Elasticity, Working Stress and Ultimate Strength Moduli of Elasticity Material Lbs. per Sq. In. Concrete Hemlock Iron (cast) Iron (wrought) . Oak Pine (white) . . Pine (yellow) . . Steel (medium) Spruce 900,000 17,500,000 29,000,000 1,500,000 1,600,000 1,600,000 30,000,000 1,600,000 Working Stresses in Lbs. per Square Inch Material Concrete Hemlock Iron (cast) . . . " (wrought) Oak Pine (white) . . " (yellow) . Steel (medium) Spuce Tension 60 600 3,000 10,000 1,200 700 1,200 1 2 ,000 800 Compression 600 W. G. 1 600 A. G. 2 150 18,000 8,000 W. G. 1,200 A. G. 500 W. G. 700 A. G. 200 W. G. 1,200 A. G. 350 12,000 W G. 800 A. G. 200 Shear 60 to 100 W. G. 100 A. G. 600 5, 000 8,000 W. G. 200 A. G. 1,000 W. G. 100 A. G. 500 W. G. 150 A. G. 1,250 12,000 W. G. 100 A. G. 750 Ultimate Strength in Lbs. per Square Inch Material Concrete Hemlock Iron (cast) . . . . " (wrought) . Oak Pine (white) . . . " (yellow) . . Steel (medium) Spruce Tension 300 6,000 18,000 50,000 1 2 ,000 7,000 12,000 60,000 8,000 Compression 3,000 W. G. 6,000 A. G. 600 90.000 40,000 W. G 7,000 A. G. 2 000 W G. 5,500 A. G. 700 W. G. 7,000 A. G. 1,400 60,000 W. G. 6,000 A. G. 700 Shear 1300 W. G. 350 A. G. 2,500 20,000 tO 30,000 35,000 to 55,000 W. G. 800 A. G. 4.000 W. G 400 A. G. 2,000 W. G. 600 A. G. 5,000 50,000 to 70,000 W. G. 400 A. G. 3,200 1 W. G. — With Grain. 2 A. G. — Across Grain. 586 GENERAL REFERENCE TABLES Table 65. Uniform Beams. Maximum Bending Moment and Deflections (Simple Cases) Case 1. Beam with ends free. Single concentrated load P in middle of span; weight of beam disregarded. The maximum moment occurs at the center of the span. P 4 The maximum deflection occurs at the center of the span. Concentrated Load in Cen- ter of span D = PP 48 EI Where D P ' I E = the deflection in inches = load in pounds = span in inches = modulus of elasticity in lbs. per sq. inch / = moment of inertia in inches 4 Mp = maximum moment in inch pounds. Cantilever Beam Case 2. Cantilever beam con- centrated load P; weight of beam disregarded. The maximum moment occurs at the support. M = PI P D 3 EI Case j. Beam with ends free. Uniformly distributed load. The maximum moment occurs at the center of the span. Wl 8 M = Uniform Load The maximum deflection occurs at the center of the span. D = j_wp 384 EI In these formulas W equals the total uniformly distributed load. UNIFORM BEAMS 587 Case 4. Cantilever beam. Uniform load W Maximum moment occurs at the point of support. m = m 2 The maximum deflection occurs at the free end. D = WP 8 EI Case 5. Beam with fixed ends, concentrated load P in center of span; weight of beam disregarded. The maximum bending moment occurs at the points of support and at the middle of the beam. if-2 D = PP 192 EI Case 6. Beam with fixed ends and a uniformly distributed load. Maximum bending moment occurs at the supports. M' Wl 12 m -m 24 Maximum deflection WP 384 EI Resisting Moment of a beam is expressed by the formula u f ~tL e Where M r = moment of resistance in inch pounds p = maximum allowable fiber stress in lbs. per sq. inch. I = moment of inertia of the beam in inches 4 e = distance in inches from the neu- tral axis to the outer fiber $88 GENERAL REFERENCE TABLES Table 66. Centers of Gravity of Ordinary Plane Figures K.Q. Squares, rectangles, parallelograms. Center of gravity is at the intersection of the diagonals or midway between the bases on a line drawn between the centers of those bases. Triangles Center of gravity is at the intersection of the medial lines a b and c d; a medial line is a line drawn from any apex to the middle of the opposite side The distance b (C. G.) = \ ab; that is, the center of gravity is on the medial line \ of the distance from the base to the apex. Trapezoid 9 -*=^-- — -iCA. Graphic Method. Prolong b a to g, making a g = c d. Pro- long c d to h, making d h = a b. Connect g h. Bisect a b at e. Bisect cd at/. Connect ef: the intersection of g h and ej is the center of gravity. The distance/ (CG.) =^X 2 f , + °f 3 ab + cd Any Quadrilateral Graphic Method. Draw the diagonals a c and b d intersecting at e. Lay off a / = e c Lay off b g = e d Bisect eg at h\ bisect ej at *. The intersection of / h and g i is the center of gravity of the figure. CENTERS OF GRAVITY 589 Circles Center of gravity at the center Semicircle, The center of gravity lies on the radius perpendicular to the diameter. The distance c (C. G.) = radius X 0.4244 a Quadrant The center of gravity lies on the radius which bisects the Z. a c b. The distances (C.G.) = radius X 0.600? Sector The center of gravity lies on the radius bisect- ing the /_a c b. The distance c (C. G.) = f radius X c chord a b radius 2 X chord arc a d b 3 X area Segment The center of gravity lies on the perpen- dicular erected at the center of the chord ab. The distance c (C G.) chord ab* 1 2 X area of segment Table 67. Moments of Inertia of Simple Sections I = Moment of Inertia 12 — ; 1 1 ■tr- 59° GENERAL REFERENCE TABLES Square / = ¥ 12 K— b — H Triangles 36 Circles / = 0.7854^ ST** i b d' 4:i 7 = bd* - Vd'* 12 APPENDIX A TRAFFIC RULES AND REGULATIONS, STATE OF OHIO FOREWORD Inasmuch as Section 249 of the Cass Law, (G. C. 7246) directing the state highway commissioner to prepare and publish a set of traffic rules and regulations, is for the protection of life and limb, it is undoubtedly the most important section of the road laws of Ohio. After much thought and investigation of rules and regulations governing traffic conditions in a number of states, we submit the following in as condensed form as explicitness will permit, having selected, as we believe, the better parts of such laws and regulations governing the traffic of other states and municipalities, and putting them into a code of rules and regulations that will fit, as nearly as may be, all conditions and localities requiring a code of regulations, which will at the same time govern traffic on all the highways of Ohio. A cursory examination of these rules may lead many to the con- clusion that unreasonable restrictions have been imposed, but we believe a thoughtful study of each section will reveal an effort on the part of the State Highway Department to furnish the public with a code of traffic regulations, permitting of the greatest amount of freedom consistent with safety first. The original draft of the following regulations was submitted to Mr. W. A. Alsdorf, Secretary of the Ohio Good Roads Federation, Mr. Harry Gordon of Cincinnati, and Mr. Fred Caley of Cleveland, who carefully studied the entire code, section by section, and sug- gested many valuable and important changes. We now put forth the result of our efforts with the belief that if the prescribed rules and regulations are followed, many embarrassing situations and distressing accidents may be averted. Clinton Cowen, State Highway Commissioner. ARTICLE I — DEFINITIONS Sec. i — The term " vehicle" shall apply to a horse being rode or led, and to any conveyance except a baby carriage or street car. Sec. 2 — The term "street car" shall apply to any conveyance confined to tracks. Sec. 3 — The term "driver" shall apply to the rider, driver, or leader of a horse, a person who pushes, draws, propels, operates, or who is in charge of a vehicle. Sec. 4 — The term "road" shall apply to that part of a street or public highway intended for vehicles. 59i 592 TRAFFIC RULES AND REGULATIONS Sec. 5. — The term "curb" shall apply to the boundary line of a road. Sec. 6 — The term "sidewalk" shall apply to a path or walk intended for pedestrians. Sec. 7 — The term "horse" shall apply to any draft animal or beast of burden. Sec. 8 — The term, "motor vehicle" shall apply to all vehicles propelled by power other than muscular, except a street car, traction engine, road roller, and police, fire or ambulance vehicles. ARTICLE II — RESPECTIVE RIGHTS AND DUTIES OF DRIVERS AND PEDESTRIANS Sec. 1 — Roads are primarily intended for vehicles, but pedestrians have the right to cross them in safety, and drivers shall exercise ail possible care not to endanger them. Sec. 2 — Pedestrians should observe the following precautions; 1st. Avoid interference with vehicular traffic, and to this end not step onto the road without first looking to see what is approaching: 2nd. Cross the road at right angles — at regular crossings where such exist, — and where a traffic officer is stationed, wait for his signal. Sec. 3 — Pedestrians will aid in expediting traffic on side-walks by keeping to the right, and when stopping for any purpose by doing so on one side and out of the way of a crossing or driveway. ARTICLE III — PASSING TURNING, STOPPING, STANDING AND STARTING Sec. 1 — A vehicle meeting another shall keep to the right, so as to leave half the road free for the coming vehicle. (6310 G. C.) Sec. 2 — A vehicle overtaking another shall pass to the left, the front vehicle giving half the road to the rear vehicle. (6310 G. C.) Sec. 3 — A vehicle turning into a road to the right shall turn the corner as near the right hand curb as practicable. This Way Sec. 4 — A vehicle turning into a road to the left shall pass around the point of intersection of the center lines of the two roads. + . This Way Not this Way TRAFFIC RULES AND REGULATIONS 593 Sec. 5 — A vehicle crossing from one side to the other of a road shall head in the same direction as the traffic on that side of the road. This Way Not this Way Sec. 6 — On heavy traffic roads, slow moving vehicles shall keep close to the right hand curb so as to leave the center of the road clear for overtaking traffic — the slower the speed the nearer the curb. Sec. 7 — A vehicle in passing around a circle shall keep to the right from entrance to exit. Sec. 8 — A vehicle on a road divided longitudinally by a parkway, walk, rope or other obstruction, shall keep to the right of such division. Sec. 9 — A vehicle shall not back to make a turn if by so doing it obstructs traffic, but shall go forward to a point where a turn can be made without backing. Sec. 10 — A vehicle shall not follow another too closely for safety. Sec. 11 — No vehicle shall stop in the road in such a position as to prevent the free passage of other vehicles in both directions at the same time. Sec. 12 — A vehicle shall not pass a street car which has stopped to receive or discharge passengers at a less distance than ten feet, nor at a greater speed than six miles per hour. A vehicle shall come to a stop if necessary to prevent interference or injury to such passengers. Sec. 13 — No horse or vehicle shall be driven, propelled or allowed to stand, on any side- walk except for purposes of crossing the same when necessary, and then only the shortest way from the road to the abutting premises. Sec. 14 — No vehicle shall stop in such a way as to interfere with the passage of pedestrians at regular crossings, or within 10 feet of a fire plug. Sec. 15 — No street car shall stop or stand within the intersection of any road. ARTICLE IV — LOADS, LOADING, WIDTH OF TIRE, TRAILERS, ETC. Sec. 1 — No traction engine or other vehicle whose wheels have tires equipped with lugs, spikes, chains or other projections seriously destructive to the surface, shall be driven over the road. (1342 1- 12 G. C.) Sec. 2 — No vehicle or load, the total width of which is greater than twelve feet, shall be operated or drawn over a road unless said greater width will leave one-half the road free for passing vehicles. Sec. 3 — No more than thirty-four hundred pounds including weight of vehicle, shall be transported over a gravel, macadam or stone road in a vehicle having a tire less than three inches in width. (7477 G. C.) 594 TRAFFIC RULES AND REGULATIONS Sec. 4 — For vehicles having tires three inches and over in width the load on any wheel per lineal inch of width of tire on any road shall not exceed six hundred pounds; and during such times as the road surface is soft, because of thawing or because of excessive rains, the load per lineal inch of width of tire on any wheel shall not exceed three hundred pounds on gravel or macadam roads. Sec. 5 — Trailers used in hauling over the road shall be so con- nected that the wheels of no two will follow in the same tracks. Sec. 6 — No vehicle carrying a load in excess of fifteen tons, including the weight of the vehicle, shall be moved over any road except under the written permission of the State Highway Commissioner. ARTICLE V— SPEEDS Sec. i — No motor vehicle shall operate on a road at a greater speed than: 8 miles per hour in the business or closely built up portions of a municipality; 15 miles per hour in other portions of a municipality; 20 miles per hour outside of municipalities. (12604 G. C.) Sec. 2 — No vehicle shall operate on a road at a speed greater than is reasonable or proper or so as to endanger the property, life or limb of any person. (12603 G. C.) Sec. 3 — No motor or other power vehicle carrying a weight in excess of four tons including a vehicle shall be operated upon any road at a speed greater than 15 miles per hour; and no such vehicle carrying a weight in excess of eight tons including the vehicle shall be operated at a speed greater than 6 miles per hour when such vehicle is equipped with iron or steel tires, nor greater than 12 miles per hour when the vehicle is equipped with tires of rubber or other similar substance. Sec. 4 — No vehicle shall cross a road or make any turn at a dangerous speed. . Sec. 5 — Where " Danger" and "Go Slow" signs appear, the speed of any vehicles shall not exceed twelve miles per hour. Sec. 6 — Trucks and heavy wagons shall not be driven recklessly so as to endanger the public. Sec. 7 — No vehicle shall emerge from an alley, stable, garage or any private drive or entrance faster than a walk or six miles per hour. Sec. 8 — A vehicle upon approaching a cross road shall slow down sufficiently to prevent any danger from meeting other vehicles on the cross road. Sec. 9 — No person shall race any horse or motor vehicle on a road whether the running, racing or trotting be for trial or speed or for the purpose of passing another horse or vehicle. Sec. 10 — A motor vehicle, road roller or traction engine shall slow down when approaching a horse, if the horse appears to be frightened, and if the driver of the horse shall signal the driver of the vehicle the latter shall be brought to a stop, and if the circum- stances require it, the engine shall be stopped, provided such signal TRAFFIC RULES AND REGULATIONS 595 be given in good faith and under circumstances of necessity. Such vehicle shall remain stationary so long as may be reasonable to allow such horse to pass. (12605 G. C.) Sec. 11 — In case of injury or damage to person or property, due to the operation of vehicle, the operator or driver of said vehicle shall stop, and, upon request of the person injured or any one present, give his name and address and that of the owner of the vehicle. (12606 G. C.) ARTICLE VI — SIGNALS, HORNS, SIGNS AND NOISES Sec. i — All motor vehicles and bicycles shall be equipped with a suitable bell or horn for signalling. (12614 G. C.) Sec. 2 — When a vehicle is slowing up or stopping, the driver shall give a timely signal to those in the rear, by raising the arm or whip vertically (preferably) or horizontally or by some other unmis- takable manner. Sec. 3 — When about to turn either from a standstill or while in motion, the driver of a vehicle shall give timely signal by hand or whip or in some other unmistakable manner, to indicate the direction of the turn. This is especially important when turning to the left. Sec. 4 — Before a vehicle is backed, the driver shall give timely warning. Sec. 5 — Sound signals are prohibited except for necessary warn- ing, and must be reserved for that purpose. Signals shall not be sounded by unauthorized persons on standing vehicles. Sec. 6 — All signs, signals and orders of a traffic officer shall be promptly complied with. Sec. 7 — Every driver of a motor vehicle shall give a timely warn- ing when overtaking a person or vehicle on a road or when approach- ing a crossing or curve where the sight of approaching vehicles may be obscured. Sec. 8 — No vehicle shall be so loaded as to cause an objectionable or unnecessary noise by parts of the load striking together or upon the vehicle. Sec. 9 — The use between the hours of 8 P. M. and 6 A. M. of the muffler cut-out or the production of any other unnecessarily loud noise on any vehicle, is prohibited within 100 yards of any residence or within such distance as might seriously disturb the in- habitants of such residence. Sec. 10 — No vehicle shall pass over any road which is closed against traffic to be repaired or constructed. A suitable sign or barricade shall be considered as sufficient evidence that such road is closed. (13421-9 G. C.) ARTICLE VII — RIGHT OF WAY Sec. 1 — Every driver of a vehicle approaching the intersection of a road where a traffic officer is not stationed, shall grant the right of way at such intersection to any vehicle approaching from his right. 596 TRAFFIC RULES AND REGULATIONS Sec. 2 — A vehicle in front of a street car shall immediately turn out upon the signal of the operator of the car. Sec. 3 — A vehicle shall not so occupy any road as to obstruct traffic. Sec. 4 — When in the performance of duty, the following vehicles shall have the right of way: Police, Fire, Fire Patrol, Ambulance, U. S. Mail; also the militia. Sec. 5 — During blockades and stoppages a clear space shall be kept open between all street cars at crossings. Sec. 6 — Pedestrians about to get on, or just having been dis- charged from, a street car shall have the right of way and vehicles shall come to a stop when necessary to give such pedestrians the right of way. ARTICLE VIII — LIGHTS Sec. i — Motor vehicles shall display between 30 minutes after sunset and 30 minutes before sunrise, two white lights in front of sufficient power to be visible 200 feet away in the direction the vehicle is moving, and one red light visible in the opposite direction; also one rear white light which shall illuminate and make plainly visible the license number tag. Provided that motorcycles need have but one front light. (12614 G. C.) Sec. 2 — During the same period given in Sec. 1, bicycles shall have a light of sufficient power to be seen 200 feet in the direction the bicycle is moving. Sec. 3 — Bright lights on any vehicle or street car operated within the limits of the right of way of any road shall be dimmed or con- trolled while approaching and passing another vehicle so as to pro- tect from the direct glare, the eyes of a driver 200 feet ahead and whose eyes are 5 feet above the road surface. Sec. 4 — In order to avoid accidents and for the purpose of secur- ing the greatest possible safeguard to human life, all drivers of horse drawn vehicles are urged and requested to display a light at night that can be seen both in front and in the rear. ARTICLE IX — MISCELLANEOUS REGULATIONS Sec. i — The unnecessary emission of dense smoke from motors is forbidden. Sec. 2 — No horse or other domestic animal shall be allowed to run loose and unattended upon the road. Sec. 3 — No vehicle shall be used or so loaded as to permit its load to be scattered over the road in such a way as to be objectionable to traffic or detrimental to the road. In no case shall ashes, garbage or other vegetable matter be scattered over the road surface. Sec. 4 — No one shall ride on any vehicle without the consent of the driver. Sec. 5 — No road shall be blocked or obstructed by any farm implements or other machinery or obstructions except when the road is legally closed for repair or construction. (13421-11 G. C.) TRAFFIC RULES AND REGULATIONS 597 Sec. 6 — No person shall operate a motor vehicle while in a state of intoxication. (1 2626-1 G. C.) Sec. 7 — Any special rules or regulations jor any road or portion of a road which are conspicuously displayed at either end of such road or have been made known to the driver of a vehicle by a road official, shall be strictly observed by all drivers of vehicles over the road. (7246 G. C.) Sec. 8 — Drivers of vehicles shall observe such care as is neces- sary to preserve the life of the road; — avoid following the tracks of a vehicle preceding them, and avoid driving in the ruts that may have started to form in the road surface. Sec. 9 — Any person operating a vehicle or moving a load, or who is responsible for the operating of a vehicle or the moving of a load over a road in violation of any of the rules and regulations apply- ing to such road, in addition to the fine imposed shall be responsible for all damage which said road may sustain as a result of such vio- lation. (13421-17 G. C.) While a good set of Traffic Rules and Regulations will be of great service to the public, yet in connection with any set of rules and regu- lations that may be in force, it will be necessary for drivers to exer- cise common sense and good judgment to avoid accidents and protect the roads. By a careful distribution of the traffic over a good road surface, roads may be made to last several times as long as they will last when all traffic concentrates in a single track. Every individual should remember that he helps build and maintain the roads regard- less of the direct taxes he pays, and hence it is to his interest that the roads be used in such a way as to preserve them and render the greatest service to the general public. APPENDIX Sec. 6290 (G. C.) ["Motor vehicle" defined.] The term "motor vehicle," as used in this chapter and in the penal laws, except where otherwise provided, shall be deemed to include all vehicles propelled by power other than muscular power, except road rollers, traction engines, police patrol wagons, police automobiles, public ambulances, vehicles run upon rails or tracks, fire engines, fire trucks or other vehicles or apparatus belonging to any police department, municipal fire department, volunteer fire company or salvage com- pany, organized under the laws of Ohio, or used by such police depart- ment, volunteer fire company, or salvage company, in the discharge of its functions or in transporting its officers, members, employes, men or articles necessary and proper for the proper discharge of such functions, to or from a fire or in response to any alarm of fire or to any other alarm or call to which it may respond. (100 v. 72; Am. 103 v. 763; 106 v. 139). Sec. 6310 (G. C.) [Meeting vehicle on highway.] A person driving a carriage or vehicle on a public turnpike, road or highway, on meeting a carriage or vehicle, shall keep to the right so as to leave half of the road free for the coming vehicle. 598 TRAFFIC RULES AND REGULATIONS A person riding on horseback or on a bicycle, tricycle, or tandem bicycle, or driving a locomobile, automobile, or any motor vehicle operated by its own power, on meeting a carriage or vehicle drawn by horses or oxen, shall keep to the right so as to leave one-half of the road free for the use of the vehicle drawn by horses, mules or oxen. A person driving a carriage, vehicle, automobile, or any kind of vehicle, who desires to pass a vehicle going in the same direction on any public road or highway shall give an alarm or demand to the person or persons driving the vehicle in front and going in the same direction, of such desire, and the person so driving the front vehicle shall immediately give the half of the road to the rear vehicle, by turning to the right, so that the rear vehicle can pass to the left of the front vehicle. [Definition; penalty.] The term vehicle herein shall apply to any vehicle propelled by its own power or drawn by horses or oxen. Any person or persons driving any vehicle, horse-drawn or otherwise, on any public road, who shall fail, in meeting or passing, to give the required road as herein stated, shall be subject to a fine in a court having jurisdiction thereof, in any sum not less than one dollar, nor more than ten dollars, and costs of prosecution. (R. S. Sec. 3490. Am. 103 v. 556). Sec. 7246 (G. C.) [Publication of traffic rules and regula- tions; special rules; enforcement.] The state highway com- missioner within sixty days after the taking effect of this act, shall prepare and publish a set of traffic rules and regulations governing the use of, and traffic on, all state roads. All rules and regulations that are to apply generally throughout the state, including those applicable to roads constructed of the various kinds of road material, shall become effective thirty days after publication. Special rules and regulations or orders, applying only to specified sections of state roads, shall become effective as soon as posted at each end, and at all road crossings on such specified section. For the purpose of carrying into effect the provisions of this section, it shall be the duty of the state highway commissioner, the county commissioners, the county highway superintendent, the township highway superinten- dent, township trustees, and all patrolmen or deputies employed on any highways within the state, to prosecute any violation of this section. It shall be unlawful for any person or persons, firm or cor- poration to enter upon, or travel over said state roads, except in accordance with the traffic rules and regulations promulgated by the state highway commissioner. Sec. 7477 (G. C.) [Weight of load and tire width pre- scribed.] No person, firm or corporation, in a county having macadamized, graveled or stone roads shall transport over such roads, in a vehicle having a tire of less than three inches in width, a burden, including weight of vehicle, of more than thirty-four hundred pounds. [Board of directors and duties; penalty.] The county com- missioners shall constitute a board of directors for their respective counties, with power to prescribe the increased gross weight in excess of thirty-four hundred pounds that may be carried, including weight TRAFFIC RULES AND REGULATIONS 599 of vehicles, in vehicles having a width of tire three inches or upwards, and cause such regulations to be recorded in their journal. Any person violating this section or any regulation duly prescribed by the board of county commissioners, made in pursuance thereof, shall be fined not less than five dollars nor more than fifty dollars. [Enforcement of traffic regulations.] The township trustees of any township and the county commissioners of any county, shall cause to be prosecuted all persons violating this section or any regu- lations prescribed by the board of county commissioners made in pursuance of the authority conferred in this section. The county commissioners within their respective counties, may appoint a suit- able person or persons to enforce such section and regulations. The person or persons so appointed shall receive for each conviction by them secured under this section, such portion of the fine or penalty as the commissioners deem just and proper. Sec. 7478 (G. C.) [Publication of traffic rules in coun- ties.] The state highway commissioner shall furnish the county highway superintendent with a copy of the rules and regulations pro- mulgated by said state highway commissioner, and applicable to his county. The county highway superintendent shall cause the rules and regulations so furnished to him by said highway commissioner to be published, at least once each week, for two successive weeks, in a newspaper published and of general circulation in said county, if there be any such paper published in said county, but if there be no newspaper published in said county then in a newspaper having general circulation in said county. When such regulations are pub- lished in the manner aforesaid, it shall be deemed a sufficient pub- lication under the provision of this act. Sec. 12603 (G- C.) [Operating motor vehicle unreasonably and improperly; penalty.] Whoever operates a motor vehicle or motorcycle on the public roads or highways at a speed greater than is reasonable or proper, having regard for width, traffic, use and the general and usual rules of such road or highway, or so as to en- danger the property, life or limb of any person, shall be fined not more than twenty-five dollars* and for a second offense shall be fined not less than twenty-five dollars nor more than fifty dollars. (99 v. 541, 543, Sees. 14, 25. Am. 103 v. 161). Sec. 12604 (G. C.) [Violation of speed limit.] Whoever operates a motorcycle or motor vehicle at a greater speed than eight miles an hour in the business and closely built-up portions of a municipality or more than fifteen miles an hour in other portions thereof or more than twenty miles an hour outside of a municipality, shall be fined not more than twenty-five dollars, and for a second offense shall be fined not less than twenty-five dollars nor more than fifty dollars. (99 v. 541, 543, Sec. 15, 25). Sec. 12605 (G. C.) [Failure to stop motor vehicle when signalled.] Whoever, operating a motor vehicle, fails to slow down and stop it when signalled so to do upon meeting or overtaking a horsedrawn vehicle or person on horseback and to remain stationary until such vehicle or person has passed, provided such signal to stop is given in good faith, under circumstances of necessity, and only 600 TRAFFIC RULES AND REGULATIONS as often and for such length of time as required for such vehicles or person to pass, whether approaching from the front or rear, shall be fined not more than twenty-five dollars, and for a second offense shall be fined not less than twenty-five dollars nor more than fifty dollars. (99 v. 541, 543. Sees. 16, 25). Sec. 12606 (G. C.) [Failure to stop motor vehicle in case of accident.] Whoever, operating a motor vehicle on a public road or highway, in case of an accident to a person or property thereon due to the operation of such motor vehicle, fails to stop upon the request of the person injured or a person present, give his name and address, and, if not the owner thereof, the name and address of such owner, shall be fined not more than twenty-five dollars, and for a second offense shall be fined not less than twenty-five dollars nor more than fifty dollars. (90 v. 541, 543, Sees. 16, 25). Sec. 12607 (G. C.) [Third or subsequent offense.] For a third or subsequent offense, a person convicted of a violation of any provision of the next four preceding sections, shall be fined not less than fifty dollars nor more than one hundred dollars or imprisoned not more than thirty days, but if such subsequent offense occurred within one year after any former offense, he shall be imprisoned not less than ten days nor more than thirty days. (99 v. 543, Sec. 25.) Sec. 1 2614 (G. C.) [Penalty for failing to provide motor vehicle with bell, brakes, and lights.] Whoever operates or drives a motor vehicle upon the public roads and highways without providing it with sufficient brakes to control it at all times and a suit- able and adequate bell or other device for signalling, or fails during the period from thirty minutes after sunset to thirty minutes before sunrise to display a red light on the rear thereof and three white lights, two on the front and one on the rear thereof, the rays of which rear white light shall shine upon and illuminate each and every part of the distinctive number borne upon such motor vehicle, the light of which front lamps to be visible at least two hundred feet in the direction in which such motor vehicle is proceeding, shall be fined not more than twenty-five dollars. Provided, that motor vehicles of the type commonly called motor cycles shall display one white light in front to be visible at least two hundred feet in the direction in which such motor vehicle is proceeding, and one rear combination red and white lights showing red in the direction from which such motor vehicle is proceeding, and such rear light to be so placed that it will reflect its white light upon and fully and clearly illuminate the distinctive license identification mark of such motor vehicle. (99 v. 540, 543, Sees. 12, 24. Am. 103 v. 766). Sec. i 2628-1 (G. C.) [Intoxicated person operating motor vehicle upon public highway or street, unlawful.] That it shall be a misdemeanor for any person to operate a motor cycle or motor vehicle of any kind upon any public highway or street while in a state of intoxication, and upon conviction he shall be subject to punish- ment by a fine not less than twenty-five dollars, nor more than one hundred dollars, or imprisonment in the county jail for not more than six months, or both. (99 v. 544, Sec. 32. Am. 103 v. 133). Sec. 13421-9 (G. C.) [Driving over closed highway ; penalty.] TRAFFIC RULES AND REGULATIONS 601 Whoever drives over, upon, along or across a public highway, or any part thereof, which has been closed, while in the process of con- struction, reconstruction or repair by order of the state highway commissioner, county highway superintendent, county commis- sioners, township trustees or other official or employe having au- thority to close such highway, shall be fined not more than fifty dollars, nor less than five dollars. Sec. 1342 i-i i (G. C.) [Placing obstruction in highway; penalty.] Whoever unlawfully places any obstruction in, or upon a public highway, shall be fined not more than fifty dollars, nor less than five. dollars. Sec. 1342 i-i 2 (G. C.) [Driving traction engine with destruc- tive tires; penalty.] • Whoever drives over the improved highways of the state, or any political subdivision thereof, a traction engine with tires of wheels equipped with lugs, spikes, chains or other pro- jections seriously destructive to such highways, or by any other means damages such highways, shall be fined for each offence not less than ten dollars nor more than two hundred dollars. Sec. 13421-14 (G. C.) [Digging, excavating, piling earth or building fence on highways; penalty.] Whoever digs up, removes, excavates or places any earth or mud upon any portion of any public highway or builds a fence upon the same without legal authority or permission so to do, shall be fined not more than two hundred dollars nor less than ten dollars. Each day that such person continues to dig up, remove or excavate any portion of the public highway shall constitute a separate offence. Sec. 1342 1-16 (G. C.) [Placing nails, tacks, glass, etc., upon highway ; penalty.] Whoever places upon any part of a public highway, lane, road, street or alley, any tacks, bottles, wire, glass, nails or other articles, except such substances as may be placed there by proper authorities for the repair or construction thereof, which may damage or injure any person, vehicle or animal traveling along or upon said public highway, shall be fined' not more than two hun- dred dollars or imprisoned not more than six months or both. Sec. 1342 i-i 7 (G. C.) [Violation of traffic rules; penalty.] Whoever enters upon, or travels over any portion of the highways, within the state, in violation of the traffic rules and regulations duly prescribed by law, or the state highway commissioner, or the county highway superintendent of any county, shall be fined not more than one hundred dollars, nor less than five dollars, and in addition thereto, such person shall be liable for all damage done to such highway. 602 TRAFFIC RULES AND REGULATIONS State of New York Highway Commission AMENDED RULES AND REGULATIONS FOR STATE AND COUNTY HIGHWAYS Adopted by the Commissioner of Highways of the State of New York Sec. i. No traction engine, road engine, hauling engine, trailer, steam roller, automobile truck, motor or other power vehicle shall be operated upon or over State or County Highways of this State, the face of the wheels of which are fitted with flanges, ribs, clamps, cleats, lugs or spikes. This regulation applies to all rings or flanges upon guiding or steering wheels on any such vehicle. In case of traction engines, road engines or hauling engines which are equipped or provided with flanges, ribs, clamps, cleats, rings or lugs, such vehicles shall be permitted to pass over such highways provided that cleats are fastened upon all the wheels of such vehicles, not less than 2\ inches wide and not more than i \ inches high, and so placed that not less than two cleats of each wheel shall touch the ground at all times, and the weight shall be the same on all parts of said cleats. The foregoing regulations relating to flanges, ribs, clamps, cleats, rings or lugs shall not apply to traction engines used solely for agri- cultural purposes, but the following requirements shall apply to such traction engines: The guide band on the front wheels shall not be less than two inches in width, but no flanges, ribs, clamps, cleats, rings or lugs shall be required upon the front wheels. The full set of cleats upon the rear wheels of the original design as furnished with the engines must be used, and no rivet heads or bolt heads shall project, and the use of such traction engines for agricultural purposes shall not permit the use for hauling purposes, excepting the hauling of threshing and other agricultural equipment necessary for threshing and agricultural purposes. This provision shall in no case relieve the owner of any traction engines from liability for damage to roads from defective wheels. ' The use also of ice picks or mud lugs shall be strictly prohibited on State and County Highways. Sec. 2. No traction engine, trailer, steam roller, automobile truck, motor or other power vehicle shall be operated upon or over the State or County Highways of this State, nor shall any object be moved over or upon any such highways upon wheels, rollers or otherwise, in excess of a total weight of fourteen tons, including the vehicle, object or contrivance and load, without first obtaining the permission of the State Commission of Highways as hereinafter provided. No' weight in excess of nine tons shall be carried on any one axle of any such vehicle. Sec. 3. The tire of each wheel of a traction engine, road engine, hauling engine, trailer, steam roller, automobile truck, motor or other power vehicle (except traction engines, road engines, and haul- ing engines) shall be smooth, and the weight of such vehicle, including TRAFFIC RULES AND REGULATIONS 603 load, shall not exceed 800 lbs. upon' any inch in width of the tire, wheel, roller or other object, and any weight in excess of 800 lbs. upon an inch of tire is pibrohited unless permission is obtained from the State Commission of Highways as hereinafter provided. Sec. 4. No motor or other power vehicle shall be operated upon any State or County Highway of a greater width than ninety inches, except traction engines which may have a width of one hundred ten inches. Sec. 5.. No traction engine, road engine, hauling engine, trailer, steam roller, automobile truck, motor, or other power vehicle, carry- ing a weight in excess of four tons, including the vehicle, shall be operated upon any State or County Highway of this State at a speed greater than fifteen miles an hour; and no such vehicle carrying a weight in excess of six tons, including the vehicle, shall be operated upon any such highway at a speed greater than six miles an hour when such vehicle is equipped with iron or steel tires, nor greater than twelve miles an hour when the vehicle is equipped with tires of hard rubber or other similar substance. Sec. 6. The State Commission of Highways of the State of New York, upon proper application in writing, may grant permission for the moving of heavy vehicles, loads, objects, or structures in excess of a total weight of fourteen tons over its State and County High- ways upon proper application in writing being made therefor, and under such restrictions as said Commission may prescribe. Sec. 7. The owner, driver, operator or mover of any vehicle over any State or County Highway shall be responsible for all damage which said highway may sustain as a result of a violation of any of the provisions of the foregoing Rules and Regulations, and the amount thereof may be recovered in an action of tort by the State Commission of Highways or by any County Superintendent of Highways of any county or by any Town Superintendent of High- ways in any town in which said violation occurs. Sec. 8. These amended regulations to take effect February 24, 1914. " Section 24 of Chapter 25 of the Consolidated Laws, entitled 'The Highway Law,' provides that any disobedience of any of the foregoing rules and regulations shall be punishable by a fine of not less than $10, and not more than $100, to be prosecuted by the Town, County or District Superintendent, and paid to the County Treasurer to the credit of the fund for the maintenance of such highways in the town where such fine is collected." INDEX A Adjustment of instruments 164 Alignment 18, 144, 206 Annual charges 108 Amiesite 82, 285, 441 Arch Culverts 48, 305 Areas, formulae 462 of cross-sections 226 right of way 261, 262 Asphalt, Rock 82, 283 Asphalt Binder (Bituminous Ma- terial A) 383-385 Asphalt Block 85, 290, 449 Asphalt pavement, Concrete foundation for . .327, 427 Topeka Mix (Mixing Method type II) 81. 101, 351, 437 Asphalt, Sheet, weight of . 352 Asphaltic Concrete _ (see Bitumi- nous Macadam mixing method). Asphalts (see Bitumens) Automobiles 18, 62, 591,602 B Banked curves 31 Beam Bridges 43, 44-47 Beams, bending moments 586 Beams, steel, properties of 59 Bench marks 146 Bitulithic Pavements 439 Bitumens . . . 130, 377, 382 Bituminous Binder, Cost of appli- cation 281 Bituminous Macadam, bottom course. .65,425 Contrac- tor'splant..3ii Costs in detail.. 266-312 crown .... 19, 80 life_ 3, 107 mainte- nance. .97, 105 mixing methods. . 81, 101, 351, 437 penetration method. . 78, 348, 433 specifica- tions. .433,437 typical section. . 23, 29, 30, 31 (605 Bituminous materials 130, 377, 382 surfacings 76, 77, 99, 332, 432 amounts required 264, 265 Blasting small boulders 278 Bonds, Highway 2 Bottom course, broken stone 65 construction of . . . 346 economic design. 73 gravel 67, 69 macadam roads. .65, 425 stone fill 71, 281 sub-base .... 24, 68, 281 Boulders, cost of sledging 278 Brick, properties 129 tests 390 Brick cube pavement 92, 284 Brick pavements 83 construction .354, 451 crown 19, 85 detail costs.;. . . 327 edging 24, 83, 328, 355, 417 estimate of cost 331 expansion joints. 83, 84, 329, 330, 452 foundation .... 84, 327, 356, 427 grout 143, 329, 330,357,375,453 life of 3, 107 maintenance. . . 102, 107 mortar bed .... 83 sand cushion. . . 83, 328. 356. 375, 452 specifications. . . 451 typical sections 24, 31, 83 Bridge rail and parapets 112 Bridges, approximate weight of plate girder 259, 260 small span 39 Burkli-Ziegler formula for run off . 36 C Calcium chloride . . . 76, 334 Cast-iron, specifications 398, 419 weight 584 Cast-iron pipe, culverts ... 3.3, 38, 58, 358 specifications. . .398, 407 weights of 54 ) 6o6 INDEX Catch basins. 116, 406 Cement, for grouting 330 specifications 372 tests and requirements 141, 373 weight 584 Center of gravity 588 Circles, areas and circumferences. . 464 Clearing and grubbing 401 Concrete, aggregates 142, 374, 444 materials, amounts re- quired . . . 289, 303, 304, 360 mixing and placing . .360, 411 sizes of stone 143 specifications for ma- terial 374 weight 584 Concrete culverts, capacities 38 construction. 350-359 costs 38, 302-305 forms. 359, 412 quantities . . . 304, 361 standards 40, 41, 44-51 Concrete foundations 84, 327, 427 Concrete Guard Rail no, 306 Concrete Masonry 410 Concrete pavement 87 construction. . 288, 443 crown 23, 87 detail costs. 288-303 expansion joints 87. 90, 91, 295, 396 forms 294 Hassam type 288, 350, 442 life 107 maintenance . . . 101, 107 plant and labor .... 289-303 specifications. . _ 442. 443 typical sec- tions 23 Construction plans 257 Conversion tables, cu. ft. to cu. yds. 230 feet to miles. . . 250 general table. . . 459 Cosecants, natural 502 Cosines, logarithmic 539 ( natural 490 Cost estimates 312 of miscellaneous minor items 320 of Roads per mile 3, 108 of staking out 340 Costs, comparative: Amiesite 81 Asphalt Block. 86 Bituminous Macadam. . . 79 Bituminous surfacing. ... 77 Brick pave- ment 84 Concrete pave- ment 87 Rock Asphalt . . 82 Costs, comparative: Stone Block Pavement. . . 85 Waterbound Macadam . . 74 Cotangents, logarithmic 539 natural 479 Cross-sections, areas . . . . f 226 notes 146, 209 templets for 221 Crowns (see type of pavement) parabolic 262 Crushing stone, cost 274 proportions of sizes 275 Cubes 91, 92, 284 cubic feet to cubic yards 230 Culling Brick 332, 356 Culverts ^ capacities 38 construction 358 standards 49 curbs 113, 115, 415 Curves, banked 31 functions of i° 170 length of 170 methods of running 199 problems 200-205 radii of 167 sharp, avoidance of 18 tangent length 197 vertical (see vertical curves) D Danger signs : . 114, 422 Deformed bars, areas and weights 57 specifications 398 Depth of pavement 64, 74^92, 345 Depreciation of Contractor's equip- ment 309 Design, economic 263 reports on 210, 255 Dicken's formula for Run Off 34 Distance, measures of 459 Ditch lining (see gutters) Ditches, function of 22 Drainage 33-60 notes 149 Driveway culverts 39 Double Meridian Distance areas. . 261 Drop inlet 39. 4°6 Durax 91 Dust layers 76, 97-100 Dustless screenings 280 E Earth, shrinkage of 219 weight ; 584 Earth excavation, computation of. 227 Earth excavation, detail costs 266 methods. . . . -34 1 . 354 specification. ... 401 table of volumes 228, 238-249 variation of. . . . i4 _I 7 INDEX 607 Edging 24, 83, 328, 355. 417 Embankments 64, 342, 403 shrinkage of earth. 2ig Equipment, contractors. . 3°Q Estimates of costs, forms for 312 Excavation (see Earth Excavation) Expanded metal 55,-398 Expansion joints, Brick Pavements 83, 84, 329, 357, 330, 452 Concrete Pave- ments 87, 90, 91, 295, 396 Guard Rail 109, no, 306, 334, 420 Guide signs 114, 421 Gutters 115, 307, 417 H Hassam concrete pavement, 288, 350, 442 Haul, computation of length 320 Hauling stone, cost 269 Highway bonds 2 Horse, tractive power of 6-10 Field stone, cost of loading 271 Fillers, bottom course, amount re- quired . . 272, 347 cost load- ing and spreading 273 require- ments 129, 426 Fills, earth 64, 219, 342, 403 stone 71, 281, 408 Financing highway improvements 2, 108 Foot, decimal equivalents of 460 Forms, concrete pavement 294 removal of 359 specifications 412 Foundation courses, macadam. . . .61-72 Foundation soils .61-72, 151, 344 Functions, Trigonometric, loga- rithmic. ..539-583 natural. . -479 - 5i3 Geological classification of rocks 125-127 Glutrin (sulphite liquor) 77, 445 Grade crossing eliminations 258 Grades 4 -1 7 controlling features 13 economy of . 13, 221 effect on hauling 10 maximum, practice as to. . 5 asphalt (3|%) asphalt block .87, 219 bituminous ma- cadam 80, 219 brick 85, 219 concrete 87, 219 rock asphalt. . . 82 stone block . .85, 219 waterbound macadam. .75, 219 woodblock .... 219 minimum 12 selection of n, 12 Grading 341 Gratings. 116, 117 Gravel, bottom course 67-69 specifications 143, 376 Gravity, center of 588 Grouting. . . 143, 329, 330, 357, 375 , 453 Igneous rocks 126 Inches as decimals of a foot 460 Interest of plant and payroll 310 Inertia, moments of 589 Inspection of construction (see Chapter XI) Instruments, adjustment of 164 Kentucky rock asphalt 82, 283 Kleinpflaster 91 Land taking surveys 154, 262 Leaching basins 116, 405 Level, adjustment of 164 notes 146, 208 Levelling, accuracy required 146 Life of pavements 3, 106-108 Lignin 77, 445 Linear measures 459 Loading stone and filler, costs .... 268, 271, 273 Logarithmic functions 539-583 Logarithms of numbers 514-538 Lumber 400, 409 M Macadam (see Waterbound and Bituminous Macadam) Machinery, contractors' 309 Maintenance 96-108, 337, 338 Manholes 407 Maps, preparation of 206-265 Masonry, concrete 410 stone 414 repointing 117 Mass diagram 251-253 Materials, strength of 585 tests 118-143 report on location 152 Maximum grades (see Grades, Max- imum) McClintock cubes 92, 284 Mesh reinforcement 55, 398 Metamorphic rocks 127 Metalling, width of 28, 263 Moduli of elasticity 585 Moment of inertia . 589 Motor truck regulations . . .62, 591, 602 6o8 INDEX N Natural cement 141 Natural trigonometric function. .479-513 Oil, table of quantities . . 264 Oiling (see Bituminous Surfacing) Overhaul 251, 404 Overhead charges 309 P Parabolic crown for pavement .... 262 Pavements, cost per mile . ._ 3, 108 economic selection ... 74, 96 life of 3> 106-108 maximum grades (see Grades, Maximum) typical sections 23-32 thickness. . . .64, 74~Q2, 345 types (see respective headings) Amiesite Asphalt Asphalt Block Asphaltic Con- crete • Bituminous Ma- cadam Pene- tration Bituminous Ma- cadam Mixed Brick Concrete Durax Kleinpflaster McClintock Cubes Rocmac Rock Asphalt Stone Block Waterbound Macadam Pavement, width 28, 263 Paving pitch, expansion joints. .330, 452 Petroleum 136, 137 Piles 408 Pipe, cast-iron, culverts. . .35, 38, 58, 358 specifications. . .398, 407 weight 54 vitrified 3°7» 400, 4°4 Pipe rail 112, 421 Pitch expansion joints 330, 452 Planimeter, use of 226 Plans, preparation of 206-265 Plants and payroll 309 Plate girder bridge weights. . . . 259, 260 Portland cement 141, 372 Posts, concrete sign 422 Profile, plotting of 209 Puddling, waterbound macadam. . 283 R Radii of curves 167 Rattler for brick tests 39° Reference points 144 Reinforcement, specifications . . .398, 418 weights and areas. 55-57 Relaying old pipe 407 Repairs (see Maintenance) Repointing masonry 117 Report on design 210, 255 on materials 152 on soil 151 Resurfacing macadam 335 Retaining walls 111-112 Rights of way . . 154, 262 Riprap 117, 409 Road materials, tests and prop- erties. . 1 18-143, 372-382 Road sections, typical 23-32 Roads, improved, advantage of . . . 10 Roadway travelled, width of 20 Rock, excavation 220, 266, 402-404 geological classification 125 properties 122-125, 128 tests 1 18-128 Rock asphalt 82, 283 Rocmac 94 Rolling, cost of 273 resistance 7, 10 Ruling grades (see Grades, Maxi- mum) Run off 34-37 S Safety measures, banked curves.. . 31 danger signs . . 114, 422 grade crossing elimination. . . 258 guard rail 109, no shoulder slope ... 19 shoulder stone . . 29. 90 sight distance on curves. . 18, 208, 225 Sand, requirements for. . . . 142, 374, 375 cushion. . . .83, 328, 356, 375, 452 Scarifying macadam 335, 431 Screed for concrete pavement 351 Screenings 128, 272, 280 Secants, natural 502 Sedimentary rocks 126 Shoulder slope 19 stone 29, 90 Shrinkage of earth . 219 Side hill roads, drainage 42 Sight distances on curves. . . 18, 208, 225 Signs, guide 114, 421 danger 114, 422 Sines, logarithmic 540 natural 490 Slag 377 Slope stakes 341 Soil 61-72, 344 Soil examination and report 151 Specifications, typical 372-458 Sprinkling 76 Square roots, table of 464 yards, table of 263 Squares, table 464 Stadia reduction tables 156 Staking out 339 Steel 398, 419, 584 INDEX 609 Steel bars, areas and weights 56 beams., properties 59 Stone, amounts required for road.. 252 costs of handling and haul- ing. 268-273 crushing, cost 274 depths 64, 74-9 2 , 345 proportions of crusher out- put 274 ratio loose to compacted depths 272 sizes for road work (see Pavement T} , pes) Stone specifications 376 tests 1 18-128 Stone block pavement 85, 91, 102, 396, 455 foundation. . 85, 327, 427 Stone fills 71, 281, 408 Stone masonry 414 Stoneway, width of 28, 263 Strength of materials 584-587 Sub-base 64, 345, 423 bottom course 24, 68, 281 Sub-grade preparation .65, 344, 354, 403 Sulphite liquor 77, 445 Surface measures 459 Surveys 144-166 Surveys accuracy required 145 equipment of party 144 rights of way 154 speed 146, 149, 151, 154 Tangents, logarithmic 539-583 natural 479-489 Tar 131, 132-135 Tar (Bituminous Material T). . .387-389 Tar surfacing (see Bituminous Sur- facing) Telford base 69, 423 Temperature units 459 Template for concrete pavement . . 351 Tests of materials .... 1 18-143, 372-382 Tile, porous. . . . : 400 vitrified 307, 400, 404 Timber 400, 409 Tire, allowable load 62-63 effect of, on tractive power. . 7, 9 Toe walls 113 Top course, economic selection ... 74, 96 Topeka mix (Mixing Method II) 81, 101, 351, 437 Topography notes 150, 208 Traction engines 4, 62, 591, 602 Tractive power 6-10 Traffic, classification of roads as to 74 regulations 62, 591, 602 report on 151 Transit, adjustment 165 points 144 Traverse computation 155 Triangles, solution of 477 Trigonometric formulae 477 Trigonometric functions, loga- rithmic 530-583 natural 479—503 Trucks, automobile 62, 271 allowable load- ing... 62, 591, 602 U Underdrains 42, 52, 405 Unloading stone, cost 268 V Velocities of flow 33, 58 Vertical curves 222, 223, 225 Vitrified pipe 307, 400, 404 Voids, determination of 89 Volume formulae 462 measures 459 W Wagon, allowable loads on. . . 62. 591, 602 Water, amount for concrete 304 puddling 284 weight 584 Waterbound macadam 23, 74, 348 Waterbound macadam, bottom course 65-70 Waterbound macadam, compara- tive cost 74 Waterbound macadam, construc- tion 348 Waterbound macadam, contractor's plant 310 Waterbound macadam, costs in de* tail 266-312 Waterbound macadam, crown . . .19, 75 Waterbound macadam, life of . . . .3, 106 Waterbound macadam, mainte- nance 97, 106 Waterbound macadam, resurfac- ing . . 335 Waterbound macadam, specifica- tions 428 Waterbound macadam, typical sections. 23-28 Weights, cast-iron pipe 54 expanded metal 55 materials 584 steel bars 56, 57 stone 252 units 459 Wheel, effect of size on tractive power 8 Width of pavement 20, 263 Wood block 219, 446 W^ork, speed of 3°8 Working stresses 585 W T rought-iron specifications. . . .398, 419 Y Yearly charges 108 NOTES A, "%. v^ ^ ■** C » i i %+* v^ , ^ ,0^ \^ X ^ ;- \ ^, ,* -p k* ^V ,^ X ^ C ^"> tP ,\V *$» \. 4 ill I LIBRARY OF CONGRESS I ■ ■ ■■ ■ i w ; ' ; t'-HB. ■ ■ M m ■ ■ mm mmmmm mm<$im mm I illlii ■ m m II ■ ^H all ■ IB EhSMI